UNDER CONSTRUCTION

Causal Sets and Quantum Gravity

Our current understanding of the physical world rests, at its most
fundamental level, on two very different conceptual structures known
respectively as ``the standard model'' and ``general relativity''.  The
former deals with subatomic particles and fields in their mutual
interactions, while the latter deals with gravity, inertia, and the
geometrical and causal properties of spacetime.  Together these two
theories cover a vast range of phenomena, extending from the
cosmological to the sub-nuclear.  However, each one of them is
incomplete in itself, and each is formulated in such a manner as to make
it incompatible with the other.  In addition, the evidence that the
expansion of the universe is speeding up rather than slowing down has
led some people to surmise that our understanding of gravity (which
normally behaves attractively) is defective in itself, even in the
domain where it was thought to be well established.

The so called problem of quantum gravity, then, is the problem of
overcoming this disunity in our most fundamental conceptions.  One might
supreficially think of it as the quest to develop a single theory
uniting general relativity with the standard model, but this would be
misleading because the standard model incorporates features that are
expected to become outmoded in a few years, as has already happened with
the massless neutrino.  What is really wanted is not a theory that would
reproduce every detail of the standard model.  Rather our task is to
discover how to combine the abstract dynamical framework on which the
standard model rests, namely quantum field theory, with the kind of
flexible spacetime structure epitomized by general relativity.  Not
having accomplished this task, we find ourselves unable to discover the
origin of the big bang, unable to interpret the accelerating Hubble
expansion of the cosmos, and unable to sort out what happens deep inside
a black hole.  It's not just that we cannot answer these questions, but
that we don't even know how to ask them properly.

In thinking about quantum gravity, it is important to realize that our
best fundamental theories are not only incomplete and difficult to
combine with each other, they also suffer from internal contradictions.
When one tries to follow the basic assumptions of the standard model to
their logical conclusions, one runs into contradictions known as
infinities or divergences.  Likewise, the Einstein equations of general
relativity give rise to singularities where the curvature becomes
infinite and spacetime cannot continue.  And yet another infinity shows
up when one tries to compute the entanglement entropy of a black hole by
combining general relativity with quantum field theory in the most
straightforward way.

If one accepts --- as very many workers do --- that these contradictions
are evidence for a discrete substratum underlying the continuous
manifold of classical spacetime, then our first task is to determine
what sort of beast that substratum is.  Following a train of thought
reviewed in the accompanying web page, I have come to believe that the
appropriate structure to fill this role is what in mathematics is called
a ``partial order'', and what in its quantum-gravitational role I will
call a causal set.

\section{The causal set programme}
%
Among the various ideas put forward in the search for a theory of
quantum gravity, the causal set hypothesis is distinguished by its
logical simplicity and by the fact that it incorporates the assumption
of an underlying spacetime discreteness organically and from the very
beginning.  The approach is conservative in that it retains to some
degree the concepts of spacetime point or {element} and of
lightcone.  It is radical in that it takes these elements to be discrete
or ``atomic'' and in that it dispenses at the fundamental level with any
notion of metric or even topology.  Instead, the spacetime continuum,
including its metric and its topology, is supposed to emerge from the
underlying causal or temporal relationships in a manner summarized in the slogan
{Geometry = Order $+$ Number}.


But how can something as elaborate as a spacetime manifold arise from
something as mathematically rudimentary as a relation of precedence
between pairs of elements?  By way of answering, let me first give some
definitions.
Mathematically, a causal set (or {causet} for short) is a set C
of elements x, y, z ...  endowed with a  discrete order-relation
x < y which is both {transitive} and {irreflexive}, where
transitivity and irreflexivity mean respectively,

  (\forall x,y,z \in C) (x <  y <  z \implies x <  z)

and   (\forall x \in C) (x \not <  x). 

These axioms, if one thinks of it, are just the attributes of the
before-after relation or of the causal relation, ``x can influence y''.
Equally they are attributes of the relation, ``x is an ancestor of y''.
Thus a causet is a sort of generalized family tree recording the
temporal or ``ancestral'' relations among the ``spacetime atoms''
composing the hypothesized substratum.
A causet can also be represented as a bit-matrix M (the ``causal
matrix''), with entries 0 or 1 defined so that M_{jk}=1 if and
only if the i^{th} element of C precedes the j^{th} element of C.
That C is discrete, or technically speaking ``locally finite'', simply
means that the elements y falling {between} any two elements x and
z are finite in number:

(\forall x, z \in C) (\card{y \in C | x <  y <  z} < \infty),

where `\card' denotes cardinality.
Presupposing a suitable cosmological intial condition, we can replace
local finiteness by the simpler condition of past-finiteness, meaning
that no element has an infinite number of ancestors.  In that case one
can define a causal set 
simply as a {past-finite partially ordered set}.



A causal set is thus a certain kind of network of primitive temporal
connections or relations of ancestry.  Roughly speaking, the reason why,
under coarse-graining, such a structure can give rise to a spacetime
(for example Minkowski space), is that the order-relation x < {y}
contains the same type of information that in spacetime defines the
lightcones: information about which elements causally precede which
other elements.  But once you know the lightcones, you also know the
lightlike vectors, meaning the vectors of zero proper length, and it's
not hard to see that from this knowledge you can deduce the
line-element, dx^2+dy^2+dz^2-dt^2, up to an overall factor.  An
absolute standard of length cannot be recovered from the lightcones, but
in the context of a {discrete} order, a measure of volume (and
thereby of length) can be obtained in another way --- by equating the
volume of a spacetime region to the number of elements in the
corresponding portion of the underlying causet.  This rule that
{\it{}number = volume} furnishes the missing standard of length.


The above sketch of how a causet can produce a spacetime is only
heuristic, but the picture it paints is borne out by
extensive work in both physics and mathematics.  This work has shown,
for example, that in any causet which can be approximated by a region of
Minkowski space, the length of the longest chain
x < {}y_1 < {}y_2\cdots < {}z 
between any two elements x and z furnishes a reliable measure of the
proper time (geodesic length) between the corresponding spacetime points
[R::longest-chain].  Similarly, one knows how to ``read out'' from the
causal matrix information about the spacetime dimensionality
[R::myrheim-meyer] and about topological parameters like the homology
groups of a Cauchy surface [R::homology].


The causal set programme has as its ultimate goal, to construct a theory
of quantum gravity based on the replacement of the spacetime continuum
by the causal set.  For this it is not enough that a causal set
{\it{can}} give rise to a spacetime; it is necessary that it actually
evolve to do so.  Whether or not this happens depends by definition on
the underlying ``quantum law of motion'' or dynamics.  Finding a
dynamics which can produce the kind of spacetime we live in is thus the
central task of the theory, just as it is the central task of any other
theory of quantum gravity that posits a more fundamental structure lying
beneath spacetime.

According to the theory as it stands at present, the dynamics would be
that of an ongoing growth process consituted by ``successive'' births of
causal set elements.


[here describe csg models or refer to below]
[also monte-carlo simulations]

In addition to reproducing a spacetime one also wants to understand in
what way the granular, atomic nature of the causal set manifests itself
in phenomena that might sooner or later be susceptible to observation.
In other words, one wants to develop the potential {phenomenology}
of causal set theory.  For this purpose it should suffice, at least
initially, to treat the causet as a fixed ``background'' in which to do
quantum field theory, or rather its discrete counterpart.  Thanks to
recent progress in setting up quantum field theory in a causal set, this
is now possible in the case of a scalar field.  Such a field can serve
as a simplified stand-in for the electromagnetic field, and it is also
precisely the type of field hypothesized in inflationary cosmology.

[also need to mention swerves, propagation on links, etc]


TO BE CONTINUED



[FROM VERY OLD VERSION:]
I will mention just two or three illustrative problems to give an idea
of the kind of work involved.

1. Djamel Dou has shown that black hole entropy, in a certain sense,
counts the spacetime ``molecules'' composing the horizon (i.e. the
surface) of the black hole.  However, this conclusion relied on an
approximate reduction of the problem to two dimensions.  To confirm the
results, one must recover them in the full four dimensional setting.

2. A major success of causal set theory was the prediction of the
accelerating cosmic expansion, long before it was established
observationally. (More precisely, the prediction was a fluctuating
``cosmological term'' in the Einstein equations.  Such a term is the
most natural explanation of the recent observations.)  Together with my
student, Maqbool Ahmed, and Fermilab collaborators Scott Dodelson and
Patrick Greene, I have extended the original prediction to a much more
detailed phenomenological model which admits more extensive comparison
with observation.  This model needs further development, especially
since extensive new data is expected soon from the PLANCK satellite and
other similar experiments.

3.  A recent breakthrough on the problem of causal set dynamics was the
introduction of the family of ``classical sequential growth'' models,
devised (by David Rideout in his thesis) as a ``toy model'' of quantum
gravity, as it would have to emerge within the causal set approach.
This model has been fruitful.  It has led in particular to a solution of
``the problem of time'' (work with Graham Brightwell, Fay Dowker, Raquel
Garci'a, Joe Henson) and to the definition of a sort of ``cosmic
renormalization transformation'' that has the potential to resolve some
of the puzzles of big bang cosmology (work with Denjoe O'Connor and
Xavier Martin, also by Avner Ash and Patrick McDonald).  The next step
would be the generalization to the quantum case.  If such a
generalization can be formulated, then we will have, for the first time,
a fully fledged candidate for quantum gravity based on causal sets.



Other collaborators in the work described above but not cited there:
Siavash Aslanbeigi, Eitan Bachmat, Dionigi Benincasa, Luca Bombelli,
Michel Buck, Adriana Criscuolo, Alan Daughton, Astrid Eichorn, Lisa
Glaser, Steven Johnston, Joohan Lee, Malwina Luczak, David Meyer, Johan
Noldus, Prakash Panangaden, Ioannis Raptis, David Reid, Rob Salgado,
Mehdi Saravani, Sumati Surya, Madhavan Varadarajan, Henri Waelbroeck,
Petros Wallden.