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OVERVIEW OF MY WORK

More than anything else, my work in physics has been guided by a desire
to overcome the disunity which characterizes our current conception of
nature and which shows itself first of all in our failure to have
reconciled our best theory of spacetime structure (general relativity)
with the best dynamical framework we know of for describing the behavior
of matter on small scales (quantum field theory).  Ever since I
encountered it in graduate school, the greater part of my efforts has
been devoted directly or indirectly to this problem, which can also be
viewed as the task of completing the twin revolutions initiated early in
the last century in connection with the study of the very small (atoms),
the very fast (light), and the very big (astronomy).

In this direction, I first explored the idea ("Regge Calculus") that
spacetime can be represented as a simplicial complex, an essentially
topological notion which today provides the basis for the causal
dynamical triangulation and spin-foam approaches to quantum gravity.
Regarded not just as a method for artificially discretizing the
continuum (in effect a finite-differencing of the Einstein equations),
but as a hypothesis about the inner basis of spacetime, this idea fit in
well with the expectation that the continuum is only an effective
description of some more fundamental -- discrete -- structure.  However,
after some work, I came to feel that if topology was to be the ultimate
source of spatio-temporal geometry, then a more flexible and natural
candidate than the simplicial complex was the finite topological space.
Such a space can also be regarded, by a kind of duality, as a partially
ordered set or "poset", and this links the idea of topology to that of
order.  But in the end the poset-type of topology seemed no more
satisfactory as a substitute for spacetime than the simplicial complex
had been.  At best it could produce a metric-space with all distances
positive, but not the kind of Lorentzian geometry which characterizes
the spacetime of general relativity.  Understood, however, not
topologically but temporally or "causally", an order-relation does
contain the seeds of Lorentzian geometry, and this led me, finally, to
the idea of the causal set as the deep structure of spacetime.  At
present, this is the road I am taking, and a separate web page describes
causal set theory in more depth, including its recent great progress and
current prospects.

Although I no longer believe that spatio-temporal topology enjoys a
fundamental status, that does not mean that topological questions are
unimportant for quantum gravity, and exploring the novel topological
possibilities inherent in curved spacetime has always been an important
strand in my work.  Emerging from a fundamental discreteness, these
possibilities will first show up on those length-scales -- possibly
still close to Planckian -- where the spacetime manifold itself begins
to make sense.  And here spacetime topology can play an important role,
first of all because it possesses kinematic resources sufficient to
reproduce all the degrees of freedom of the standard model.  For
example, extra spatial dimensions could give rise to gauge fields in the
"Kaluza-Klein" manner, while topological geons (quasi-local excitations
of the spatial topology) could play the role of quarks, at least as far
as kinematics is concerned.  In particular, geons can exhibit fermionic
statistics purely for topological reasons, and without the need for any
fundamental fermionic fields.  (In this way, the causal set in its role
as the source of an emergent spacetime topology could in principle be
all there is.)

Second of all, the study of spacetime topology, and especially of its
temporal variation, seems to me to have furnished important clues to the
nature of quantum gravity in general.  For example, it led me to
conclude that one can expect the spin-statistics correlation to persist
for geons only if quantum gravity provides for transitions from one
topology to another.  In the contrary case of frozen spatial topology,
geons (including Kaluza-Klein monopoles) could not be pair-produced, and
with the loss of this process one would lose the source from which,
plausibly, the correlation between spin and statistics springs in
general.  Recovery of spin-statistics for geons thus points at the
necessity for topology change, and in the context of the gravitational
path integral, it turns out to place definite constraints on the
amplitudes associated with the cobordisms involved in pair-creation.  No
less importantly, the need for topology-change reinforces the belief
that a path-integral formulation of quantum mechanics is the only one
that can meet the needs of quantum gravity, an encouraging conclusion
from the point of view of causal sets, which seem to admit no other form
of quantal dynamics.

[mention skyrmions here?]
[anyons?]

Along with topology change, black hole entropy has long interested me as
a potential clue to the nature of quantum gravity.  When I first learned
that a black hole carries entropy and that (up to a coefficient of
proportionality of order unity) this entropy equals the surface-area of
the black hole in Planck units, the thought that suggested itself almost
immediately was that this was telling us something important about the
micro-structure of spacetime because the most natural reading of such an
area law is that the horizon is somehow carrying about one bit of
information per unit (Planck) area; and this in turn is most easily
explained if the deep structure of spacetime is discrete so that the
horizon is actually composed of large but still finite number of
"molecules" of Planckian size.  In relation to causal sets, the lesson
is that the fundamental discreteness scale must be near to 1e-32 cm,
consistent with many other indications that sizes shorter than this lack
physical meaning.  (The subsequently popular idea of "holography" must
have sprung from a train of thought similar to mine, but it has led in
different directions and does not seem to have shed much light on the
structure of the horizon, or on how the entropy can be "located there".)
A correct counting of horizon "degrees of freedom" must of course be
conducted quantum mechanically.  To that end, I computed what is now
called the entanglement entropy across the horizon, regarded as a
surface separating the outside of the black hole from the inside.  The
calculation did indeed yield an entropy proportional to the area, or
rather an entropy given by the horizon area in units of the cutoff that
one needed to render finite an otherwise divergent answer.  This result,
together with others of a similar nature, furnishes to my way of
thinking the single best argument in favor of discreteness, and possibly
the most direct indication of where the discreteness-scale lies.  If,
for example, the kinematic counting of "horizon molecules" in causal set
theory were to find a dynamical justification, one would be able to
derive the coefficient of proportionality relating the nominal Planck
length of 0.81019e-32 cm to the true discreteness scale.

[mention fractal horizon? entropy as shapes?]

[mention cosmol cst prediction?]
[unimodular gravity?]

The last strand of my work that I will sketch here concerns the
path-integral and its interpretation.  Or maybe it would be better to
say that it concerns the possibility of re-formulating quantum mechanics
entirely as a theory of quantal histories, without ever needing to call
on state vectors, measurements, or external agents as fundamental
notions.  

But what purpose would such a re-formulation serve?  In what way would
it aid the quest for quantum gravity?  Part of the answer, as already
alluded to above, is that quantum gravity seems to resist the
Hamiltonian framework of "canonical quantization", and indeed any other
dynamical framework which is not based on histories.  This reflects
first of all the so-called problem of time and the need to deal with
diffeomorphism-invariant quantities.  But when we add to the mix not
only spatial but temporal discreteness, the contradiction with the
Hamiltonian as a generator of continuous time-evolution becomes severe.
And when we add topology change -- or do away with the continuum
altogether -- any thought of quantum gravity as a theory of
operator-valued fields on a fixed manifold also fades out.  A second
part of the answer might not appear to concern gravity at all; rather it
harks back directly to the question of the unity of physics.  The lack
of a theory of quantum gravity is an important symptom of disunity in
our present understanding, but it is not the only symptom one can point
to.  Another notorious instance is the "cut" which quantum mechanics
supposedly forces on us, that severs the macroscopic world of classical
physics from the microscopic world of quantal objects and processes.
(Some authors have even taken this cut so seriously as to deny reality
to the micro-world altogether.)  But is this issue of the quantal cut as
remote from quantum gravity as it might appear to be at first sight?
Insofar as the contradiction, quantum vs. gravity, reduces to the
opposition micro vs. macro, we are dealing with something rather similar
to the quantal cut; hence one might suspect that the two problems are in
fact closely related.  Even if such suspicions are mistaken, though, the
practical need for a histories formulation remains, and further reasons
to desire a more realistic re-formulation of quantum mechanics crop up
when one tries to generalize to the quantal case the condition of "Bell
Causality" that figures so heavily in connection with classical models
of a "sequential growth dynamics" for causal sets.

How then does the path-integral provide an alternative to the textbook
formalism of state-vectors, Hamiltonians, and external observers?  The
short answer is that from the path integral one can derive a functional
mu_quantum -- the quantal measure -- which directly furnishes the
probability of any desired "instrument-event" E.  (The quantal measure
is essentially equivalent to the so called decoherence functional.)  In
saying this, I am presupposing that the Born rule (or rule of thumb!) is
correct, and then just taking note of the fact that the Bornian
probabilities for any specified set of "pointer readings" are furnished
directly by mu_quantum, without any appeal to Schroedinger evolution of
the wave-function or its "collapse" during measurement.  In this way
mu_quantum is exactly analogous to the probability measure mu_classical
that furnishes the classical probability of a set of histories -- an
"event" -- in the case of a purely classical stochastic process like
diffusion or Brownian motion.  If one construes the path-integral in
this way, namely as a generalized measure on a space of "classical
histories", then one sees not only how quantal processes differ from
classical stochastic processes, but also how closely the two resemble
each other, the primary difference being simply that mu_classical and
mu_quantum satisfy different sum-rules.  The former obeys a "2-slit"
sum-rule that expresses the absence of interference between alternative
histories, the latter obeys a weaker, "3-slit" sum-rule that expresses
the merely pairwise nature of this interference.  (Recently this
latter sum-rule, which derives from the quadratic nature of Bornian
probabilities, has been tested directly in a literal 3-slit experiment
with individual photons.)

The formal framework I have just sketched rests mathematically on a
space of histories and a notion of integration thereon that allows one
to compute the quantal measure mu(A) for any (sufficiently regular)
subset of the history-space.  Satisfactory as this framework is in some
ways, the interpretation of mu(A) as a probability still does not take
us beyond the realm of "pointer events" that occur in laboratory
instruments or other macroscopic objects.  But the laboratory, or even
the observatory, is not the place where we expect to encounter quantum
gravity. There are no laboratories deep inside black holes or in the
very early universe.  Not instrument-events, but events like the "big
bang" are the natural domain of quantum gravity, and one would like to
be able to reason about these events in direct physical terms.  When one
ventures into such realms, however, the physical significance of the
quantal measure becomes uncertain: the Born rule loses its force and,
because of interference, one can no longer interpret mu(A) as a
probability in any ordinary sense.  The question then is whether mu
admits of a broader interpretation that remains serviceable "outside of
Copenhagen".  (If it does not, then perhaps an entirely different type
of dynamical framework will be called for.)

My current belief is that the quantal-measure (meaning in effect the
path-integral) does have a microscopic meaning, not as a probability per
se, but as an arbiter of whether a given event can occur at all.  In
other words, we concede that an event A of vanishing measure mu(A) will
never occur.  Drawing out the implications of this "preclusion
principle", one soon finds that it conflicts with the classical
conception of reality as a single history (a single point in the
"history space" over which the path-integral integrates).  Of course,
such a contradiction with classical conceptions of reality is just what
one might have expected, but an escape route also opens up.  It turns
out that -- at least so far -- the preclusion principle does not seem to
conflict with a modified conception of reality that replaces the single
classical history with a set of one or more histories.  Such a set is a
special case of what is called an "anhomomorphic coevent", and the new
point of view is that reality is best described by such a coevent.  (In
the resulting re-formulation of quantum theory, it is most natural to
reason about events using rules of inference that differ from those of
classical logic.  One could even regard this as the essence of the new
formulation, but one should not confuse these modified inference-rules
with what has previously been called "quantum logic".)  We thus acquire
an interpretation of the quantal measure in conjunction with a
particular answer to the question, What is quantum mechanics telling us
about the nature of reality?  As a byproduct of this development, one
solves the "measurement problem", or more properly, one obtains a
solution of that problem if instrument-events can be proved to obey a
certain separability condition.  It remains to be seen whether this
condition can be established in sufficient generality.  It also remains
to be seen whether the new formulation will be be able to accomplish the
task for which it was ultimately intended, namely to provide a more
precise framework for thinking about quantum gravity, and thereby to
help clear away some of the obstacles standing between us and that
theory.

[mention QBC (Quantal Bell Causality) here or above?]