[Most references still to be added]
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 present conception of
nature, and which shows itself most obviously 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 addressing this "quantum gravity" problem, 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 an artificial finite-differencing
of the Einstein equations, but as a hypothesis about the inner basis of
spacetime, Regge Calculus 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 structure than the simplicial complex was
the finite topological space. By a kind of duality, such a space can
also be regarded as a partially ordered set or "poset", and this links
topology to the arguably more basic concept 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 special and
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 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 an underlying discreteness, potential topological
effects will first arise on those length-scales -- possibly still close
to Planckian -- where the spacetime manifold itself begins to make
sense. In this regime spacetime topology 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 serve as quarks,
at least as far as kinematics is concerned. In particular, geons can
exhibit fermionic statistics for purely topological reasons, without the
need to introduce any fundamental fermions into the story. In this way,
the causal set in its role as the source of an emergent spacetime
topology could in principle be all there is.
Beyond the potential to reproduce well established types of fields and
particles, spacetime topology might also give rise to novel behaviours
not so far observed or even theorized in other contexts, for example
certain types of exotic statistics. But the study of spacetime
topology, and especially of its temporal variation, also seems to me to
have furnished important clues to the nature of quantum gravity in
general. For example, my work with geons led me to believe that the
spin-statistics correlation will break down unless 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 that 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.
In this connection, I should also mention Skyrmions and anyons, two
other instances of topological phenomena I have worked on. In the case
of Skyrmions, the topology in question is that of the group SU2 or SU3,
regarded as the "target manifold" of a meson field. The Skyrmion itself
is a topological excitation of the vacuum to be identified with the
nucleon or one of the other baryons, and fermionic statistics arises in
close analogy to how it does for topological geons in quantum gravity.
Even dynamically, the analogy with geons remains close, with the
exception that (thanks to the absence of gauge-issues) one can actually
compute the energy-spectrum and understand how the uncertainty principle
can stabilize the Skyrmion against collapse. The resulting
zero-parameter fit to the low lying baryon masses turned out to be
surprisingly accurate.
Thinking about geon statistics also led me to try to analyze the meaning
of particle-statistics in general, as a kind of nonlocal interaction
among particles which is expressed topologically. As a byproduct I
noticed the possibility of so-called anyons, which actually had been
pointed out a few years earlier by Leinaas and Myrheim.
Along with topology and topology-change, black hole entropy has long
interested me as another 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 magnitude. 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 spatio-temporal discreteness, and the most direct indication so
far 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 true discreteness scale to
the nominal Planck length of 0.81019e-32 cm.
[mention fractal horizon, entropy as shapes]
[mention Lambda prediction, or refer to accompanying web page on causets]
[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, and would it aid in
the quest for quantum gravity? Part of the answer which I've already
alluded to above is that quantum gravity seems to resist the Hamiltonian
framework of "canonical quantization". This statement reflects first of
all the so-called problem of time and the need to deal with
diffeomorphism-invariant quantities, but when, as with causets, we add
to the mix a discreteness which is not only spatial but temporal, the
contradiction with the Hamiltonian as a generator of continuous
time-evolution becomes severe. And when we allow for 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 harks back directly to the question of the
unity of physics. The lack of a theory of quantum gravity is an obvious
symptom of the 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 taken this cut so
seriously as to deny reality to the micro-world altogether.) This issue
of the quantal cut might appear to be remote from quantum gravity, but
insofar as "quantum vs. gravity" reduces to "micro vs. macro", we are
dealing with something rather similar in both situations; 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 "sequential
growth" for causal sets.
How then does the path-integral offer an alternative to the textbook
formalism of state-vectors, Hamiltonians, and external observers? A
first 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. (This measure is
closely related 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 the measurement. In this way
mu_quantum is analogous to the classical measure mu_classical that
furnishes the 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 "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 absence of "higher order"
interference beyond pairwise. (This 3-slit sum-rule, which reflects the
quadratic nature of Bornian probabilities, has now 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 the
city limits 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 direct microscopic significance, 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 would 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 these modified rules as
the essence of the new formulation, but one should not confuse them 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?]