Causal Set Phenomenology

C AUSAL S ET P HENOMENOLOGY Lydia Philpott arXiv:1009.1593v1 [gr-qc] 8 Sep 2010 Imperial College London A thesis submitted for the degree of Doctor of Philosophy of the University of London and the Diploma of Imperial College London 2010  To my grandfather, Alan William Conway 22 January 1924 – 9 May 2008. 2 D ECLARATION I declare that all work in this thesis is my own except where specifically mentioned otherwise. Lydia Philpott September 9, 2010 3 A BSTRACT Central to the development of any new theory is the investigation of the observ- able consequences of the theory. In the search for quantum gravity, research in phe- nomenology has been dominated by models violating Lorentz invariance – despite there being, at present, no evidence that Lorentz invariance is violated. Causal set theory is a Lorentz invariant candidate theory of quantum gravity that seeks not to quantise gravity as such, but rather to develop a new understanding of the universe from which both general relativity and quantum mechanics could arise separately. The key hypothesis is that spacetime is a discrete partial order: a set of spacetime events where the partial ordering is the physical causal ordering between the events. This thesis investigates Lorentz invariant quantum gravity phenomenology motivated by the causal set approach. Massive particles propagating in a discrete spacetime will experience diffusion in both position and momentum in proper time. This thesis considers this idea in more depth, providing a rigorous derivation of the diffusion equation in terms of observable cosmic time. The diffusion behaviour does not depend on any particular underlying particle model. Simulations of three different models are conducted, revealing be- haviour that matches the diffusion equation despite limitations on the size of causal set simulated. The effect of spacetime discreteness on the behaviour of massless particles is also investigated. Diffusion equations in both affine time and cosmic time are derived, and it is found that massless particles undergo diffusion and drift in energy. Constraints are placed on the magnitudes of the drift and diffusion parameters by considering the blackbody nature of the cosmic microwave background. Spacetime discreteness also has a potentially observable effect on photon polarisation. For linearly polarised photons, underlying discreteness is found to cause a rotation in polarisation angle and a suppression in overall polarisation. 4 ACKNOWLEDGEMENTS Special thanks to my supervisor, Fay Dowker, without whose support this work would not have been possible. Thanks are also due to Rafael Sorkin, whose ideas provided much of the inspiration for this work. I would like to thank Steven Johnston, Joe Hen- son, and Sumati Surya for helpful discussions throughout the course of this research. Thanks are also due to Steven Johnston for proofreading parts of this thesis. Thanks to David Rideout for making the Cactus CausalSets arrangement available and for taking the time to assist me with its use. This research was funded by a Tertiary Education Commission of New Zealand Top Achiever Doctoral Scholarship and an FfWG Foundation Main Grant. Finally, many thanks to my parents for their unfailing support, and to Richard for proofreading large parts of this work and for all his patience, support, and encourage- ment. 5 C ONTENTS Declaration 3 Abstract 4 Acknowledgements 5 Notation 10 1 Introduction 11 2 A brief introduction to causal set theory 15 2.1 What is a causal set? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The causal set hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Sprinklings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Causal set phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Swerves 22 3.1 Models for massive particle propagation on a causal set . . . . . . . . . . 24 3.1.1 The swerves model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Intrinsic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 The swerves diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Stochastic evolution on a manifold of states . . . . . . . . . . . . . 29 3.2.2 Diffusion in proper time . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Diffusion in cosmic time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Finiteness of the diffusion constant . . . . . . . . . . . . . . . . . . . . . . 39 3.4.1 Proofs of Lemmas 3.4.1 and 3.4.2 . . . . . . . . . . . . . . . . . . . 40 3.5 Consequences and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.6 Simulations of particle models . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6.1 Example trajectories in 1+1 dimensions . . . . . . . . . . . . . . . 55 3.6.2 A close investigation of Model 1 . . . . . . . . . . . . . . . . . . . 59 6  Contents 7 3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 Massless particles 71 4.1 Diffusion in affine time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Diffusion in cosmic time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.1 Power law equilibrium solutions . . . . . . . . . . . . . . . . . . . 77 4.3 Diffusion and drift in the cosmic microwave background . . . . . . . . . 79 4.3.1 Modelling diffusion in the CMB . . . . . . . . . . . . . . . . . . . 80 4.3.2 The expanding universe . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 The shifting of spectral lines . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5 Polarisation 95 5.1 The polarisation state space . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.1 Polarisation under Lorentz transformation . . . . . . . . . . . . . 99 5.1.2 Stokes parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Polarisation diffusion in affine time . . . . . . . . . . . . . . . . . . . . . . 104 5.3 Polarisation diffusion in cosmic time . . . . . . . . . . . . . . . . . . . . . 107 5.3.1 The solution to the diffusion equation . . . . . . . . . . . . . . . . 108 5.4 Observational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.4.1 Constraining the drift parameter with radio galaxy data . . . . . 116 5.4.2 Polarisation diffusion and drift in the cosmic microwave back- ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6 Conclusions and future directions 126 A Explicitly Lorentz invariant polarisation diffusion 129 A.1 Invariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A.2 The diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Bibliography 133 L IST OF F IGURES 2.1 Hasse diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 A diamond lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 An example sprinkling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 The swerves model for particle propagation . . . . . . . . . . . . . . . . . 25 3.2 Intrinsic models for particle propagation . . . . . . . . . . . . . . . . . . . 28 3.3 Flowlines in M′ = M4 × H3 × R . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Hyperbolic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5 An example of a Model 1 particle trajectory in 3+1 dimensions . . . . . . 53 3.6 Examples of intrinsic particle trajectories in 3+1 dimensions . . . . . . . 54 3.7 An example of a Model 1 particle trajectory in 1+1 dimensions . . . . . . 56 3.8 An example of a Model 2 particle trajectory in 1+1 dimensions . . . . . . 56 3.9 Examples of Model 3 particle trajectories in 1+1 dimensions . . . . . . . 57 3.10 Multiple Model 3 trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.11 An invalid Model 1 trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.12 Example position and momentum histograms for Model 1 . . . . . . . . 64 3.13 k vs. τf and ln k vs. ln τf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.14 ln k vs. ln dpl and k vs. m2 /τf5 . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Example Planck spectrum evolved according to the diffusion equation . 83 4.2 Varying d2 with d1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Varying d1 with d2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Varying both d1 and d2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Varying both d1 and d2 for an expanding universe . . . . . . . . . . . . . 88 4.6 Example quasar spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.1 The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 Polarisation of radio galaxies . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 CMB correlation spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8  LIST OF FIGURES 9 5.4 T B and EB spectra under rotation . . . . . . . . . . . . . . . . . . . . . . 122 5.5 T E, EE, and BB spectra under rotation . . . . . . . . . . . . . . . . . . . 123 N OTATION ≺ The ‘precedes’ relation on a causal set. ≺∗ The ‘link’ relation on a causal set. d(x, y) The length of the longest chain between elements x and y on a causal set. x˙ The derivative of x with respect to time. A∗ , A† , ∗A The complex conjugate, adjoint, and Hodge dual of A, respectively. M4 4-dimensional Minkowski spacetime with metric ηµν . H3 The mass shell, momentum state space for massive particles. H30 The massless particle momentum state space. B The Bloch sphere polarisation state space. A metric signature (− + ++) is used throughout. Planck units with c = h = G = 1 and Boltzmann constant kB = 1 are used. 10  Chapter 1 I NTRODUCTION . . . I shall admit a system as empirical or scientific only if it is capable of being tested by experience. Karl Popper [1] Is there really a singularity inside a black hole? Did the universe have a beginning? These are just two of the many questions that can’t be answered until a theory of quantum gravity is developed. Quantum gravity isn’t demanded by any unexplained experimental results. The predictions of general relativity and quantum theory agree with experiments wherever they have been tested. The main motivation behind the long standing search for quantum gravity is a philosophical desire for a single unified approach to physics. General relativity and quantum theory provide entirely different world views: is the universe a dynamic four dimensional Lorentzian manifold, or is there a fixed background spacetime on which fields exist? Or is reality stranger than either of these options? Not only are our main theories of physics seemingly incompatible, they contain within themselves the signs of their own breakdown. The singularity inside a black hole is not so much a prediction of general relativity as an indication that we are attempting to use the theory in a realm where it doesn’t apply. Renormalisation and the associated problems in quantum theory suggest to me, at least, that a key part of our understanding is lacking. The search for quantum gravity has become not merely a search for a mathemat- ical method of tying the theories together, but a search for a completely new under- standing of the fundamental nature of spacetime. A number of approaches to quan- tum gravity exist. Most, including the prominent approaches of string theory and loop quantum gravity, give some preference to quantum theory, attempting to liter- ally ‘quantise gravity’ to fit it into a unified framework. Others, such as the causal 11  12 set approach that will be discussed in this thesis, begin with a completely new fun- damental structure and hope that general relativity and quantum theory will arise in the appropriate limits of the theory. No approach to quantum gravity, as yet, can be considered complete. This thesis will focus on the possibility that spacetime may be fundamentally dis- crete and, more importantly, whether such a hypothesis is testable. If the explanation of experimental results has not yet required any theory of quantum gravity, how are we to test the many theories that are being developed? If no effort is made to test developing theories they risk becoming only interesting mathematical constructs and losing any connection with a description of reality. As research areas expand and it becomes more and more difficult for an individual to comprehend an entire subject, it is crucial to remind ourselves that if we claim to be physicists we should be attempting to check our theories against the real world at every step. Since theories of quantum gravity must reproduce the results of general relativity and quantum theory in all areas where they have been shown to hold, and introduce new phenomena only on very small spacetime scales (or large energy scales), directly testing them is, at least in the near future, impossible. To test quantum gravity we need to seek ways in which such small effects could become amplified and result in deviations from the standard predictions of general relativity and quantum theory. It may seem strange to seek a signature for fundamental spacetime discreteness in large scale phenomena. Consider, however, the discreteness of matter. When Einstein provided an explanation for Brownian motion, he provided the last piece of evidence necessary to convince any doubters of the atomic nature of matter. That matter was made of atoms and molecules was, of course, not a new idea at that point. Einstein did not resolve the question of the discreteness of matter by directly observing atoms or molecules, but by recognising that the already observable phenomenon of Brownian motion demonstrated their existence. Likewise, to determine if spacetime is discrete we should not look to magical future technology that may let us see spacetime ‘atoms’ directly, but rather seek some currently observable phenomena that reveals the an- swer. Investigations into quantum gravity phenomenology have thus far focused over- whelmingly on producing and observing small violations of Lorentz invariance, often by introducing modified dispersion relations. As yet there is no evidence that Lorentz  13 invariance is violated at all. This is not to say that Lorentz violating quantum grav- ity phenomenology should not be explored: Lorentz violations can be investigated by current experiments and thus offer a very useful way of testing quantum gravity theories. It is often unclear, however, whether Lorentz violations are a necessary con- sequence of any particular quantum gravity theory, or simply a possible outcome that happens to be the only one that can currently be tested. Observations have forced very tight constraints on Lorentz violating models, for a recent review see [2]. Violations of Lorentz invariance can never be ruled out, as experiments can only constrain the effects to be smaller. While it is important to explore these constraints to experimen- tal limits, it is clear that Lorentz invariant quantum gravity phenomenology deserves more attention than it currently receives. This thesis attempts to begin to remedy this problem. Causal set theory, a Lorentz invariant, discrete theory of quantum gravity, provides the primary motivation for the phenomenology discussed in this thesis. Although the work in this thesis is discussed within the framework of causal set theory, it should be noted that many of the conclusions are more generally applicable. To investigate Lorentz invariant phenomenology, I focus on the effect spacetime discreteness would have on the propagation of particles. Particles travelling through a spacetime with an underlying discreteness are expected to deviate from the contin- uum geodesics predicted by general relativity. For massive particles this was first con- sidered by Dowker et al. [3], who found the propagation of massive particles could be described by a diffusion equation in the proper time of the particle. This thesis extends the work of Dowker et al. to derive a diffusion equation in observable lab- oratory time (‘cosmic time’) and also numerically investigates causal set models for massive particle propagation. A phenomenological model for massless particles trav- elling in discrete spacetime is also developed and astrophysical and cosmological data allow the free parameters in the model to be constrained. The models discussed here do not, unfortunately, make falsifiable predictions. Stronger and stronger constraints could be placed on the free parameters without ever ruling out the effects, unless, of course, there is an indisputable observation of Lorentz invariance violation. It is hoped that future developments in causal set theory will allow ‘natural’ values for the phenomenological parameters to be derived. Observations that conclusively rule out the models would then be possible. In the meantime the models help us determine  14 where to look for the signature of discreteness. The questions raised by the investiga- tions into phenomenology will hopefully also contribute to progress in the underlying theory. Chapter 2 introduces causal set theory and provides the basic definitions that will be required later in the thesis. Chapter 3 discusses the effect of spacetime discreteness on the propagation of massive particles. This chapter begins by reviewing the swerves diffusion model proposed by Dowker et al. in [3], while the remainder of the chapter consists of original work. In Chapter 4 massless particles are considered and found to also experience diffusion. Chapter 5 develops the work on massless particles further by considering the effect of discreteness on photon polarisation. Portions of Chapter 3, together with a large part of the content of Chapter 4 appear in F. Dowker, L. Philpott, and R. Sorkin. Energy-momentum diffusion from spacetime discreteness. Phys. Rev., D79:124047, 2009, 0810.5591. Section 3.6 appears in L. Philpott. Particle simulations in causal set theory. Class. Quantum Grav., 27:042001, 2010, 0911.5595. The majority of the work in Chapter 5 appears in C. Contaldi, F. Dowker, and L. Philpott. Polarisation Diffusion from Space- time Uncertainty, 1001.4545. Throughout this thesis Planck units defined by c = h = G = 1 will be used. Unless otherwise specified Boltzmann’s constant is also set to one: kB = 1.  Chapter 2 A BRIEF INTRODUCTION TO CAUSAL SET THEORY . . . in a discrete manifold the principle of metric relations is already con- tained in the concept of the manifold, but in a continuous one it must come from something else. Therefore, either the reality underlying space must form a discrete manifold, or the basis for the metric relations must be sought outside it, in binding forces acting upon it. Riemann (translated in [4]) The continuity of space apparently rests upon sheer assumption unsup- ported by any a priori or experimental grounds. Whitehead [5] Given that both general relativity and quantum theory are based on continuum spacetimes, it may seem slightly strange that spacetime discreteness is invoked in at- tempts to reconcile the two. Why throw away the one thing these theories have in common? In brief, the singularities and associated infinite curvatures of general rel- ativity could indicate that the continuum is not a good description of spacetime on very small scales; in quantum field theory discreteness would give a natural cutoff and allow issues with renormalisation to be avoided. Discrete spacetime is by no means a new idea. Riemann (as quoted above) points out the elegance of a discrete manifold: a discrete manifold actually contains more information than a continuum. Einstein, in view of the discrete nature of matter, sug- gested that “the continuum of the present theory contains too great a manifold of pos- 15  16 sibilities”. 1 Schrodinger ¨ also discusses the subject, stating that the continuum “may very well turn out to be out of place for physical space and physical time” [7]. A comprehensive review of the history of discrete spacetime would fill a thesis in and of itself; for an interesting discussion of early ideas see [8]. One further historical aspect should be mentioned briefly: the first Lorentz invariant ‘quantised spacetime’ was de- scribed by Snyder in 1946 [9]. It is, however, a rather different approach to the one discussed here. A considerable difficulty in formulating a discrete theory lies in the fact that many of the mathematical tools we are accustomed to using in the description of physics rely on the existence of the continuum. What is the analogue of a differential equation of motion in a discrete spacetime? Developing a discrete theory of spacetime involves not only new physics, but a new language to describe that physics. Causal set theory, first proposed by Bombelli et al. in 1987 [10], is a discrete, Lorentz invariant approach to quantum gravity. Although there are a number of approaches to quantum gravity that are in some sense discrete (Regge calculus, causal dynamical triangulations, loop quantum gravity, to name a few), the exact nature of the discrete- ness varies greatly. Causal set theory draws its primary motivation from theorems proved by Malament [11] and Levichev [12], and based on the results of Hawking, King, and McCarthy [13]: for a past and future distinguishing spacetime, the causal ordering together with a four-dimensional volume element provides sufficient infor- mation to determine all metric, topological, and differentiable structure of a spacetime. If spacetime is discrete, this result translates to (in a phrase coined by Sorkin): order + number = geometry. Number here is just the equivalent of volume – in a discrete spacetime the volume of a region is determined by counting the elements within it. In other words, a collection of elements endowed with a causal ordering should be all that is needed to describe all the complexities of spacetime to the discreteness scale. Early proposals for a theory based on discrete spacetime, endowed only with a causal ordering of events, were made independently by both Myrheim [14] and ’t Hooft [15] in 1978. These theories remained apparently undeveloped until causal set theory was put forward in 1987. For comprehensive reviews of the field see, for example, [16, 17, 18, 19, 20]. 1 Einstein to Walter D¨allenbach, November 1916, excerpts published in [6].  §2.1 What is a causal set? 17 2.1 What is a causal set? Specifically, a causal set is a set C endowed with a binary relation ≺ satisfying: 1. transitivity: if x ≺ y and y ≺ z then x ≺ z, ∀x, y, z ∈ C; 2. reflexivity: x ≺ x, ∀x ∈ C; 3. acyclicity: if x ≺ y and y ≺ x then x = y, ∀x, y ∈ C; 4. local finiteness: ∀x, z ∈ C the set {y | x ≺ y ≺ z} of elements is finite. The relation ≺ is commonly called ‘precedes’. This definition seems a little more in- tuitive if one considers, for a moment, a standard continuum Lorentzian manifold, M. The usual causal future relation, J + , satisfies the first three of the above points. The causal future J + (x), x ∈ M, is the union of x with the set of all y ∈ M that can be reached from x by a future-directed, non-spacelike curve in M. This relation is clearly transitive, reflexive (x ∈ J + (x)) and acyclic (provided the spacetime has no closed causal curves). The condition that the causal future relation does not satisfy is local finiteness. Local finiteness provides the discreteness of the theory – in any causal interval of the causal set there are only finitely many elements. Causal sets can be visualised using Hasse diagrams (see Figure 2.1). Here time runs vertically, elements are drawn as points and relations as lines. Relations implied by transitivity are omitted to avoid over-complicating the diagram. Even so, visualising a causal set using a Hasse diagram is only feasible for very small causal sets. Although Hasse diagrams will not be used in the remainder of this thesis, Figure 2.1 provides useful examples for the following important definitions. Let C be a causal set. 1. A chain is a totally ordered subset of C, e.g. α ≺ β ≺ ζ in Figure 2.1. 2. A longest chain between two elements x, y ∈ C is a chain whose length is longest amongst chains between those endpoints, e.g. α ≺ β ≺ δ ≺ ζ is a longest chain between α and ζ. There may be more than one longest chain between two ele- ments. The length (i.e. number of steps) of the longest chain between elements x, y ∈ C will be denoted d(x, y). For example d(α, ζ) = 3.  §2.2 The causal set hypothesis 18 ζ γ δ time ǫ β α η Figure 2.1: An example Hasse diagram of a 7 element causal set. 3. A link is an irreducible relation: elements x and y are linked if and only if d(x, y) = 1, e.g. α ≺ β is a link. If two elements x, y ∈ C are linked, it will be denoted x ≺∗ y. 4. A path is a chain consisting of links, e.g. α ≺∗ β ≺∗ γ. 2.2 The causal set hypothesis Above I provided a mathematical definition of a causal set. Causal set theory is based on the hypothesis that spacetime is a causal set (or, indeed, a quantum sum-over- causal sets). The observed continuum Lorentzian manifold, it is assumed, arises as an approximation to an underlying causal set. The partial order gives rise to the causal ordering of events in the approximating continuum spacetime, and the number of ele- ments comprising a spacetime region gives the volume of that region in fundamental units. The fundamental volume unit is expected to be of the order of the Planck vol- ume (where necessary in the later chapters of this thesis, it will be assumed that the fundamental volume is equal to the Planck volume). Note that any continuum man- ifold that approximates a causal set is necessarily Lorentzian, as only a Lorentzian metric can give rise to a partial order on spacetime points. The central, and as yet unproved, conjecture of causal set theory is that if two continuum manifolds are in some sense a ‘good approximation’ to a given causal set,  §2.3 Sprinklings 19 Figure 2.2: A diamond lattice no longer appears uniform when viewed in a boosted frame. then those two continuum manifolds should themselves be approximately the same. The idea here is simply that although multiple manifolds could approximate a given causal set, the manifolds should capture all the significant information from the causal set and only vary between themselves on very small scales. It is far from simple to mathematically define what is meant by ‘approximate’ in this situation. Some intuitive understanding can be gained by working backwards: constructing causal sets from manifolds via a process called ‘sprinkling’. 2.3 Sprinklings A causal set can be constructed by selecting points from a Lorentzian manifold. The causal relations between the points on the manifold induce the order relations between the elements in the causal set. The most obvious way to select points from a manifold is to construct a regular lattice, but this does not provide the features needed in causal set theory. Consider choosing a frame and selecting points from two dimensional Minkowski spacetime in a diamond lattice. If a Lorentz boost is applied the points are no longer a good sampling of the manifold, as is shown in Figure 2.2. In the boosted frame the number of elements in any region is clearly not a good approximation of the volume of that region. Consider, instead, selecting points at random from a manifold via a Poisson pro- cess in which the probability measure is equal to the spacetime volume measure in some fundamental units – this process is called ‘sprinkling’. The number of points chosen from any region of the manifold will be approximately equal (up to Poisson fluctuations) to the volume of that region in fundamental units. If a Lorentz boost is  §2.4 Causal set phenomenology 20 1 1.8 0.9 1.6 0.8 1.4 0.7 1.2 0.6 1 0.5 0.8 0.8 0.8 0.4 0.75 0.6 0.3 0.7 0.2 0.4 0.7 0.1 0.2 0.65 0 0.6 0.6 0.65 0.7 0.75 0.8 0 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.6 0.65 0.7 0.75 0.8 Figure 2.3: A causal set produced by sprinkling into a region of two dimensional Minkowski spacetime remains uniform density under a Lorentz boost. The empty regions in the boosted case are edge effects due to sprinkling into a finite region. applied to points selected this way, the distribution remains of uniform density (see Figure 2.3). For a proof that the sprinkling process is Lorentz invariant, see [21]. Sprin- klings prove very useful in analytic and numeric studies of causal sets – Chapter 3 will make use of them in both ways. Earlier in this section, the longest chain between two elements x, y in a causal set was defined. For a causal set generated by sprinkling into Minkowski spacetime, the length of the longest chain, d(x, y), is found to be proportional to the proper time between x and y in the limit of large distances [22]. The constant of proportionality is dependent on the spacetime dimension. A longest chain in a causal set is thus a close approximation to a timelike geodesic. For a numerical investigation of this correspondence in conformally flat spacetimes see [23]. 2.4 Causal set phenomenology As discussed in Chapter 1 the majority of current research in quantum gravity phe- nomenology focuses on violations of Lorentz invariance. Causal set theory, on the other hand, provides a way of investigating Lorentz invariant quantum gravity phe- nomenology. The difficulty of course, is that causal set theory is incomplete, and ob- taining testable predictions from an incomplete theory is far from simple. Although much will have to wait until the development of a quantum dynamics for causal sets, the concreteness and simplicity of the causal set hypothesis allow considerable progress to be made even at our early state of research.  §2.4 Causal set phenomenology 21 Before moving on in Chapter 3 to the particle phenomenology that forms the basis of this thesis, another significant result in causal set phenomenology should be men- tioned. As early as 1997 Sorkin [24] suggested that causal set theory could give rise to a cosmological constant of the order 10−120 – a number that matches current ob- servations (see also [25, 26]). This prediction for, in fact, a non-constant cosmological constant relies on the simple idea that a discrete spacetime leads to fluctuations in vol- ume. As discussed above, the sprinkling procedure places N points in a continuum √ volume V with fluctuations of the order N . Although there are questions that re- main to be addressed [27, 28] this model illustrates the powerful results that can be obtained from causal set theory.  Chapter 3 S WERVES quare etiam atque etiam paulum inclinare necessest corpora; nec plus quam minimum, ne fingere motus obliquos videamur et id res vera refutet. Lucretius, De Natura Rerum, ll. 243-245 Wherefore it is necessary that bodies swerve a little perpetually, Not more than a little, lest we seem to imagine sideway motions, and the facts confute our conjecture.1 General relativity predicts that a free, massive particle will travel along a timelike geodesic – a prediction well tested by observations. If the underlying structure of spacetime is discrete rather than continuous, particles will no longer be able to travel along perfect geodesics. Discreteness, one expects, will introduce some random fluc- tuations into a particle’s trajectory. Of course, given how well tested general relativity is, if such fluctuations exist they must be so small as to have escaped notice. Space- time discreteness is likely to occur on the Planck scale, so it is certainly reasonable to suppose that an effect on particle trajectories exists that has not yet been observed. In [3] Dowker et al. consider this problem in the context of causal set theory. Moti- vated by a simple model of particle motion on a causal set (see Section 3.1) they sug- gest that particles travelling through a discrete spacetime will be subject to swerves: random fluctuations in momentum. From a simple random walk on R we can derive the standard diffusion equation in the continuum limit: similarly, a basic stochastic process that gives random fluctuations in momentum is found in the continuum limit to lead to a diffusion equation. 1 Translation courtesy of Dr. R. M. Pollard, who notes that inclinare would be better translated as ’move back and forth’, which is somewhat less succinct yet even more fitting for our context. 22  23 The swerves diffusion equation given in [3] does not depend on any specific un- derlying particle model – nor, in fact, does it explicitly depend on the hypothesis that spacetime is a causal set. The power of the swerves diffusion equation lies in it be- ing the unique Markovian Poincar´e invariant diffusion equation on the state space. Any underlying process that gives small random fluctuations in momentum will be described by this equation in the continuum limit. A full derivation of the swerves diffusion equation, first stated in [3], is given in Section 3.2. The diffusion equation, as derived by Dowker et al. in terms of the proper time of the particle, is not the most useful form. To make contact with observations an equation in terms of an observable cosmic time is needed. In [3] Dowker et al. suggest the form of such an equation in the special case of a homogeneous distribution of particles. Here this work is extended and a full derivation of the inhomogeneous case is given in Section 3.3. I begin, however, by investigating models for particle propagation on causal sets more fully. Three models, including the model first described in [3], are discussed in Section 3.1 below. The connection between fundamental models and the diffu- sion equation is investigated in Section 3.4, where it is shown that the swerves model on the causal set gives rise to a finite nonzero diffusion constant. The work in Sec- tions 3.1,3.2,3.3 appears in [29] and together with Section 3.4 is work done in collabo- ration with Fay Dowker and Rafael Sorkin. Section 3.5 discusses the observable con- sequences of the swerves diffusion equation and reviews the bounds placed on the diffusion constant in [3, 30]. Lastly, the models introduced in Section 3.1 are investi- gated numerically in Section 3.6, and the behaviour is compared with that predicted by the swerves equation. The work in Section 3.5 appears in [31].  §3.1 Models for massive particle propagation on a causal set 24 3.1 Models for massive particle propagation on a causal set What is a particle in causal set theory? It must be admitted that this is not a question that can currently be answered. In general relativity matter determines spacetime cur- vature. In the context of a causal set this curvature must be encoded in the structure of the causal links – perhaps a particle is some pattern or knot of causal links? Of course, if the mass of the particle is a manifestation of a tangle of links between causal set elements, what about the other particle properties, the charge or spin? Approaching from the direction of quantum field theory we could assign amplitudes for a particle to be ‘on’ a particular causal set element and to move from one element to another. For work on developing quantum field theory on causal sets, see [32, 33]. An understanding of particles in causal set theory will really require a theory of quantum causal set dynamics (QCSD). If QCSD takes the form of a ‘sum-over-causal sets’, will a particle trajectory be some sum over all trajectories in all possible causal sets? Even without a theory of QCSD an intuition into the effect of discreteness can be gained by considering simple models. Massive particles will be considered as clas- sical point particles located ‘on’ a causal set element, a particle trajectory will con- sist of jumps from one element to another. The models discussed in this section are clearly not physically realistic, but they do capture the important aspects of the causal set approach and therefore provide one very useful way of investigating observable consequences of discrete spacetime. The models introduced in this section will be investigated numerically in Section 3.6. 3.1.1 The swerves model This model was introduced in [3]. Model 1 Construct a causal set C by sprinkling into Minkowski spacetime. A massive particle trajectory is taken to be a chain of elements en in the causal set, i.e. a linearly ordered subset of C. It is assumed that the trajectory’s past determines its future, but that only a certain proper time τf into the past is relevant. When this ‘forgetting time’ τf is small the process should be approximately Markovian. τf is also assumed to be much larger than the discreteness scale. The particle trajectory is constructed iteratively. Suppose  §3.1 Models for massive particle propagation on a causal set 25 t τf en+1 pn+1 x en pn en−1 Figure 3.1: The swerves model for particle propagation on a causal set. This illustration is not to scale: τf should be much greater than the discreteness scale. the particle is currently located ‘on’ an element en , with a four-momentum pn . The next element, en+1 is chosen such that • en+1 is in the causal future of en and within a proper time τf of en , • the momentum change |pn+1 − pn | is minimised. Here the momentum pn+1 is defined to be proportional to the vector between en and en+1 , and thus this model relies on knowledge of the embedding of the causal set into Minkowski spacetime. The constant of proportionality is chosen such that the mass of the particle remains fixed, i.e p2 = −m2 . This method for determining the trajectory is illustrated in Figure 3.1, where the frame has been chosen to be the rest-frame of the particle at en . Heuristically, the requirement that the momentum change be minimised creates a trajectory that stays as straight as possible at each step. If we work in the rest- frame of the particle, this method chooses the next element to be as close as possible to the t axis within the forgetting time. At each step there is a small random fluctuation in the momentum of the particle. There is no direct dependence on the mass of the particle in this model. Any nonzero mass will result in the same trajectory, although we could imagine that the forgetting time τf is mass dependent.  §3.1 Models for massive particle propagation on a causal set 26 This model is clearly not a realistic fundamental law of motion for a particle on a causal set. It requires information not contained in the causal set – the embedding of the elements into the approximating Minkowski spacetime. This issue is overcome in the ‘intrinsic’ models described below. All these models treat particles as classical and zero size. 3.1.2 Intrinsic models For a model of particle propagation to be intrinsic to a causal set, it cannot refer to a continuum forgetting time τf . Instead, a ‘forgetting number’ (an integer nf ) must be defined. The forgetting number can be interpreted as the number of discrete steps into the past of the trajectory that are relevant in determining the future trajectory (although in Model 2 below this number is in fact 2nf ). If the embedding into a con- tinuum spacetime was known, nf could be roughly written nf = τf /dpl , where dpl is the discreteness scale. It is important to note, however, that the models are defined on a general causal set and it is not assumed here that the underlying causal set can be faithfully embedded into a ‘reasonable’ spacetime. In defining these intrinsic models there is no reference to the mass of the particle. As above, the forgetting number could be taken to be mass dependent. The models discussed cannot, however, be considered as models of particle propagation for mass- less particles – these models give trajectories whose long time behaviour approximates a timelike geodesic, rather than the null geodesic appropriate to massless particles. Two slightly different intrinsic models will be described in this section, to give an idea of the wealth of possibilities available. First recall some causal set definitions from Chapter 2: • d(a, b) denotes the length of the longest chain between two elements a and b, • a path is a chain consisting of links a ≺∗ b, i.e. d(a, b) = 1. Model 2 A massive particle trajectory is taken to be a chain of elements . . . en−2 ≺ en−1 ≺ en . . .. Given a partial particle trajectory . . . en−1 , en the next element en+1 is chosen such that • d(en−1 , en+1 ) ≤ 2nf ,  §3.1 Models for massive particle propagation on a causal set 27 • d(en , en+1 ) is maximised subject to d(en , en+1 ) ≤ nf , (see Figure 3.2(a)). These requirements do not guarantee the existence of a unique en+1 . There will, however, almost surely be finitely many eligible elements and the trajectory can be constructed by choosing an element uniformly at random from these. Note that this model is slightly different from the first intrinsic model presented in [29] (where equalities in the above conditions were given). Model 1 of [29] does not guar- antee the existence of an en+1 under reasonable conditions. Under this model the particle trajectory should swerve a little, but remain approx- imately straight so long as nf is large. It is easiest to see this if we consider the ‘ideal’ case where there exist en−1 , en , en+1 such that d(en−1 , en ) = nf and d(en , en+1 ) = nf . The elements have been chosen such that d(en−1 , en+1 ) ≤ 2nf ; since d(en−1 , en ) = nf and d(en , en+1 ) = nf we know there exists some chain between en−1 and en+1 of length 2nf and hence the equality d(en−1 , en+1 ) = 2nf must hold. In other words, the chain we have chosen between en−1 and en+1 is a longest chain, and thus the trajectory is approximately geodesic over {en−1 , en+1 } segments. In practice it will not always be possible to choose an en+1 such that d(en , en+1 ) = nf and d(en−1 , en+1 ) = 2nf , neces- sitating the inequalities. In this model the trajectory can be considered as composed of just the elements . . . en−1 , en , en+1 . . . or of the ‘filled in chain’ consisting of a (randomly chosen) longest chain (of length ≤ nf ) between en−1 and en , another between en and en+1 , and so on. Variations on this model can also be constructed. One possibility is to choose the forgetting number at random at each step from a distribution with a mean nf and some fixed variance. Model 3 The trajectory is constructed as a path in this model, i.e. d(en , en+1 ) = 1 for any en , en+1 . Given a partial particle trajectory . . . en−nf , . . . , en−1 , en the next element en+1 is chosen such that • d(en , en+1 ) = 1, • d(en−nf , en+1 ) + . . . + d(en−1 , en+1 ) + d(en , en+1 ) is minimised, (see Figure 3.2(b)). Note that this minimisation does not necessarily yield a unique en+1 , in which case we construct the trajectory by choosing an element uniformly at  §3.1 Models for massive particle propagation on a causal set 28 en+2 en+2 d(n + 1, n + 2) ≤ nf d(n + 1, n + 2) = 1 en+1 d(n, n + 2) ≤ 2nf en+1 d(n, n + 1) ≤ nf d(n, n + 1) = 1 d(n − 1, n + 1) ≤ 2nf en en d(n − 1, n) ≤ nf d(n − 1, n) = 1 d(n − 1, n + 2) ≥ 3 en−1 en−1 (a) Model 2. (b) Model 3. Figure 3.2: Trajectories constructed using intrinsic models. random from those eligible. Also, if the trajectory has length less than nf the minimi- sation is done over all elements available. In this model each element is linked to the previous, i.e. d(en−1 , en ) = 1, and thus we know there exists a chain (our trajectory) of length nf between en−nf and en . The longest chain length, d(en−nf , en ), must therefore be greater than or equal to nf . If we choose en to minimise d(en−nf , en ) we ask that the trajectory be as close as possible to geodesic between en−nf and en while fulfilling d(en−1 , en ) = 1. Minimizing the sum of the partial lengths distributes the geodesic property along the path.  §3.2 The swerves diffusion equation 29 3.2 The swerves diffusion equation Although the intrinsic models described in Section 3.1 are defined on any causal set, the process is really only of interest when the causal set is a good approximation to Minkowski spacetime (or another physically relevant spacetime, such as Friedmann- Robertson-Walker). The universe we live in is approximately Minkowskian, and thus even though the behaviour of particles on nonembeddable causal sets may be very in- teresting, it is of little relevance when investigating the observable phenomenology of discrete spacetime. If it is assumed that the causal set can be produced by sprinkling into Minkowski spacetime, the particle models can be thought of as defining piece- wise linear curves in Minkowski spacetime. The particle’s momentum – which cannot (currently) be defined in a manner intrinsic to the causal set – is then defined every- where on the curve except at the vertices. It is clear here that in the continuum limit the intrinsic models lead to random fluctuations of momentum (defined, as usual, in Minkowski spacetime). Rather than choosing a specific particle model on a causal set and investigating its continuum behaviour, an entirely general situation will be considered. Motivated by the models for particle trajectories on a causal set, it is assumed that there exists some underlying Markovian, Poincar´e invariant process that causes random fluctuations in momentum (and consequently position). The process takes place in the proper time of the particle. Note that the models described above are approximately Markovian when the forgetting time (number) is small. If the discreteness scale is of the order of the Planck scale, the forgetting time can be many orders of magnitude greater than the discreteness scale and yet still small compared to the trajectory length. 3.2.1 Stochastic evolution on a manifold of states The swerves equation is obtained using the general formalism of [34] for stochastic evolution on a manifold of states. Chapters 4 and 5 also draw on this framework, and thus for completeness the main points in this work will be discussed here. Consider a ‘mesoscopic’ process occurring on a manifold of states M. The current state of the system is described by a probability density ρ. It is assumed that the future of the physical system depends only on its present state and not its past, i.e. the evo- lution is a Markov process. It is also assumed that the system traces a continuous path  §3.2 The swerves diffusion equation 30 through M. Such a process can be described by a linear, first order in time equation ∂ρ = K AB ∂A ∂B ρ + B A ∂A ρ − Cρ , (3.1) ∂T where ρ = ρ(x, T ), the probability density for the system to be in a state x ∈ M at time T , is a scalar density in M. The indices A, B = 1, . . . , dim(M). K AB = K BA , B A , C are functions of x (and possibly T ). For extensive justification of this assumption see [34]. Note that if Equation 3.1 were to contain higher order terms it would allow unphysical negative probability densities, ρ < 0, proof of this is given in Appendix A of [34]. The requirement that the probability density ρ be nonnegative leads to a constraint on K AB in the above, general equation. Suppose the initial distribution, in some neighbourhood of the origin, is given by 1 ρ(x, 0) = ZAB xA xB , (3.2) 2 where Z must be positive by the requirement ρ ≥ 0. Clearly ρ(0, 0) = 0 and in order for ˙ 0) ≥ 0. From Equation 3.1 we can conclude ZAB K AB ≥ ρ(0, T ) ≥ 0 we must have ρ(0, 0. Suppose ZAB = zA zB . This implies zA zB K AB ≥ 0, i.e. K AB is a (symmetric) positive semidefinite matrix (a matrix A is positive semidefinite if xT Ax ≥ 0 ∀x ∈ Rn ). We can also impose the conservation of probability Z ρ˙ dx = 0 . (3.3) With ρ given by Equation 3.1, integration by parts implies C = ∂A ∂B K AB − ∂A B A . Substituting this expression back into Equation 3.1 gives ∂ρ = K AB ∂A ∂B ρ + B A ∂A ρ − ∂A ∂B K AB − ∂A B A ρ
∂T = ∂A ∂B (K AB ρ) + ∂A (B A ρ) − 2∂A ∂B K AB ρ − 2∂B K AB ∂A ρ = ∂A ∂B (K AB ρ) + ∂A (B A ρ) − 2∂A (∂B K AB ρ). (3.4) Equation 3.1 can therefore be written in terms of a current, J A , and a continuity  §3.2 The swerves diffusion equation 31 equation: J A = −∂B (K AB ρ) + v A ρ , (3.5) ∂ρ = −∂A J A , (3.6) ∂T where v A = 2∂B K AB − B A . Note that J A is a vector density (see pg. 124 of [34]) and the vector density transformation implies that while K AB is a tensor, v A is not. The above equations can be expressed in terms of a true vector uA if the density of states, n, is introduced. The vector uA is defined by uA = v A − ∂B K AB − K AB ∂B s , (3.7) where s is the entropy scalar s = kB log n . (3.8) Here kB is the Boltzmann constant (note that in the sections that follow kB = 1). Equations 3.5 and 3.6 can then be written in the form ∂ρ = ∂A K AB DB ρ − uA ρ ,
(3.9) ∂T where DA ρ = ∂A ρ − ρ∂A s . (3.10) In the sections that follow, it will often be useful to use the following relations for K AB and v A : ∆xA ∆xB   AB K = lim , (3.11) ∆T →0 2∆T ∆xA   vA = lim . (3.12) ∆T →0 ∆T Here <> denotes the expectation value. These equations give a way of relating the abstract objects K AB and v A to the basic properties of the physical stochastic process – the ‘spatial’ step ∆x and the time step ∆T . Equations 3.11 and 3.12 are derived in Appendix B of [34].  §3.2 The swerves diffusion equation 32 3.2.2 Diffusion in proper time The state space, M, of a swerving particle of mass m is M = M4 × H3 , where H3 is the mass shell. A point on M thus represents a position in M4 and a momentum in H3 . The coordinates on M4 are the usual Cartesians {xµ }, µ = 0, 1, 2, 3 and indices are raised and lowered with ηµν , the Minkowski metric. The spatial coordinates on M4 will be written as {xi }. Cartesian coordinates in momentum space are pµ , where pµ is subject to the constraint that it lies on the mass shell, i.e. the hyperboloid H3 in momentum space defined by pµ pµ +m2 = 0. p0 = E is the energy (taken to be positive) p and p = p21 + p22 + p23 is the norm of the three momentum. The three coordinates on H3 will be written abstractly as pa . Coordinates on M are denoted collectively as X A = {xµ , pa } and in what follows capital letters A, B will be used to indicate general indices on M; µ, ν = 0, 1, 2, 3 are indices on M4 ; i, j = 1, 2, 3 are spatial indices on M4 ; a, b = 1, 2, 3 are indices on H3 . The metric on M is the product of the Minkowski metric ηµν on M4 and the Lobachevski metric gab on H3 . This is the unique Poincar´e invariant metric on M (up to an overall constant). The ‘density of states’, n, plays a role in the formalism of [34], and by symmetry, it must be proportional to the volume measure on M, so √ n ∝ g where g = det(gab ). As described above, a process that undergoes stochastic evolution on a manifold of states, M, in time parameter T , can be described by a current, J A and a continuity equation [34]: J A = −∂B K AB ρ + v A ρ ,
(3.13) ∂ρ = −∂A J A , (3.14) ∂T or alternatively by the equation ∂ρ  AB ρ A  = ∂A K n ∂B −u ρ , (3.15) ∂T n where uA = v A − ∂B K AB − K AB ∂B s . (3.16) To find the diffusion equation for the swerves particle process, it is necessary to deter-  §3.2 The swerves diffusion equation 33 mine K AB and uA . The requirement that the equation be Poincar´e invariant is a very stringent condition and allows K AB and uA to be determined up to the choice of one constant parameter. Consider the process in terms of τ , the proper time along the worldline of the particle. From Equation 3.11 in Section 3.2.1 ∆X A ∆X B   AB K = lim . (3.17) ∆τ →0+ 2∆τ Recall K AB is a symmetric, positive semidefinite matrix, and here is also required to be Poincar´e invariant. Looking first at the spacetime component of this matrix, ∆xµ ∆xν   µν K = lim . (3.18) ∆τ →0+ 2∆τ 1 µ 1 ∆xµ = m p ∆τ at every step of the process and so K µν = 2 lim∆τ →0 pµ pν ∆τ = 0. Given K µν = 0, K µa = 0 is required by the condition that K AB be positive semidef- inite. To see this, first recall that a matrix is positive semi-definite if all the principal minors of the matrix are nonnegative. Consider a symmetric 2x2 matrix of the form   0 a  . (3.19) a b To be positive semi-definite the matrix must have −a2 ≥ 0, i.e. a = 0. Similarly, beginning with   0 K µa K AB =  , (3.20) K aµ K ab we can consider all the 2 × 2 principal minors and determine K µa = K aµ = 0. The only Lorentz invariant tensor on H3 is proportional to the metric, gab , and the coefficient is independent of xµ by translation invariance, thus K ab = kg ab , where k > 0 is a constant. This gives   0 0 K AB =  . (3.21) 0 kgab  §3.2 The swerves diffusion equation 34 To determine uA first recall Equation 3.12: ∆X A   A v = lim . (3.22) ∆τ →0+ ∆τ The spacetime component ∆xµ   µ v = lim , (3.23) ∆τ →0+ ∆τ is simply v µ = pµ /m. The components of the true vector uµ are equal to v µ since K µA = 0. There is no Lorentz invariant vector on H3 and so ua = 0, giving uA = (pµ /m, 0) . (3.24) The proper time diffusion equation can now be written down from Equation 3.15:    ∂ρτ ab √ ρτ 1 = k ∂a g g∂b √ − pµ ∂µ ρτ . (3.25) ∂τ g m √ This can be seen to be equivalent to Equation 1 of [3] if a scalar ρ¯ = ρτ / g is defined: ∂ ρ¯ 1 = k ∇2H ρ¯ − pµ ∂µ ρ¯ , (3.26) ∂τ m where ∇2H is the Laplacian on H3 .  §3.3 Diffusion in cosmic time 35 3.3 Diffusion in cosmic time Given an initial distribution of particles, for instance from an astronomical source, the equation derived above is not very useful for predicting the results of observations. Even if particles all leave the source at the same time with the same momentum, the momentum variation induced by the swerves will result in particles arriving after dif- ferent proper times and at different observatory times. The proper time that elapses along the particles’ worldlines from source to detector is not observable. To compare the swerves model with experiment and observation it is necessary to describe the evolution of the distribution in time in the rest frame of our detector, which time will be referred to as cosmic time. One may ask why the equation was derived in terms of proper time if it is more useful to obtain a cosmic time equation. The reason be- comes clear if one considers that when working with proper time, the explicit Lorentz invariance makes it simple to determine K AB and uA . The result can then be easily transformed to a specific frame, as shown here. A first step in this direction was to look at the nonrelativistic limit of the proper time diffusion equation, when proper time and cosmic time are comparable. The non- relativistic limit in fact proves sufficient to place very strong bounds on the value of the diffusion constant and severely limit any observable effects (see [3] and [30] and Section 3.5). In the fully relativistic case, Dowker et al. wrote down the diffusion equation in terms of cosmic time for the special case of an initially spatially homogeneous distri- bution [3]. The derivation of the cosmic time evolution equation for the general case of a spatially inhomogeneous distribution will be given here. The conversion between proper time and cosmic time is possible because both are good time parameters along all possible particle worldlines, which are causal. If the diffusion process is visualised as a collection of worldlines through spacetime and momentum space, both cosmic time, t = x0 , in our chosen frame and proper time τ increase monotonically along each trajectory. Assume that the particle starts at param- eter τ = 0 and cosmic time t = 0. Proper time can be added to the state space and the process is represented by flowlines in M′ = M4 × H3 × R (see Figure 3.3). Along each flowline, both τ and t are good time parameters. The proper time diffusion equation, Equation 3.25, describes the evolution of the distribution on constant τ hypersurfaces  §3.3 Diffusion in cosmic time 36 t onstant τ onstant t τ x Figure 3.3: Particle trajectories as flowlines in M′ = M4 × H3 × R (where the momentum and two spatial dimensions have been suppressed). in M′ . What is needed is a diffusion equation for evolution of the distribution on constant t hypersurfaces integrated over all proper times. First, working in the larger space M′ , define a new current component J τ (t, xi , pa , τ ) = ρτ . (3.27) If coordinates on this extended space M′ = M × R, are denoted by X α = {X A , τ } then the continuity equation (3.6) can be written ∂α J α = 0 . (3.28) Using Equation (3.5) (and still treating τ as the time parameter) the t component of the current can be expressed in terms of J τ (equivalently ρτ ). J t (t, xi , pa , τ ) = −∂B K tB J τ + v t J τ
= vt J τ = γJ τ , (3.29) where γ = ∂t/∂τ is the usual relativistic gamma factor.  §3.3 Diffusion in cosmic time 37 The remaining components of the current can now be written in terms of J t . The spatial components are: J i (t, xi , pa , τ ) = −∂B K iB J τ + v i J τ
= vi J τ pi J t = . (3.30) m γ In the case of the p components the algebra is simpler if Equation 3.5 is expressed in the form (see Equation 3.15) ρ J A = −K AB n ∂B + ρuA , (3.31) n and so Jt   J a (t, xi , pa , τ ) = −kgab n ∂b γn  t  √ J = −kg ab g ∂b √ . (3.32) γ g The metric gab that appears here is the Lobachevski metric on H3 . Since τ is unobservable we need to integrate J over τ : the current J describes the flow through a region ∆t∆x∆p at any point (t, xi , pa , τ ); what is needed is the cumulative flow through ∆t∆x∆p at any given (t, xi , pa ) over all proper times. The ¯ Integrating the t component of the current over integrated current will be denoted J. proper time from zero to infinity gives the probability density on a hypersurface of constant t: ρt = J¯t (xi , pa , t) Z ≡ J t dτ . (3.33)  §3.3 Diffusion in cosmic time 38 The components of the new current can be written: Z J¯i (xi , pa , t) ≡ J i dτ pi J t Z = dτ mγ pi J¯t = m γ pi ρt = , (3.34) mγ Z J¯a (xi , pa , t) ≡ J a dτ  ¯t  J ab = −kg n∂b γn   ρt = −kgab n∂b . (3.35) γn Integrating the continuity equation over τ gives [J τ ]∞ ¯t ¯i ¯a 0 + ∂t J + ∂i J + ∂a J = 0 . (3.36) J τ |τ =0 is zero for all t > 0 and J τ tends to zero as τ goes to infinity for finite t. So for all t > 0 ∂t J¯t + ∂i J¯i + ∂a J¯a = 0 . (3.37) Finally, substituting Equations 3.34 and 3.35 into Equation 3.37 and recalling J¯t = ρt , gives the cosmic-time diffusion equation pi    ∂ρt ab √ ρt =− ∂i ρt + k ∂a g g∂b √ . (3.38) ∂t mγ γ g This is a powerful phenomenological model because it depends on only one parame- ter, the diffusion constant k. Data can therefore strongly constrain k.  §3.4 Finiteness of the diffusion constant 39 3.4 Finiteness of the diffusion constant The swerves diffusion equation derived in Section 3.2 is independent of the details of the underlying particle model. To make clearer how the underlying model on the causal set gives rise to a diffusion equation on M4 × H3 the swerves model (Model 1) will be investigated more fully. Specifically, in this section it will be shown that Model 1 gives rise to a finite nonzero diffusion constant in the macroscopic limit. Recall that a process undergoing stochastic evolution on a manifold of states can be described by J A = −∂B K AB ρ + v A ρ ,
(3.39) ∂ρ = −∂A J A , (3.40) ∂T where K AB is a symmetric, positive semi-definite matrix given by ∆X A ∆X B   AB K = lim . (3.41) ∆T →0+ 2∆T In Section 3.2 it was shown that due to the constraint of Lorentz invariance K AB has the form:   0 0 K AB =  , (3.42) 0 kgab where k is some constant, irrespective of the underlying model. It is not immediately clear, however, that the constant of proportionality, k, need be finite and nonzero for any particular model. For the specific case of the swerves model, the component K ab is indeed nonzero and finite (and thus K AB is given by Equation 3.42). Demonstrating this is the task of this section. In the swerves model there are two length scales: the discreteness scale dpl , and the forgetting time τf , where τf >> dpl . The trajectory length (or total proper time of the particle) must also be much greater than τf . It is only in the continuum limit dpl → 0 that the model can be described by a diffusion equation. Suppose, for the swerves model, that K ab = kgab . From Equation 3.41 we also have ∆pa ∆pb   ab K = lim . (3.43) ∆τ →0+ 2∆τ  §3.4 Finiteness of the diffusion constant 40 Here ∆pa is the change in the momentum component pa in a single swerves step and ∆τ is the change in proper time for that step. Note that in the swerves model ∆τ is not constant for each step: it is a random variable in the range 0 < ∆τ < τf . We wish to determine K ab as dpl → 0. First note the following two lemmas for the swerves model: D E ∆pa ∆pb Lemma 3.4.1. If dpl → 0 and τf is fixed then 2∆τ → 0. D E ∆pa ∆pb Lemma 3.4.2. If dpl is fixed and τf → 0 then 2∆τ → ∞. Section 3.4.1 below contains the proofs of these lemmas. From these two lemmas it is clear that as we take the continuum limit dpl → 0 we can also take τf → 0 (and D a bE thus ∆τ → 0) in such a way that ∆p2∆τ ∆p remains fixed at some finite nonzero value. Thus, if K ab = kgab the diffusion constant k is finite and nonzero, i.e. the swerves model results in the diffusion equation. 3.4.1 Proofs of Lemmas 3.4.1 and 3.4.2 Hyperbolic coordinates For ease of calculation, hyperbolic coordinates (η, ξ, θ, φ) in four dimensional Minkowski spacetime can be defined: η 2 = t2 − r 2 , (3.44) r  ξ = tanh−1 , (3.45)  zt θ = arccos , (3.46) ry  φ = arctan , (3.47) x p where r = x2 + y 2 + z 2 . These coordinates are illustrated in Figure 3.4: η defines hyperbolae and ξ defines radial directions, varying from ξ = 0 on the t-axis to ξ = ∞ at t = r. Calculating volumes in hyperbolic coordinates Before addressing the Lemmas 3.4.1 and 3.4.2 it is useful to calculate the volume of the shaded region shown in Figure 3.4: the volume of 4-dimensional Minkowski space- time such that 0 < η < η ′ and 0 < ξ < ξ ′ . The first step is to determine the vol- ume element in coordinates (η, ξ, θ, φ). It is easiest to begin with spherical coordinates  §3.4 Finiteness of the diffusion constant 41 Ξ=Ξ' Η=Η' Figure 3.4: An illustration of hyperbolic coordinates η and ξ, with the angular dimensions suppressed. (t, r, θ, φ) defined by t = t, (3.48) x = r sin θ cos φ , (3.49) y = r sin θ sin φ , (3.50) z = r cos θ. (3.51) The spherical coordinates can then be expressed in terms of hyperbolic coordinates t = η cosh ξ , (3.52) r = η sinh ξ , (3.53) θ = θ, (3.54) φ = φ. (3.55)  §3.4 Finiteness of the diffusion constant 42 The Jacobian determinant for this change of variables is    ∂t ∂t ∂t ∂t   ∂η ∂ξ ∂θ ∂φ     ∂r ∂r ∂r ∂r  =  ∂η ∂ξ ∂θ ∂φ   det J  ∂θ  ∂η ∂θ ∂θ ∂θ  ∂ξ ∂θ ∂φ     ∂φ ∂φ ∂φ ∂φ    ∂η ∂ξ ∂θ ∂φ    cosh ξ η sinh ξ 0 0       sinh ξ η cosh ξ 0 0    =     0 0 1 0       0 0 0 1  = η cosh2 ξ − η sinh2 ξ = η. (3.56) The volume element is thus given by dt dx dy dz = r 2 sin θ dt dr dθ dφ = η 3 sinh2 ξ sin θ dη dξ dθ dφ . (3.57) The volume of the shaded region is Z 2π Z π Z ξ′ Z η′ V = η 3 sinh2 ξ sin θ dη dξ dθ dφ 0 0 0 0 Z ξ ′ Z η′ = 4π η 3 sinh2 ξ dη dξ 0 0 Z ξ′ 4 = πη ′ sinh2 ξ dξ 0 ′ sinh 2ξ ′  ξ ′4 = πη − + . (3.58) 2 4 This result will be used repeatedly in the calculations below. To prove Lemmas 3.4.1 and 3.4.2 the following property of sprinklings is required: if points are sprinkled with a density ρ into a region, the probability that n points are sprinkled into a volume V is given by the Poisson distribution (ρV )n e−ρV Prρ = . (3.59) n!  §3.4 Finiteness of the diffusion constant 43 For example, the probability that no points are sprinkled into the shaded region in Figure 3.4 is simply Prρ = e−ρV  ′ sinh 2ξ ′   ′4 ξ = exp −ρπη − + . (3.60) 2 4 To prove Lemmas 3.4.1 and 3.4.2 it is necessary to calculate the expectation value D E ∆pa ∆pb 2∆τ . In fact, since it is known that K ab ∝ gab it suffices to calculate a single component, K pp . Working in terms of polar coordinates {p, θ, φ} on H3 , the metric gab is m2 ds2 = dp2 + p2 (dθ 2 + sin2 θ dφ2 ) . (3.61) m2 + p 2 Consider a swerves trajectory in the rest frame of the particle after n steps (see Fig- m2 +p2 ure 3.1). In this frame the metric component gpp = m2 = 1 and thus K pp = k. Since the three momentum pn is zero, the change in momentum in the next step is simply pn+1 . Working in Cartesian coordinates for a moment, the momentum at step en+1 is proportional to the coordinates X µ of the point en+1 in this frame: pµn+1 = m µ ηX , where η is the proper time between en and en+1 . Thus the magnitude of the three momentum is given by q pn+1 = p21 + p22 + p23 m q = X12 + X22 + X32 η m = r η = m sinh ξ. (3.62) The expectation value   τf ∞   ∆p∆p ∆p∆p ∆p∆p Z Z = P dη dξ, (3.63) 2∆τ 0 0 2η 2η   ∆p∆p where P = P 2η is the probability of the particular value occurring. The proba- bility P is the probability that the element en+1 has (hyperbolic) coordinates η ′ < η < η ′ + dη, ξ ′ < ξ < ξ ′ + dξ. Recall that in the swerves model en+1 is chosen  §3.4 Finiteness of the diffusion constant 44 such that η < τf and the momentum change is minimised. P is thus the probability that there are no points in the region {η < τf , ξ < ξ ′ } and there is a point in the region {η ′ < η < η ′ + dη, ξ ′ < ξ < ξ ′ + dξ}. The probability that there are no points in the region {η < τf , ξ < ξ ′ } is simply ξ ′ sinh 2ξ ′    exp −ρπτf4 − + , (3.64) 2 4 from Equation 3.60 above. The probability that there is a point in the region {η ′ < η < η ′ + dη, ξ ′ < ξ < ξ ′ + dξ} is 3 4πρη ′ sinh2 ξ ′ dη dξ , (3.65) from the integral in Equation 3.58 and Equation 3.59 (the exponential term can be neglected as the volume is very small). The expectation value is therefore (dropping the primes on ξ and η)   τf ∞    ∆p∆p ξ sinh 2ξ Z Z 2 2 4 = 2πρm η sinh ξ exp − + −ρπτf4 dη dξ 2∆τ 0 0 2 4 Z ∞    2 2 3 4 4 ξ sinh 2ξ = πρm τf sinh ξ exp −ρπτf − + dξ . (3.66) 3 0 2 4 Integrating by parts, this reduces to 4m2 ∞      ∆p∆p ξ sinh 2ξ Z = cosh ξ sinh ξ exp −ρπτf4 − + dξ . (3.67) 2∆τ 3τf 0 2 4 The proofs of the Lemmas now follow without difficulty. Proof of Lemma 3.4.1 If we take the discreteness scale dpl to zero, while keeping the forgetting time, τf , fixed, then D a bE ∆p ∆p 2∆τ → 0. Taking dpl → 0 is equivalent to taking the sprinkling density ρ to infinity. To deter- mine the effect of ρ → ∞ on Equation 3.67, the integral can be divided into two parts. There exists some ξ¯ such that, for all ξ > ξ, ¯ cosh ξ < 2 sinh ξ. Let   ∆p∆p = I1 + I2 , (3.68) 2∆τ  §3.4 Finiteness of the diffusion constant 45 where 4m2 ∞    ξ sinh 2ξ Z I1 = cosh ξ sinh ξ exp − + −ρπτf4 dξ , 3τf ξ¯ 2 4 Z ¯ 4m2 ξ    4 ξ sinh 2ξ I2 = cosh ξ sinh ξ exp −ρπτf − + dξ . (3.69) 3τf 0 2 4 I1 can be written 4m2 ∞    ξ sinh 2ξ Z 2 I1 ≤ 2 sinh ξ exp − + −ρπτf4 dξ 3τf ξ¯ 2 4 ∞ 4m2 −2     4 ξ sinh 2ξ ≤ exp −ρπτf − + 3τf ρπτf4 2 4 ξ¯ 8m 2   ¯ ξ sinh 2ξ¯ ≤ 5 exp −ρπτf4 − + . (3.70) 3ρπτf 2 4 Thus lim I1 = 0 . (3.71) ρ→∞ Now consider the integral I2 and the standard result Z a |f (x)|dx ≤ max |f (x)|(b − a). (3.72) b x∈[a,b] R ξ¯ Let I2 = 0 f (ξ)dξ, noting that f (ξ) > 0. For any ǫ > 0 there exists some ρ large enough that f (ξ) < ǫ/ξ¯ for all 0 ≤ ξ ≤ ξ. ¯ Thus Z ξ¯ I2 = f (ξ)dξ 0 ǫ¯ ≤ ξ ξ¯ < ǫ. (3.73) I2 therefore also tends to zero in the limit ρ → ∞. Thus in the limit ρ → ∞, the D E expectation value ∆p∆p 2∆τ → 0, proving Lemma 3.4.1.  §3.4 Finiteness of the diffusion constant 46 Proof of Lemma 3.4.2 D E ∆pa ∆pb If we take the forgetting time τf → 0 and keep the discreteness scale dpl fixed then 2∆τ → ∞. First note that for all ξ, cosh ξ ≥ sinh ξ. The expectation value, Equation 3.67, can therefore be written 4m2 ∞      ∆p∆p ξ sinh 2ξ Z 2 4 ≥ sinh ξ exp −ρπτf − + dξ 2∆τ 3τf 0 2 4 ! ∞ 4m2   −1 4 ξ sinh 2ξ ≥ exp −ρπτf − + 3τf ρπτf4 2 4 0 4m2 ≥ . (3.74) 3ρπτf5 D E ∆p∆p Thus in the limit τf → 0, the expectation value 2∆τ → ∞ proving Lemma 3.4.2.  §3.5 Consequences and bounds 47 3.5 Consequences and bounds It is important to reiterate that the swerves diffusion equation is very general: any Lorentz invariant process for massive particles that results in fluctuations in energy- momentum will be governed by this equation. Deriving such an equation is of lit- tle use, however, unless we investigate the observable consequences. Ideally, a new phenomenological model should agree with all current observational data and, in ad- dition, make predictions for new, feasible, experiments. In reality, models will al- ways contain free parameters that limit their predictive powers. The swerves diffu- sion equation contains only one free parameter, the diffusion constant k, and is thus a strong phenomenological model. The usefulness of this model is unfortunately still limited. It will never be possible to prove that swerves do not occur – null obser- vations only allow the parameter k to be constrained to smaller values. It is hoped, of course, that experiments will actually observe the swerves effect; or that swerves will provide an explanation for some already observed, unexplained phenomena. The difficulty is (as discussed in the Introduction) that quantum gravitational effects are inherently very small and to make positive observations we must think of ways in which these tiny effects could have been amplified. For swerves the first step is thus to consider how the magnitude of the diffusion constant can be constrained by current data. Strong constraints have been placed on k by both Dowker et al. [3] and Kaloper and Mattingly [30]. These constraints are reviewed in this section. Lacking the in- spiration of an astrophysical or cosmological problem swerves may solve, it seems unnecessary to seek tighter bounds at this stage. For simplicity and consistency with [3, 30] I will work in terms of a scalar dis- √ tribution ρ¯ = ρt / g rather than the scalar density ρt . In terms of the scalar ρ¯ the homogeneous cosmic time diffusion equation becomes:    ∂ ρ¯ k ab √ ρ¯ = √ ∂a g g∂b . (3.75) ∂t g γ The swerves model predicts the spontaneous heating of a nonrelativistic gas. To see this, consider the nonrelativistic limit of Equation 3.75. Working in polar coordi-  §3.5 Consequences and bounds 48 nates on H3   m2 0 0  m2 +p2  gab =  0 p2 0 , (3.76)     0 0 p2 sin2 θ and thus    ∂ ρ¯ k ab √ ρ¯ = = √ ∂a g g∂b ∂t g γ p " p !! k m2 + p2 ∂ p2 sin θ m2 + p2 ∂ m¯ ρ = mp2 sin θ p ∂p m ∂p m2 + p 2 !! ∂ m sin θ ∂ m¯ρ + p p ∂θ m + p ∂θ 2 2 m + p2 2 !!# ∂ m ∂ m¯ρ + p p . (3.77) ∂φ sin θ m2 + p2 ∂φ m2 + p 2 In the nonrelativistic limit p << m this becomes    ∂ ρ¯ k ∂ 2 ∂ ρ¯ = p sin θ ∂t p2 sin θ ∂p ∂p     ∂ ∂ ρ¯ ∂ 1 ∂ ρ¯ + sin θ + ∂θ ∂θ ∂φ sin θ ∂φ      1 ∂ 2 ∂ρ ¯ 1 ∂ ∂ ρ¯ = k 2 p + 2 sin θ p ∂p ∂p p sin θ ∂θ ∂θ 2  1 ∂ ρ¯ + 2 2 p sin θ ∂φ2 = k∇2 ρ¯ , (3.78) where ∇2 is the standard Laplacian on R3 , i.e. in the nonrelativistic limit the swerves diffusion equation reduces to the standard three dimensional diffusion equation. The general causal solution to Equation 3.78 can be derived using Greens’s function tech- niques. The result is ! ∞ 1 (p − p′ )2 Z ρ¯(p, t) = exp − f (p′ )d3 p′ , (3.79) −∞ (4πkt)3/2 4kt where f (p) is the initial distribution f (p) = ρ¯(p, 0). Consider an initial distribution of particles at rest in our frame, i.e. f (p) = δ3 (p). In this case, the solution of the diffusion  §3.5 Consequences and bounds 49 equation simplifies to ! ∞ 1 (p − p′ )2 Z ρ¯(p, t) = 3/2 exp − δ3 (p′ )d3 p′ −∞ (4πkt) 4kt p2   = (4πkt)−3/2 exp − . (3.80) 4kt 2kt This is just the Maxwell-Boltzmann distribution with a temperature T = mkB , where kB is the Boltzmann constant. The temperature is dependent on the length of time the diffusion process has been occurring – in other words, the nonrelativistic limit of the swerves diffusion equation describes a spontaneous heating of a thermal nonrel- ativistic gas. It is also clear that an initially thermal distribution will remain thermal. Suppose the initial distribution is a thermal distribution with a temperature T0 : p2   −3/2 f (p) = (2πmkB T0 ) exp − . (3.81) 2mkB T0 The solution is then ! ∞ 1 (p − p′ )2 Z ρ¯(p, t) = 3/2 exp − f (p′ )d3 p′ −∞ (4πkt) 4kt ! ∞ (p − p′ )2 p′ 2 Z 2
−3/2 = 8π ktmkB T0 exp − − d3 p′ −∞ 4kt 2mk B 0T 2  
−3/2 p = 8π 2 ktmkB T0 exp − 2mkB T Z ∞  2 ! −T T0 × exp ′ p − p d3 p′ −∞ 4ktT 0 T −p2   −3/2 = (2πmkB T ) exp , (3.82) 2mkB T where T ≡ T0 + 2kt/mkB , i.e. a thermal distribution remains thermal, with the tem- perature changing at a rate of dT /dt = 2k/mkB . Dowker et al. [3] placed a bound on the diffusion constant k by considering the spontaneous heating (or rather, the lack of spontaneous heating) of hydrogen gas in the laboratory. They assumed that a heating rate of 10−6 degrees a second would have  §3.5 Consequences and bounds 50 already been detected if it existed. This resulted in the bound mkB dT k = 2 dt ≤ 10−56 kg2 m2 s−3 , (3.83) or equivalently k ≤ 10−44 GeV 3 . (3.84) The average energy of a thermal distribution at temperature T is given by 3kB T hEi = . (3.85) 2 Constraints on k can thus also be expressed in terms of the maximum energy gain allowed m∆hEi k ≤ . (3.86) 3∆t To more tightly constrain k it is therefore necessary to consider systems that have been around for a long time without much gain in energy. Kaloper and Mattingly [30] looked at astrophysical clouds that have existed for roughly the age of the Milky Way without significant energy gain. They found, however, that energy loss due to radi- ation cannot be neglected and the result is a bound on k roughly comparable to that given above. They obtained a much stronger bound of k < 10−61 GeV 3 by considering cosmic neutrinos. When heated by swerves, the cosmic neutrino background would appear as hot dark matter. The diffusion constant is thus bounded by observations that suggest dark matter is mainly cold. In attempting to bound the diffusion parameter, k, it is assumed that k is the same for all particles. Other than for simplicity, there is no reason to believe this is the case. A strong bound on k for neutrinos does not, therefore, rule out observable swerves for other particles. Without an underlying model for particles on a causal set (a model for the particles themselves, not just their propagation) it is unlikely any progress will be made determining the dependencies of k. It is also interesting to note that, although Lorentz invariant, swerves could give rise to an apparent signal of Lorentz violation [35]. This is because many investiga-  §3.5 Consequences and bounds 51 tions into Lorentz violation rely not only on the details of the Lorentz violating model, but also on the assumption of energy-momentum conservation. A violation of energy- momentum conservation could be incorrectly interpreted as a signal of Lorentz viola- tion. Consider, for example, a time-of-flight experiment where a particle is measured to arrive with energy E. If the particle is subject to swerves, the arrival energy, E, is not the energy of the particle throughout its propagation, and the propagation time will therefore be miscalculated if swerves are neglected. Mattingly [35] concludes, however, that any false signal of Lorentz violation that appears due to the energy- momentum violation of swerves would be considerably smaller than current limits of experimental sensitivity.  §3.6 Simulations of particle models 52 3.6 Simulations of particle models The diffusion equations derived in this chapter do not rely on any particular model for particle propagation on a causal set. The diffusion parameter k is, however, likely to depend on the underlying propagation mechanism. Simulating the particle models discussed in Section 3.1 shows explicitly that the models lead to diffusion like be- haviour. It also allows an initial investigation into how the continuum diffusion pa- rameter might depend on characteristics of the underlying physical process. To simulate the particle models causal sets are constructed via sprinkling into Minkowski spacetime (as discussed in Section 2.3), using the Cactus numerical rela- tivity code [36] and the CausalSets arrangement written by David Rideout. In this en- vironment, code was developed to implement the three particle trajectory algorithms discussed in Section 3.1. The causal sets generated in this manner can of course be considered as partially ordered sets independent of Minkowski spacetime. Here the information about the coordinates of the points in Minkowski spacetime will be re- tained to allow the causal sets to be easily visualised. The particular frame to which the coordinates of the sprinkled points in Minkowski spacetime refer will be called the embedding frame. The results discussed here are, for the most part, in 1+1 dimensions. The code is, however, completely general and can also be run in 3+1 dimensions (example trajecto- ries in 3+1 for each particle model are shown in Figures 3.5 and 3.6). While simulations in 3+1 are more physically relevant, they require much larger causal sets to obtain the same discreteness scale and the necessary time and computer resources were unfortu- nately not available during the writing of this thesis. Simulations in 1+1 dimensions in fact prove to be more than adequate to investigate the link between the continuum and discrete processes. Several important aspects of the simulations need to be noted. Firstly, an element is added to the sprinkling at the origin to allow a fixed beginning point for the trajec- tories. Also, the models all assume a partial particle trajectory already exists and is being extended – in these simulations the trajectories start with a single element and thus the development of the first part of the trajectory needs to be carefully considered. For Model 1, the swerves model, the matter is very simple. An initial momentum, p0 is defined and no further information is needed to construct the future trajectory. In  §3.6 Simulations of particle models 53 1 0.8 0.6 0.4 0.2 0.4 0.2 0-0.5 -0.4-0.3 0 -0.2-0.1 0 0.1 -0.2 0.2 0.3 -0.4 0.4 0.5 Figure 3.5: An example trajectory using Model 1 in 3+1 dimensions (one spatial dimension is suppressed). The causal set here has size N = 32768. The trajectory is constructed using τf = 0.37 in embedding coordinates. these simulations p0 is taken as p0 = 0, i.e. the particle is initially at rest in the embed- ding frame. Model 2 relies on knowledge of en−1 as well as en to determine en+1 . Unfortu- nately it is not possible to just define an e−1 to begin the trajectory. In this case, when constructing the first step, the condition d(en−1 , en+1 ) ≤ 2nf is simply ignored. The element e1 is chosen such that it maximises d(e0 , e1 ) ≤ nf . Points satisfying this con- √ dition will be close to the hyperbola defined by a proper time τ = t2 − x2 = nf dpl , where dpl is the discreteness scale of the causal set. Again, the particle should be ini- tially at rest (or as close as possible) in the embedding frame. To impose this, instead of choosing e1 at random from those eligible, the element with the smallest time coor- dinate in the embedding is chosen. The rest of the trajectory is constructed as per the algorithm. Model 3 presents a slightly more complicated situation. Here the future trajectory from e0 depends on elements e−nf , . . . , e−1 . In simulations of Model 3, I first construct a longest chain between the element at t = 0, x = 0 and an element at t = 0.25, x = 0  §3.6 Simulations of particle models 54 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.5 0.4 0.1 0.3 0.2 0-0.5 0.1 -0.4-0.3 0 -0.2-0.1 -0.1 0 0.1 -0.2 0.2 0.3 -0.3 -0.4 0.4 0.5 -0.5 (a) Model 2, nf = 5. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.5 0.4 0.1 0.3 0.2 0-0.5 0.1 -0.4-0.3 0 -0.2-0.1 -0.1 0 0.1 -0.2 0.2 0.3 -0.3 -0.4 0.4 0.5 -0.5 (b) Model 3, nf = 5. Figure 3.6: Example intrinsic trajectories in 3+1 dimensions (one spatial dimension is sup- pressed). The causal set here has size N = 32768.  §3.6 Simulations of particle models 55 (also artificially added to the sprinkling). This longest chain forms the beginning of the trajectory: essentially, it imposes the condition that the beginning of the trajectory is as close to geodesic as possible, and again the particle is initially at rest in the embedding frame. The rest of the trajectory is constructed as per the algorithm. Note that the initial longest chain may have less than nf elements, in which case minimisation is over all elements until nf elements are reached. A second important point to note about the simulations is scale: the causal set models give rise to the continuum diffusion equation when 1 << nf << l where l is the trajectory length. In other words, the causal set must be large enough that the trajectory involves many nf steps, and nf itself must be much larger than 1. This is quite difficult to obtain. Suppose nf = 10 and we want the trajectory length to be 10nf steps (or in the case of Model 3, 100 elements). In 1+1 dimensions this requires a causal set of roughly 10000 elements – quite manageable. If nf = 100, however, a causal set of roughly 1 × 108 is needed for l = 100nf .2 This is certainly not manageable with the current code, so it is not really possible to explore realistic values of nf and trajectory length. Despite this limitation the simulations clearly show diffusion behaviour, as will be demonstrated below. 3.6.1 Example trajectories in 1+1 dimensions Simply constructing an example trajectory for each of the three models in 1+1 dimen- sions reveals something of the relationship between the parameters of the model and the diffusion parameter in the continuum. Figure 3.7 shows a trajectory constructed in a causal set of size N = 32768, using the swerves momentum model, Model 1.3 For this trajectory τf = 0.1105 in embedding coordinates, and the final length of the trajectory is l = 16 elements. Clearly this trajectory does not have a forgetting time, τf , many orders of magnitude greater than the discreteness scale, or a total trajectory length much greater than τf . The result is, however, very very close to the expected straight line in the continuum. Model 2 also gives a reasonably straight line for comparable parameters. Figure 3.8 shows a trajectory constructed in a causal set with N = 32768 and nf = 20. Note that 2 In 3+1 dimensions you would need the absurd number of 1 × 1016 elements. 3 The causal sets are generated by a Poisson process with a mean of N , the actual number of elements varies.  §3.6 Simulations of particle models 56 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3.7: Example trajectory using Model 1 in 1+1 dimensions: N = 32768, τf = 0.1105, l = 16. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3.8: Example trajectory using Model 2 in 1+1 dimensions: N = 32768, nf = 20, l = 13.  §3.6 Simulations of particle models 57 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 (a) Model 3, nf = 20, l = 163. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 (b) Model 3, nf = 40, l = 154. Figure 3.9: Example Model 3 trajectories in 1+1 dimensions: N = 32768.  §3.6 Simulations of particle models 58 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3.10: 30 Model 3 trajectories in 1+1 dimensions: N = 32768, nf = 40. nf = 20 is approximately equivalent to τf = 0.1105 for a causal set of size N = 32768. Recall, however, that Model 2 actually relies on information 2nf into the past of the trajectory: it would be more correct to compare a nf = 10 Model 2 trajectory with a τf = 0.1105 Model 1 trajectory. Regardless, it is clear that with roughly comparable forgetting times Model 2 results in more fluctuations in position and momentum. A comparison of the two intrinsic models, Model 2 and Model 3, is also interesting. In Figure 3.9 two trajectories constructed using Model 3 are shown: one with nf = 20 and one with nf = 40. As mentioned above, Model 2 relies on information 2nf steps into the past, so we might expect similar trajectories from nf = 20 Model 2 and nf = 40 Model 3. The trajectories from Model 3, however, swerve much more. Given that the values of nf and l are not really in the realm 1 << nf << l, where the continuum diffusion is expected to hold, care should be taken in drawing conclusions about the continuum behaviour of Models 2 and 3. Even so, it seems reasonable to conclude that a simple universal relation between the forgetting number of an intrinsic particle model and the diffusion parameter in the continuum does not exist (a concrete relation between the forgetting time of Model 1 and the diffusion parameter will be found in Section 3.6.2). Of course, the models here are not physically realistic. It seems likely,  §3.6 Simulations of particle models 59 however, that when physically realistic models are developed, the determination of the diffusion constant k from the ‘fundamental’ parameters of the underlying model on the causal set, will be model dependent. The diffusion-like behaviour of the models is most obvious if multiple trajectories are generated. Recall that the intrinsic models do not define unique trajectories – an element is chosen at random from the eligible elements at each step. For a fixed causal set and initial trajectory, multiple trajectories can thus be generated. Figure 3.10 shows 30 (distinct) Model 3 trajectories in a fixed causal set, N = 32768, nf = 40. Note that from t = 0 to t = 0.25 the trajectory is a fixed longest chain for all 30 trajectories. Although diffusion-like behaviour is clear, it is difficult to determine whether it is exactly the diffusion given by the swerves diffusion equation. The values of N and nf are nowhere close to the continuum limit, and 30 trajectories is simply insufficient. To compare more directly with the swerves diffusion equation, I will look instead at the non-intrinsic model, Model 1. 3.6.2 A close investigation of Model 1 Simulating trajectories Unlike the intrinsic models, for a given causal set and fixed initial position and mo- mentum Model 1 defines a unique trajectory. To say that Model 1 results in diffusion is, in a sense, saying that the exact underlying causal set is unknown. To simulate this I generate many different sprinklings into the same region of Minkowski spacetime, and calculate the unique trajectory in each. As above, the starting point of each tra- jectory is fixed at x = 0, t = 0, an extra element artificially added to each sprinkling. For each trajectory a final position and momentum can be calculated at tf inal = 0.95. This is done by identifying elements en and en+1 in the trajectory such that en is the last element with t < tf inal and en+1 is the first element with t > tf inal . Note that there is a vanishingly small probability that any element has t = tf inal . The final position is found by linear interpolation between en and en+1 . The final momentum is simply the usual momentum at en+1 , i.e. it is proportional to the vector between en and en+1 . The first step in the analysis is to reject some trajectories – if a trajectory is close to the boundary of the region of Minkowski spacetime at any point, it will ‘bounce’ back and distort the results. An example of such a trajectory is given in Figure 3.11.  §3.6 Simulations of particle models 60 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 3.11: An invalid Model 1 trajectory, N = 32768, τf = 0.02. Provided only a small fraction of the trajectories are invalid, the invalid ones can be safely rejected without much loss of information about the trajectory distribution. The parameter values investigated in this section are chosen such that there are indeed few, if any, invalid trajectories. Once the invalid trajectories have been removed, histograms of final position and final momentum can be generated from the remaining trajectories. These histograms clearly show diffusion behaviour, but to truly test the model the results must be com- pared with the distributions expected from the swerves diffusion equation. The swerves equation The inhomogeneous, 1+1-dimensional swerves diffusion equation is p ! ∂ρt −p ∂ρt ∂ m2 + p2 ∂ρt =p +k . (3.87) ∂t m + p ∂x 2 2 ∂p m ∂p As this does not have an exact solution, it must be evolved numerically. For the results shown here this numerical evolution is conducted with NDSolve in Mathematica. The initial distribution of the simulated trajectories is a δ-function: all trajectories be-  §3.6 Simulations of particle models 61 gin with x = 0 and p = 0. To numerically evolve Equation 3.87 with a δ-function initial distribution is not possible – instead, the initial distribution is taken to be a highly peaked Gaussian x2 p2   ρt (x, p, 0) = exp − 2 − 2 . (3.88) σx σp Here, σx2 and σp2 are not fixed: they are chosen such that Equation 3.88 is a good approx- imation to a histogram of the initial position-momentum data constructed with the same bin size as used for the final data, and such that Equation 3.87 evolves smoothly without numerical artifacts. To obtain final position and momentum distributions the evolved ρt (x, p, tf inal ) is summed over momentum and position, respectively. The matter of units To compare the simulated trajectories with the swerves diffusion equation a consis- tent set of units needs to be chosen: Equation 3.87, above, is expressed in terms of Planck units; the simulated trajectories are expressed in terms of ‘embedding’ coordi- nates. As mentioned earlier in this thesis, the discreteness scale in causal set theory is expected to be of the order of the Planck scale. For concreteness I assume the two are the same, and thus Equation 3.87 can be considered as expressed in ‘discreteness p units’. A discreteness length dpl = V /N , where V is the volume of the region of Minkowski spacetime in embedding units and N is the mean causal set size, can be calculated from the simulated causal sets. The position, time, momentum, and mass in the simulations are then re-expressed in terms of discreteness units. Comparing the simulations to the swerves equation Equation 3.87 is evolved for a range of values of k and the resulting distributions are compared to the position and momentum histograms for given values of the model parameters (τf , N , and particle mass m). A best fit value of k for each set of model parameters is found by minimising the reduced χ2 : X (Oi − Ei )2 χ2 = , (3.89) Ei i 2 χ2red = χ /f, (3.90)  §3.6 Simulations of particle models 62 where Oi is the observed frequency, Ei is the expected frequency (i.e. that given by the evolution of the diffusion equation), and f is the number of degrees of freedom (here, f = number of data points - 1). The reduced χ2 is calculated for the distribution in momentum rather than the full momentum-position distribution. The momentum distribution is chosen for comparison as it drives the position diffusion. Care must be taken when calculating the value of the reduced χ2 – if a significant proportion of the frequency values are less than five, χ2 is no longer a good measure of fit (see, for example [37]). In situations where the frequency in a given bin is less than five, multiple bins are combined to resolve this problem. A reduced χ2 value of χ2red ∼ 1 is usually taken to indicate a good fit (see, for example [38]), i.e. the evolved diffusion equation is a good description of the simulated trajectories. Results The goal of these simulations is not just to demonstrate that the model gives diffusion behaviour, but also to determine the relationship between the continuum diffusion parameter k and the parameters of the underlying discrete model. The relationship between k and the forgetting time τf can be determined by running simulations for a range of values of τf (with fixed N and m) and determining the best fit value of k in each case. A causal set of size N = 32768 was used in these simulations. The particle mass in embedding units is taken to be m = 1. Five hundred trajectories were evolved for each value of τf in τf = {0.03, 0.035, 0.04, 0.045, 0.05, 0.055, 0.06, 0.07, 0.08, 0.09, 0.1} (in embedding units). The results are summarised in Table 3.1. Several things need to be considered when interpreting these results, as discussed below. Table 3.1 shows the simulated values of τf in both embedding units and multiples of the discreteness length dpl . In units of dpl , τf varies between 5.4 and 18 – clearly not the many orders of magnitude greater than dpl that is expected in reality. The average trajectory length (this is the number of ‘steps’ in the trajectory rather than the number of elements) is also well out of the realistic range, reaching a maximum of only 48. Unfortunately realistic values are simply not feasible with the current computing facilities. What is remarkable is that, even with these values of τf and N , the distribution of the simulated trajectories is extremely well modelled by the diffusion equation. The best fit values of the diffusion parameter and corresponding  §3.6 Simulations of particle models 63 Number of invalid Average trajectory τf τf (dpl ) best fit k χ2red trajectories length 0.03 5.4 10 48 1.2 × 10−8 1.5 0.035 6.3 2 42 5.5 × 10−9 1.0 0.04 7.2 0 37 2.8 × 10−9 0.59 0.045 8.1 0 33 1.5 × 10−9 0.64 0.05 9.1 0 30 9.2 × 10−10 1.0 0.055 10 0 27 5.0 × 10−10 2.4 0.06 11 0 25 3.7 × 10−10 1.6 0.07 13 0 22 1.7 × 10−10 0.56 0.08 14 0 19 9.9 × 10−11 0.55 0.09 16 0 17 4.7 × 10−11 2.0 0.1 18 0 16 2.6 × 10−11 2.4 Table 3.1: Analysis of Model 1 trajectories for various values of τf , with fixed N and m. χ2red values are given in Table 3.1. As mentioned earlier, χ2red ∼ 1 is usually considered a good fit. It is easy to see just how good the fit is when the evolved diffusion equation is plotted together with the simulation histograms. Examples of both the position and momentum histograms for four of the values of τf are shown in Figure 3.12. The particular values of τf shown are not special in any way, and are merely chosen to illustrate the results. As Table 3.1 shows, invalid trajectories are an issue only for the smallest two values of τf and even in the worst case only comprise 2% of the 500 trajectories. As such, they don’t influence the results given here. They do, however, become significant for values of τf smaller than 0.03 lessening the utility of simulations for those values. Although the best fit values of k and corresponding χ2red calculated here are fairly robust, it should be noted that the best fit values of k are determined by calculating χ2red for various values of k and choosing the one that minimises χ2red . A truly system- atic search of the parameter space is not undertaken, and a measure of the error in k is not calculated. The exact values of χ2red depend slightly on the bin width chosen when generating the histograms. They also vary slightly if σx and σp in the initial Gaussian distribution are changed. A further point is that for some τf there are a range of values of k that give χ2red ∼ 1 and thus all could be considered a good fit. For the results that I have obtained, k vs. τf is plotted in Figure 3.13. The dependence of k on τf is clear in the ln − ln plot, where the best fit line to the data, ln k = −5.02 ln τf − 9.8 is also shown. Despite the considerations mentioned above, it seems clear that k ∼ 1/τf5 .  §3.6 Simulations of particle models 64 60 60 50 50 40 40 30 30 20 20 10 10 -60 -40 -20 0 20 40 60 -0.003 -0.0015 0 0.0015 0.003 (a) τf = 0.04. 80 60 50 60 40 40 30 20 20 10 -30 -20 -10 0 10 20 30 -0.0014 -0.0007 0 0.0007 0.0014 (b) τf = 0.05. 60 80 50 60 40 30 40 20 20 10 -20 -10 0 10 20 -0.001-0.0005 0 0.0005 0.001 (c) τf = 0.06. 80 50 40 60 30 40 20 20 10 -10 -5 0 5 10 -0.0004 -0.0002 0 0.0002 0.0004 (d) τf = 0.08. Figure 3.12: Example histograms with best fit numerical solutions. The position distribution is shown on the left, momentum on the right. .  §3.6 Simulations of particle models 65 -9 7·10 -9 6·10 -9 5·10 -9 4·10 k -9 3·10 -9 2·10 -9 1·10 0 0 2.5 5 7.5 10 12.5 15 17.5 Τ f Hin units of d pl L (a) k vs. τf . -18 -19 -20 ln k -21 -22 -23 -24 1.6 1.8 2 2.2 2.4 2.6 2.8 ln Τ f Hin units of d pl L (b) ln k vs. ln τf with the best fit line −5.02 ln τf − 9.8. Figure 3.13: Comparison of the diffusion parameter k and forgetting time τf , for fixed N and m.  §3.6 Simulations of particle models 66 Number of invalid Average trajectory N dpl best fit k χ2red trajectories length 4096 0.016 12 19 0.034 0.64 8192 0.011 0 19 0.0088 1.0 16384 0.0078 0 19 0.0021 0.45 32768 0.0055 0 19 0.00056 0.55 Table 3.2: Analysis of Model 1 trajectories for various values of dpl (in embedding units). The diffusion parameter k also depends on the particle mass m. In Model 1 the trajectory is independent of the particle mass – the mass is only used to correctly normalise the momentum. The momentum histogram for trajectories of a particle of mass m′ = αm can therefore be obtained simply by scaling the original momentum histogram by α. Looking at Equation 3.87, if both m and p are scaled by α and k is scaled by α2 the equation remains unchanged. Thus it can be determined, without reanalysing the simulations, that k ∼ m2 . There is one remaining parameter that k can depend on: the discreteness length itself. From dimensional analysis of Equation 3.87 it can be deduced that k has units of L−3 (since Planck units are used, L = T = M −1 ). Since k ∼ m2 τf−5 the dependence on dpl must be k ∼ d4pl . This dependence can be checked through simulations. In this case everything needs to be expressed in terms of embedding units rather than discreteness units. Five hundred trajectories are calculated for four different values of N (and thus dpl ) and the same values of forgetting time τf = 0.08 and mass m = 1. The value of τf was chosen to avoid too many invalid trajectories in the causal sets considered. As above, the best fit values of k and the corresponding χ2red can be calculated. The results are summarised in Table 3.2. Again, a ln − ln plot (see Figure 3.14(a)) allows the dependence of k on dpl to be determined. The result is as expected: k ∼ d4pl . The analysis of the simulations leads to the relation k ∼ m2 d4pl /τf5 . The constant of proportionality can in fact also be determined. Looking back at the results for varying τf and noting that in terms of discreteness units m = 0.0055 and dpl = 1 the relation k vs. m2 /τf5 can be plotted (see Figure 3.14(b)). The best fit line is found to have the equation k = 1.9 m2 /τf5 − 2.5 × 10−11 . Thus, it can be concluded that k ≈ 2m2 d4pl /τf5 . (3.91) Causal set theory contains not only a discreteness scale, but also a hypothesised  §3.6 Simulations of particle models 67 -3 -4 -5 ln k -6 -7 -8 -5.2 -5 -4.8 -4.6 -4.4 -4.2 -4 ln d pl (a) ln k vs. ln dpl with the best fit line 4.0 ln dpl + 13. -8 1.2·10 -8 1·10 -9 8·10 -9 k 6·10 -9 4·10 -9 2·10 0 0 1·10 -9 -9 2·10 3·10 -9 4·10 -9 5·10 -9 -9 6·10 m2 Τ f 5 (b) k vs. m2 /τf5 with the best fit line 1.9 m2 /τf5 − 2.5 × 10−11 . Figure 3.14: The relationship between the diffusion parameter k and underlying model pa- rameters dpl , m, and τf .  §3.6 Simulations of particle models 68 ‘nonlocality’ scale below which physics cannot remain local [39, 40]. The forgetting parameter that appears in the massive particle models may be a measure of this non- locality scale. Sorkin [39] estimates the nonlocality scale to be of the order 1020 in Planck units. Although the simulations here use forgetting times nowhere near this estimate, and there is no theoretical reason why the forgetting time should be exactly the nonlocality scale, it is interesting to consider the value of the diffusion constant that arises if the above results are extrapolated. Taking the particle to be a proton with mass m = 10−20 , τf = 1020 , and dpl = 1, Equation 3.91 leads to the estimate k ≈ 10−140 . As discussed in Section 3.5, tight constraints have already been placed on k through observations. In particular (see Equation 3.83) k ≤ 10−56 kgm2 s−3 ∼ 10−102 due to the lack of spontaneous heating of hydrogen in the laboratory. The diffusion parameter estimated from the nonlocality scale is thus many orders of magnitude smaller than the current constraint (the factor of two mass difference between the proton and a hy- drogen molecule is inconsequential here). This rough calculation should not be given too much weight, but it does suggest that k could be well outside our current observ- able range.  §3.7 Concluding remarks 69 3.7 Concluding remarks Following the proposal by Dowker et al. of the swerves model, and their derivation of the proper time swerves diffusion equation, the question arose of how an underlying model of particle behaviour on a causal set could be formulated without relying on information about the approximating continuum spacetime. The association between longest chains in a causal set and geodesics in the continuum allowed such models to be created, two examples were given in Section 3.1. The intrinsic models described in this thesis are by no means the only options, and simply give an idea of the wealth of possibilities available. The models proposed here (including the original swerves model) make no attempt to be physically realistic. A true description of particle mo- tion in causal sets will have to await many further developments in the theory. The first step might involve developing a quantum mechanical rule for the trajectory evo- lution, or modelling particles as wavepackets. The models proposed do capture the crucial features of causal set theory – they could, in a sense, be interpreted as a coarse grained description. Just as modelling a gas as an unrealistic collection of hard spheres proved enormously useful, so too do these models prove useful in determining con- tinuum behaviour. The proper time swerves diffusion equation, first given in [3] and derived in full in Section 3.2, is the unique continuum diffusion equation given an underlying Marko- vian, Poincar´e invariant process causing random fluctuations in momentum. It is in- dependent of the particular underlying models discussed – those models and many more would result in continuum behaviour described by the swerves diffusion equa- tion. One query that remained, was whether a given model was guaranteed to result in a finite diffusion constant. In Section 3.4 it was shown analytically that the swerves model does lead to a finite diffusion constant. The swerves diffusion equation, expressed in terms of proper time, is difficult to compare with observations. Although bounds had been placed on the diffusion pa- rameter by considering the non-relativistic limit, bounds in the fully relativistic case could not formerly be considered. The full inhomogeneous swerves diffusion equa- tion in terms of cosmic time was derived in Section 3.3. With this equation complete comparison with observations will be possible. After exploring swerves analytically, the second task of this chapter was to explore  §3.7 Concluding remarks 70 the models numerically. The underlying models give rise to diffusion in the contin- uum limit. With simulations limited to quite small causal sets (far from the infinite sprinkling density of the continuum limit) it was not expected that the full diffusion behaviour of the models would be apparent. The results were, in fact, far better than anticipated. All three models clearly show diffusion like behaviour. A close investiga- tion of the swerves model shows the model behaviour perfectly matches the swerves diffusion equation. It was even possible to determine an expression for the continuum diffusion parameter in terms of the parameters of the underlying causal set model. That the diffusion behaviour occurs even this far from the continuum limit is very interesting. There are also wider implications for causal set simulations – the results here show that despite computational limits on causal set sizes, useful information about the continuum appearance of discrete spacetime can be obtained.  Chapter 4 M ASSLESS PARTICLES Die ganzen 50 Jahre bewusster Grubelei ¨ haben mich der Antwort der Frage Was sind Lickhtquanten“ nicht n¨aher gebracht. Heute glaubt zwar jeder ” Lump, er wisse es, aber er t¨auscht sich. A total of fifty years of careful reflection has not brought me any closer to answering the question ‘what are the quanta of light?’. It’s true, today, that many a fool believes he knows the answer, but he’s mistaken. Einstein to Besso, 12th December 19511 An underlying spacetime discreteness results in diffusion in momentum and space- time for massive particles. Does a similar diffusion occur for massless particles? For massive particles concrete models of particle propagation on a causal set were con- structed, motivating the derivation of the swerves diffusion equation. For massless particles such underlying models are unfortunately elusive. Consider a sprinkling into Minkowski spacetime: for any given element, p, there will almost surely be no element sprinkled on the future lightcone of p. A trajectory constructed, as in the mas- sive case, by jumping from one element p to another q, where p ≺ q, will rarely be close to null. A close analogue to the future lightcone of p can, however, be defined simply as the set of all elements q linked to p: p ≺∗ q. These elements q lie near the hyperboloid with proper time one Planck unit from p. This suggests that the simplest analogue of a null ray is a single link. More generally, p ≺ q is a good approximation to a section of a null ray if the causal interval between p and q is totally ordered [40, 12], but it is still hard to see how a discrete Markovian process resulting in a close-to-null trajectory might be defined. It may be that options such as allowing particles to also 1 Published in [41], English translation from the French translation in [41] courtesy of Dr. R. M. Pollard. 71  72 move backward in time need to be considered. More generally, it is not surprising that attempts to model massless particles as classical point particles run into trouble. It is hoped that the study of massless fields on a causal set [39, 42] and general quantum field theory on a causal set [32, 33] will eventually lead to a model for massless particle propagation at a discrete level. The current lack of an underlying model does not, fortunately, exclude the inves- tigation of the potential effect of discreteness on photons in the continuum approxi- mation. A massless diffusion equation can be derived in one of two ways: following the stochastic evolution on a manifold of states procedure, as for the massive case; or, simply taking a m → 0 limit of the massive particle diffusion equation. As will be discussed in the following section, it turns out that the second method gives an incomplete result. In Section 4.1 the massless particle diffusion equation in terms of affine time is derived. As for massive particles, in order to make comparisons with observations an equation in terms of cosmic time is needed – such an equation in derived in Section 4.2. The resulting massless particle model has two free parameters. In Section 4.3 the effect of massless particle diffusion on the cosmic microwave background is studied, and bounds are placed on the values of the two free parameters. Finally, the conse- quences of the model for astrophysical spectra are discussed in Section 4.4, where it is found that the strong bounds from the cosmic microwave background rule out ob- servable effects. The work in Sections 4.1, 4.2, 4.3 was conducted in collaboration with Fay Dowker and Rafael Sorkin, and appears in [29]. Note that the massless particles considered in this chapter are assumed to have no polarisation. The issue of polarisation is addressed in depth in Chapter 5.  §4.1 Diffusion in affine time 73 4.1 Diffusion in affine time The state space in the massless case differs from the massive case. For massive parti- cles there is a probability distribution on M4 × H3 . For massless particles H3 becomes the light cone in momentum space defined by pµ pµ = 0. This cone will be denoted H30 . Hereafter the momentum of a massless particle will be denoted kµ . Assuming that the photons under consideration are well described in a geometrical optics approxima- tion (and thus have definite spacetime worldlines and momenta), the state space will be M4 × H30 . Since proper time vanishes along a lightlike worldline, it is clearly not a suitable time parameter for a massless diffusion process. Instead, an affine time, λ, defined by dxµ = kµ dλ , (4.1) is used to parametrise the photon worldline. In the massive particle case the density of microstates, n, was proportional to the √ determinant of the metric on the state space: n ∝ g. In the massless case, the induced metric on H30 is degenerate. A volume element on H30 can, however, still be defined. The four dimensional volume element d4 k of momentum space, together with the masslessness constraint, kµ kµ = 0, gives the invariant volume element d4 k δ(kµ kµ ) = d3 k/2k0 on H30 , i.e. n ∝ 1/k 0 in Cartesian coordinates. It turns out to be more useful, however, to work in polar coordinates on H30 : {k, θ, φ} where k is the magnitude of the three momentum and θ and φ are the usual polar angles in momentum space. In these coordinates, the density of states is n ∝ k sin θ. As with the massive particle case, the derivation of the massless diffusion equation begins with the current and continuity equations, Equation 3.5 and Equation 3.6. The first step is to determine K AB and uA . To begin, note that there is a unique, up to a constant factor, invariant vector field on H30 which is the momentum itself, ka , i.e. the vector with components (k, 0, 0) in polar coordinates. This is absent in the massive case, where the momentum vector does not lie in the mass shell. Also, although there is no invariant metric on H30 , there is an invariant symmetric two-tensor, ka kb , again unique up to a constant factor.  §4.1 Diffusion in affine time 74 From Equation 3.11, the spacetime component of K AB is ∆xµ ∆xν   µν K = lim ∆λ→0+ 2∆λ µ ν = lim k k ∆λ ∆λ→0+ = 0. (4.2) K AB is positive semidefinite so K µa = 0. Finally, the requirement that K AB be Lorentz invariant and translation invariant fixes the remaining component K ab to be propor- tional to ka kb :   0 0 K AB =  , (4.3) 0 d1 ka kb where d1 ≥ 0 is a constant. The vector uA is easily determined by looking individually at the components in spacetime and momentum space. As in the massive particle case, the spacetime com- ponent uµ = v µ by Equation 3.16, and v µ = kµ by Equation 3.12. In contrast to the massive case, there can be nonzero components of uA in the momentum space direc- tions because the momentum itself is an invariant vector. The momentum direction components are thus given by ua = d2 ka , where d2 is a constant. Working in polar coordinates the ‘position’ vector ka on the cone H30 is simply (k, 0, 0) where k2 = k0 2 . Thus uA = (k0 , k1 , k2 , k3 , d2 k, 0, 0) on M4 × H30 . Substituting the forms for K AB and uA into Equation 3.15 gives the massless par- ticle affine time equation: ∂ρλ  ρ  λ  = ∂A K AB n∂B − uA ρλ ∂λ n ∂ρ ∂  ρ  µ λ 3 ∂ λ = −k µ + d1 k ∂x ∂k ∂k k ∂ −d2 (kρλ ) . (4.4) ∂k Lorentz invariance constrains the diffusion process such that, although there is diffusion in the energy of the photon, the direction of the photon is unchanged and it continues to propagate at the speed of light. Note that there are two parameters, d1 and d2 , in this phenomenological model, rather than the single parameter of the massive particle model. There is not only a diffusion term but an independent drift  §4.1 Diffusion in affine time 75 term, arising from the existence of an invariant vector on H30 . It will be shown further below that this extra term leads to the existence of power law equilibrium solutions of the diffusion equation. It is not unreasonable to expect that a diffusion equation for massless particles could be obtained simply by taking the m → 0 limit of the massive particle diffusion equation, Equation 3.25. This, however, results in Equation 4.4 with d2 = 0, as there is no invariant vector in the massive case. It is interesting to note that there is this discon- tinuity in behaviour between very highly relativistic massive particles, and massless particles. It suggests that massive and massless particles are fundamentally different objects and will need to be described in completely different ways in causal set theory.  §4.2 Diffusion in cosmic time 76 4.2 Diffusion in cosmic time As with massive particles, the massless particle diffusion equation is more useful for comparison with observations when expressed in terms of cosmic time. The massless cosmic time diffusion equation can be obtained by following the same argument as in the massive case (Section 3.3). It is assumed that the massless particle trajectory begins at λ = 0 and t = 0. The first step is to define a new larger state space incorporating the affine time: M′ = M4 × H30 × R. In this larger space, a new current component can be defined: J λ (t, xi , ka , λ) = ρλ . (4.5) The t component of the current J can be expressed in terms of J λ , using Equations 3.5 and 4.5. J t (t, xi , ka , λ) = kJ λ , (4.6) Jt ⇒ ρλ = . (4.7) k The remaining components of the current can now be expressed in terms of J t . ki t J i t, xi , ka , λ =
J , (4.8) k Jt   J a t, xi , ka , λ = −d1 ka kb k sin θ ∂b
k2 sin θ Jt + d2 ka , (4.9) k where Equation 3.31 has been used. In polar coordinates the vector ka = (k, 0, 0), and a thus there is only one nonzero component of J k in the radial (energy) direction: ∂J t J k t, xi , ka , λ = −d1 k + (2d1 + d2 ) J t .
(4.10) ∂k Since the affine time along the particle trajectories is unobservable, it can be integrated ¯ can be defined. Integrating over the t compo- over, and a new integrated current, J, nent gives Z ∞ J¯t (t, xi , ka ) = J t (t, xi , ka , λ)dλ. (4.11) 0  §4.2 Diffusion in cosmic time 77 The remaining current components can be integrated and re-expressed in terms of J¯t : ki ¯t J¯i (t, xi , ka ) = J , (4.12) k ∂ J¯t J¯k (t, xi , ka ) = −d1 k + (2d1 + d2 ) J¯t . (4.13) ∂k In the extended space M′ , the continuity equation becomes ∂α J α = 0 , (4.14) where X α = {X A , λ} are coordinates on M × R. Integrated over λ the continuity equation becomes [J λ ]∞ ¯t ¯i ¯a 0 + ∂t J + ∂i J + ∂a J = 0 . (4.15) J λ |λ=0 is zero for all t > 0 and J λ tends to zero as λ approaches infinity for finite t. So for all t > 0 ∂t J¯t + ∂i J¯i + ∂a J¯a = 0 . (4.16) If J¯t is renamed ρt the massless particle cosmic time equation can be written: ∂ρt = −∂i J¯i − ∂a J¯a ∂t ki ∂ρt ∂ 2 ρt = − ∂i ρt − (d1 + d2 ) + d1 k 2 . (4.17) k ∂k ∂k 4.2.1 Power law equilibrium solutions It is interesting to note that negative values of the drift parameter d2 allow for power law equilibrium solutions of the massless particle cosmic time diffusion equation. For homogeneous distributions Equation 4.17 becomes ∂ρt ∂ρt ∂ 2 ρt = − (d1 + d2 ) + d1 k 2 . (4.18) ∂t ∂k ∂k Equilibrium solutions of this equation must satisfy ∂ρt ∂ 2 ρt 0 = − (d1 + d2 ) + d1 k 2 . (4.19) ∂k ∂k  §4.2 Diffusion in cosmic time 78 Equation 4.19 has a power law solution 2d1 +d2 ρt ∝ k d1 . (4.20) If a small energy cut off kmin is assumed, this solution is normalisable when the ex- ponent is less than -1. Since the diffusion parameter d1 is positive this can only occur for negative values of d2 . It can be conjectured that if (2d1 + d2 )/d1 < −1 (equiva- lently |d2 | > 3d1 ) any normalised distribution will tend to this power law equilibrium solution at late times.  §4.3 Diffusion and drift in the cosmic microwave background 79 4.3 Diffusion and drift in the cosmic microwave background In developing a phenomenological model the aim is to provide a model for currently unexplained observations or to suggest new observations that might be made to test a theory. Before proposing new observations or exotic experiments, however, one should constrain the model as tightly as possible based on what is already known. The massless particle momentum diffusion discussed in the preceding sections has two free parameters: a positive diffusion constant d1 and a ‘drift’ constant d2 , which may be either positive or negative. If the photons we encounter in our everyday lives experienced large amounts of diffusion or drift, it would of course have already been noticed. The motivation for the massless momentum diffusion model, however, lies in the underlying discreteness of causal set quantum gravity – the resulting effects are expected to only be apparent on very small scales, outside the range of current exper- iments. How then can the parameters be constrained? The key is to look at photons that have been travelling long enough for the diffusion and drift to ‘accumulate’ to an observable level. The cosmic microwave background (CMB) seems an ideal testing ground: not only are its photons the ‘oldest’ we can observe, but its spectrum has been determined with great precision. The photons in the CMB have been ‘free streaming’ since the surface of last scattering at a redshift of z ∼ 1100 – approximately 13.7 billion years. When the universe became transparent at recombination, the photons that are now observed as the CMB would have had a blackbody spectrum with a temperature of 3000 K (see for example [43]). As the universe expanded the photons were both stretched and diluted, a process that neatly preserves the blackbody nature of the spectrum and simply lowers the temperature. Current observations of the CMB yield a temperature of 2.728 ± 0.004 K and measure the spectrum to be Planckian (blackbody) over the 2 − 21cm−1 frequency range to within a weighted root mean square (rms) deviation of only 50 parts per million (ppm) of the peak brightness [44]. Any diffusion and drift in energy would have distorted the CMB spectrum – the fact that the CMB photons have travelled so far but remained so perfectly thermal allows the parameters in the model to be constrained very tightly.  §4.3 Diffusion and drift in the cosmic microwave background 80 4.3.1 Modelling diffusion in the CMB The derivation of the massless cosmic time diffusion equation assumed spacetime to be Minkowskian. Although the expansion of the universe can be incorporated into the model (and will be in Section 4.3.2 below) a first bound on the parameters will be obtained by simply ignoring the expansion. In a non-expanding universe the temper- ature of the CMB would have remained constant from the surface of last scattering to today. To allow comparison with data, it will in fact be assumed that the CMB was emitted with the temperature observed today, T = 2.728K. If a blackbody spectrum is evolved according to the massless cosmic time diffu- sion equation, the final distribution will no longer be blackbody. The deviation from blackbody can be determined and, by requiring it to be within the current deviation allowed by observations, bounds can be placed on the drift and diffusion parameters. Specifically, the initial Planckian spectrum, expressed as a number density of photons per unit spatial volume per unit energy, is k2 ρ(k, t = 0) = 8π k
, (4.21) exp T −1 with a temperature T = 2.728K. This distribution is evolved over a time interval equal to that since the surface of last scattering (1×1060 in Planck units) via the homogeneous massless cosmic time diffusion equation ∂ρt ∂ρt ∂ 2 ρt = − (d1 + d2 ) + d1 k 2 , (4.22) ∂t ∂k ∂k using the MATLAB numerical pde solver pdepe. Boundary conditions Some questions must be addressed when choosing a boundary condition at k = 0 for this integration. What happens to a photon as its momentum approaches zero? Do photons leak away through the tip of the null cone in momentum space? Physically, the photon concept employed by the model breaks down as the wavelength tends to infinity, because the geometrical optics approximation fails. This suggests that an ‘absorbing boundary condition’, ρ(E) = 0, is appropriate at E = 0. As it happens, the  §4.3 Diffusion and drift in the cosmic microwave background 81 current is ∂ρt J = (2d1 + d2 )ρt − d1 k , (4.23) ∂k and thus (so long as ∂ρt /∂k remains finite) the absorbing boundary condition ρt = 0 is equivalent to the ‘reflecting boundary condition’, J = 0, at E = 0. In the simulations that follow, a reflecting boundary condition is used at the upper boundary k = kmax . Calculating the deviation The evolved spectrum was converted from a number density per unit volume per unit frequency to a spectral radiance (energy per unit area per unit time per unit frequency per steradian) as used in the analysis of the COBE FIRAS data. This allowed the devi- ation from Planckian to be compared with the quoted 50ppm of the peak brightness. The first step was to fit a Planck spectrum to the evolved spectral radiance using the least squares method. Specifically, the MATLAB function fminsearch was used to find the temperature that minimised the sum of the squared differences between the evolved distribution and the fitted Planck spectrum. Looking for the best fit Planck spectrum rather than comparing with the initial 2.728K spectrum, allowed for the possibility that the diffusion changes the temperature of the CMB in a way that may be reconciled with observation. As it happens, it was found that the temperature of the best fit Planck spectrum was very close to the initial temperature in cases where the deviation is within the allowed tolerance. For example the choice of parameters d1 = 5 × 10−97 and d2 = 1 × 10−96 gives a best fit temperature of 2.7281K, indistin- guishable from the current observed temperature of 2.728 ± 0.004K. Finally, the rms deviation between the fitted Planckian spectrum and the evolved spectrum rP i (si− fi )2 rms = , (4.24) N where si is the value of the evolved spectrum at a point i, fi is the value of the fitted spectrum, and N is the total number of data points, was calculated over the frequency range 2−21cm−1 (energy range 4×10−23 −4×10−22 J) with all points weighted equally. After calculating the peak brightness of the evolved spectrum, this rms deviation can be compared to the allowed tolerance of 50 parts per million of the peak brightness. This process was repeated for a range of values of the parameters d1 and d2 .  §4.3 Diffusion and drift in the cosmic microwave background 82 Results For the evolved spectrum to be close enough to Planckian to be within the allowed tol- erance, the deviation must be essentially invisible to the eye. Before placing bounds on d1 and d2 it is useful therefore to get a general feel for how a blackbody spectrum evolves under Equation 4.22 by looking at values of d1 and d2 that result in deviations well outside the allowed tolerance. Figure 4.1 shows the evolved distribution and best fit Planck spectrum for d1 = 1 × 10−93 , d2 = 1 × 10−93 . The positive drift parameter d2 clearly shifts the spectrum to the right. With these large values of d1 and d2 the difference between the best fit Planck spectrum (2.851 K) and the initial Planck spec- trum (2.728 K) is quite considerable. However, no amount of temperature change will disguise the translation of the Planck spectrum – the diffusion and drift are clearly visible. To place bounds on the diffusion and drift constants they were first examined sep- arately, varying d1 with d2 = 0 and varying d2 with d1 = 0. When d1 = 0 Equation 4.22 can be solved exactly: ρt (k, t) = ρ0 (k − d2 t) , (4.25) i.e. the spectrum just translates at a constant speed. For d2 negative, this is inconsistent with the boundary condition ρt = 0 at k = 0. However, in this case one can implement an absorbing boundary condition trivially: simply cut off the translated distribution at k = 0. The exact solution is also useful as a method of checking the level of nu- merical error in the simulations. Since the deviations from Planckian that are being investigated are so small there is the possibility that the rms deviation calculated is significantly affected by numerical error. Fortunately, this is not the case – the differ- ence between the exact solution and the numerically evolved solution for d1 = 0 is several orders of magnitude smaller than the deviation from Planckian.2 For example, if d2 = ±4 × 10−96 the rms deviation from the best fit Planck spectrum is 5 × 10−101 (5 × 10−5 peak brightness) for both the exact and the numerical solution. The rms de- viation between the exact and numerical solutions is 4 × 10−104 . This also demonstrates that the ρ = 0 boundary condition imposed on the numerical solution, although in- 2 Rerunning all the simulations in this section using the Mathematica numerical solver NDSolve yields identical results further suggesting that numerical error and integration step size do not have a significant effect on the bounds.  §4.3 Diffusion and drift in the cosmic microwave background 83 −96 x 10 1.6 1.4 1.2 1 spectral radiance 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −31 k x 10 (a) The initial Planck spectrum at 2.728 K (dotted line), evolved spectrum (dashed line) and best fit Planck spectrum at 2.851 K (solid line). −97 x 10 3 2 1 difference in spectral radiance 0 −1 −2 −3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −31 x 10 k (b) The difference between the initial spectrum and the evolved spectrum (dotted line), and the best fit spectrum and the evolved spectrum (dashed line). Figure 4.1: Comparison of evolved and best fit Planck spectrum for d1 = 1 × 10−93 , d2 = 1 × 10−93 . Energy k and spectral radiance are in Planck units.  §4.3 Diffusion and drift in the cosmic microwave background 84 −3 x 10 9 8 7 6 rms deviation 5 4 3 2 1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −93 x 10 d2 Figure 4.2: The rms deviation of the simulated spectrum from Planckian as a proportion of the spectrum peak, varying d2 , d1 = 0. consistent with the exact solution when d2 < 0, does not introduce noticeable errors for the values of d2 that are of concern here. The rms deviation over a range of values of d2 with d1 = 0 is shown in Figure 4.2 (plotted from the exact solution). For d2 = 0 and d1 > 0 the equation can only be solved numerically. The results in this case are shown in Figure 4.3. In both cases the deviation from Planckian increases approximately linearly with increasing magnitude of the parameters. The simulations suggest that for the deviation from Planckian of the CMB to be within the allowed 5 × 10−5 of the peak brightness the diffusion constant d1 must be less than approximately 7 × 10−97 if d2 = 0, and the drift parameter d2 must fall within the range −5 × 10−96 < d2 < 5 × 10−96 if d1 = 0. Converting to SI units this gives the bounds: d1 < 3 × 10−44 kgm2 s−3 , (4.26) −2 × 10−43 < d2 < 2 × 10−43 kgm2 s−3 . (4.27) If both d1 and d2 are nonzero the picture is somewhat more complicated, but sim- ilar bounds result. The results in this general situation are shown in Figure 4.4. The allowed region (within the 5e − 005 contour) is seen to be not quite symmetric in the drift parameter d2 . Placing concrete bounds on d1 and d2 from Figure 4.4 is difficult,  §4.3 Diffusion and drift in the cosmic microwave background 85 0.7 0.6 0.5 rms deviation 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −92 x 10 d1 Figure 4.3: The rms deviation of the simulated spectrum from Planckian as a proportion of the spectrum peak, varying d1 , d2 = 0. since the allowed values of d1 depend on the value of d2 and vice versa – the bounds are, however, of the order of magnitude given above. 4.3.2 The expanding universe To determine the bounds on the drift and diffusion parameters, given in Section 4.3.1 above, the effect of the expansion of the universe on the CMB was ignored – it was assumed that the CMB had remained at a temperature of ∼ 2.7 K from the surface of last scattering up to today. This is of course not the case. At the surface of last scattering the CMB had a temperature of about 3000 K. As the universe expanded the individual photons were stretched along with the space, and correspondingly di- luted, leaving the 2.7K spectrum observed today. It turns out that the expansion has essentially no effect on the model: the distribution in the expanding universe can be deduced easily from the nonexpanding one and the bounds derived from the nonex- panding simulation change only slightly. Before including the effect of cosmological expansion in the diffusion equation, a differential equation that describes the cooling of the CMB needs to be found; the following is a standard result in cosmology. Consider the usual continuity equation ∂ρ = −∂i (ρv i ) , (4.28) ∂t  §4.3 Diffusion and drift in the cosmic microwave background 86 −97 x 10 6e− 005 8e −0 9 5 0 0 5 −0 05 5e−0 6e 5e −0 8 05 6e 4e−005 −0 05 7 5e−005 6 4e−005 5e 1 5 d −0 4e 05 −0 05 6e −0 4 05 3e−005 3 4e− 5e− 5e−005 2 2e 3e 005 05 05 −0 005 −0 4e−005 −0 05 3e−0 6e−005 05 2e 1 −8 −6 −4 −2 0 2 4 6 d x 10 −96 2 Figure 4.4: The rms deviation of the simulated spectrum from Planckian as a proportion of the spectrum peak, varying both d1 and d2 . Values of d1 and d2 within the 5e−005 contour give a spectrum that is Planckian to within 50ppm of the peak brightness. for flux ρv i . In this case we are interested in the flux in momentum space of the CMB photons and thus v has a single component in the energy direction v = ∂k/∂t. This gives the continuity equation   ∂ρt ∂ ∂k =− ρt . (4.29) ∂t ∂k ∂t Under expansion the CMB photons are stretched such that k ∼ 1/a, where a is the ˙ 2 = −ka/a, scale factor. Thus, ∂k/∂t = −a/a ˙ and the continuity equation becomes   ∂ρt ∂ a˙ = − −ρt k ∂t ∂k a a˙ ∂ = (ρt k) . (4.30) a ∂k Consider the distribution 8π a3 k 2 ρ0 (k, t) = , (4.31) a30 exp ak − 1   a0 T0 where a0 is the scale factor of the universe today. Photons in the CMB are not only stretched, but also diluted by the expansion of the universe. The distribution in Equa-  §4.3 Diffusion and drift in the cosmic microwave background 87 tion 4.31 can be seen as a Planck spectrum where the normalisation depends on the scale factor a, to take into account the photon dilution. Equation 4.31 is indeed a solu- tion to the expansion equation, Equation 4.30. This confirms, as mentioned earlier, that the blackbody nature of the CMB is preserved during the expansion of the universe. If the expansion effect is included in the diffusion equation, Equation 4.22, the equation becomes ∂ρt ∂ρt ∂ 2 ρt a˙ ∂ = −(d1 + d2 ) + d1 k 2 + (ρt k) . (4.32) ∂t ∂k ∂k a ∂k Now consider defining a new variable k˜ = k a/a0 and a new density function ρ˜(k) ˜ = a0 ρt (k)/a. Expressed in these variables the spectrum, Equation 4.31, becomes ˜ t) = 8π k˜2 ρ˜0 (k,  ˜  , (4.33) k exp T0 −1 which is constant in time. Expressing Equation 4.32 in terms of ρ˜ and k˜ gives ! ∂ k˜ ∂  ∂ a LHS = + ρ˜ ∂t ∂t ∂ k˜ a0    ∂ a˙ ˜ ∂ a = + k ρ˜ ∂t a ∂ k˜ a0 a˙ a ∂ ρ˜ a˙ ∂ ρ˜ = ρ˜ + + k˜ , a a0 ∂t a0 ∂ k˜ a2 ∂ ρ˜ a2 ∂ 2 ρ˜ a˙ ∂  ˜  RHS = −(d1 + d2 ) 2 + d1 k˜ 2 + kρ˜ , a0 ∂ k˜ a0 ∂ k˜2 a0 ∂ k˜ ∂ ρ˜ a ∂ ρ˜ a ∂ 2 ρ˜ ⇒ = −(d1 + d2 ) + d1 k˜ . (4.34) ∂t a0 ∂ k˜ a0 ∂ k˜2 Defining a new time variable t′ such that dt′ /dt = a/a0 reduces this to ∂ ρ˜ ∂ ρ˜ ˜ ∂ 2 ρ˜ = −(d1 + d2 ) + d1 k , (4.35) ∂t′ ∂ k˜ ∂ k˜2 exactly the same form as the non-expanding diffusion equation, Equation 4.22. 2/3 For a matter dominated FRW universe a ∼ t2/3 , i.e. a(t) = a0 t2/3 /t0 , where t0 is  §4.3 Diffusion and drift in the cosmic microwave background 88 −97 x 10 16 8e 6e −0 −0 05 05 0.0001 14 5e−005 12 6e −0 5e 05 −0 4e 05 8e 05 −0 −0 05 −0 10 05 5 4e 5e−00 1 8 05 d 005 −0 3e 6e 6e −0 −0 3e− 05 05 5e −0 6 05 4e −0 05 4e− 005 5e− 4 005 3e −0 05 05 5 −0 0 2 3e− −0 6e 05 05 e−0 2e −0 5e− 005 6e− 005 4e 5 05 8e− 005 −0 005 8e −1 0 1 d x 10 −95 2 Figure 4.5: The rms deviation of the simulated spectrum from Planckian as a proportion of the spectrum peak, varying both d1 and d2 in the expanding universe. Values of d1 and d2 within the 5e−005 contour give a spectrum that is Planckian to within 50ppm of the peak brightness. the current value of t. Since dt′ /dt = a/a0 integration gives 3 t5/3 t′ = + const . (4.36) 5 t2/3 0 Since the time since the surface of last scattering (t = 1060 ) is of the same order as the age of the universe t0 , this effectively gives t′ ∼ 3/5t. In other words, including expansion is essentially the same as running the non-expanding simulations for 3/5 of the time. As such, the bounds found in Section 4.3.1 would not be expected to change greatly. For completeness the simulations have been run for the expanding case, and the results are shown in Figure 4.5 – the order of magnitude of the bounds is indeed the same.  §4.4 The shifting of spectral lines 89 4.4 The shifting of spectral lines The blackbody nature of the cosmic microwave background placed strong bounds on the magnitude of the diffusion and drift parameters. It is worthwhile, however, looking at other potential astrophysical consequences of the diffusion and drift: are observable effects ruled out by our current bounds, or do other sources allow even stronger bounds to be derived? Here the effect of drift and diffusion on the spectral lines of distant sources will be investigated. Diffusion will result in a broadening of spectral lines. The energy dependence of the diffusion term in Equation 4.17 implies higher energy lines will experience greater broadening than low energy lines. The difficulty is, however, that spectral lines from distant sources will experience line broadening due to a number of other effects. For example, the varying velocities of atoms in a gas cloud source will result in thermal Doppler broadening of the spectral lines. A calculation of the exact amount of broad- ening due to diffusion would not be trivial. Given the very tight bound on the diffu- sion parameter d1 < 7 × 10−97 , together with broadening due to other effects and the inherent spectral resolution (discussed further below), it seems unlikely that broaden- ing due to diffusion would be observable. In this section I will instead concentrate on drift. The drift term will shift all lines in a spectrum by a fixed amount. Spectral lines from astrophysical sources are, of course, already shifted with respect to laboratory spectra due to the expansion of the universe. The expansion of the universe results in lines shifted by a fixed ratio, the redshift z: λobs 1+z = , (4.37) λe where λobs is the wavelength we observe, and λe is the wavelength at which the light was emitted. Since the drift is independent of wavelength, this effect could be easily distinguished from redshifting. In fact, if drift occurs, the calculated redshift for a source would depend on which spectral line was used. Assuming that the diffusion constant is zero, the drift will shift spectral lines by an energy ∆k = d2 t, where t is the time of travel of the photons. If the line has also been redshifted the observed line will  §4.4 The shifting of spectral lines 90 −17 x 10 2.5 2 flux (erg cm−2s−1 Angstrom−1) 1.5 1 0.5 0 Lyβ Lyα −0.5 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 λ (Angstroms) Figure 4.6: The spectrum of quasar SDSSp J083643.85+005453.3 at redshift z = 5.82.3 have energy Eobs = Eredshif t + d2 t Eemit = + d2 t . (4.38) 1+z If drift occurs, but is not taken into account, observations would lead to the calculation of an apparent redshift z ′ that differed from the actual redshift: Eemit 1 + z′ = Eobs Eemit = (1 + z) . (4.39) Eemit + (1 + z)d2 t As an example, consider the quasar SDSSp J083643.85+005453.3 with redshift z = 5.82±0.02 [45], whose spectrum is shown in Figure 4.6. A prominent line in the spectra of quasars is the hydrogen Lyman-α line, which has a wavelength of 121.6nm. For a quasar at z = 5.82 this line is redshifted to 829.3nm, or working in energies Eemit = 1.634 × 10−18 J, (4.40) Eredshif t = 2.395 × 10−19 J . (4.41) 3 Many thanks to Rick White (Space Telescope Science Institute) and Bob Becker (University of Cali- fornia Davis) for providing the data to plot this spectrum.  §4.4 The shifting of spectral lines 91 To determine how much the Lyman-α line would be shifted by the drift, it is nec- essary to know the time it has taken for the light to travel from the quasar. The cosmic time-redshift relation 2 tu = 1/2 , (4.42) 3H0 Ωm (1 + z)3/2 (see for example [46]) gives the approximate age of the universe when photons at a redshift of z were emitted. Take the Hubble parameter to be H0 = 70.5kms−1 M pc−1 , the matter density to be ΩM = 0.274 and the current age of the universe to be t0 = 13.72 × 109 years [47]. The age of the universe when the photons from the quasar were emitted is tu = 9.64 × 108 years (2.25 × 1059 in Planck units), and thus they have been travelling for t0 − tu = 1.28 × 1010 years (2.98 × 1060 in Planck units). In Section 4.3 the drift parameter was constrained to be −5 × 10−96 < d2 < 5 × 10−96 . These values give a drift of magnitude ∆E = d2 t = 1.49 × 10−35 = 7.30 × 10−26 J . (4.43) This drift is seven orders of magnitude smaller than the redshifted energy 2.395 × 10−19 J. When measuring the spectra of the quasar, Fan et al. [45] note they have a ˚ A remeasurement of the spectra [48] gives a much spectral resolution of about 20A. ˚ If the measured Lyman-α line is 8293 ± 2A, better resolution of about 2A. ˚ this corre- sponds in energy to approximately 2.395 × 10−19 ± 1 × 10−22 J. The drift is thus much smaller than even the improved spectral resolution. If z = 5.82 is the true redshift of the quasar, the measured redshift from Equa- tion 4.39 due to the shift in the Lyman-α line would be imperceptibly different. If one were to also consider the Lyman-β line at 102.5nm, the difference between the redshifts calculated using Lyman-α and Lyman-β would be of the order 10−7 , much smaller than the stated uncertainty of 0.02. The effect of drift would be more apparent for more distant objects. The calcu- lations given here for the high redshift quasar at z = 5.82 do however seem to rule out any observation of the drift in spectra in the near future. It is certainly clear that the bounds placed on the drift parameter through consideration of the CMB are much  §4.4 The shifting of spectral lines 92 stronger than any that could currently be obtained from spectral lines. This is of course hardly surprising given the combination of the very accurate measurements of the CMB and the redshift of z ∼ 1000.  §4.5 Concluding remarks 93 4.5 Concluding remarks The assumption of Lorentz invariance is very powerful: despite the lack of an under- lying microscopic model for massless particles, Lorentz invariance allows a concrete diffusion model to be derived in the continuum limit. Such models were derived in Sections 4.1 and 4.2 in terms of affine time and cosmic time respectively. For mass- less particles without internal degrees of freedom, the diffusion model is found to have two free parameters – making the phenomenological model, although powerful, slightly less powerful than the massive particle case. A distribution of massless particles initially peaked at some energy will undergo a diffusion in energy at a rate governed by the diffusion parameter d1 , and also drift to a higher or lower energies depending on the parameter d2 . The direction of the photon is unchanged, and light still travels at the speed of light. The values of the two parameters will depend on properties of the, as yet un- known, microscopic model. For massive particles the diffusion parameter depended on the forgetting time, τf – for massless particles it is likely there will also be some dependence on a non-locality scale, such as that discussed in [39]. The parameters could also depend on properties of the wave packet associated with the photon. Fu- ture work on scalar field propagation may lead to progress in this area. In the mean- time, constraints can be placed on the values of the parameters by comparison with observations. The CMB, consisting as it does of the ‘oldest’ photons, provides the perfect labora- tory for constraining the drift and diffusion. As shown in Section 4.3, the blackbody nature of the CMB constrains d1 and d2 very strongly. The tightness of the constraints is due not only to time the photons have been propagating, but also to the accuracy with which the CMB spectrum has been measured. Conceivably, future measure- ments could narrow the deviation from blackbody even further and thus result in even tighter bounds on d1 and d2 . The drift and diffusion in energy of massless particles could have other observable consequences. One possibility, as discussed in Section 4.4, is the broadening and shift of spectral lines. The tight constraints from the CMB do, however, appear to rule out the detection of the effect in spectral lines with current technology. As mentioned earlier, the work up to this point has considered only particles with  §4.5 Concluding remarks 94 no internal degrees of freedom. The ideas here can, however, be applied more gen- erally. In particular, as will be seen in the next chapter, the effect of an underlying discreteness on photon polarisation can be modelled.  Chapter 5 P OLARISATION In the previous chapter the effect of spacetime discreteness on the energy of massless particles was discussed, but the possible polarisation of the particles was neglected. Here that omission is remedied. The study of polarisation is of particular relevance at present – the polarisation of the cosmic microwave background (CMB) was first detected as recently as 2002 by DASI. It has since been observed by a number of experiments. Current and future experiments, such as the recently launched Planck, will hopefully reveal even further information. As such, the CMB polarisation is an ideal testing ground for quantum gravity phenomenology at the moment. If spacetime is discrete it is expected that the polarisation of a photon travelling through spacetime will be subject to some fluctuations. Unfortunately there is, again, no underlying model for polarisation on a causal set. It is hoped that such a model will be developed in the future. In the meantime, the expected continuum behaviour of a polarised photon in an underlying discrete spacetime can be found by following the stochastic evolution on a manifold of states procedure. Before deriving a diffusion equation for polarisation it is necessary to choose, from the multitude of possibilities, a suitable way to describe the polarisation. This choice is discussed in Section 5.1. In Section 5.2 the affine time polarisation diffusion equa- tion is derived. It is found that restricting to linear polarisation is necessary for useful concrete phenomenology. The cosmic time linear polarisation diffusion equation is derived in Section 5.3, and it is found that polarised light will experience both a sup- pression in polarisation fraction and a rotation in polarisation angle. Section 5.4 discusses possible ways to constrain the free parameters of the model. Rough constraints from radio galaxy polarisation data are discussed, as is the effect of rotation and suppression on the CMB. The majority of the work in this chapter appears in [49]. 95  §5.1 The polarisation state space 96 5.1 The polarisation state space The polarisation of a massless particle with momentum kµ can be described by a com- plex four-vector aµ , that satisfies kµ aµ = 0 and a∗µ aµ = 1. This description is not, however, gauge invariant: if aµ satisfies these conditions then a′ µ = aµ + λk µ , for any complex number λ, will describe the same polarisation state. To avoid this problem, the polarisation can be described in terms of a complex two-form Pµν = kµ aν − aµ kν . This Pµν satisfies the following Lorentz invariant conditions: 1. Pµν = −Pνµ ; 2. Pµν P µν = 0 and Pµν ∗ P µν = 0 ; 3. P µν kµ = 0 ; 4. P µν∗ Pµσ = kν kσ . Note that it is also the case that any Pµν satisfying these conditions can be expressed in terms of some aµ in the form above. The antisymmetry condition (1) reduces Pµν to six complex (12 real) degrees of freedom. The constraints (2) reduce it further to 10 real degrees of freedom. P µν kµ = 0 and P µν∗ kµ = 0 give six independent con- straints, leaving only four degrees of freedom. Finally, the constraint (4) is only one new condition, leaving Pµν with three real degrees of freedom. Consider the specific case kµ = sµ where sµ = (1, 0, 0, 1). The condition kµ aµ = 0 requires aµ = (a0 , a1 , a2 , −a0 ), and thus: Pµν = sµ aν − aµ sν   0 −a1 −a2 0   a1 0 0 −a1    =   ,  (5.1) a2 0 0 −a2    0 a1 a2 0 where |a1 |2 +|a2 |2 = 1. The overall phase of the complex vector v = (a1 , a2 ) ∈ C2 is not physically relevant for the description of the polarisation state of a single photon (see for example [50]), leaving a state space with only two real dimensions. This state space is the Bloch (or Poincar´e) sphere, which will be denoted B. The diffusion equations in the following sections will be derived in terms of the angles on the Bloch sphere rather  §5.1 The polarisation state space 97 than the polarisation two-form Pµν . The resulting equations are of course still Lorentz invariant, despite this not necessarily being immediately apparent. For further discus- sion of the explicitly Lorentz invariant Pµν formulation see Appendix A. The next step is to show that the state space given by v = (a1 , a2 ) is indeed the Bloch sphere. Let m+ = √1 (1, i) and m− = √1 (1, −i), corresponding to left and right 2 2 circularly polarised states, respectively. The vector v = (a1 , a2 ) can be expanded in terms of {m+ , m− }, and an overall phase extracted to leave the coefficient of m+ real and positive:   π  π  v = eiγ cos χ + m+ + e2iψ sin χ + m− , (5.2) 4 4 e iγ h  π   π i ⇒ a1 = √ cos χ + + e2iψ sin χ + , (5.3) 2 4 4 ieiγ h  π  π i a2 = √ cos χ + − e2iψ sin χ + , (5.4) 2 4 4 where χ ∈ [−π/4, π/4] and ψ ∈ [0, π]. Note that this slightly unnatural choice of angles is made to match the standard Poincar´e sphere description of polarisation shown in Figure 5.1 (see, for example [51]). The north and south poles (χ = ±π/4) of the Bloch sphere correspond to  π eiγ+2iψ v χ= = √ (1, −i) , (5.5) 4 2  π eiγ v χ=− = √ (1, i) , (5.6) 4 2 right and left circularly polarised states, respectively. States on the equator (χ = 0) have eiγ   1 2iψ 1 a1 = √ √ +e √ 2 2 2 iψ+iγ = e cos ψ , (5.7) iγ   e 1 2iψ 1 a2 = i√ √ −e √ 2 2 2 iψ+iγ = e sin ψ , (5.8) and thus the equator consists of the linear polarisation states. The remainder of the sphere describes elliptical polarisation.  §5.1 The polarisation state space 98 right ir ular polarization 2χ 2ψ linear polarized states left ir ular polarization Figure 5.1: The Bloch/Poincar´e sphere description of polarisation. General kµ The above calculations showed that for the specific case of kµ = sµ = (1, 0, 0, 1) the po- larisation state can be described by the angles χ and ψ on the Bloch sphere. It remains to show that for a general kµ the polarisation state Pµν can be expressed in terms of the Bloch sphere. This can be done by transforming Pµν by the Lorentz transforma- tion that takes kµ to sµ and then simply using the Bloch sphere description for the special sµ case. In general the Lorentz transformation mapping kµ to sµ is not unique. A standard Lorentz transformation taking sµ to kµ can, however, be defined by (see, for example [50]): ˆ z (|k|) , L(k) := R(k)B (5.9) ˆ where Bz (|k|) is the boost along the z-direction that takes sµ to (|k|, 0, 0, |k|), and R(k) ˆ If k is the standard rotation that takes the z-axis into the direction of the unit vector k. ˆ has spherical polar coordinates θ , φ then the standard rotation is defined by ˆ := Rz (φ)Ry (θ) , R(k) (5.10)  §5.1 The polarisation state space 99 i.e. a rotation by angle θ around the y-axis followed by a rotation by φ around the z-axis. The transformation L(k)−1 takes kµ to sµ . Under the same transformation the polarisation state, Pµν , gives P (k)µν := (L(k)−1 )ρµ (L(k)−1 )σν Pρσ . (5.11) P (k) now has the same form as Equation 5.1, with components P (k)µν = sµ aν − aµ sν , where a1 = P (k)10 and a2 = P (k)20 . Equation 5.11 will henceforth be written P (k) = L(k)−1 P for simplicity. Thus, a general state (kµ , Pµν ) is specified by coordinates (kµ , χ, ψ), where the angles χ and ψ are determined by first Lorentz transforming Pµν into the form of Equation 5.1, and then using Equations 5.3 and 5.4. 5.1.1 Polarisation under Lorentz transformation Before moving onto the derivation of the diffusion equation, it is necessary to know how the polarisation state (χ, ψ) changes under Lorentz transformation. The pho- ton state (kµ , Pµν ) transforms as usual under a Lorentz transformation Λ: (kµ , Pµν ) → (k′ µ , P ′ µν ) = (Λµσ kσ , Λρµ Λσν Pρσ ). For simplicity this will be written (k′ , P ′ ) = (Λk, ΛP ). Under Lorentz transformation (a1 , a2 ) → (a′1 , a′2 ) = (P ′ (k′ )10 , P ′ (k′ )20 ) and P ′ (k′ ) = L(k′ )−1 P ′ = L(Λk)−1 ΛP = L(Λk)−1 ΛL(k)P (k) = W (Λ, k)P (k) , (5.12) where W (Λ, k) = L(Λk)−1 ΛL(k) is an element of the little group of the Lorentz trans- formations that leaves sµ fixed. The little group is isomorphic to the Euclidean group ISO(2): the set of rotations and translations of the plane. Following Weinberg [50], an element of the little group can be written W (θ, α, β) = S(α, β)R(θ) , (5.13)  §5.1 The polarisation state space 100 where S is the ‘translation’   1 + ζ α β −ζ    α 1 0 −α    µ Sν (α, β) =   ,  (5.14)  β 0 1 −β    ζ α β 1−ζ α and β are real numbers and ζ = (α2 + β 2 )/2. R(θ) is the usual rotation   1 0 0 0   0 cos θ − sin θ 0   Rνµ (θ) =   .  (5.15) 0 sin θ cos θ 0   0 0 0 1 To see how the translations S(α, β) act on the polarisation consider the infinitesimal transformation   1 α β 0   α 1 0 −α   µ ∆Sν (α, β) =   .  (5.16) β 0 1 −β    0 α β 1 From Equation 5.1 it is clear that ∆S can be written ∆Sνµ (α, β) = δνµ + sµ bν − bµ sν , (5.17) where sµ bµ = 0 and thus bµ = (b0 , α, β, −b0 ). Recall P (k)µν = sµ aν − aµ sν . Thus ′ ′  ′ ′ ′  ′ ′ ′  ∆Sµµ ∆Sνν P (k)µ′ ν ′ = δµµ + sµ bµ − bµ sµ δνν + sν bν − bν sν
× sµ′ aν ′ − aµ′ sν ′ = s µ aν − aµ s ν , (5.18) using the relations sµ sµ = 0, sµ aµ = 0 and sµ bµ = 0. The ‘translations’ therefore leave the polarisation unchanged. The rotation R acts as usual, thus under a Lorentz transformation (a1 , a2 ) → (a′1 , a′2 ) = (a1 cos δ − a2 sin δ, a1 sin δ + a2 cos δ) , (5.19)  §5.1 The polarisation state space 101 where δ = δ(Λ, kµ ). The transformation of the Bloch sphere angles can be determined from Equations 5.3, 5.4: eiγ h  π  π i a′1 = √ cos χ + + e2iψ sin χ + cos δ 2 4 4 ieiγ h  π  π i + √ cos χ + − e2iψ sin χ + sin δ 2 4 4 eiγ h  π  π i 1  iδ  = √ cos χ + + e2iψ sin χ + e + e−iδ 2 4 4 2 ie iγ h  π   π −i  iδ i  + √ cos χ + − e2iψ sin χ + e − e−iδ 2 4 4 2 eiγ+iδ h  π   π i = √ cos χ + + e2i(ψ+δ) sin χ + , (5.20) 2 4 4 eiγ h  π  π i a′2 = √ cos χ + + e2iψ sin χ + sin δ 2 4 4 ieiγ h  π  π i + √ cos χ + − e2iψ sin χ + cos δ 2 4 4 eiγ h  π  π i −i  iδ  = √ cos χ + + e2iψ sin χ + e − e−iδ 2 4 4 2 ie iγ h  π   π 1  iδ i  + √ cos χ + − e2iψ sin χ + e + e−iδ 2 4 4 2 ieiγ+iδ h  π   π i = √ cos χ + − e2i(ψ+δ) sin χ + , (5.21) 2 4 4 thus (χ, ψ) → (χ′ , ψ ′ ) = (χ, ψ + δ) , (5.22) i.e. Lorentz transformations act on the Bloch sphere as rotations around the north- south polar axis. 5.1.2 Stokes parameters The Stokes parameters are a common method of describing polarisation, particularly in cosmology and astrophysics. They have the advantage of being quantities that can be directly measured by experiments. As such, it is useful to understand the relation- ship between the Bloch sphere coordinates χ , ψ and the Stokes parameters I , Q , U , V . The first Stokes parameter, I, is the total intensity of the light. The Bloch sphere, in fact, only parametrises polarisation of a fixed intensity (or more correctly a fixed polarisation intensity Ip where p is the polarisation fraction). The parameters Q and U describe linear polarisation. Consider a set of orthogonal axes {x, y} in the plane  §5.1 The polarisation state space 102 perpendicular to the direction of propagation of the light, and another set of orthog- onal axes {a, b} at 45◦ in the plane to {x, y}. Q is the difference between the intensity transmitted by a polariser that accepts light polarised in the x-direction, and the in- tensity transmitted by one that accepts light polarised in the y-direction. U is given by the difference between the intensity transmitted by a polariser that accepts light in the a-direction, and the intensity transmitted by one that accepts light in the b-direction. Note that the {x, y} can be chosen freely within the plane perpendicular to the prop- agation of the light. In practice there are several standard coordinate conventions, for example the International Astronomical Union coordinate convention defines the z- axis as pointing along the line of sight towards the observer, the x-axis points north and the y-axis, east. The remaining Stokes parameter, V , measures circular polarisa- tion. The Stokes parameters obey the inequality I 2 ≥ Q2 + U 2 + V 2 . (5.23) The polarisation fraction p is defined by p Q2 + U 2 + V 2 p= . (5.24) I Consider a perfectly polarised beam with fixed intensity normalised to one. The relationship between the Bloch sphere angles and the Stokes parameters can be found by calculating the density matrix S = vv † from the vector v = (a1 , a2 ). Equating   1 I +Q U − iV S = , (5.25) 2 U + iV I −Q (see, e. g. [52]), gives Q = cos 2ψ cos 2χ , (5.26) U = sin 2ψ cos 2χ , (5.27) V = sin 2χ . (5.28) More generally, if a collection of photons has a distribution of polarisations given  §5.1 The polarisation state space 103 by a probability density ρ(χ, ψ) on the Bloch sphere, B, the Stokes parameters are Z Q = ρ(χ, ψ) cos 2ψ cos 2χ dχ dψ , (5.29) ZB U = ρ(χ, ψ) sin 2ψ cos 2χ dχ dψ , (5.30) ZB V = ρ(χ, ψ) sin 2χdχdψ . (5.31) B Note that a given set of Stokes parameters Q , U , V does not correspond to a unique distribution. For example, unpolarised light Q = U = V = 0 could be modelled as a uniform distribution of linearly polarised states, or a uniform distribution on the two circularly polarised states alone – or any other distribution uniform in ψ and symmetric in χ.  §5.2 Polarisation diffusion in affine time 104 5.2 Polarisation diffusion in affine time The state space of a photon with polarisation is M = M4 × H30 × B. The affine time λ is again the suitable time parameter for the trajectories. Recall that the stochastic process on M can be described by the equation (Equa- tion 3.15): ∂ρλ  ρ  λ  = ∂A K AB n ∂B − uA ρλ , (5.32) ∂λ n and thus to derive the affine time diffusion equation the Lorentz invariant, symmetric positive semi-definite matrix K AB and the Lorentz invariant vector uA need to be determined. The microscopic density of states, n, on M must also be known. Unfortunately there are just too many choices when it comes to determining in- variant objects on the Bloch sphere. As discussed earlier, Lorentz transformations act on the Bloch sphere as polar rotations. Thus, any tensor, vector, or scalar that does not depend on ψ is Lorentz invariant. For example, K could be the metric on a sphere, or an oblate or prolate spheroid, or indeed any other such strange shape generated by rotating a curve around the polar axis. It need not even be invariant under reflec- tion in the equatorial plane. The invariant vector, u, can be any linear combination of f (χ)∂/∂χ and g(χ)∂/∂ψ, where f and g are arbitrary functions of χ. All these free functions mean the model is not much use for phenomenology. Predictive power can be regained if the model is restricted to linear polarisation only. For astrophysical and cosmological applications this is precisely the area of interest – the CMB, for exam- ple, is linearly polarised. The matter is not quite so simple, however, as assuming the initial distribution has no circular polarisation. An initially linearly polarised distri- bution could become elliptically (or circularly) polarised if K or g(χ) are not invariant under reflection in the equatorial plane. To allow progress to be made, it will be as- sumed from here on that circular polarisation can be neglected. It is hoped that it will be possible to determine if this assumption is justified when the microscopic physics is better understood. In the meantime, useful results can still be obtained. Restricting to linear polarisation, the polarisation state space is just the equator of the Bloch sphere: the unit circle, with coordinate ψ. In a slight abuse of notation the state space with linear polarisation only will also be denoted M. There is one Lorentz invariant vector, ∂/∂ψ, on the space of linear polarised states. The Lorentz invariant density of states n is constant on the circle. As in Chapter 4, it is convenient to work  §5.2 Polarisation diffusion in affine time 105 in polar coordinates (k, θ, φ) on the momentum space H30 . The coordinates on the state space M will be denoted X A = (xµ , k, θ, φ, ψ). As shown Section 4.1 there is the invariant vector field ka = (k, 0, 0), and invariant two-tensor ka kb on H30 . The density of states on H30 is k sin θ. We also know from Section 4.1 that the spacetime components of K AB are zero, and the spacetime component of uA is simply kµ . Let v1 and v2 be the invariant vectors v1A = (0, k, 0, 0, 0) and v2A = (0, 0, 0, 0, 1) on M. The most general symmetric, positive semi-definite invariant two-tensor on the full state space M is K AB = av1A v2A + b(v1A v2B + v2A v1B ) + cv2A v2B xµ k θ φ ψ   xµ 0 0 0 0 0   k  ak 2 0 0 bk   0    = θ  0 0 0 0 ; (5.33)   0     φ  0 0 0 0 0    ψ 0 bk 0 0 c the invariant vector is uA = (kµ , dk, 0, 0, e) . (5.34) Here a ≥ 0, c ≥ 0, b, d, e are real constants. In order for K AB to be positive semi- definite the constants must also satisfy ac − b2 ≥ 0. On M the density of states is simply n = k sin θ. Substituting these forms for K AB and uA into Equation 5.32 above gives the affine time linear polarisation diffusion equation ∂ρλ  ρ  λ  = ∂A K AB n∂B − uA ρλ ∂λ n ∂ρλ ∂ 2 ρλ ∂ρλ = −kµ µ + ak 2 + (a − d)k − (a + d)ρλ ∂x ∂k2 ∂k ∂ 2 ρλ ∂ 2 ρλ ∂ρλ +c + 2bk −e . (5.35) ∂ψ 2 ∂ψ∂k ∂ψ At first glance this equation appears unwieldy, with still too many free parameters to be useful for phenomenology. Recall, however, the massless particle affine time  §5.2 Polarisation diffusion in affine time 106 diffusion equation, Equation 4.4:   ρ  ∂ρλ ∂ρλ ∂ 3 ∂ λ ∂ = −kµ µ+ d1 k − d2 (kρλ ) ∂λ ∂x ∂k ∂k k ∂k ∂ρλ ∂ 2 ρλ ∂ρλ = −kµ µ + d1 k2 + (d1 − d2 )k − (d1 + d2 )ρλ . (5.36) ∂x ∂k 2 ∂k The parameters a and d in Equation 5.35 are just d1 and d2 renamed. In Section 4.3 very tight constraints were placed on the values of d1 and d2 . Because d1 and d2 are constrained to be so small, they can be neglected in this discussion of linear polarisa- tion. Taking a = d = 0 further requires b = 0. Finally, the homogeneous polarisation affine time equation reduces to ∂ρλ ∂ 2 ρλ ∂ρλ =c 2 −e . (5.37) ∂λ ∂ψ ∂ψ A linearly polarised photon in an underlying discrete spacetime therefore experiences a diffusion, determined by the parameter c, and a drift, given by e, in its angle of polarisation. Although Equation 5.37 is Lorentz invariant, the Lorentz invariance is not immediately obvious. In Appendix A the linear polarisation diffusion equation is formulated in an explicitly Lorentz invariant way, using the polarisation two-form Pµν . The resulting equation, although equivalent to the equation above, provides less immediate physical insight.  §5.3 Polarisation diffusion in cosmic time 107 5.3 Polarisation diffusion in cosmic time Again, to compare with observations the affine time polarisation diffusion equation must be re-expressed in terms of cosmic time. Following the arguments given in Sec- tions 3.3 and 4.2 a larger state space incorporating the affine time is defined: M′ = M × R. The probability density ρλ can be written as a component of the current in M′ : J λ (t, xi , ka , ψ, λ) = ρλ . (5.38) The t component of the current is J t (t, xi , ka , ψ, λ) = kJ λ , (5.39) where Equation 3.5 has been used. The remaining components of the current, ex- pressed in terms of J t are: ki t J i (t, xi , ka , ψ, λ) = J , (5.40) k ∂J t t   ∂ J J k (t, xi , ka , ψ, λ) = −ak 3 − b + dJ t , (5.41) ∂k k2 ∂ψ c ∂J t ∂ Jt   e J ψ (t, xi , ka , ψ, λ) = − − bk 2 + Jt . (5.42) k ∂ψ ∂k k2 k The unobservable affine time can be integrated over, and a new integrated current J¯ defined: Z ∞ J¯t (t, xi , ka , ψ) = J t (t, xi , ka , ψ, λ)dλ , (5.43) 0 ki J¯i (t, xi , ka , ψ) = J¯t , (5.44) k  ¯t  ¯ 3 ∂ J ∂ J¯t k i a J (t, x , k , ψ) = −ak − b + dJ¯t , (5.45) ∂k k2 ∂ψ c ∂ J¯t  ¯t  ¯ 2 ∂ J e ψ i a J (t, x , k , ψ) = − − bk 2 + J¯t . (5.46) k ∂ψ ∂k k k The continuity equation on M′ can also be integrated over λ to give: h i∞ Jλ + ∂t J¯t + ∂i J¯i + ∂k J¯k + ∂ψ J¯ψ = 0 . (5.47) 0 J λ |λ=0 is zero for all t > 0 and J λ tends toward zero as λ approaches infinity for finite  §5.3 Polarisation diffusion in cosmic time 108 t. Thus, renaming J¯t as ρt , the polarisation cosmic time equation is ∂ρt = −∂i J¯i − ∂k J¯k − ∂ψ J¯ψ ∂t ki ∂ρt ∂ 2 ρt ∂ρt ∂ 2 ρt = − + ak − (a + d) + 2b k ∂xi ∂k2 ∂k ∂k∂ψ 2 c ∂ ρt 2b ∂ρt e ∂ρt + − − . (5.48) k ∂ψ 2 k ∂ψ k ∂ψ Neglecting the energy drift and diffusion as discussed in Section 5.2 (a = d = 0 ⇒ b = 0), and looking at the homogeneous case gives ∂ρt δ1 ∂ 2 ρt δ2 ∂ρt = − , (5.49) ∂t k ∂ψ 2 k ∂ψ where, for convenience, the diffusion parameter, c, and drift parameter, e, have been renamed δ1 and δ2 respectively. From this equation we see that the rates of drift and diffusion in linear polarisation angle are energy dependent: the higher the energy of the photon, the slower the rate of drift and diffusion. This result is particularly inter- esting because, although there are other models that predict drift in linear polarisation angle, most do not have this particular energy dependence. This will be discussed fur- ther in Section 5.4. 5.3.1 The solution to the diffusion equation Equation 5.49 is a standard equation describing diffusion with drift on a circle. Solu- tions to this type of equation are well known, see for example [53]. Here I will solve the equation using Fourier series. First, note that the diffusion occurs on a half circle rather than a full circle: ψ ∈ [0, π]. Physically, a polarisation angle ψ and a polarisation angle ψ + π are equivalent. For simplicity I define a new variable x = ψ/π, 0 ≤ x < 1. The distribution ρt (t, ψ) is a scalar density, rather than a scalar, and thus under this rescaling transforms as dψ ρ¯t (t, x) = ρt (t, ψ) dx 1 = ρt (t, ψ) . (5.50) π  §5.3 Polarisation diffusion in cosmic time 109 Equation 5.49 becomes 1 ∂ ρ¯t δ1 ∂ 2 ρ¯t δ2 ∂ ρ¯t = 3 2 − 2 , (5.51) π ∂t kπ ∂x kπ ∂x ∂ ρ¯t δ1 ∂ 2 ρ¯t δ2 ∂ ρ¯t ⇒ = 2 2 − . (5.52) ∂t kπ ∂x kπ ∂x Expand ρ¯t in a Fourier series: ∞ X ρ¯t (t, x) = cn (t)e2πinx , (5.53) n=−∞ Z 1 cn (t) = ρ¯t (t, x)e−2πinx dx. (5.54) 0 The diffusion equation can thereby be written as an ordinary differential equation 1 ∂ ρ¯t −2πinx Z c˙n = e dx 0 ∂t Z 1 δ1 ∂ 2 ρ¯t  δ2 ∂ ρ¯t = 2 ∂x2 − e−2πinx dx. (5.55) 0 kπ kπ ∂x Integrating by parts and recalling ρ¯t (x = 1) = ρ¯t (x = 0) and e−2πin = 1 gives 1 δ1 ∂ 2 (e−2πinx ) δ2 ∂(e−2πinx ) Z  c˙n = ρ t + ρt dx 0 kπ 2 ∂x2 kπ ∂x 2   4δ1 n 2iδ2 n = − − cn , (5.56) k k 4δ1 n2 2iδ2 n     ⇒ cn (t) = cn (0) exp − + t . (5.57) k k Let the initial distribution be f¯(x) = ρ¯t (0, x), then Z 1 cn (0) = f¯(y)e−2πiny dy , (5.58) 0 X∞ ⇒ ρ¯t (t, x) = cn (t)e2πinx n=−∞ ∞ Z 1 4δ1 n2 2iδ2 n X     = f (y)e dy exp − −2πiny + t e2πinx n=−∞ 0 k k Z 1 X∞ 4δ1 n2 2iδ2 n     = exp − + t e2πin(x−y) f¯(y)dy 0 n=−∞ k k = p¯ ◦ f¯ , (5.59)  §5.3 Polarisation diffusion in cosmic time 110 where ∞ 4δ1 n2 2iδ2 n X    p¯(x) = exp − + t e2πinx , (5.60) n=−∞ k k is the fundamental solution (see for example [53]). To be consistent with the notation that appears in other papers (e.g. [54, 55]) p¯ can be expressed in a slightly different way. Making use of the Jacobi identity ∞ ∞ (x − n)2 √ X   X exp −2π 2 n2 t e2πinx ,   exp − = 2πt (5.61) n=−∞ 2t n=−∞ p¯ can be written ∞      X δ2 t 2δ1 t p¯(x) = exp −2π 2 n2 2 exp 2πin x − n=−∞ kπ kπ   2  δ2 t  x − kπ − n  r ∞ πk X = exp − 4δ1 t , (5.62) 4δ1 t n=−∞ kπ 2 1 p(ψ) = p¯(x) π   2  δ2 t  ψ− − πn r ∞ k X k = exp − . (5.63)  4πδ1 t n=−∞ 4δ1 t k The full solution to Equation 5.49 is thus   2  δ2 t r πk Z 1 ∞ X π(x − y) − k − nπ ¯ ρ¯t (t, x) = exp −  f (y)dy  4δ1 t 4δ1 t 0 n=−∞ k   2  δ2 t  π(x − y) − r ∞ n+1 πk Z X k ¯ = exp − 4δ1 t  f (y)dy 4δ1 t n=−∞ n k dt 2 "
# ∞ r πk π(x − y) − Z = exp − 4δ1 t k f¯(y)dy , (5.64) −∞ 4δ1 t k 1 ⇒ ρt (t, ψ) = ρ¯t (t, x) π   2  ′ δ2 t  ψ−ψ − Z ∞r k k ′ ′ = exp −  f (ψ )dψ . (5.65)  4πδ1 t 4δ1 t −∞ k  §5.3 Polarisation diffusion in cosmic time 111 In the case of zero diffusion, i.e. δ1 = 0, the fundamental solution, Equation 5.60, becomes ∞    X δ2 t p¯(x) = exp 2πin x − , (5.66) n=−∞ πk i.e. an initial distribution of δ(x) will become rotated by δ2 t/πk, corresponding to an angle of δ2 t/k. For an arbitrary initial distribution f¯(x), this gives a solution 1 ∞    δ2 t Z f¯(y)dy , X ρ¯t (t, x) = exp 2πin (x − y) − (5.67) 0 n=−∞ πk Z π X∞    ′ δ2 t ρt (t, ψ) = exp 2in (ψ − ψ ) − f (ψ ′ )dψ ′ . (5.68) 0 n=−∞ k Thus, the drift term in the linear polarisation diffusion equation simply rotates the plane of polarisation of light by an angle α: ψ → ψ ′ = ψ + α, where α = δ2 t/k. In terms of the Stokes parameters, V = 0 since there is no circular polarisation and Q and U become: Q → Q′ = Q cos 2α − U sin 2α , (5.69) U → U ′ = U cos 2α + Q sin 2α . (5.70) The diffusion term acts as an overall suppression of polarisation. To see this, first note that given a collection of photons with a linear polarisation distribution ρ(ψ), the Stokes parameters are Z π Q = ρ(ψ) cos 2ψdψ , (5.71) Z0 π U = ρ(ψ) sin 2ψdψ . (5.72) 0  §5.3 Polarisation diffusion in cosmic time 112 Consider a source with an initial distribution ρi (ψ) = δ(ψ). This implies Qi = 1, Ui = 0, and the light is fully polarised, i.e. the polarisation fraction is p = 1. Suppose ρi evolves under some process, such as the diffusion process, to a distribution p(ψ), the fundamental solution of the process. The Stokes parameters are then Z π Qf = p(ψ) cos 2ψ dψ , (5.73) 0 Z π Uf = p(ψ) sin 2ψ dψ . (5.74) 0 The polarisation fraction is now p2 = Q2f + Uf2 Z π 2 Z π 2 = p(ψ) cos 2ψ dψ + p(ψ) sin 2ψ dψ 0 0 ≤ 1. (5.75) Assuming that the process is homogeneous, i.e. δ(ψ − ψ ′ ) → p(ψ − ψ ′ ), an arbitrary Rπ initial distribution ρt (0, ψ) will evolve to ρt (t, ψ) = 0 ρt (0, ψ ′ )p(ψ−ψ ′ )dψ ′ . The Stokes parameters are therefore Z π Qi = ρt (0, ψ) cos 2ψ dψ , (5.76) Z0 π Ui = ρt (0, ψ) sin 2ψ dψ , (5.77) Z0 π Z π Q(t) = ρt (0, ψ ′ )p(ψ − ψ ′ ) cos 2ψ dψ ′ dψ , (5.78) Z0 π Z0 π U (t) = ρt (0, ψ ′ )p(ψ − ψ ′ ) sin 2ψ dψ ′ dψ . (5.79) 0 0 Consider the linear polarisation diffusion equation with drift δ2 = 0. For zero drift the fundamental solution, p(ψ), is given by Equation 5.60 with δ2 = 0: ∞ −4δ1 n2 t 2inψ   1 X p(ψ) = exp e . (5.80) π n=−∞ k  §5.3 Polarisation diffusion in cosmic time 113 The Q Stokes parameter is therefore π π ∞ −4δ1 n2 t 2in(ψ−ψ′ )   1 X Z Z ′ Q(t) = ρt (0, ψ ) exp e cos 2ψ dψ ′ dψ 0 0 π n=−∞ k ∞ Z π Z π −4δ1 n2 t  1 X ρt (0, ψ ′ ) cos 2n ψ − ψ ′ cos 2ψ dψ ′ dψ
= exp π n=−∞ k 0 0 ∞ −4δ1 n2 t   1 X = exp π n=−∞ k Z πZ π ρt (0, ψ ′ ) cos 2nψ cos 2nψ ′ + sin 2nψ sin 2nψ ′ cos 2ψ dψ ′ dψ   × 0 0 ∞ Z π Z π −4δ1 n2 t  1 X = exp cos 2nψ cos 2ψ dψ ρt (0, ψ ′ ) cos 2nψ ′ dψ ′ π n=−∞ k 0 0 ∞ Z π Z π −4δ1 n2 t  1 X + exp sin 2nψ cos 2ψ dψ ρt (0, ψ ′ ) sin 2nψ ′ dψ ′ . π n=−∞ k 0 0 (5.81) The second sum vanishes and only the n = ±1 terms in the first sum contribute, leaving  Z π −4δ1 t Q(t) = exp ρt (0, ψ ′ ) cos 2ψ ′ dψ ′ k   0 −4δ1 t = exp Qi . (5.82) k Similarly, the U Stokes parameter is π π ∞ −4δ1 n2 t 2in(ψ−ψ′ )   1 X Z Z ′ U (t) = ρt (0, ψ ) exp e sin 2ψ dψ ′ dψ 0 0 π n=−∞ k ∞ Z π Z π −4δ1 n2 t  1 X ρt (0, ψ ′ ) cos 2n ψ − ψ ′ sin 2ψ dψ ′ dψ
= exp π n=−∞ k 0 0 ∞ −4δ1 n2 t   1 X = exp π n=−∞ k Z πZ π ρt (0, ψ ′ ) cos 2nψ cos 2nψ ′ + sin 2nψ sin 2nψ ′ sin 2ψ dψ ′ dψ   × 0 0 ∞ Z π Z π −4δ1 n2 t  1 X = exp cos 2nψ sin 2ψ dψ ρt (0, ψ ′ ) cos 2nψ ′ dψ ′ π n=−∞ k 0 0 ∞ π Z π −4δ1 n2 t   1 X Z + exp sin 2nψ sin 2ψ dψ ρt (0, ψ ′ ) sin 2nψ ′ dψ ′ . π n=−∞ k 0 0 (5.83)  §5.3 Polarisation diffusion in cosmic time 114 Here the first sum vanishes and the n = ±1 terms of the second sum give  Z π −4δ1 t U (t) = exp ρt (0, ψ ′ ) sin 2ψ ′ dψ ′ k 0   −4δ1 t = exp Ui . (5.84) k The polarisation suppression due to the diffusion is thus p2f p2 = p2i Q(t)2 + U (t)2 = , (5.85) Q2i + Ui2   −4δ1 t ⇒ p = exp . (5.86) k Both the suppression and rotation are independent of the current polarisation an- gle, and thus in the general case of nonzero δ1 and δ2 a linearly polarised beam in a spacetime with an underlying discreteness will experience a rotation in polarisation angle and a suppression of its polarisation fraction.  §5.4 Observational constraints 115 5.4 Observational constraints The model discussed here is by no means the only mechanism for producing rotation of the plane of linear polarisation. Such rotations, when discussed in the context of the polarisation of astrophysical or cosmological sources, are usually referred to as ‘cosmological birefringence’. Rotation of polarisation is not merely hypothetical – the polarisation from cosmological sources is well known to be subject to rotation, namely the Faraday effect. Light travelling through the interstellar medium is subject to ro- tation of the plane of polarisation, because the presence of a magnetic field results in different refractive indices for left and right circularly polarised modes. The angle of rotation, α, is dependent on the wavelength: α ∝ λ2 . Proposed methods for generating additional cosmological birefringence include a loop quantum gravity motivated modification of the dispersion relation [56, 57]. This modification leads to circular polarisation modes travelling at different velocities and thus can give birefringence. In this case the rotation angle α ∝ 1/λ2 ∝ k2 . Quantum gravity motivated birefringence proportional to k2 is also discussed in [58] and [59], where polarisation data from the Crab Nebula and CMB respectively are used to con- strain the model parameter. Birefringence also arises in discussions of dark energy, specifically the quintessence approach. The coupling of the quintessence scalar field to the pseudoscalar of elec- tromagnetism leads to birefringence independent of wavelength [60, 61, 62]. Vio- lating CPT symmetry through the addition of a Chern-Simons term to the Maxwell Lagrangian also results in rotation of the plane of polarisation independent of wave- length [63, 64, 65]. Constraints on such polarisation rotation come from radio galaxy polarisation data and CMB polarisation. Methods of detecting polarisation rotation through radio galaxy data and the CMB will be discussed further below. None of the proposed effects discussed above have the α ∝ 1/k dependence of the polarisation diffusion model. One source of a 1/k rotation has been suggested previously: according to Prasanna and Mohanty [66], polarisation rotation could be induced by a gravitational wave propagating in the same direction as the photons. This effect would only be present where there is a source of strong gravitational ra- diation, such as a binary pulsar, and thus it is unlikely to be confused with the effect studied here. Note also that most models that predict birefringence violate Lorentz  §5.4 Observational constraints 116 invariance as well as parity. The polarisation diffusion model of this chapter is partic- ularly interesting because it preserves Lorentz invariance. Before discussing the bounds on the model parameters, it should be noted that special care needs to be taken when searching for changes in polarisation. Unless the method of generation of polarisation in a source is completely understood, natural variations in polarisation at the source could be mistaken for effects occurring over the travel time of the light to us. 5.4.1 Constraining the drift parameter with radio galaxy data Radio galaxies emit linearly polarised light and often have an elongated structure defining a natural axis against which their polarisation angle can be measured. Af- ter subtracting the effect of Faraday rotation it is found that the polarisation angle for radio galaxies is highly correlated with the observed galaxy axis (radio axis), peak- ing at 90◦ to the radio axis (see, for example, [67, 64, 68] and Figure 5.2). The linear polarisation of radio galaxies is due to the synchrotron process. Although the cause of the correlation between polarisation angle and radio axis does not seem to be fully understood it is likely there is a physical basis for the alignment of the magnetic field. The correlation holds over a range of redshifts and thus provides a method of con- straining a birefringence effect. If it is assumed that the light has a polarisation of 90◦ at the source, the difference from 90◦ in the measured polarisation angle constrains the amount of rotation, and in the case of the model discussed here, the magnitude of the drift parameter. Data also exists for polarisation fraction for radio galaxies. Un- fortunately there seems to be less understanding of what the polarisation fraction is at the source and thus the change in polarisation fraction (and the diffusion parameter) cannot be constrained. In this section I will use data from one particular galaxy to place an order of mag- nitude bound on the drift parameter. In [68] the galaxy MRC 2025-218 was stud- ied. Cimatti et al. deduced a polarisation angle (with respect to the galaxy axis) of θ = 93 ± 8◦ and a polarisation fraction of p = 8.3 ± 2.3%. They concluded that there is “strong evidence that the plane of polarisation is not rotated by more than 10 degrees when the radiation travels from z = 2.63 to us”. Recall from Section 5.3 that the angle of rotation is related to the drift parameter by α = δ2 t/k. The time of travel t can be calculated from the redshift, z = 2.63, of the galaxy. As mentioned in Section 4.4 the  §5.4 Observational constraints 117 15 10 number of galaxies 5 0 0 20 40 60 80 100 120 140 160 180 polarisation − position angle Figure 5.2: The correlation between radio galaxy axis and polarisation angle. Data from 71 galaxies with redshift z > 0.3 is shown here. This data is taken from [64, 67]. cosmic time - redshift relation, 2 tu = 1/2 , (5.87) 3H0 Ωm (1 + z)3/2 gives the approximate age of the universe when photons at a redshift of z were emit- ted. Taking the Hubble parameter to be H0 = 70.5kms−1 M pc−1 , the matter density to be ΩM = 0.274, and the current age of the universe to be t0 = 13.72 × 109 years (see [47]), gives tu = 2.55 × 109 years, and thus the photons have been travelling for a cosmic time of t = 11.17 × 109 years. Cimatti et al. observed the galaxy MRC 1015-218 in the R band, corresponding to a ˚ rest frame spectral region (i.e. non-redshifted wavelength) of ∆λrest = 1600 to 2100A. In the calculation below, I will assume the photons were emitted with a wavelength ˚ As the polarisation rotation angle, α, depends on the wavelength, some of λ = 1850A. error is introduced by taking λ to be a fixed value within the bandwidth. The variation of α over this wavelength range is not expected to be significant however, and should not affect the order of magnitude bound obtained. The polarisation diffusion model assumes Minkowksi spacetime, and thus to cor- rectly bound the drift parameter the change in energy of the photons from the galaxy due to the expansion of the universe must be taken into account. The inclusion of  §5.4 Observational constraints 118 expansion into the model will be discussed more generally in Section 5.4.2, here I will only note that the wavelength of photons in an expanding universe is given by a(t) λ(t) = λe , (5.88) ae where λe is the wavelength the photons were emitted at, a(t) is the scale factor, and ae is the scale factor at the time when the photons were emitted. Assuming a matter 2/3 dominated FRW universe, the scale factor goes as a ∼ t2/3 , i.e. a(t) = t2/3 /t0 where t0 is the current age of the universe and the scale factor today is taken to be a0 = 1. The total angle of rotation of the polarization is thus Z t0 ∆α = δ2 λ(t)dt te Z t0 a(t) = δ2 λe dt te ae t0 t2/3 Z = δ2 λe 2/3 dt te te ! 5/3 3δ2 λe t0 = 2/3 − te . (5.89) 5 te The drift parameter can be constrained to be δ2 < 7 × 10−90 < 2 × 10−37 kgm2 s−3 . (5.90) 5.4.2 Polarisation diffusion and drift in the cosmic microwave background The effect of polarisation rotation on CMB data has been well studied. Suppression of polarisation, however, seems to have been little considered. Before discussing the CMB we need to fully understand how the expansion of the universe changes the diffusion equation. In fact, an equation that includes expansion can be determined in a similar manner to that discussed in Section 4.3.  §5.4 Observational constraints 119 Including the expansion of the universe The redshifting of the photons is given by ae k(t) = ke , (5.91) a(t) where ke is the emitted frequency, a(t) is the scale factor, and ae is the scale factor at the time the photons were emitted. Note that the expansion of the universe also dilutes the photons, but this dilution does not change the diffusion equation. As mentioned in 2/3 the previous section, if a matter dominated FRW universe is assumed, a(t) = t2/3 /t0 , where t0 is the current age of the universe and the scale factor today is taken to be a0 = 1. If the form for k is substituted into the diffusion equation, Equation 5.49, it becomes ∂ρt δ1 a(t) ∂ 2 ρt δ2 a(t) ∂ρt = − ∂t ke ae ∂ψ 2 ke ae ∂ψ 2/3 2   t δ1 ∂ ρt δ2 ∂ρt = − . (5.92) t 2/3 ke ae ∂α2 ke ae ∂ψ 0 A new time variable t′ can be defined by dt′ = a(t), (5.93) dt and thus ∂ρt t2/3 ∂ρt = , (5.94) ∂t′ t 2/3 ∂t 0 ∂ρt ∂ρt dt′ ⇒ = ∂t ∂t′ dt t2/3 ∂ρt = 2/3 ∂t′ , (5.95) t0 ∂ρt δ1 ∂ 2 ρt δ2 ∂ρt ⇒ ′ = 2 − . (5.96) ∂t ke ae ∂ψ ke ae ∂ψ If we further note ke /k0 = a0 /ae , where k0 is the energy observed today, the equation becomes simply ∂ρt δ1 ∂ 2 ρt δ2 ∂ρt ′ = 2 − . (5.97) ∂t k0 ∂ψ k0 ∂ψ  §5.4 Observational constraints 120 In other words, the change in energy due to expansion can be compensated for by fix- ing the energy at the observed energy and simply transforming to a new time variable 3 t5/3 t′ = + const. (5.98) 5 t2/3 0 For analysis of the CMB this is approximately equivalent simply to evolving the equa- tion for 3/5 the time since t is so close to t0 . The polarisation of the CMB The cosmic microwave background is expected to be partially polarised due to Thom- son scattering off free electrons at the surface of last scattering. Thomson scattering can produce only linear polarisation (and only do so when there is a quadrupole anisotropy in the incident radiation) and thus the CMB is expected to have no circular polarisation. The CMB polarisation has now been detected by multiple experiments, including WMAP [69], BICEP [70], and QUaD [71]. The polarisation of the CMB is commonly discussed not in terms of Stokes param- eters, or angles on the Bloch/Poincar´e sphere, but rather in terms of E and B modes (see, e. g. [72]). Instead of describing the polarisation of a single photon, E and B modes describe how the polarisation varies over a small area. The names ‘E’ and ‘B’ mode are drawn from an analogy with electric and magnetic fields: the polarisation pattern on the sky (direction and intensity) is decomposed into a curl-free component (E mode) and a divergence free component (B mode). The exact method by which these components are calculated is not important for the current discussion. As men- tioned above, a quadrupole anisotropy in the radiation at last scattering is necessary for there to be polarisation in the CMB. This anisotropy can arise from scalar, vec- tor, or tensor perturbations. Scalar perturbations, due to density fluctuations in the plasma, give rise to E mode polarisation; vector perturbations, due to vorticity in the plasma, are expected to be negligible; tensor perturbations can be caused by gravita- tional waves, and would give rise to B mode polarisation. Analysis of the CMB is usually in terms of correlations between E mode, B mode, and temperature T . In all, there are six cross-correlation spectra that can be examined: T T, T E, T B, EE, EB, BB. In the absence of a parity violating effect, the correlators T B and EB are expected to vanish. The rotation of the polarisation angle can thus  §5.4 Observational constraints 121 100 5000 50 €€€€€€€€€€€€€€€€€€€€€€ ClTE 4000 €€€€€€€€€€€€€€€€€€€€€€ ClTT 0 l Hl + 1L l Hl + 1L 3000 2Π 2Π -50 2000 1000 -100 -150 0 500 1000 1500 2000 0 500 1000 1500 2000 l l (a) TT. (b) TE. 0.07 40 0.06 0.05 €€€€€€€€€€€€€€€€€€€€€€ ClEE €€€€€€€€€€€€€€€€€€€€€€ ClBB 30 0.04 l Hl + 1L l Hl + 1L 2Π 2Π 20 0.03 0.02 10 0.01 0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 l l (c) EE. (d) BB. Figure 5.3: CMB T T, T E, EE and BB correlation spectra produced using CAMB with stan- dard parameters. l is the multipole moment. be detected through the presence of a nonzero T B and EB. More specifically, for an angle of rotation ∆α the exact change in the correlators can be written as (see, for example [62, 73, 63]): TB C ′l = ClT E sin 2∆α , (5.99) EB 1 C ′l ClEE − ClBB sin 4∆α ,
= (5.100) 2 TE C ′l = ClT E cos 2∆α , (5.101) EE C ′l = ClEE cos2 2∆α + ClBB sin2 2∆α , (5.102) BB C ′l = ClBB cos2 2∆α + ClEE sin2 2∆α , (5.103) where Cl are the spectra when there is no rotation. Note that the T T correlation is unchanged by rotation. For the polarisation diffusion and drift model discussed here the angle ∆α is dependent on the energy, and thus the expected spectra above will depend on the frequency band one is working in. Example correlation spectra can be produced using the Code for Anisotropies in the Microwave Background (CAMB).1 Figure 5.3 shows T T, T E, EE and BB spectra 1 Available at http://camb.info/.  §5.4 Observational constraints 122 75 50 €€€€€€€€€€€€€€€€€€€€€€ C¢ TB 25 l 0 l Hl + 1L 2Π -25 -50 -75 -100 0 500 1000 1500 2000 l (a) TB. 20 €€€€€€€€€€€€€€€€€€€€€€ C¢ lEB 15 l Hl + 1L 2Π 10 5 0 0 500 1000 1500 2000 l (b) EB. Figure 5.4: CMB T B and EB correlation spectra when polarisation is rotated by an arbitrary angle ∆α = π/8. for standard parameters, without any polarisation rotation (recall T B and EB van- ish). Figure 5.4 shows the T B and EB spectra that would result if a rotation of po- larisation occurs; an arbitrary angle ∆α = π/8 has been chosen to illustrate the effect. The changes in T E, EE, and BB spectra under this rotation are shown in Figure 5.5. Note that there is some difficulty determining what values of ∆α are allowed by cur- rent data. For example, Xia et al. [74] give differing (and seemingly contradictory at 68% C.L.) values of ∆α = −2.62 ± 0.87◦ , ∆α = 0.59 ± 0.42◦ , and ∆α = 0.09 ± 0.36◦ depending on which subset of CMB experiments they analyse. Note that as they as- sume ∆α is frequency independent these values cannot be used to constrain the drift parameter δ2 . Recall that α = δ2 t′ /k when expansion of the universe is included. Tak- ing t′ = 3/5 × 1060 , and choosing one of the WMAP observation frequencies, 40GHz, shows that the (probably too large) angle ∆α = π/8 in fact corresponds to δ2 ∼ 10−93 , already several orders of magnitude smaller than the bound placed on δ2 with radio galaxy data in Section 5.4.1.  §5.4 Observational constraints 123 100 50 €€€€€€€€€€€€€€€€€€€€€€ ClTE 0 l Hl + 1L 2Π -50 -100 -150 0 500 1000 1500 2000 l (a) TE. 40 €€€€€€€€€€€€€€€€€€€€€€ ClEE 30 l Hl + 1L 2Π 20 10 0 0 500 1000 1500 2000 l (b) EE. 20 €€€€€€€€€€€€€€€€€€€€€€ ClBB 15 l Hl + 1L 2Π 10 5 0 0 500 1000 1500 2000 l (c) BB. Figure 5.5: CMB T E, EE, and BB correlation spectra when polarisation is rotated by an ar- bitrary angle ∆α = π/8 (dashed lines) compared to the spectra in the unrotated case (solid lines).  §5.4 Observational constraints 124 The discussion above has all been concerned with the effect of the rotation of the plane of polarisation on CMB data. Effects such as the suppression of polarisation predicted by the polarisation diffusion equation do not seem to have been discussed much in the literature. In terms of the correlation spectra above, the diffusion will sim- ply reduce the amplitude of each spectra by some fixed, energy dependent, amount. To properly constrain the drift and diffusion parameters, CMB data from a single fre- quency band must be compared with the corresponding predicted spectra generated by CAMB and modified for drift and diffusion. Comparing over a range of drift and diffusion parameters will allow the values that best fit current CMB observations to be found. Unfortunately this process requires modifying the Markov Chain Monte Carlo code, CosmoMC, and is outside the scope of this thesis. This work has recently been undertaken by Carlo Contaldi and the results appear in [49].  §5.5 Concluding remarks 125 5.5 Concluding remarks When studying massless particles with polarisation in a discrete spacetime, the as- sumption of Lorentz invariance turns out to be insufficient to completely constrain the phenomenological model. To make concrete predictions it is necessary to restrict attention to linear polarisation – fortunately it is precisely linear polarisation that is of interest in astrophysics and cosmology. As yet there is no underlying microscopic model for polarised photons on a causal set (or even, as discussed in Chapter 4, unpolarised photons). When particles in causal set theory are better understood it is hoped that useful models including circular po- larisation will be able to be developed and the validity of the results obtained in the current restriction to linear polarisation checked. For linear polarisation a polarisation diffusion equation was derived in Sections 5.2 and 5.3 in terms of affine and cosmic time respectively. The tight constraints on en- ergy diffusion and drift obtained in Chapter 4 allow the polarisation diffusion equa- tion to be simplified to a diffusion equation in terms of polarisation angle with two free parameters. A distribution of linearly polarised photons is found to experience a suppression in polarisation determined by the diffusion parameter δ1 and a rotation of polarisation determined by the drift parameter δ2 . The suppression of polarisation over the travel time of photons from distant sources to us does not seem to have been much considered. Unfortunately it is, for the most part, a difficult effect to constrain since the polarisation fraction at the source must first be well understood. There are a number of different proposed models that would generate polarisation rotation over the travel time of photons. The model proposed here is distinct, how- ever, in the 1/k dependence of its rotation and in that it preserves Lorentz invariance. The polarisation rotation is easily constrained by existing data. A natural correlation between the polarisation angle and the axis of radio galaxies allowed a constraint of δ2 < 2 × 10−37 kgm2 s−3 to be placed in Section 5.4. The photons from the CMB, having had more time to experience the underlying discreteness, impose a tighter constraint on the rotation and also constrain the suppression. These constraints are discussed in [49].  Chapter 6 C ONCLUSIONS AND FUTURE DIRECTIONS Lorentz invariance is not optional in the causal set approach to quantum gravity. It is an essential feature of the theory and proves a very powerful constraint in the devel- opment of causal set phenomenology. A surprising amount can be deduced beginning with only the simple assumptions that spacetime is discrete and Lorentz invariant. The behaviour of massive particles in a discrete spacetime was shown to be de- scribed by a diffusion equation with one free parameter in the continuum limit. The derivation of this equation did not rely on any underlying particle model, but sim- ply on the assumptions of discreteness and Poincar´e invariance. It could be argued that the derivation does not guarantee a finite diffusion constant and thus a truly ob- servable effect. To rebut this objection it was shown analytically that the continuum diffusion parameter was necessarily finite for the swerves model of particle propaga- tion on a causal set. Numerical simulations of the swerves model and two ‘intrinsic’ models gave results that clearly demonstrated diffusion behaviour, and in the case of the swerves model it was possible to determine the relationship between the diffusion parameter and the underlying model parameters. In the causal set approach, spacetime discreteness is expected to be of the order of the Planck scale. The constraints of computer memory limit causal set sizes simu- lated on a standard computer to around 217 elements. We could question the point of even attempting to investigate large scale behaviour in causal sets that correspond to such absurdly small regions of spacetime (even a volume of 1mm3 × 1s would con- tain on the order of 10137 elements). Yet remarkably, the simulations do agree with the expected continuum behaviour. The results suggest that causal set simulations will have an important role to play in the further development of the theory and its 126  127 phenomenology. Lorentz invariance also allowed a useful phenomenological model for massless particles in discrete spacetime to be developed. Massless particles were shown to ex- perience a drift and diffusion in energy in the continuum limit. Crucially, all massless particles still travel at the speed of light in this model. The accurate data available on the cosmic microwave background allowed both the drift and diffusion parameters to be tightly constrained. Whether the constraints are so tight as to exclude all possibility of future observations of the effect remains to be seen. The common word ‘particle’ erroneously suggests massless and massive particles are variations of the same fundamental object. The diffusion experienced by massless particles is not simply the massless limit of the massive particle diffusion, suggesting that massive and massless particles will appear in quite different forms in any future complete causal set theory. How exactly will particles fit into the causal set approach? The massive particle models considered here were classical, point particle models and, as yet, we have no candidate models at all for massless particles. To produce any underlying model for massless particles, it seems likely quantum behaviour will have to be considered. Placing particles ‘on’ a fixed causal set seems highly unsatisfactory. The causal set should be a dynamic structure, growing in number of elements. On large scales the theory must reproduce the relationship between mass and curvature, suggesting that the presence of particles will influence the growth of the causal set. The need to reproduce quantum effects suggests assigning amplitudes to each possible step in the causal set growth. It would be nice if we could just devise a theory now that satisfies all our requirements, but the problem is simply too big. Progress is made by looking at what can be learned from phenomenology, by developing quantum field theory on causal sets, by looking at the structure of curves spacetime in causal sets, and so on. Steps in the direction of a complete theory will hopefully lead to further phenomenology and, ultimately, falsifiable predictions. The power of the Lorentz invariance assumption has its limitations. When con- sidering the polarisation of massless particles, Lorentz invariance proved insufficient to constrain the model. Identifying the assumptions necessary to constrain the model further will have to wait for the next step in understanding particles in causal set theory. In the meantime, restricting to linear polarisation has allowed progress to be  128 made. In a discrete spacetime linearly polarised photons were found to experience rotation in polarisation angle and suppression in polarisation fraction. Rotation of po- larisation angle is an effect predicted by a number of other approaches to quantum gravity, but the particular energy dependence of this model is unique. Again, cosmo- logical data allowed the free parameters of the model to be constrained. It is important to remember that the results discussed here do not, for the most part, call on specific features of causal set theory. They rely on the assumptions of Lorentz invariance and discreteness and may, hopefully, be generalisable to other quantum gravity theories. A lot has been achieved looking at the large scale behaviour of particles in discrete spacetime, despite the incomplete nature of the theory. Where next can we look for signatures of spacetime discreteness? The importance of phenomenology in the search for a theory of quantum gravity should not be underestimated. It is crucial in the development of the theories and, more importantly, reminds us to look out at the universe once in a while and remem- ber why it is we seek such a theory to begin with.  Appendix A E XPLICITLY L ORENTZ INVARIANT POLARISATION DIFFUSION The state space for a photon with polarisation is M = M4 × H30 × P, where P is the polarisation state space. For an explicitly Lorentz invariant formulation of the polari- sation diffusion equation, P is the polarisation state space with coordinates Pµν , where Pµν = kµ aν − aµ kν is the complex two-form satisfying the conditions set out in Sec- tion 5.1. If we wish to discuss only linear polarisation, Pµν can be taken to be a real two-form satisfying the conditions 1. Pµν = −Pνµ ; 2. Pµν P µν = 0 ; 3. P µν kµ = 0 ; 4. P µν Pµσ = kν kσ . Again, any Pµν satisfying these conditions can be expressed as Pµν = kµ aν − aµ kν , where kµ aµ = 0 and aµ aµ = 1. aµ is now a real four-vector. Condition (1) above provides six constraint equations, condition (2) a further one constraint. Although condition (3) is four equations, it gives only three new independent constraints. Fi- nally condition (4), although 16 equations, gives only one new constraint. The state space P therefore has only one degree of freedom. A.1 Invariant vectors Before writing down the diffusion equation, K AB , uA , and the density of states n must be known. Consider the submanifold H30 × P. There are two Lorentz invariant vector 129  §A.1 Invariant vectors 130 fields on the submanifold: v1A = (kµ , P µν ) and v2A = (0, ∗P µν ), where ∗P µν is the Hodge dual defined in the usual way as ∗P µν = −1 ρσµν 2 ǫ Pρσ . To see this, first consider the more general vector fields w1A = (α1 kµ , β1 P µν ) and w2A = (α2 kµ , β2 ∗ P µν ). These vectors are clearly Lorentz invariant, as is the submanifold H30 × P. It remains to show that the vectors are tangent to the submanifold. The normals to the submanifold are derived from the constraint equations in the usual way: ∂ (kν kν ) ∂ (kν kν )   µ k kµ = 0 ⇒ n1A = , ∂k µ ∂P ρσ = (2kµ , 0) , (A.1) ρσ µν   ∂ (P Pρσ ) ∂ (P Pµν ) P µν Pµν = 0 ⇒ n2A = , ∂k µ ∂P ρσ = (0, 2Pρσ ) , (A.2) να να   ∂ (P kν ) ∂ (P kν ) P µα kµ = 0 ⇒ nαA = , ∂kµ ∂P ρσ α α = (Pµ , δ σ kρ ) , (A.3)   ∂ ∂ P µν Pµσ = kν kσ ⇒ nνσA = α (P µν Pµσ − kν kσ ) , (P µν Pµσ − kν kσ ) ∂k ∂P αβ = −gσα kν − δαν kσ , δβν Pασ + gσβ Pα ν .
(A.4) The vector w1 is clearly orthogonal to the first three normals: n1A w1A = 2α1 kµ kµ = 0, (A.5) n2A w1A = 2β1 Pρσ P ρσ = 0, (A.6) nαA w1A = α1 Pµ α kµ + β1 δα σ kρ P ρσ = 0. (A.7) For the last normal nνσA w1A = (−gσα kν − δαν kσ ) α1 kα + δβν Pασ + gσβ Pα ν β1 P αβ
= −α1 (kν kσ + kν kσ ) + β1 (Pασ P αν + Pα ν P α σ ) = 2 (−α1 kν kσ + β1 Pασ P αν ) . (A.8)  §A.2 The diffusion equation 131 This equals zero only if α1 = β1 . Thus the vector v1A = (kµ , P ρσ ) is normal to the surface. w2 is also obviously orthogonal to the first three normals if it is first noted that Pµν ∗ P µν = 0 and ∗P µν kµ = 0 if both Pµν P µν = 0 and P µν kµ = 0. These results can be shown simply if Pµν is expressed as Pµν = kµ aν − aµ kν . n1A w2A = 2α2 kµ kµ = 0, (A.9) n2A w2A = 2β2 Pρσ ∗ P ρσ = 0, (A.10) nαA w2A = α2 Pµ α kµ + β2 δα σ kρ ∗ P ρσ = 0. (A.11) The remaining normal gives nνσA w2A = (−gσα kν − δαν kσ ) α2 kα + δβν Pασ + gσβ Pα ν β2 ∗ P αβ
= −α2 (kν kσ + kν kσ ) + β2 (Pασ ∗ P αν + Pα ν ∗ P α σ ) = 2 (−α2 kν kσ + β2 Pασ ∗ P αν ) . (A.12) It turns out that Pασ ∗ P αν = 0, this is easily checked by choosing a specific frame, say kµ = (k, 0, 0, k). Therefore w2 is only tangent to the surface if α2 = 0. Thus v2A = (0, ∗P µν ) is an invariant vector. A.2 The diffusion equation Given the above invariant vectors v1 and v2 , the most general Lorentz invariant, sym- metric, positive semi-definite matrix K AB on the submanifold H30 × P is K AB = av1A v1B + b v1A v2B + v2A v1B + cv2A v2B ,
(A.13) the general invariant vector is uA = dv1A + ev2A , (A.14)  §A.2 The diffusion equation 132 where a, b, c, d, e are constants and ac − b2 ≥ 0. The spacetime components of K AB and uA on the full state space M are unchanged from the zero spin massless particle case: K µA = 0, uµ = kµ . The diffusion equation can thus be written down from Equation 3.15: ∂ρλ  ρ  λ  = ∂A K AB n∂B − uA ρλ ∂λ n ∂ρ λ = −kµ µ ∂x  ∂ ∂  ρλ  ∂  ρλ  + µ akµ kν ν + (ak µ P ρσ + bk µ ∗ P ρσ ) n ρσ − dk µ ρλ ∂k ∂k n ∂P n  ∂ ∂  ρλ  + ρσ (ak µ P ρσ + bk µ ∗ P ρσ ) n µ ∂P ∂k n     ∂  ρλ  + aP ρσ P γδ + b P ρσ ∗ P γδ + ∗P ρσ P γδ + c ∗ P ρσ ∗ P γδ n γδ  ∂P n − (dP ρσ + e ∗ P ρσ ) ρλ . (A.15) Although it explicitly demonstrates the Lorentz invariance of the process, this equation is hardly user-friendly. To understand the physical consequences of the dif- fusion, the equation derived in Section 5.2 using the Bloch sphere as the polarisation state space is far more useful. It is important to confirm however, that these equa- tions are in fact equivalent. First consider polar coordinates on H30 : {k, θ, φ}. In these coordinates the vector kµ has components (k, 0, 0). As mentioned earlier, the polarisa- tion state space P has only one degree of freedom – specifically, it can be shown that the constraints on P µν reduce the polarisation state space to a circle. The density of states, n, on this circle is simply a constant and thus the density of states for the full state space M is n = k sin θ. The submanifold H30 × P is thus the momentum space, a null cone, with a copy of the polarisation space, a circle, at each point. The invariant vector v1A = (kµ , P µν ) is simply the vector at a point on the cone that points along the cone, away from the tip. The invariant vector v2A = (0, ∗P µν ) is the tangent vector to the polarisation circle. 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