# Matter in Discrete Space-Times

P. P. Divakaran
(Email<EMAIL_ADDRESS>
###### Abstract

In the Einsteinian model of space-time as a 4-dimensional pseudo-Riemannian
manifold, special relativity holds exactly in the tangent space of every
point. A quantised matter field of a given mass and spin, corresponding to an
elementary particle of matter, is then to be regarded as being defined by a
unitary representation (UR) of the Poincaré group at each point. This
Wignerian viewpoint leads to a more general reformulation of the equivalence
principle as the unitary equivalence of these URs as the point is varied. In
this background, the main question addressed in these notes is whether, as a
necessary first step in a discretisation of gravity, the Wigner construction
can be carried over to a model space-time which is a 4-dimensional lattice
embeddable in real Minkowski space with a distance function inherited from it
(but physically not so embedded). Working with a hypercubic lattice, it is
shown in full mathematical detail that the Wigner paradigm continues to be
valid but with some exotic new features. The description of spin is
essentially the same. In contrast, the momentum space is the 4-torus,
identified as the Brillouin zone of space-time where all physical phenomena
occur: 4-momentum is defined and conserved only modulo a reciprocal lattice
vector, implying that there is no notion of an invariant mass except when it
vanishes. Nevertheless, massless particles continue to have a constant
invariant speed (the ‘speed of light’), a result of crucial importance for the
viability of discrete relativity. A massive particle in contrast is
characterised by a rest mass and under large boosts it will pass through
phases of superluminal propagation. If the lattice spacing is taken as a fixed
fundamental length of the order of the Planck length, such effects can be
observed only in the evolution of the very early regime of the conventional
big bang universe, of which the two most dramatic manifestations are i) cosmic
Umklapp processes, leading to a degradation of energies of individual
particles, as a possible source of ultra-high energy cosmic rays and ii)
primordial superluminal expansion as a contribution to or even as the root
cause of cosmic inflation. A fundamentally discrete space-time is not in
conflict with known physics; it may in fact be of help in explaining some
otherwise mysterious aspects of early cosmology.

##  1 A fundamental length?

Explorations of the possibility that space or space-time may have a discrete
structure on a scale much smaller than currently accessible have a long
history. Though the motivations for considering such a fine structure have
changed somewhat over time, they have a common origin in the old observation
of Planck that the fundamental constants $c$ and $\hbar$, which are
independent of dynamics, can be combined with the coupling strength $G_{N}$ of
the gravitational interaction of matter in the Newtonian limit to produce a
unit of length $L=(\hbar G_{N}/c^{3})^{1/2}$, of a far smaller scale than any
length we can conceivably measure. The expectation, supported by ingenious
thought experiments and some theoretical considerations, is that in a quantum
theory of gravity (if and when we succeed in constructing one), or even in a
semi-classical approximation to it, distances of the order of $L$ or smaller
cannot have an operational significance.111There is an extensive review of
work in this area, particularly valuable for the historical and motivational
background, in [1]. It is also very useful for its comprehensive bibliography
as of the time of its writing. New work continues to be produced in profusion
but it seems fair to say that there has been no significant recent
breakthrough.

We may then consider two physically distinct types of models of space-time as
posssible ways of accommodating a fundamental length. The more popular is one
in which space-time is still a pseudo-Riemannian manifold but does not admit
physical measurements of lengths below a finite length much smaller than any
that is accessible to present day methods (which we may take, at least
tentatively, to be the Planck length $L$). Models of this type have problems
of reconciliation with principles we hold to be inviolable, such as the
observer-dependence of the magnitude of $L$, making its interpretation as a
fundamental characteristic of space-time itself somewhat questionable. There
are many variants of such models, not always mutually compatible, but all of
them have to invoke some sort of deformation of special relativity, whose
consequences are not fully understood.

A more ambitious approach is to take space-time to be fundamentally discrete,
not embedded physically in a manifold (but of course, mathematically, so
embeddable). The fundamental dynamical variables – the counterparts of local
fields on manifolds – are then to be defined on the points of a discrete set
(and only on them because that is all there is) endowed with a notion of
ordering, a lattice, derived perhaps from the metric in an embedding manifold.
It will obviously be a formidable undertaking, already at the classical level,
to transcribe the geometry of general relativity to lattice space-times.
Different types of lattices have been studied in the literature and it is fair
to say that it is too early for a prognosis of where these studies might lead,
see the review [1] and the many references therein. To then extend it to the
quantum domain will be an even greater challenge. An idea of the conceptual
and technical difficulties that have to be overcome may be had from, for
example, the reviews [2] and [3] (written some years ago; the situation is
much the same today).

There are very good reasons, nevertheless, for hoping that the successful
incorporation of a fundamental length in physics at very short distance
(reciprocally, very large momentum) scales can be a major step forward in an
eventual quantisation of gravity. First, it is the gravitational constant
$G_{N}$ itself that allows the introduction of a natural lattice structure for
space-time through $L$. Indeed, in a world without gravitation, the Planck
length vanishes; conversely and more speculatively, if $L$ is taken to 0
keeping $c$ and $\hbar$ finite, the space-time manifold will tend to flatness.
Secondly, in a quantum or semiclassical framework, $L^{-1}$ provides a
momentum and energy cutoff that makes possible the calculation of finite
gravitational effects, conventionally unrenormalisable;222The first attempts
at using a discrete space-time to make perturbatively finite (special)
relativistic quantum field theory date from the 1930s ([1], section 2), well
ahead of the time standard perturbative renormalisation theory came to be
developed. Subsequently, much effort has gone into exploring the lack of
renormalisability of gravitationsl interactions and its possible cures. And,
finally, current attempts at quantising the geometry of space-time, even
without a fundamental length introduced ab initio, seem to point to the need
for imposing a discrete structure (as in loop gravity for instance).

The Einstein equations not only have a left side originating in the geometry
of the space-time manifold ${\cal M}$. They have also a right side concerned
with matter. At the fundamental level – the level at which ideas such as
quantisation can be meaningfully addressed – this takes the form of
contributions of various matter fields to the energy-momentum density. The
description of matter fields involves only flat space-time, the tangent spaces
to ${\cal M}$, essentially because of the equivalence principle (as briefly
recollected in the next section). Therefore any discretisation of ${\cal M}$
will entail a discretisation of flat Minkowski space $M$ in its guise as a
tangent space and it becomes necessary then to study lattices in $M$ and to
ask whether matter fields can be defined on them in physically satisfactory
and mathematically unambiguous terms. To take a first serious look at this
particular question – a necessary step and one which is much less daunting
than the discretisation of gravity – is the purpose of this article. Although
there are some mathematical issues which are yet to be fully resolved, the
first answers are affirmative: ‘local’ matter fields can be defined on regular
lattices, without contradicting physics as we know it at length scales that
are accessible today, but are still enormously large in comparison to $L$. As
is to be expected, intriguing new physics, with possible (and perhaps
desirable) cosmological consequences, does emerge at the scale of $L$.

Even though the discrete space-time envisaged in this paper is not to be
thought of as a subset of a true physical continuum space-time, the working
out of the discrete physics will often use some of the very detailed knowledge
we have of the continuum physics as a model. For that reason, there will be
occasional recapitulations of well-known facts from standard theory, sometimes
in a more general form than we are used to. This is especially the case in the
next section on the relevance of the equivalence principle in defining matter
fields and in the summary (section 5) of their construction as representations
of the group of special relativity. Also, for the benefit of readers who may
not be interested in the unavoidable but often tedious arguments in the main
part of the paper, I have given in the next few sections a schematic outline
of some of the novel points that arise.

Throughout this paper, ${\cal M}$ and $M$ have dimension (1,3) with the metric
having the signature $(+,-,-,-)$.

##  2 Matter fields: tangent spaces and a general equivalence principle

In general relativity, the bridge between the two sides of the Einstein
equation is the classical equivalence principle, usually stated simply as the
equality of the inertial mass and the gravitational mass. A natural starting
point for the description of matter within general relativity is therefore the
equivalence principle itself,333Rewritings of the standard field equations,
e.g., the Dirac equation, so as to make them consistent with GR have a long
history; most of them do not explicitly invoke the equivalence principle.
suitably reinterpreted where necessary. In this section, I recall in a
qualitative way the relevant issues. At a fundamental (‘elementary particle’)
level, there are two steps involved. Firstly, matter fields contributing to
the energy-momentum are defined not globally on ${\cal M}$ but locally at each
point $x\in{\cal M}$ – specifically, on the tangent space $T_{x}{\cal
M}=:M_{x}$ which is by defintion flat and whose spatial projection is the
local inertial space at $x$. The reason of course is that an invariant mass is
an attribute associated with the Poincaré group $P$ operating on flat
Minkowski space. Thus a complete theoretical understanding of the principle as
first formulated by Einstein had to wait till Wigner’s landmark paper ([4],
see also [5]), according to which an invariant mass is one of the parameters
labelling irreducible unitary representations (URs for short) of the Poincaré
group $P$.444Several variants of this foundational principle, ranging from the
descriptive to the abstractly mathematical, are still current in the
literature, more than a century after Einstein first proposed his version of
it. The specal theory is not primarily just a weak field approximation to the
general theory but a constituent part of it that holds exactly in every
tangent space, thereby defining precisely the intuitive notion of an inertial
frame; that is the content of the equivalence principle. The formulation given
here is a modern, physically and mathematically sharp version of the original
Einstein formulation that takes account of some key insights that came later,
such as the significance of symmetries in quantum theory. (Precise
characterisations of this and other relevant groups will be given below as and
when they are needed). It follows that a logical precondition for the
formulation of the equivalence principle is a precise notion of an equivalence
of the tangent spaces as $x$ is varied, and a consequent criterion for the
independence of the mass of a test particle of the point $x$ in space-time
where it is measured. It is to be noted that this condition requires only the
special theory for its formulation. Only after its validity is accepted can
the second step, the assertion of the equality of this common inertial mass
with the gravitational mass, which is otherwise a property deriving from the
global geometry of ${\cal M}$, be taken.

As everyone knows, Wigner’s main result is that an irreducible UR of the
universal covering group $\bar{P}$ of $P$, satisfying certain physically
reasonable conditions (absence of tachyons and boundedness of helicities) is
characterised by a fixed non-negative real number (the square of the mass) and
a fixed set of integral or half-integral helicities (‘spin’). In other words,
if we ignore ‘internal quantum numbers’ (‘charges’), such an irreducible UR of
$\bar{P}$ (a Wigner UR in short) can be identified with an elementary particle
with a given mass (including vanishing mass) and spin. The Wigner construction
has been much studied in the literature (and will be very briefly reviewed
below as it is the basis on which the description of matter in discrete space-
times rests). Here I make two qualitative observations which can be expected
to have a bearing on the general philosophy of matter fields in a theory of
gravity already in the ‘continuum limit’.

1\. Wigner’s construction is the foundation of a quantum description of
matter. The Hilbert space ${\cal H}^{1}$ on which an irreducible UR of
$\bar{P}$ is realised is the 1-particle state space. An $n$-particle state
then belongs to the $n$th tensor power $\otimes^{n}{\cal H}^{1}=:{\cal H}^{n}$
(symmetrised or antisymmetrised according to whether the particle is a boson
or a fermion) and a general state in a quantum field description of the system
is a linear sum of tensor product states (with decay conditions on the
coefficients to ensure that they form a Hilbert space ${\cal H}$) – creation
and annihilation operators are essentially operators which map ${\cal H}^{n}$
to ${\cal H}^{n+1}$ and ${\cal H}^{n-1}$ respectively. This is standard and
the point of bringing it up is to highlight the fact that in a putative
quantum theory of gravity, the description of matter à la Wigner can be
expected to remain valid. Indeed, we have no other way of characterising
matter at the quantum level, a recognition that is implicit in all current
work.

2\. To give meaning to the notion of the identity of a matter particle
independently of its gravitational environment – to be able to say as is
commonly done that an electron is an electron whether in a vanishingly weak
gravitational field or near a black hole – it is necessary to formulate the
equivalence principle somewhat more generally. A reasonable and obvious
generalisation would be to postulate that the Wigner UR $U$ corresponding to a
given particle (not just its partial attribute, the mass) is abstractly the
same at all points $x\in{\cal M}$. The qualification ‘abstractly’ is necessary
since, even though the group $\bar{P}_{x}$ is, abstractly, the same at all
$x$, as a transformation group on the tangent space $M_{x}$ it depends on $x$
and so does the UR $U_{x}$ as the unitary group of ${\cal H}^{1}_{x}$ (where I
have distinguished structures localised at $x$ by a subscript). This
$x$-dependence of the representation as a whole (unlike its Casimir, the mass,
which is supposed constant over ${\cal M}$), leads to an immediate natural
linkage between matter fields and gravity: specifying a connection (or some
other equivalent object such as the covariant derivative) on the tangent
bundle of ${\cal M}$ allows the parallel transport of a frame in $M_{x}$ to
$M_{y}$ (together with their induced Minkowski metric), and hence of their
group of isometries: $\bar{P}_{x}$ to $\bar{P}_{y}$, and hence of a given UR:
$U_{x}$ to $U_{y}$. Since these representations are abstractly identical, they
will correspond to the same mass and spin. The equivalence can in fact be
stated in a form respecting the quantum nature of the Wigner description of
matter: there is a unitary map $W(x,y):{\cal H}_{x}^{1}\rightarrow{\cal
H}_{y}^{1}$ such that $W(x,y)U_{x}=U_{y}W(x,y)$.

The setting best suited to the exploration of the interrelationship between
gravity and matter – equivalently, of the geometry of ${\cal M}$ and the
representation theory of the isometries of $M_{x}$ – is thus that of the
tangent bundle over the space-time manifold. Its full working out will be a
major undertaking. What is already clear from the above very preliminary
remarks is that the matter side of the equation is not only an essential part
of such a project, but may even be a good starting point in the search for a
quantum theory of gravity, much as the formulation of quantum electrodynamics
is most profitably approached by starting with the invariance properties of
charged fields.

In any case, independently of how such a programme will work out, one cannot
avoid dealing with the representation theory underpinning the description of
matter. In the continuum case, the representations are very well understood
from Wigner’s work. Their significance as the foundation of a quantum
definition of matter is also well known. It is less well appreciated that in
that role it is central to the description of matter in the general theory as
well, by enabling the original formulation of the equivalence principle to be
generalised and completed as indicated above. The purpose of the present work
is however more limited: to show how that description can be adapted to the
case of space-time being discrete and to bring out the ways the results
deviate from standard wisdom. The most crucial of these deviations will turn
out to result in a departure from causal propagation of massive matter at
extremely high (Planck scale?) speeds, perfectly acceptable in our present
state of knowledge, perhaps even desirable in current models of very early
cosmology.

##  3 Discrete Minkowski space and its symmetries

The core of this paper is concerned with the question of whether and how
elementary matter fields can be associated with URs of the restriction of the
Poincaré group to a suitably chosen discrete subgroup. Answering the question
will involve the following steps:

1\. The choice of a discretisation of $M$, i.e., ${\bf R}^{4}$ with the
Minkowski metric. The simplest choice is the hypercubic lattice ${\bf Z}^{4}$
of points in ${\bf R}^{4}$ with integral coordinates with respect to a fixed
set of axes. This means in particular that space and time coordinates are
related by the speed of light, assumed to be a fundamental constant and put
equal to unity. (Or, equivalently, the speed of light is defined as the ratio
of the spatial and temporal lattice spacing – but measured in what units?).
Thus there is a unique lattice spacing which is the unit of length and time,
also put equal to unity in most of what follows. (Where necessary, the Planck
length will be brought in to play that role). From the point of view of
symmetries, this is the simplest choice; other regular lattices will not pose
any conceptual problems but will add to the technical and computational
burden. Random lattices are excluded from consideration since they will entail
randomly distributed lattice spacings and cannot naturally accommodate a
unique fundamental length.

A distance function is defined on ${\bf Z}^{4}$ by restricting the Minkowski
metric in ${\bf R}^{4}$ to its integral points: if $X=\\{X_{\mu}\in{\bf
Z};\mu=0,1,2,3\\}$ is a point of ${\bf Z}^{4}$, its (length)2 is given by
$X_{\mu}X_{\mu}:=X_{0}^{2}-X_{1}^{2}-X_{2}^{2}-X_{3}^{2}=:X^{2}$ (and
similarly for the (distance)2 between two points $X$ and $Y$). The discrete
set ${\bf Z}^{4}$ together with this distance function is our discrete
Minkowski space and will be denoted by $M({\bf Z})$ in what follows.

2\. The identification of the lattice Poincaré group. In the continuum, the
relativity group is $P({\bf R})=L({\bf R})\vec{\times}T({\bf R})$, where
$L({\bf R})=SO(3,1,{\bf R})/\\{\pm 1\\}$ is the connected (proper,
orthochronous) Lorentz group, $T({\bf R})$ $(\sim{\bf R}^{4})$ is the
translation group (in a more explicit notation than in the introductory
remarks) and $\vec{\times}$ denotes the semidirect product, the arrow
indicating that (the quotient group) $L$ operates on (the normal subgroup)
$T$. The connectedness requirement on $SO(3,1)$ (the quotienting by $\\{\pm
1\\}$) keeps out reflections which are not, observationally, symmetries of
matter field interactions among themselves even though they leave the metric
invariant. The discrete Lorentz group is then the subgroup of $L({\bf R})$
obtained by restricting every $4\times 4$ matrix $\lambda\in L({\bf R})$ to
have integral entries: $L({\bf Z}):=SO(3,1,{\bf Z})/\\{\pm 1\in SO(3,1,{\bf
Z})\\}$. Our discrete Poincaré group $P({\bf Z})\subset P({\bf R})$ is
therefore the semidirect product of this group with the discrete translation
group $T({\bf Z})$ $(\sim{\bf Z}^{4}\subset{\bf R}^{4}$). It is an interesting
fact that, while $L({\bf Z})$ (generalised in the obvious manner) is the
2-element group in 1 + 1 dimensions, it is an infinite group in all higher
dimensions.

3\. Determination of the appropriate representations of $P({\bf Z})$. The
guiding spirit in identifying and constructing the representations will be the
work of Wigner on the corresponding problem for $P({\bf R})$. Technically,
this will be the major concern of the present paper. Here I limit myself to
describing the physics and mathematics background to the identification of the
relevant representations. The starting point is the recognition (due, also, to
Wigner [6]) that the group of symmetries of a quantum system is represented on
its state space by projective URs. Wigner ([4]) first establishes the result
that every continuous projective UR of $P({\bf R})$ lifts to a continuous UR
of its universal covering group $\bar{P}({\bf R})=\bar{L}({\bf
R})\vec{\times}T({\bf R})$, with $\bar{L}({\bf R})=SL(2,{\bf C})$; i.e., given
a projective UR of $P({\bf R})$, we can find a UR of $\bar{P}({\bf R})$ whose
projection onto the quotient group $P({\bf R})$ is the given projective UR.
This key result has the following ingredients: i) though $T({\bf R})$ has
nontrivial projective URs (i.e., projective URs which are not equivalent to
URs of itself), they do not extend to the whole of $P({\bf R})$ as nontrivial
projective URs and can be ignored; ii) though semidirect product Lie groups
$G\vec{\times}A$ with $A$ abelian can in general have nontrivial projective
URs which restrict to $A$ as URs, this does not happen for $P({\bf R})$ on
account of the semisimplicity of $L({\bf R})$; and iii) every projective
representation (not necessarily unitary) of $L({\bf R})$ lifts to a (linear)
representation of its universal cover $SL(2,{\bf C})$, again because of
semisimplicity. The realisation of an irreducible PUR for a particle of a
given mass and a given spin, integral or half-odd-integral, then involves the
choice of a mass shell (an orbit of $L({\bf R})$ in momentum space) and a
finite dimensional (necessarily non-unitary) representation of $SL(2,{\bf C})$
which determines the spin.

These results, in particular the assertion iii), are obviously specific to Lie
groups and are therefore not directly applicable to the projective URs of
their discrete subgroups. There are however theorems relating projective
representations (not limited to projective URs) of any group to linear
representations of related ‘universal’ groups other than the universal cover.
Specifically, given a group $G$, we can construct a group $\hat{G}$, called a
universal central extension of $G$, of which $G$ is a quotient group, and
having the property that every projective representation of $G$ is the
projection of a linear representation of $\hat{G}$.555Even for Lie groups,
$\hat{G}$ is not necessarily $\bar{G}$. Questions regarding projective
representations of $G$ are most efficiently addressed in the language of the
theory of central extensions of $G$ by appropriate abelian groups and the
associated group cohomology theory. For a clear and thorough treatment of the
topic, including the construction of universal central extensions, see
Raghunathan ([7]) and, for a physicists’ version with applications to many
examples, see Divakaran ([8]). The theory can in fact be used to reformulate
the process of quantisation of a system entirely in terms of its symmetries in
a manner free from commonly encountered ambiguities ([9]); in particular, the
superselection structure of the state space is seen to be of cohomological
origin, a fact which may play a role in the specification of particle states
in discrete relativity (see section 7 below). Wigner’s work ([4]) was of
course the first to determine the state space of an elementary particle
explicitly as a projective UR of $P({\bf R})$ but he did not connect it to the
consequent superselection rule, that of univalence. The statements i) to iii)
above are in fact specialisations of properties of universal central
extensions to Lie groups having different structural properties. In
particular, iii) follows from a theorem which says that a connected semisimple
Lie group has a unique universal central extension and that it is the same as
its universal cover; so equivalence classes of its projective representations
are classified by the Pontryagin dual (the group of 1-dimensional
representations or characters) of its fundamental group. This is the reason
why it is legitimate to work with $\hat{L}({\bf R})=\bar{L}({\bf R})=SL(2,{\bf
C})$.666Many commonly met groups in physics serve as examples of the
distinction between $\hat{G}$ and $\bar{G}$. Thus, the universal cover of the
2-dimensional rotation group $SO(2)$ is the real line but its universal
extension is itself (it has no nontrivial projective UR – hence no non-
integral spin) while the vector (translation) group ${\bf R}^{n},\,n>1$ has
nontrivial projective URs but is its own universal cover. The indiscriminate
substitution of $G$ by $\bar{G}$, rather than by the always correct $\hat{G}$
has led to much misunderstanding in the physics literature, see [8].

To deal with the discrete groups of interest to us with anything like this
degree of completeness is not a feasible option, primarily because of the lack
of a physically satisfactory criterion (such as continuity) for acceptable
representations. And it is, to a great extent, unnecessary for our purpose;
physically, it is sufficient to note first that the discrete groups of our
interest are subgroups of the corresponding Lie groups by construction and
then to find those representations which, in the limit, approach in a well-
defined sense the physically acceptable representations of the embedding Lie
groups. That is possible thanks to the fact that $L({\bf Z})$ and
$\hat{L}({\bf Z})$, as subgroups of the embedding Lie groups, have a property
known as Borel density which enables them to inherit several useful results
regarding their representations from the Lie groups (see below for details).
This is in fact one of the mathematical inputs that make our project at all
feasible.

4\. Interpreting representations as particles. The implementation of the
programme outlined above presents some (though surprisingly few) serious
mathematical obstacles. The resulting physical picture too, naturally, differs
in some significant respects from the continuum theory. Firstly, since the
momentum space (the space of characters of the translation group) is now the
4-torus ${\bf T}^{4}$ rather than ${\bf R}^{4}$, momentum itself is defined
and conserved only modulo a reciprocal lattice vector (which is the same as
the momentum cut-off, the Planck momentum by choice). This has consequences
somewhat like the familiar momentum space properties of an electron moving in
a crystal; in particular, the mass shell is the Minkowski metric analogue of
the Fermi surface of an empty lattice. Secondly, the spin of a representation
can no longer be defined generally as an attribute of rotation invariance as
the discrete ‘rotation’ group, being a discrete subgroup of the compact Lie
group $SO(3,{\bf R})$, is a finite group with a finite set of inequivalent
irreducible URs. But this is not a serious handicap since it turns out
(essentially because of the Borel-density property of the discrete $SL(2)$)
that spin, both integral and half-integral, can be defined by reference to the
Lorentz group alone (in the continuum, the two ways of defining spin are of
course equivalent).777Another respect in which the identification of
representations with particle states in the discrete world differs from
continuum relativity is that they may apparently be chosen to be (highly)
reducible. Whether this freedom is physically significant is at present
unclear, see the discussion in section 7 below. If we accept these deviations
from the received wisdom of continuum relativity – which, we shall see, are
not in contradiction with our current state of knowledge – projective URs of
$P({\bf Z})$, of a certain general type, have a perfectly reasonable
interpretation as elementary particles.

Of the deviations from standard lore, the more dramatic are those having their
origin in the compactness of the momentum space. These include in particular
possible apparent violations of energy-momentum conservation in elementary
processes – the analogue of the Umklapp processes of crystal physics –
involving energies of the order of the Planck mass. More intriguingly, the
distinction between time-like and space-like momenta is no more an invariant
concept. ‘Massive’ orbits of the discrete Lorentz group in momentum space do
not have an invariant mass associated to them and have tachyonic branches that
begin to sprout around the Planck scale: the (energy-momentum)2 can be
negative even when the (rest-mass)2 (which, being a zero-momentum attribute,
is a valid concept) is positive. The light cone itself is well-defined as a
closed hypersurface in the momentum space ${\bf T}^{4}$; zero mass orbits lie
within it and have no tachyonic branches. (This is a result of independent and
fundamental importance, as described in a separate added note.) The general
scheme for the construction of URs following from these considerations will be
described later on with a degree of mathematical detail, as well as,
qualitaively, some of the unfamiliar physical consequences of discrete
relativity. What is certain is that the exotic features that emerge have no
impact on elementary particle phenomena at energies presently accessible to
experiments or many orders of magnitude higher; they will, however, have
cosmological implications which also will be touched upon at the end.

The main results of this work, then, hold no bad surprises: the description of
elementary matter as founded on special relativity survives discretisation,
subject to some reinterpretations which are capable of being tested. But these
are only the first steps and much still needs to be done. Moreover, beyond the
continued validity of the Wigner definition of elementary particles, a lattice
structure for space-time as put forward here will have other macroscopic
manifestations, ‘macroscopic’ in the present context meaning (here as
elsewhere in this article) length scales characterising the structure and
interactions of the currently known particles and greater: issues such as
possible deviations from isotropy, the accommodation of truly macroscopic (in
dimension and/or mass) classical systems etc. They have been much written
about in the literature (see [1] and the many references therein) and this
paper will have not much to add to it.

These introductory sections are meant to bring out, more or less
qualitatively, the following points: a) the critical importance of a good
description of matter fields – including a suitable formulation of the
equivalence principle – before we can think of bringing together quantum
mechanics and general relativity; b) the feasibility of such a programme in a
discrete space-time; and c) a brief foretaste of the necessary (but not widely
known in the physics literature) mathematical material, described in the next
few sections, that it entails.

##  4 The discrete Lorentz and Poincaré groups and their central extensions

In the rest of this paper, the connected real Lorentz group $L({\bf R})$ will
be called simply the Lorentz group without any qualifiers and its discrete
subgroup $L({\bf Z})$ the discrete Lorentz group (and corresondingly for the
Poincaré groups). As noted in section 3, the semisimplicity of the Lorentz
group has the consequence that the universal central extensions of $L({\bf
R})$ and $P({\bf R})$ are in fact their universal covering groups and hence
that all their projective representations can be obtained as projections of
true (linear) representations of the universal covers. To repeat, this is the
reason why $L({\bf R})$ is replaced by $\bar{L}({\bf R})=SL(2,{\bf C})$ in the
determination of physical (i.e., projective) URs of the full symmetry group of
special relativity and, hence, for their interpretation as the state spaces of
elementary particles.

Lacking the completeness of such Lie-theoretic concepts and results, the
method followed here has the limited aim of finding certain finite-dimensional
projective representations of $L({\bf Z})$ (and, eventually, of projective URs
of $P({\bf Z})$) which are inherited naturally from those of $L({\bf R})$. In
other words, we shall look for a certain subgroup $\hat{L}({\bf Z})$ of
$SL(2,{\bf C})$ having the property that every projective representation of
$L({\bf Z})$ that is the restriction of a projective representation of $L({\bf
R})$ lifts to a linear representation of $\hat{L}({\bf Z})$. This requirement
is met if $\hat{L}({\bf Z})$ has a central ${\bf Z}_{2}$ subgroup such that
$\hat{L}({\bf Z})/{\bf Z}_{2}=L({\bf Z})$, exactly as in the corresponding Lie
group case where it is a standard construction found in text books (see for
example [10]). In fact, the result in the discrete case is a direct
transcription of this standard construction which I therefore recall.

Denote by $H(2,{\bf C})$ the real vector space of $2\times 2$ complex
hermitian matrices and by $\tau_{i},i=1,2,3$, the Pauli spin matrices. The
association of $x\in M$ to the matrix $\mathrm{x}:=x_{\mu}\tau_{\mu}$
($\tau_{0}=$ unit matrix), with
$x_{\mu}=(1/2)\mathrm{tr}(\tau_{\mu}\mathrm{x})$, is a bijection of $M$ and
$H(2,{\bf C})$, such that $x^{2}=\mathrm{det(x)}$. $SL(2,{\bf C})$ has an
action on $H(2,{\bf C})$ preserving the determinant:
$\mathrm{x}\rightarrow\alpha\mathrm{x}\alpha^{*},\alpha\in SL(2,{\bf C})$.
Correspondingly, for any $\alpha$ there is a $\lambda\in L({\bf R})$ such that
$(\lambda x)_{\mu}\tau_{\mu}=\alpha\mathrm{x}\alpha^{*}$ and hence a
homomorphism $SL(2,{\bf C})\rightarrow L({\bf R})$ whose kernel is easily seen
to be the central subgroup ${\bf Z}_{2}=\\{\pm 1\in SL(2,{\bf C})\\}$.

Given this explicit identification of $SL(2,\bf C)$ as the (unique) nontrivial
central extension of $L(\bf R)$ by $\bf Z_{2}$, its adaptation to the discrete
case is straightforward, thanks to the fact that the basis $\\{\tau_{\mu}\\}$
of $H(2,\bf C)$ are actually matrices over the ring ${\bf Z}[i]$ of Gaussian
integers, i.e., complex numbers whose real and imaginary parts are integers.
Replacing $M$ by its discrete counterpart $M({\bf Z})$ therefore gives a
bijection of $M({\bf Z})$ and $H(2,{\bf Z}[i])$ exactly as in the real case
(even though they are no longer vector spaces), $X\rightarrow
X_{\mu}\tau_{\mu}=:\mathrm{X}$, such that $X^{2}=\mathrm{det(X)}$. And, as
before, i) the group $SL(2)$ over Gaussian integers, $SL(2,{\bf Z}[i])\subset
SL(2,\bf C)$, acts on $H(2,{\bf Z}[i])$ by $\mathrm{X}\rightarrow
A\mathrm{X}A^{*}$; ii) given any $A\in SL(2,{\bf Z}[i])$, there is a discrete
Lorentz transformation $\Lambda$ such that $(\Lambda
X)_{\mu}\tau_{\mu}=A\mathrm{X}A^{*}$ and a homomorphism $SL(2,{\bf
Z}[i])\rightarrow L({\bf Z})$; and iii) the kernel of this homomorphism is
${\bf Z}_{2}=\\{\pm 1\in SL(2,{\bf Z}[i]\\}$. In other words, $SL(2,{\bf
Z}[i])$ is a nontrivial central extension of $L({\bf Z})$ by ${\bf
Z}_{2}:SL(2,{\bf Z}[i])/{\bf Z}_{2}=L({\bf Z})$; every finite dimensional
representation of $SL(2,{\bf Z}[i])$ will project to $L({\bf Z})$ as a
projective representation, either as a true representation or as a
‘representation up to sign’. This is the reason why this group is denoted by
$\hat{L}({\bf Z})$; it plays the same role in the theory of projective
representations in discrete relativity as $\hat{L}({\bf R})=SL(2,\bf C)$ does
traditionally. In particular, it will allow for the presence of states of
half-integral helicities in discrete quantum relativity as we shall see below.

As noted in section 3 above, the full discrete relativity group – the discrete
Poincaré group – is the semidirect product group $P({\bf Z})=L({\bf
Z})\vec{\times}T({\bf Z})$, $T({\bf Z})\sim{\bf Z}^{4}$ being the discrete
translation group; it is this group whose projective URs we would like to
interpret as the state spaces of elementary matter. By the general theory of
central extensions, they can be obtained as the (linear) URs of the
corresponding central extension888In general, inequivalent central extensions
of any group $G$ are classified by the second cohomology group $H^{2}(G)$ with
appropriate coefficients. $H^{2}$ of groups having a semidirect product
structure $G\vec{\times}A$, $A$ abelian, can have contributions other than
just $H^{2}(G)$. They are absent in the continuum case (see section 3) and,
for that reason, will be ignored (if they are present; the answer is not known
to me) in the discrete case as well. See, however, the discussion in section 7
on the role of irreducibility in the particle interpretation of URs of
$\hat{P}({\bf Z})$. For completeness I add that, in the semidirect product,
the action of $\hat{L}({\bf Z})$ on $T({\bf Z})$ is the same as that of
$L({\bf Z})$, extended by letting the central ${\bf Z}_{2}$ act trivially, as
in the real case. $\hat{P}({\bf Z})=SL(2,{\bf Z}[i])\vec{\times}{\bf Z}^{4}$.

##  5 Unitary representations of $\hat{P}(\bf R)$ – an overview

For the construction of physically acceptable URs of $\hat{P}({\bf Z})$ the
model will be Wigner’s method of ‘inducing from little groups’ for the
corresponding Lie group [4].999Wigner’s pioneering work was given a general
treatment, in particular as it applies to semidirect product groups, by Mackey
([11]). There are many subsequent accounts of the method in the literature.
For the positive mass URs the summary given here is based on the description
in [8] and, for the massless URs, the treatment below of the group-theoretic
origin of the subsidiary condition appears to be new. The generality of the
method allows room for its adaptation, with suitable adjustments, to the
discrete case. More importantly, the method naturally highlights the physical
attributes, mass and (Lorentz) helicity, that allow a direct association of
elementary quantum fields with URs, thereby serving as a model for deciding
which URs of $\hat{P}({\bf Z})$ can be considered ‘physical’. The following is
a compressed description of the essential elements of the Wigner construction
in the continuum case.

The momentum space is the dual group of the translation group $T(\bf R)$,
isomorphic also to ${\bf R}^{4}$ and denoted by $M^{*}$ with coordinates
$\\{p_{\mu}\\}$. Let $O$ be an orbit of $\hat{L}(\bf R)$ in $M^{*}$ for the
natural action of $L(\bf R)$, lifted to $\hat{L}(\bf R)$ by letting the
central ${\bf Z}_{2}$ act trivially,101010The action of $SL(2,\bf C)$ on
$M^{*}$ will often be denoted as $p\rightarrow\alpha p$ (as though $\alpha\in
L(\bf R)$) and likewise in the corresponding discrete case. No
misunderstanding is likely to arise. and let $S$ be the little group
(stabiliser) of any point in $O$. $O$ can then be identified with
$\hat{L}({\bf R})/S$. Suppose given a finite dimensional representation $\rho$
of $\hat{L}(\bf R)$ on a Hilbert space $V$ with the property that the
restriction of $\rho$ to $S$ is unitary. Denote by $\pi$ the projection of
$\hat{L}(\bf R)$ onto $O$ and by $\sigma$ a section of $\pi$, i.e., a map
$O\rightarrow\hat{L}(\bf R)$ such that $\pi(\sigma(p))=p$ for all $p\in O$.
Finally, let ${\cal H}_{O,V}$ be the space of vector valued functions
$\phi,\psi:O\rightarrow V$, square-integrable with respect to the (positive)
$\hat{L}(\bf R)$-invariant measure $\omega$ on $O$.

On ${\cal H}_{O,V}$, define a bracket $\langle\;,\;\rangle$ by

$\langle\phi,\psi\rangle=\int_{O}d\omega(p)\langle(\rho(\sigma(p)^{-1})\phi(p),(\rho(\sigma(p)^{-1})\psi(p)\rangle_{V},$

$\langle\;,\;\rangle_{V}$ being the scalar product on $V$. If $\sigma$ and
$\sigma^{\prime}$ are two sections of $\pi$, it follows from
$\pi(\sigma(p))=\pi(\sigma^{\prime}(p))\;(=p)$ that
$\sigma(p)^{-1}\sigma^{\prime}(p)$ is in $S$. And, since $\rho$ restricts to
$S$ unitarily, the bracket $\langle\;,\;\rangle$ is independent of the section
used to define it, making it a scalar product. Moreover, it follows from the
positivity of the measure that the norm of $\phi$ vanishes if and only if
$\phi=0$ identically, making (the completion of) ${\cal H}_{O,V}$ a Hilbert
space.

With these notions in place, one verifies first that the action of
$\hat{P}(\bf R)$ on ${\cal H}_{O,V}$ given by

$(U(\alpha,x)\phi)(p):=\chi_{p}(x)\rho(\alpha)\phi(\alpha^{-1}p),\hskip
14.22636pt\alpha\in\hat{L}({\bf R}),x\in T({\bf R}),$

where $\chi_{p}(x)=\exp(ip_{\mu}x_{\mu}$) is the character of $T(\bf R)$
corresponding to $p$, is a representation. It is in fact unitary because: i)
we have

$\|U(\alpha,x)\phi\|^{2}=\int d\omega(p)\|\rho(\sigma(\alpha
p)^{-1})\rho(\alpha)\phi(p)\|^{2}_{V}$

using the invariance of the measure under $p\rightarrow\alpha p$; ii)
$\sigma(\alpha p)$ and $\alpha\sigma(p)$ have the same projection onto $O$ and
hence differ by an element of $S$, implying $\rho(\sigma(\alpha
p)^{-1})\rho(\alpha)=\rho(s)\rho(\sigma(p)^{-1})$ for some $s\in S$; and iii)
$\rho$ is unitary on $S$. It is irreducible whenever $V$ is irreducible under
$\hat{L}(\bf R)$. In the language of induced representations, it is the UR
induced by the (unitary) restriction of $\rho$ to $S\subset\hat{L}(\bf R)$.
Physically, it is helpful to refer to it as the UR supported on a given mass
shell (the orbit $O_{m}$) and ranging over a given spectrum of helicities (the
representation $V$).

When $O$ is a positive (mass)2 positive (or negative) energy mass shell of
mass $m$ – i.e., the orbit $O_{m}$ through any point $p$ with $p^{2}=m^{2}$,
in particular through $(m,0,0,0)$ – the stabiliser $S_{m}$ is isomorphic to
$SU(2)$ and every irreducible representation of $\hat{L}(\bf R)$ on a Hilbert
space ${\cal H}_{m}$ constructed as above (dropping the subscript $V$)
restricts to $S_{m}$ as an irreducible UR. Hence the helicity spectrum of
$U_{m}$ is determined equivalentely and alternatively by the $\hat{L}(\bf R)$
or $S_{m}(=SU(2))$ transformation properties of the functions $\phi$. The spin
determines the helicity; in particular the number of distinct helicity states
in $U_{m}$ is $\dim V$. Moreover, the condition that $S_{m}$ fixes $p$
translates as the condition

$(U_{m}(s,x)\phi)(p)=\chi_{p}(x)\rho(s)\phi(p)$

for all $s\in S_{m}$. This, or rather its Lie algebra version, is the
‘invariant wave equation’ or the free field equation corresponding to the UR
$U_{m}$ of $\hat{P}({\bf R})$ ([5]). In the discrete case, it is (the discrete
form of) this condition which will replace the wave equation.

Since all known elementary particles have non-negative (mass)2 and finite sets
of helicities, it is customary to reject URs of $\hat{P}(\bf R)$ not having
these two properties as unphysical. As will be seen below, in the discrete
context there is no ‘invariant mass’ – a fact that has to be physically
interpreted with care – though the notion of a rest mass $m$ still makes
sense; a physical UR will then have to be characterised as one supported on an
orbit passing through $(m,0,0,0)$ with $m^{2}\geq 0$ and ranging over a finite
dimensional representation of $\hat{L}(\bf Z)$. The condition on the helicity
spectrum is a powerful one already at the continuum level: in the cases where
$S$ is a non-compact (Lie) group, it puts strong restrictions on its
admissible URs from which the induction process may be initiated (as also will
be seen below).

Apart from the one-point orbit that is the origin of the momentum space
$M^{*}$, there are two nontrivial $m=0$ orbits, the open upper and lower half
light cones in $M^{*}$. To construct physical URs of $\hat{P}(\bf R)$
supported on the upper half light cone $C_{+}$ for example, consider the
stabiliser $S_{0}$ of the representative point $(p_{0},0,0,p_{0}),\;p_{0}>0$,
consisting of the upper triangular matrices of $SL(2,\bf C)$ which we may
parametrise as

$s(\theta,z)=\left(\begin{array}[]{cc}\exp(i\theta)&z\exp(-i\theta)\\\
0&\exp(-i\theta)\end{array}\right),\;0\leq\theta<2\pi,\;z\in{\bf C}.$

The group law in $S_{0}$:
$s(\theta_{1},z_{1})s(\theta_{2},z_{2})=s(\theta_{1}+\theta_{2}\;\mathrm{mod}\,2\pi,z_{1}+z_{2}\exp(2i\theta_{1}))$,
identifies it as the Euclidean group in 2 dimensions $E(2,{\bf R})=SO(2,{\bf
R})\vec{\times}{\bf R}^{2}$, with ${\bf
R}^{2}=\\{(\mathrm{Re}\,z,\mathrm{Im}\,z)\\}$ on which the $SO(2)$ subgroup
acts as the 2-fold cover of the circle group; physically this $SO(2)$ is in
fact the group of rotations about the 3rd axis111111That it covers the circle
twice is the reason why massless particles can have half-integral helicities.
The Lie algebra of our $E(2)$ has $J_{3},J_{1}+K_{2},J_{2}-K_{1}$ as a basis,
the $J$s and the $K$s being generators of rotations and boosts in the standard
physics terminology. Obviously, the ${\bf R}^{2}$ subgroup does not relate to
physical translations. (more generally the direction of the momentum vector).
Its characters in a (massless) UR of $\hat{P}({\bf R})$ constitute what is
generally called its helicity spectrum. In contrast to the massive case, this
(Lorentz) helicity cannot be defined through the full rotation group, which is
natural since a massless state cannot be transformed to rest.

Now, a finite dimensional UR of $E(2,\bf R)$ is necessarily non-faithful;
indeed the only such URs are characters of $SO(2,\bf R)$ and have the normal
subgroup ${\bf R}^{2}$ as kernel.121212A thorough account of the unitary
representation theory of $E(2,\bf R)$ is available in [12]. Hence $\hat{L}(\bf
R)$, being simple, cannot have any finite dimensional representation
restricting to $E(2,\bf R)$ unitarily. This in turn means that the procedure
of inducing from the stabiliser no longer works as directly as in the massive
case and has to be modified suitably. The well known way to do this ([12],
[5]) is to specialise the space ${\cal H}_{0,V}$ of functions
$\phi:C_{+}\rightarrow V$ as defined earlier to the subspace ${\cal
H}^{\prime}_{0}$ (dropping again the subscript $V$) on which the action of
${\bf{R}}^{2}(\subset E(2,{\bf R})\subset SL(2,\bf C))$ is trivial:

$\rho(r)\phi(r^{-1}p)=\phi(p),\>r\in{\bf R}^{2}.$

The Lie algebra form of this condition encodes the familiar subsidiary
conditions satisfied by massless fields. A character of $SO(2,{\bf
R})=E(2,{\bf R})/{\bf{R}}^{2}:\rho(\theta)\phi(p)=\exp(im\theta/2)\phi(p)$
with $\theta/2\in SO(2,{\bf R}),m\in\bf Z$, then induces a UR of $\hat{P}(\bf
R)$ on ${\cal H}^{\prime}_{0}$. The fact to be noted, especially relevant in
the context of discrete relativity, is that massless finite helicity URs exist
because the appropriate stabiliser has a normal subgroup with compact quotient
group. It also follows that a physically acceptable irreducible UR has
precisely one (Lorentz) helicity, namely the character of $SO(2,{\bf R})$ to
which $\rho$ restricts.

As for representations supported on orbits with (mass)${}^{2}<0$, they are
doubly unphysical. Not only will they correspond to particles which are
tachyonic at all momenta, they will suffer from unphysical helicities as well:
the stabiliser, which is $SL(2,\bf R)$, has no finite dimensional nontrivial
URs at all.

This summary of Wignerism is meant also to remind us that the general theory
of quantum fields having a particle interpretation is no more than the
representation theory of the group of relativistic symmetries, subject to
certain physical criteria. The significance of this foundational construction
as the prelude to any attempt to quantise gravity has been noted in sections 1
and 2 above. The question now is whether this identification of particles and
representations can be carried over realistically to space-times which are
discrete.

##  6 Masses and helicities in discrete relativity

We turn first to an examination of how the fundamental notions of mass and
helicity survive the reduction of the group of symmetries from $P(\bf R)$ to
$P(\bf Z)$, both for their intrinsic significance and as preparation for the
construction of physically acceptable projective URs of the latter.

The momentum space of discrete space-time is the dual group of the discrete
translation group $T(\bf Z)$ ($\sim{\bf Z}^{4}$), namely the 4-torus, denoted
by $B$ from now on. As in other familiar lattice problems, it is useful to
think of $B=M^{*}/{\bf Z}^{4}$ as the fundamental domain for the action of
$\bf Z^{4}$ on $M^{*}$ where $M^{*}$ $(\sim{\bf R}^{4})$ as earlier is the
momentum space of the continuum translation group and ${\bf Z}^{4}$ is to be
identified with the reciprocal lattice. In terms of coordinates
$\\{p_{\mu}\\}$ in $M^{*}$, $B$ is thus the cube
$\\{-\pi<P_{\mu}\leq\pi,\;P_{\mu}:=p_{\mu}\;\mathrm{mod}\;2\pi\\}$, i.e., the
unit cell of the reciprocal lattice, the relativistic analogue of the
Brillouin zone or, in its cosmological role, the Brillouin zone of the
universe (called simply the Brillouin zone from now on). Physically, this
means of course that the momentum components are defined and conserved modulo
$2\pi$, a circumstance that has far more fundamental consequences here than in
the crystal physics context (keeping in mind that it is only a minuscule part
of the interior of this space-time Brillouin zone, far from its boundary, that
terrestrial experiments and observations and the theories that deal with them
can explore). Nevertheless, as in crystal physics, we can study the action of
the discrete group $\hat{L}(\bf Z)$ on $B$ by starting with its action on
$M^{*}$ and then translating the coordinates of the image of a point in $B$
considered as a point of $M^{*}$ back to $B$ by some integral multiples of
$2\pi$. Under this projection $M^{*}\rightarrow B$, $P_{\mu}=\pi$ and
$P_{\mu}=-\pi$ get identified for each $\mu$.

Thus a generic orbit $O_{B}({\bf Z})$ of $\hat{L}({\bf Z})$ in $B$, through a
given point $P\neq 0$, can be determined by first finding the orbit $O(\bf Z)$
through $P$ of $\hat{L}({\bf Z})\subset\hat{L}(\bf R)$ in all of $M^{*}$ and
then translating the points of $O({\bf Z})$ outside $B$ back to $B$. It is to
be expected that, generically, the orbits will be quite ‘wild’.131313The study
of the action of discrete subgroups of Lie groups on manifolds is an active
field of research and is potentially of great use in the physics of discrete
systems in general.  $O({\bf Z})$ being a subset of the orbit of $\hat{L}(\bf
R)$ through $P$ considered as a point of $M^{*}$, a good starting point then
is the projection onto $B$ of the relevant orbits in the standard continuum
picture.

Consider first the orbit $O_{m}$ of $\hat{L}(\bf R)$ through
$(m,0,0,0),\;m\neq 0$, i.e., the set of points $\\{p_{\mu}\\}$ with
$p_{\mu}p_{\mu}=m^{2}$ (the positive energy mass shell of rest mass $m$). It
projects on to $B$ as the set of points $\\{P_{\mu}=p_{\mu}$ mod $2\pi\\}$,
the ‘torus mass shell’ of rest mass $m$. Figure 1 is a depiction of this (for
a value of $m$ chosen to be very large so as to bring out its complicated
structure) as its intersection with, say, the (0,1) plane. The corresponding
orbit $O_{B,m}(\bf Z)$ of $\hat{L}(\bf Z)$ for any rest mass $0<m<\pi$, our
objects of interest, will be discrete subsets of such torus mass shells.

Figure 1: The intersection of a torus mass shell with the (0,1) plane. The
arrows indicate that the particle is being boosted from rest in the positive 1
direction. The dotted segments illustrate the first few superluminal phases.

At the classical level, the torus mass shell summarises the kinematics – the
relationships among energy, momentum and velocity – of a particle of a given
rest mass moving in discrete space-time. The following qualitative remarks are
self-evident.

Within the Brillouin zone, i.e., where $P_{\mu}=p_{\mu}$, kinematics is
entirely as in continuum relativity; in particular, the rest mass $m$ is the
invariant mass for transformations in $M^{*}$ which keep the particle within
the Brillouin zone. But when a particle initially at rest is subjected to
larger and larger boosts, say along the positive direction 1, there occur a
sequence of critical points on the boundary of $B$ at which the energy or the
momentum appears to undergo a discontinuous change of magnitude $2\pi$ in our
units (or $2\pi/L$ in terms of the Planck length141414For numerical guidance
we may take it to be of the order of the Planck mass, $10^{20}$ GeV or so.),
induced by the identification of the points $p_{\mu}$ and $p_{\mu}+2\pi
n_{\mu}$ in $M^{*}$ for arbirary integers $n_{\mu}$. The first critical point
occurs at $(P_{0}=\pi,P_{1}=\sqrt{\pi^{2}-m^{2}}$). A further boost in the
same direction will give rise to the segment of a hyperbola (restricting the
torus mass shell to the (0,1) plane as in the figure) connecting the translate
$(-\pi,\sqrt{\pi^{2}-m^{2}})$ of this point with the next critical point,
namely the translate $(\sqrt{\pi^{2}+m^{2}}-2\pi,\pi)$ of the point
$(\sqrt{\pi^{2}+m^{2}},\pi)$ (which is outside $B$). And so on ad infinitum.
In all the segments except the initial one, $P_{0}^{2}-P_{1}^{2}\neq m^{2}$.
More generally, $P_{\mu}P_{\mu}$ is not invariant under the action of the
Lorentz group, continuous or discrete; ‘mass’ in the term ‘torus mass shell’
refers to the rest mass as the parameter labelling distinct mass shells. This
is just a reflection of the fact that there are no $\hat{L}(\bf Z)$ (and hence
no $\hat{L}(\bf R)$) invariant non-zero periodic functions on $M^{*}$ and
stems from the periodicity of the momentum components in $M^{*}$.

It is also clear, most simply and graphically from Figure 1, that all segments
of the mass hyperbola except the first have parts with $P_{0}^{2}-P_{1}^{2}$
negative, where the particle’s speed is greater than 1: under large boosts the
particle behaves intermittently as a tachyon with an imaginary effective mass
$m_{\mathrm{eff}}=\sqrt{P_{\mu}P_{\mu}}$. In the limit of an infinitely large
boost, the effective mass tends to zero and the mass shell for any non-zero
rest mass tends to the torus light cone. The full torus mass shell embedding a
massive orbit thus has a complicated structure.To be noted in particular is
that the speed of light is not an impassable barrier: a particle in a
tachyonic kinematic regime can always be deboosted to rest.

For ease of reference, I will refer to the part of a torus mass shell on which
the rest mass is also the invariant mass as the ‘conventional part of the mass
shell’ or ‘the conventional regime’. It consists of the orbit through
$(m,0,0,0)$ before it hits the first critical point
$(P_{0}=p_{0}=\pi,|\vec{P}|=|\vec{p}|=\sqrt{\pi^{2}-m^{2}})$; none of the
exotic aspects of the new kinematics manifest themselves when the energy and
momentum are constrained to be in the conventional part of a mass shell.

In contrast, consider the light cone $C$ of $M^{*}$. Under projection onto $B$
(translations by multiples of $2\pi$), its image $C_{B}$ remains in the torus
light cone, the part of the light cone lying in $B$ (with the usual boundary
conditions), In consequence, any boost of a point in $C_{B}$ will take it
first to a point in $C$ and then, on translation back into $B$, to a point in
$C_{B}$ itself. Masslessness is an invariant property in discrete relativity –
the polynomial $p_{\mu}p_{\mu}$ is periodic and invariant as long as it
vanishes. The orbit $O_{B,0}$ of any point in $C_{B}$ for the action of
$\hat{L}(\bf Z)$ on $B$ is a set of discrete points in $C_{B}$.

As will be seen below, this is a property of capital importance.

Next is the problem of whether and how helicities can be defined in discrete
space-time. They cannot be defined as arising from the URs of the rotation
group by restriction since $SO(3,\bf R)$ or $SU(2)$, being compact, has only
finite groups as discrete subgroups and therefore cannot accommodate general
spins. The way out is to define helicity relativistically, as arising directly
from the Lorentz group, without invoking the rotation group and without
transforming to rest. (Recall the discussion in section 5 of how massless
helicities are defined in the Wigner construction). We have in fact the key
result:

Every finite dimensional irreducible representation of $\hat{L}(\bf R)$
restricts to its discrete subgroup $\hat{L}(\bf Z)$ as an irreducible
representation.

This is a special case of a general theorem, the density theorem of A. Borel
([13]; see also [14] for an account of the general theoretical framework), on
representations of discrete subgroups of non-compact semisimple Lie groups. A
general formulation of the theorem is as follows. Let $G$ be a semisimple Lie
group none of whose factors is compact and $\Gamma$ a discrete subgroup of $G$
having the property that the quotient $G/\Gamma$ has finite volume (i.e.,
$\Gamma$ is sufficiently dense in $G$). Then every finite dimensional
irreducible representation of $G$ restricts to $\Gamma$ irreducibly.151515Work
subsequent to [13] and [14] has extended this result in several directions.
For our purpose, the original density theorem is enough. The group $SL(2,{\bf
Z}[i])$ has finite covolume in $SL(2,\bf C)$ and hence meets the conditions of
the theorem, thereby enabling the taking over of the helicity content of any
finite dimensional irreducible representation of $SL(2,\bf C)$ as that of the
(irreducible) representation of $SL(2,{\bf Z}[i])$ to which it restricts.

An easy illustration of the density theorem is provided by the important
special case of spin 1/2. The defining (left-chiral) representation $\rho_{L}$
of $SL(2,{\bf C}):\rho_{L}(\alpha)=\alpha\in SL(2,{\bf C})$ restricts to
$SL(2,{\bf Z}[i])$ as $\rho_{L}(A)=A\in SL(2,{\bf Z}[i])$. Let $K$ be an
operator on ${\bf C}^{2}$, the representation space of $\rho_{L}$, that
commutes with $\rho_{L}(A)$ for all $A\in SL(2,{\bf Z}[i])$. Each of the Pauli
matrices (multiplied by $i$) is in $SL(2,{\bf Z}[i])$ and so will commute with
$K$ by assumption. Hence $K$ is a multiple of the unit operator and, by
Schur’s lemma, $\rho_{L}$ remains irreducible when restricted to $SL(2,{\bf
Z}[i])$. The same conclusion – and the same argument – holds for the conjugate
(right-chiral) representation. We may note incidentally that any two of the
Pauli matrices can be picked to belong to the generators of $SL(2,{\bf
Z}[i])$([15]).

The density theorem is critically important because of our general condition
that only those representations are physically relevant whose ‘continuum
limits’ exist and are the Lie group representations we are used to. This is
ensured if only those irreducible representations of $\hat{L}({\bf Z})$ are
considered physical which are restrictions of (continuous) finite dimensional
irreducible representations of $\hat{L}({\bf R})$.161616It is useful to
remember that even in the standard Wigner philosophy, certain URs of the
Poincaré group are excluded from physics solely on the ground of lack of
experimental support, e.g., those corresponding to imaginary mass or
infinite/continuous spin. These criteria will be assumed to be valid in the
discrete case as well. Thus URs of $\hat{P}({\bf Z})$ which cannot be
deboosted to rest – corresponding to tachyons in continuum relativity – are
excluded.

##  7 Matter fields as unitary representations of $\hat{P}({\bf Z})$

Having seen how the basic notions of momentum, mass and helicity carry over to
a discrete space-time, we can turn now to our primary objective, that of
identifying the elementary consistuents (particles) of matter with the fields
that constitute Hilbert spaces of those irreducible URs of the discrete
Poincaré group that are subject to the two physical criteria introduced in the
last section. These are, to repeat, i) the orbit supporting a UR must be
massless (which is an invariant property) or must have a real rest mass; and,
ii) the representation of $\hat{L}(\bf Z)$ over which the UR ranges must be
the restriction of a representation of $\hat{L}(\bf R)$. To these must be
added a third, apparently more technical, criterion which arises as follows.

A point $P$ of the Brillouin zone $B$ will be said to be a rational point if
its coordinates $\\{P_{\mu}\\}$ are all rational multiples of $\pi$:
$P_{\mu}=(q_{\mu}/d_{\mu})\pi$, with $q_{\mu},d_{\mu}\in\bf Z$,
$-|d_{\mu}|\leq q_{\mu}\leq|d_{\mu}|$, and an irrational point otherwise.
Every point in the orbit of $\hat{L}(\bf Z)$ through a rational (irrational)
point is rational (irrational), since $\hat{L}(\bf Z)$ acts on $B$ by integral
linear transformations followed by shifts by integral multiples of $\pi$.
Consider rational orbits first. Reexpress the coordinates
$\\{q_{\mu}/d_{\mu}\\}$ (dropping the factor $\pi$ for the time being) in
terms of the lowest positive common multiple $D$ of {$d_{\mu}\\}$ , i.e.,
$P_{\mu}=Q_{\mu}/D$ with $-D\leq Q_{\mu}\leq D$ and no positive integer
$D^{\prime}<D$ exists such that $P_{\mu}=Q^{\prime}_{\mu}/D^{\prime}$ for any
$\\{Q^{\prime}_{\mu}\\}$ with $-D^{\prime}\leq Q^{\prime}_{\mu}\leq
D^{\prime}$; $\\{Q_{\mu}/D\\}$ will be referred to as the standard coordinates
of the rational point $P$.

Suppose now that the transform by $A$ of $P$ in standard form is not in
standard form, i.e., $(AQ)_{\mu}$ has a common factor with $D$ for each $\mu$.
Then the standard coordinates of $AP$ will have a denominator $D_{A}$ strictly
less than $D$; if there is no common factor, the denominator of course remains
unchanged. But the same argument applies also to $A^{-1}$ acting on $AP$,
implying that $D$ cannot be greater than $D_{A}$. It follows that $\hat{L}(\bf
Z)$ acts on every rational point without changing the denominator of its
standard coordinates. Since the numerators $\\{Q_{\mu}\\}$ lie between $-D$
and $D$, we can conclude that a rational orbit of $\hat{L}(\bf Z)$ in $B$ is a
finite set.

Recalling the identification of the orbit with the quotient of $\hat{L}(\bf
Z)$ by the stabiliser, we see thus that the stabiliser of any rational point
is a subgroup of finite index, in other words almost all of $\hat{L}(\bf Z)$.
The situation is not very different from that of the one-point orbit
consisting of the origin $P=0$ – the stabiliser has no finite dimensional UR
from which to induce a UR of $\hat{P}(\bf Z)$ with a finite helicity spectrum
(even when subjected to a finite set of subsidiary conditions, see the
discussion of the massless URs of $\hat{P}(\bf R)$ in section 5). So our third
criterion is: rational orbits must be excluded from consideration in
constructing physically reasonable URs of $\hat{P}(\bf Z)$.

The next task is to determine the stabiliser of an irrational point $P$ for
the action of $\hat{L}(\bf Z)$ on $B$. This group is

$\Sigma_{P}:=\\{A\in\hat{L}({\bf Z}):(AP)_{\mu}=P_{\mu}\;\mathrm{mod}\;\bf
Z\\}.$

It is most simply characterised in terms of the discrete Lorentz group $L(\bf
Z)$ as the subgroup satisfying the conditions

$(\Lambda P)_{\mu}=P_{\mu}+N_{\mu},\hskip 14.22636pt\Lambda\in L(\bf Z),$

for arbitrary integers $\\{N_{\mu}\\}$. For an irrational $P$, these
conditions put strong restrictions on the admissible values of $N_{\mu}$:
writing them as $(\Lambda-I)^{-1}N=P$ ($I$ is the unit matrix), we see that
$P$ will be a rational point (since $(\Lambda-I)^{-1}$ has rational entries
($\Lambda-I$ is integral)) unless all $N_{\mu}$ vanish.

It follows that the stabiliser $\Sigma_{P}$ of an irrational $P$ for the
action of $\hat{L}(\bf Z)$ on $B$ coincides with its stabiliser for the
$\hat{L}(\bf Z)$ action on the whole of $M^{*}$; it is determined by the
condition $(\Lambda P)_{\mu}=P_{\mu}$ exactly as in the continuum case. Thus,
if $P$ is in a massive orbit (of rest mass $m$) of $\hat{L}(\bf Z)$, the
corresponding stabiliser $\Sigma_{m}$ consists of all unitary matrices in
$SL(2,{\bf Z}[i])$. It is a finite (of course) group of order 8, isomorphic to
the quaternion group: $\Sigma_{m}=\\{\pm 1,\pm i\tau_{i}\\}$ where
$\\{\tau_{i}\\}$ as earlier are the Pauli matrices.

Also as in continuum relativity, when $P$ is in a massless orbit, its
stabiliser $\Sigma_{0}$ is the subgroup of $SL(2,{\bf Z}[i])$ consisting of
upper triangular matrices (for a suitable choice of basis in $M^{*}$)

$s(\zeta,Z)=\left(\begin{array}[]{cc}\zeta&\zeta^{-1}Z\\\
0&\zeta^{-1}\end{array}\right);\;\zeta,\zeta^{-1},Z\in{\bf Z}[i].$

The only elements of ${\bf Z}[i]$ with inverses in ${\bf Z}[i]$, namely the
units, being $\zeta=\pm 1,\pm i$, the subgroup of diagonal matrices is the
cyclic group ${\bf Z}_{4}$ and the subgroup of elements $s(1,Z)$ is the planar
lattice of points $(\mathrm{Re}\;Z,\mathrm{Im}\;Z)$. Composition in
$\Sigma_{0}$ is given by
$s(\zeta_{1},Z_{1})s(\zeta_{2},Z_{2})=s(\zeta_{1}\zeta_{2},Z_{1}+{\zeta_{1}}^{2}Z_{2})$,
confirming that $\Sigma_{0}$ is indeed the discrete Euclidean group $E(2,{\bf
Z})=SO(2,{\bf Z})\vec{\times}{\bf Z}^{2}$, with $\zeta\in{\bf Z}_{4}$ acting
on ${\bf Z}^{2}$ by multiplication by $\zeta^{2}$ – which, as in the continuum
case, is a reminder that $\hat{L}(\bf Z)$ covers $L(\bf Z)$ twice, thereby
accommodating representations of half-integral helicities.

We are thus confirmed in our expectation that the stabilisers are just the
natural discretisations of their Lie group counterparts. The exercise also
reinforces the choice of $SL(2,{\bf Z}[i])$ as the correct replacement for
$SL(2,\bf C)$.

In constructing URs of $\hat{P}({\bf Z})$, whether massive or massless, we can
now try to imitate Wigner’s method in the continuum case, keeping in mind that
all orbits are now discrete sets. Define momentum space fields as functions
$\phi:O_{B}\rightarrow V$, where $O_{B}$ is $O_{B.m}$ or $O_{B,0}$ as the case
may be and $V$ as before is the space of an irreducible representation $\rho$
of $\hat{L}(\bf R)$ (and hence, by the density theorem, of $\hat{L}(\bf Z)$).
Given a set-section $\sigma:O_{B}\rightarrow\hat{L}({\bf Z})$, such fields
form a Hilbert space ${\cal H}_{O_{B}}$ with scalar product

$\langle\phi,\psi\rangle=\sum_{P\in
O_{B}}\langle\rho(\sigma(P)^{-1})\phi(P),\rho(\sigma(P)^{-1})\psi(P)\rangle_{V}$

(subject to the condition $\langle\phi,\phi\rangle<\infty$). On this Hilbert
space, we have a UR $U_{O_{B}}$ of $\hat{P}(\bf Z)$ given by

$(U_{O_{B}}(A,X)\phi)(P)=\exp(iP_{\mu}X_{\mu})\phi(A^{-1}P),\>A\in\hat{L}({\bf
Z}),X\in T({\bf Z}).$

To repeat, all this is no more than a direct adaptation of the Wigner method
to the discrete situation. It is to be noted in particular that the Fourier
transform $\phi^{*}$ of $\phi$ is the field defined on space-time; since
$\phi$ is a function on ${\bf T}^{4}$, $\phi^{*}$ is supported on ${\bf
Z}^{4}$.

But there are also some significant differences. The less serious one
conceptually is that the discrete counterparts of the conventional field
equations, which are Lie-algebraic statements to the effect that the
stabiliser fixes points in an orbit (see section 5), have of course no
infinitesimal version. Nor has the subsidiary condition which, in the massless
case, enforces the requirement that admissible representations must have the
vector subgroup of the Euclidean group as kernel. More seriously, while the
orbit of $\hat{L}(\bf R)$ through $p\in M^{*}$ is the whole mass shell,
massive or massless, containing $p$ – i.e., $\hat{L}(\bf R)$ operates
transitively on the submanifold of $M^{*}$ defined by a constant $p^{2}$ –
that is not the case for the action of $\hat{L}(\bf Z)$ on the torus mass
shell; momenta which are linearly independent over the rationals cannot be
connected by a discrete Lorentz transformation even if they are both massless
or have the same rest mass. We can then take the orbit through a point
$P^{\prime}$ that is not in the orbit through $P$ (i.e., $P^{\prime}$ and $P$
are rationally independent) but is in the same torus mass shell, and construct
another irreducible UR of $\hat{P}(\bf Z)$, and so on. All of them will have
the same stabiliser and the same rest mass and the same set of helicities; the
Wignerian association of an elementary particle of a given mass and a given
spin with an irreducible UR of $\hat{P}(\bf R)$ does not hold in discrete
relativity.

There is then a choice to be made. One option is to associate each of the
irreducible URs as constructed above with a particle type. Since all of them
have the same rest mass and spin, distinguishing among them will require the
introduction of new ‘quantum numbers’ which, however, must still have their
origin in the kinematics of discrete space-time itself. What might they be? It
is unlikely that they can be related to any of the charges (‘internal quantum
numbers’) of models currently in favour; among the more pragmatic reasons, we
know of no infinite multiplets of particles of different charges all having
the same mass and spin.

An alternative possibility is that the different irreducible URs are
superselection sectors. Recall that superselection rules are vetoes on the
unrestricted superposability of states; they decompose the total state space
into a collection of sectors whose direct sums do not represent states. An
immediate physical consequence is that no observable can connect states from
two distinct sectors (which in fact was the formulation they were first given
[16]). The vetoes arise ([8,9]) from the non-additivity of inequivalent
projective URs of a symmetry group and are determined by its 2nd cohomology
group with suitable coefficients. In the present context, the symmetry group
is $P(\bf Z)$ and its 2nd cohomology group is determined by that of its
Lorentz subgroup $L({\bf Z})$. Group cohomolgy of discrete subgroups of Lie
groups is a very active field but I have not been able to find the specific
result needed here in the literature. (See also the discussion in section 3).
Space-time symmetries can certainly give rise to superselection rules, the
prime example being univalence, the rule that forbids the superposition of
integral and half-integral spins, both in the continuum and, as we have seen
in this paper, discrete situations. The question is whether in the discrete
case the relevant cohomology group, which must contain the univalence ${\bf
Z}_{2}$, is in fact larger.171717Very elementary examples of discretisations
of Lie groups giving rise to additional nontrivial projective URs with
interesting physical applications are provided by the symmetries of
2-dimensional spaces. Thus the cylinder group ${\bf S}^{1}\times{\bf R}$ (the
natural group of an infinitely long strip with the two edges identified) has
vanishing 2nd cohomolgy but its subgroup ${\bf S}^{1}\times{\bf Z}$ has
nontrivial projective URs [17]. Similarly, the torus group ${\bf T^{2}}$
(acting on a rectangular space with pairs of opposite edges identified) has
vanishing 2nd cohomology but its discrete subgroups ${\bf Z}_{n_{1}}\times{\bf
Z}_{n_{2}}$ for positive $n_{1}$ and $n_{2}$ have non-zero cohomologies. The
corresponding nontrivial projective URs are the underlying causes of phenomena
related to quantum Hall effects ([18,19]).

If it turns out that there is no extra superselection structure a third option
will be to give up the Wignerian criterion of associating a particle to an
irreducible UR of $\hat{P}({\bf Z})$. We are then free to assign a particle to
a UR constructed, at least formally to start with, as the space of functions
from the union of all irrational orbits of a given rest mass – i.e., all of a
given torus mass shell omitting rational orbits – to a given irreducible
representation of $\hat{L}(\bf R)$ (and hence, by the density theorem, of
$\hat{L}({\bf Z})$). Though highly reducible, such a UR of $\hat{P}({\bf Z})$
will still be characterised by a unique rest mass and a unique spin.
Transformations belonging to $\hat{L}(\bf R)$ but not to $\hat{L}({\bf Z})$
will connect different irreducible components of this UR which, in the
continuum limit in some suitable sense, should tend to an irreducible UR of
$\hat{P}(\bf R)$.

Which of these options is the right one is a well-posed mathematical problem
that is unsolved at present but solvable in principle. Until that is done, it
seems prudent not to favour one over the others.

##  8 Physical effects: generalities

The picture that has emerged can be summarised as follows. A discrete
Minkowskian space-time incorporating a fundamental length $L$ – to be thought
of as one of the primordial constants of nature – is fully capable of
supporting the Wignerian correspondence between projective URs of its group of
isometries and the quantum fields of elementary particles as defined by their
masses and spins, subject to certain criteria of physical admissibility.
Unsurprisingly, discretisation of special relativity makes no difference to
the physics involving these particles as long as they are massless or are not
subjected to large boosts taking them outside the conventional part of their
mass shells. At energies and momenta resulting from large boosts there are new
effects. This section and the next are devoted to a preliminary survey, mostly
qualitative and occasionally speculative, of some such effects. (It is
implicit that $L$ is (of the order of) the Planck length.) It is meant to be
no more than an introduction to the issues involved and to convince the reader
that the idea of a fundamentally discrete world is not to be rejected out of
hand; it is premature to think of detailed quantitative computations.

Virtually all these deviations from conventional wisdom have their origin in
the compactness of momentum space.181818This is a very general feature,
independent of any particular discretisation: the Pontryagin dual of a
discrete abelian group is a compact abelian group. In general terms, its basic
consequence is that energy and momentum are defined and conserved only modulo
reciprocal lattice vectors, implying in turn that the notion of an invariant
mass needs a reformulation. The details are easiest to describe for the
hypercubic discretisation employed here, for which the reciprocal lattice is
also hypercubic, with the (Minkowskian) Brillouin zone given by (restoring its
physical role to the lattice spacing $L$) $B=\\{-\pi/L<P_{\mu}\leq\pi/L\\}$.
Firstly, masslessness of a field/particle as an invariant concept survives
discretisation: the torus light cone is mapped into itself under all boosts
$P\rightarrow P^{\prime}$. To remind ourselves, this property derives
fundamentally from the absolute constancy of the speed of light, which itself
is just the ratio of the spatial and temporal lattice spacings in conventional
units (see section 3).191919There is a subtle point of principle involved
here. The general result, free from conventions, is that massless URs of
$\hat{P}({\bf Z})$ exist and that the speed of propagation of the
cirresponding particles, in particular the photon, is constant and equal to
the ratio of the spatial and temporal lattice spacings. That our initial
definition of the lattice using the speed of light as a conversion factor
guarantees this outcome is far from obvious at the outset. A reassessment of
the fundamental significance of the speed of light, both logically and
historically, will be found in a review now in preparation. Next, though a
non-zero mass remains invariant under ‘small’ boosts $p\rightarrow p^{\prime}$
with $p,p^{\prime}\in B$, that is not the case for ‘large’ boosts taking $p$
out of $B$, resulting in discontinuous changes in $P$ at the boundary of $B$.
The physically meaningful (and useful) notions are that of a rest mass and of
a mass shell of rest mass $m$, the orbit of the Lorentz group in $B$ through
the point $(m,0,0,0)$ (the torus mass shell of section 7). Consequently, when
a particle of rest mass $m$ undergoes large boosts, it traverses
intermittently regions of the mass shell where its speed becomes superluminal.
In the limit of an infinitely large boost, the speed will tend to the speed of
light, passing through, in the process, an infinite number of superluminal
phases (Figure 1). It is also clear that the momentum intervals in which the
particles have a transluminal speed grow as its rest mass $m$ increases
(reminder: Figure 1 is for $m$ of the order of $L^{-1}$).

In looking at the observable effects of these unfamiliar kinematical
properties, a reservation will be in order: our inability, as of now, to
resolve the question of reducibility/degeneracy of URs of $\hat{P}(\bf Z)$,
discussed at the end of section 7. As long as the identity of an elementary
particle is primarily decided by its mass and spin – i.e., assuming, as seems
justified in our present state of knowledge, that internal charges and the
concomitant gauge interactions have an origin outside space-time geometry –
irreducibility by itself may not be a critically important criterion. Such a
position would require us to accept that matter considered elementary at
currently accessible scales, including the invisible quarks and gluons, will
continue to be describable as elementary quantum fields of the discrete
Poincaré group. That would be a big extrapolation; on the natural scale of the
Planck mass, the known elementary particles are massless to an extraordinarily
good approximation, to an accuracy of some eighteen orders of magnitude. Aside
from the cosmological constant problem, there are no other examples in
fundamental physics of such an enormous deviation from naturalness. One can
wonder whether there actually are – or were – elementary particles of mass
comparable to $L^{-1}$ but, as of now, that would be idle speculation. So, in
checking the exotic effects of discreteness against facts on the ground – or,
rather, in the heavens – no specific assumption will be made here regarding
the possible presence of such ultra-heavy matter particles through the history
of the universe. (Modulo poorly understood phenomena such as dark matter or
conjectural entities like the inflaton, modern cosmological theory, including
that of the very early universe, has no place for any exotic particles).

Of the direct effects of discreteness, testing for deviations from homogeneity
and isotropy of space is no more or less than determining its ‘crystal
structure’. Intuitively it is clear that any such procedure will need to
measure spatial (and, in the case of homogeneity of time, temporal)
resolutions finer than $L$ (the Planck length by default). For illustration,
for a beam of particles (emitted somewhere in the universe) interacting with
the fields $\phi^{*}$ concentrated at the ‘lattice sites’ of space to produce
a measurable (macroscopic) diffraction pattern, the wavelength of the particle
has to be of the order of the lattice spacing, just from Bragg’s law. There is
no need to add that such tests are out of reach by very many orders of
magnitude.

An essentially similar conclusion holds for tests of isotropy, even though
angles (as opposed to lengths) are not restricted to be discrete. For a
periodic lattice such as the cubic one considered here, a reasonable statement
of isotropy would appear to be that an arbitrary straight line through any
chosen origin 0 (‘the line of sight’) should pass through an infinite number
of lattice points $lX$, $l=0,1,2,\cdots;\;X\in M(\bf Z)$. (It is enough that
it passes through one such point, say $l=1$). This sharp formulation however
does not quite work. When the line of sight through 0 is, for example, in the
1-2 plane: $X=(0,X_{1},X_{2},0)$, and makes an angle $\theta$ with the 1 axis,
$\tan\theta=X_{2}/X_{1}$ will be rational whereas there is a general result
(part of what is known as Niven’s theorem [20]) which says that the only
rational values of the tangent function at rational angles (rational multiples
of $\pi$) are $0,\pm 1$. In other words, if the line of sight is at a rational
angle (except for $0$ and $\pi/4$ in the first quadrant) it will pass through
no lattice points at all. The remedy suggests itself: the continuity of the
tangent function implies that, given a point $X$, there are angles $\theta$
such that the corresponding lines of sight pass as close as desired to the
points $lX$ for all $l$ less than any given finite integer. This slightly
weaker formulation of isotropy is perfectly satisfactory both theoretically
and observationally.

##  9 Physical effects: some qualitative cosmology

The kinematical effects of the boundedness of momentum and energy seem,
likewise, to be far beyond the capabilities of any controllable experiment
even in a remote future: how to boost particles to energy/momentum of
magnitudes of the order of $10^{20}$ GeV, i.e., to the edge of the Brillouin
zone? That leaves cosmic observations as the only realistic test of
discreteness, especially those which are sensitive to the physics of processes
which we believe happen in the early universe. One such is the
deboosting202020Throughout this article, I have avoided using the terms
‘acceleration’ and ‘deceleration’ in favour of ‘boost’ and ‘deboost’ for the
reason that energy and momentum are always within $B$ and change
discontinuously when they hit its boundary. Velocity, defined as
$\vec{V}=\vec{P}/P_{0}$ so as to be consistent with the continuum limit, also
has discontinuities. Boost operations are well-defined in $B$ and, along any
of the three spatial axes, can be ordered by the value of the parameter
(sometimes called rapidity) they depend on. The essential point is that an
infinite boost does not result in infinite energy, momentum or velocity. For
the same reason, the evolution of the discrete universe from the big bang to
the present is to be thought of as parametrised by a monotonically decreasing
(average) boost parameter of matter particles rather than energy or velocity.
of matter in the initial hot dense ‘stuff’ that leads to expansion and cooling
(in the accepted, ‘standard’, cosmological model). In the discrete world the
process will go through a series of steps in which every particle of matter
crosses the light barrier, into and out of the light cone, alternately (Figure
1 with the arrows reversed shows the final stages of the process) until it
makes its final transit into the interior (to the conventional part of its
mass shell), after which it will behave exactly as in conventional special
relativity. The question is how these repeated transluminal episodes fit in
with the generally agreed post-big-bang picture of the early cosmos.

Considering that all transits including the final one into the conventional
regime, the subluminal world of our experience, took place when particle
energies were close to $L^{-1}$ (see Figure 1 and recall again that what
decreases monotonically with time as the universe evolves is not energy but
the boost parameter) and therefore in the very remote past about which we know
little, the answer to the question has to be that cosmology as we know it
today is definitely not in conflict with the above picture. Indeed, any
possibility of identifying superluminal expansion originating in discreteness
is likely to be subsumed in the by-now standard – though still mysterious –
cosmic inflation. The two differ in their origins. The superluminal
propagation arises here kinematically and in a purely special relativistic
context – gravity plays no role – whereas the root cause of inflation remains
unsettled despite the enormous amount of work, mixing the usual quantum field
theory ideas with general relativity, that has gone into it. The details of
the process are also different: episodic with a long series of superluminal
phases in our case and one big blowout in inflation. One may then ask: if the
world is really discrete, can one do without inflation in its current
formulation(s)? In partial answer, let us note the encouraging fact that the
energy of a massive particle vanishes – the torus mass shell intersects the
cube that is the subspace $P_{0}=0$ of the Brillouin zone (the horizontal axis
in Figure 1 is its projection) – in a repeated series of events until it makes
its final transit into the conventional regime. Such events occur during
highly superluminal phases: when $P_{0}=0$, the velocity is infinite.212121The
sequential superluminal expansion implied by discreteness and advocated here
has nothing conceptually in common with models, some of them very explicit (an
early example is [21]), which postulate that the speed of light itself has
changed enormously over cosmic time scales, cause unknown. In the discrete
universe the agency of homogenisation is not radiation (massless particles
have an absolutely constant, conventional, speed) but massive matter. I add
that cosmological issues related to inflation continue to see an enormous
amount of activity, as regards both its conceptual moorings and their
empirical validation; for a thorough review of the current situation, see
[22].

Aside from global consequences like superluminal expansion, the lack of strict
energy-momentum conservation – conservation only modulo $2\pi/L$ – also leads
directly to a class of Planck-scale Umklapp processes, namely interactions of
individual elementary particles in which some of them in the initial or/and
final states would have energies or momenta or both outside the Brillouin zone
if they were strictly conserved, i.e., before enforcing the periodicity of
reciprocal space. Umklapp phenomena in crystal physics222222An excellent
general reference is the book of Peierls ([23]) who first recognised the
phenomenon and its origin in the discrete symmetry groups of crystals. concern
only the momenta and velocities of the relevant elementary excitations (mainly
phonons and electrons) – energy is always strictly conserved since time is not
discrete – and they influence chiefly the transport properties in crystalline
media. Moreover, the medium itself is not discrete – only the symmetry group
is – and is of finite size. Consequently, the effect of periodicity is just to
modulate the 1-particle wave function (Bloch’s theorem) and the apparent
violation of momentum conservation is only an artefact of translating all
momenta back to the Brillouin zone. Physically, the momentum of the total
system consisting of the elementary excitations and the crystal is conserved;
the crystal as a whole recoils to balance the loss or gain of momentum by the
crystalline equivalent of matter.

In contrast, when the ‘crystal’ is space-time itself, the violation of energy-
momentum conservation must be regarded as a real physical phenomenon: there is
no sensible physical meaning that can be given to the notion of Minkowskian
space-time having kinematical attributes like energy and momentum.
Nevertheless, since very few processes in solids are measurably sensitive to
crystal recoil,232323Exceptions would be resonance absorption of photons and
neutrinos. we can adapt the methodology of crystal physics – extending them to
include energy in addition to momentum – for a first orientation on recoilless
space-time Umklapp interactions. An interaction in which the initial state has
particles each of which is in its conventional regime, with momenta
$\\{P_{\mu}^{(i)}\\}$, can result in two general classes of final states:
either i) all of the particles in the final state, with momenta
$\\{P_{\mu}^{{}^{\prime}(j)}\\}$, are in their conventional regimes – normal
processes in the language of crystal physics; or ii) some of the particles
will go outside the Brillouin zone $B$ if energy and momentum were strictly
conserved, $\sum_{i}P_{\mu}^{(i)}=\sum_{j}P_{\mu}^{{}^{\prime}(j)}$, which
have then to be translated back to $B$ – Umklapp processes. The general
conservation law, always valid, is

$\sum_{i}P_{\mu}^{(i)}-\sum_{j}P_{\mu}^{{}^{\prime}(j)}=\frac{2\pi
N_{\mu}}{L}$

for some $N_{\mu}\in\bf Z$, depending on the values of $P_{\mu}^{(i)}$ and
$P_{\mu}^{{}^{\prime}(j)}$.

As the simplest possible example of the new kinematics, consider first the
decay at rest of a particle of rest mass $M$ into two particles of equal rest
mass $m$. Assuming strict momentum conservation, the daughter particles will
have momenta in opposite directions, say along the 1-axis, of equal magnitude
$k$. Assuming also strict energy conservation, we have $4(k^{2}+m^{2})=M^{2}$
or $k=\sqrt{M^{2}/4-m^{2}}$. From the reality of $k$ (for the decay to be
kinematically possible), $m<M/2$. And since $M<\pi/L$ (from the definition of
the rest mass), it follows further that $k<M/2<\pi/2L$. No energy or momentum
is outside the Brillouin zone and the decay is a normal process. No surprise
here.

But this apparently trivial example becomes interesting when we consider its
inverse reaction: the collision of two equal rest mass particles to produce a
final state whose centre-of-mass is at rest, including the possibility of a
single particle at rest. Assuming strict conservation, the total energy of the
final state, $E=2\sqrt{k^{2}+m^{2}}$ (which is also its invariant mass), will
exceed the Brillouin zone bound $\pi/L$ if either $k>\pi/2L$ or $m>\pi/2L$ (or
both). For such values of $k$ and $m$ strict conservation cannot hold: an
Umklapp shift down has to be made (one shift is enough since $k$ and $m$ are
bounded above by $\pi/L$), resulting in a final state energy of
$2\sqrt{k^{2}+m^{2}}-2\pi/L$.

The example can be extended to more general kinematic situations in a
straightforward manner though the energy-momentum book-keeping can get quite
involved. The inescapable fact is that dramatic degradations of energy and
momentum in interactions of particles of Planck scale masses and energies are
a generic consequence of discrete special relativity. Once again, the only
realistic possibilities of testing the effect would seem to involve the
physics of the very early cosmos but the fact that the shift can result in
relatively moderate final energies gives some hope. Perhaps ultra high energy
cosmic rays (with energies some 8 or 9 orders of magnitude smaller than $1/L$)
are the final result of such degradation, tantalising relics from the early
history of the universe, rather than the product of some unknown cosmic
mechanism involving extreme acceleration.

While on the subject of possible future lines of work, we should also note
that the breakdown of conventional energy momentum conservation – but in a
precisely quantifiable way – has other consequence which are relevant to the
study of the early universe, specifically in its hot dense phase. In general
terms, what is required is a reformulation of thermodynamics in extreme
conditions in which the kinematics of binary collisions is governed by the new
conservation laws. That is a well-posed problem but, obviously, a very
challenging one. We can surely expect a reliable theory incorporating them to
lead to deviations from currently popular cosmological models of the young hot
dense universe. Whether they will turn out to be theoretically and
observationally acceptable is for the future to decide. What is generally
gratifying in the meantime is that the knowledge we already have accommodates
the effects of discreteness in the structure of space-time quite comfortably
without any extra assumptions. Theoretically, the deep connection between the
existence and properties of elementary particles and the symmetries of space-
time survives intact and, observationally, none of our hard-won insights into
the mechanisms of the physical world is put at risk. One can then hope that
the hypothesis of a fundamental length may, through the novel kinematics it
entails, provide an opening into some of the more ad hoc features of current
cosmological theory and into fundamentally kinematic phenomena for which there
is no convincing basis yet (such as, obviously, dark matter and dark energy).
To conclude, it is worth repeating that these novel effects do not invoke the
curvature of space-time in any way; the consequences of the gravitational
aspects of general relativity will be above and beyond them.

It seems fitting to end with Riemann’s much-cited exhortation on the need to
respect both experience and logic in studying the nature of space (in his
famous Habilitation lecture “On the Hypotheses which Lie at the Foundation of
Geometry”, 1854):

> . . . it is a necessary consequence that . . . those properties which
> distinguish space from other conceivable triply extended (3-dimensional)
> quantities can only be deduced from experience. Thus arises the problem of
> seeking out the simplest data from which the metric relations of space can
> be determined . . . These data, like all data, are not logically necessary,
> they are hypotheses; one can therefore investigate their likelihood, . . .
> and afterwards decide on the legitimacy of extending them beyond the bounds
> of observation, both in the direction of the immeasurably large and (my
> italics) in the direction of the immeasurably small . . .

I acknowledge with warm thanks the invaluable advice of M. S. Narasimhan, M.
S. Raghunathan, Parameswaran Sankaran and T. N. Venkataramana on the
mathematical aspects of this work, first addressed, in a different context, in
an unpublished manuscript ([24]). On questions related to the current state of
knowledge of early cosmology, especially of the role of inflation, I have
benefited greatly from the explanations of Swagat Mishra. For that, and for
his help with Figure 1, I express my gratitude. Finally, I acknowledge here
the hospitality of the Inter-University Centre for Astronomy and Astrophysics
(Pune) over the years that helped bring this work to a (tentative) close.

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