UNDER CONSTRUCTION

Causal Sets and Quantum Gravity

Our current understanding of physical reality rests, at the most
fundamental level, on two very different theories, one known as
``general relativity'' (the theory of gravity and spacetime structure)
and the other as ``the standard model'' (the theory of subatomic
particles and fields).  Together these two theories cover a vast range
of phenomena, from sub-nuclear particles to the cosmos on the largest
scales observable so far.  However, each of these theories is incomplete
and each is formulated in such a manner as to make it incompatible with
the other.  Moreover the fact that the expansion of the universe is
speeding up is interpreted by some workers as evidence that something is
wrong with our understanding of gravity (which normally should be
attractive), even in the domain where it was thought to be well
established.  

Not having resolved these difficulties, we find ourselves unable to
explain the big bang, or discover the reason for the accelerating
expansion of the cosmos, or say what happens deep inside a black hole.
The long sought theory that would unify our understanding of nature, and
thereby let us answer some of these questions, is known as ``quantum
gravity''.  This problem has been my primary interest in physics ever
since my student days.

In considering how to approach this problem, it is important to realize
that current theories are not only incomplete, they contradict
themselves if one takes their basic assumptions to their logical
conclusions.  These contradictions are known as singularities and
infinities.  If one accepts, as very many workers do, that these
failures are telling us that a discrete substratum must underlie the
continuous manifold of classical spacetime, then the first task is to
determine what sort of mathematical object that substratum is.

After exploring several possibilities, I now believe that the correct
mathematical structure is a {\it causal set}, a kind of ``family tree''
for the ``spacetime atoms'' composing the substratum.  More precisely, a
causal set is a {\it locally finite partially ordered set} whose order
relation represents a fundamental temporal connection between its
elements.  The ultimate goal of this approach is to construct a theory
of quantum gravity based on the replacement of spacetime by a causal
set.  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: Luca Bombelli, Alan
Daughton, Joohan Lee, David Meyer, David Reid, Rob Salgado.