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\centerline{\bf RESEARCH INTERESTS \& GUIDE TO PUBLICATIONS}
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\centerline{\it Rafael D. Sorkin}
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My interests in physics are broad, and my work has addressed a large range
of individual problems, touching General Relativity, Particle Theory,
Astrophysics, Cosmology, Condensed Matter Theory, Information
Theory/Statistics, Computational Physics, the Foundations of Quantum
Mechanics and others.  But this variety has not been an unfocused one.
Rather it has been dictated by my training in particle theory and general
relativity, and my consequent desire to help clarify and resolve what have
seemed to me to be the fundamental questions within those two disciplines.
In the following essay, I try to show how the greater part of my work has
in fact been shaped by concern with a single such question, that of
``quantum gravity''.

Although physics in this century has discovered new forms of matter
(electrons, nucleons, the metric field, quarks, neutrinos, $\ldots$)
and new types of dynamical principle (quantum mechanics, quantum field
theory), it has not yet succeeded in attaining a unified conceptual
framework which would embrace the full range of these discoveries.  The
central obstruction to such a synthesis seems to be a contradiction between
the two partial syntheses which have been achieved; or more specifically an
incompatibility between the mathematical structures which underly General
Relativity on one hand, and the dynamical rules embodied in Quantum Field
Theory on the other hand.  Actually, neither of these theories is free from
its own internal contradictions (known as singularities or infinities), and
one may hope that the attempt to unite the two into what is usually
called a theory of quantum gravity would lead to an overcoming of their
internal difficulties as well.

Ever since graduate school, my work in physics has pursued this
perspective, and the problems I have addressed have primarily (though of
course not exclusively) been ones which I saw as bearing either directly or
indirectly on the larger question of quantum gravity.  A question as broad
as this one will in general benefit from exploration along more than one
direction, of course, and correspondingly my work is comprised of
complementary components.

First, one can attempt to clarify and deepen one's own understanding of
quantum mechanics and general relativity separately, paying special
attention to concepts which are likely to be important for the deeper
synthesis, and to new phenomena which are likely to bring the two theories'
essential structures into sharper focus.  Much of my work on the meaning of
conserved quantities in General Relativity (Papers 4, 5, 47, 66, 83 and 85
on the enclosed list; related: 6; 68, 72, 79) and on higher-dimensional
gravity (Papers 43 and 50) has been of this nature, as was, for example, my
work on the time-energy uncertainty principle in quantum mechanics (Paper
9).  For example, the work on conserved quantities obtained a deeper
understanding of the variations defining such quantities, as representing
physically a motion of an isolated subsystem with respect to an implicit
environment, and this in turn yielded a technically improved expression for
the conserved quantities themselves.

Second, one can try to reformulate, extend, or modify either quantum
mechanics or general relativity, so as to render it more compatible with
its ``opponent'', or even with general philosophical principles which one
may hold.  Most of my work on developing a sum-over-histories
interpretation for quantum theory (Papers 61, 63, 76, 80, 84, 89; related:
27, 74; see also the review 82) has been of this character [as also was
earlier unpublished work on formulating General Relativity directly in
terms of the curvature tensor, and work seeking a spinor field which could
exist independently of a metric field (Paper 13)].  The main recent fruit
of the former work is the discovery of a sum-rule which allows one to
interpret the sum-over-histories as a species of generalized measure
theory, and the consequent introduction of a ``preclusion criterion''
which, if it works out, will offer a realistic and objective interpretation
of quantum mechanics in its sum-over-histories formulation.

Third, and most straightforwardly, one may seek out physical situations in
which both quantum mechanics and general relativity will be simultaneously
relevant, hoping either to deepen the contradiction between them, or to
effect a partial union of the two theories.  In this way one can hope to
arrive at physical conclusions which will anticipate results of the
ultimate synthesis and furnish clues to help guide the formulation of the
deeper theory.  My work on black hole entropy (Papers 11, 18, 31, 32, 39,
40; related: 19, 21, 35, 83, 85) answers to both poles of this description,
since it on one hand brings out a new contradiction (a divergence in the
horizon entropy) strongly indicating the need for a fundamental spacetime
discreteness, and on the other hand it shows what features of the deeper
theory would allow one to recover the Second Law of Thermodynamics for
systems including black holes (the existence of an autonomously evolving
density matrix describing the region outside the black holes).  With the
recent interest in black hole thermodynamics among particle physicists, the
ideas of this work have become the basis of much current activity.

The greater part of my work on topological geons also belongs in this third
category since it deals with quantum properties of particle-like structures
built into the topology of spacetime; and the possibility of non-Euclidean
topology is one of the most characteristic innovations of General
Relativity.  In 5-dimensions one such topological structure which I
discovered is the Kaluza-Klein monopole, described in Papers 29, 30, 33,
50.  Back in 4-dimensions, some new phenomena which emerged were spin-$1/2$
geons (as states of pure quantum gravity) and, more generally, a new kind
of ``quantum multiplicity of states'' associated with non-abelian
representations of the diffeomorphism group (see Papers 15, 20, 28; 52, 55,
56; and related work in Papers 7, 10.)  Aside from these results, the most
important finding was that {\it violations} of the spin-statistics
correlation occur in a manner that suggests they will be cured only when
the new physical process of topology-change is properly taken into account
(cf. Paper 88).  Thus, the theory of geons yields the important insight
that quantum gravity must incorporate topology-changing processes in order
to be physically self-consistent.  (Reviews of some of this work are in
Papers 36 and 54 and 57.  Related matter is in Papers 12, 14, 42, 56.)

This led me to study possible mechanisms for topology-change, in order to
see whether any characteristic features would turn up, which could furnish
further ``phenomenological clues'' to the nature of the deeper theory.  In
fact, I found a selection rule which, in odd spacetime dimensions, forbids
the Euler characteristic to change via Lorentzian cobordism.  Since this
would preclude physically natural processes like pair production of
Kaluza-Klein monopoles, a natural conclusion to draw is that pure
Lorentzian cobordism (with its built-in causality violations) is not the
right mechanism for topology-change.  Instead, ``causal cobordism'', which
preserves causality at the slight cost of isolated zeroes in the metric,
seems promising.  (Papers 37, 38, 78, 95, 97; reviewed in: 57)

I also returned to the study of the spin-statistics correlation itself,
hoping to find topological aspects which could help bring out the suspected
role of topology-change in the case of geons.  In fact, a purely
topological spin-statistics theorem did emerge, which also has application
to a large range of extended objects such as ``anyons''. (Papers 23, 24;
25; 46; 60, 75, 70, 86; related: 69, 71.)

In parallel work I sought to understand energy level spectrum of the
Skyrmion (a topological structure representing the nucleon) as a first step
toward understanding the analogous but conceptually more complicated
quantum dynamics of geons.  Of course the problem of predicting the mass of
the nucleon was of interest in itself, independent of any implications for
quantum gravity!  The results were encouraging: the choice of a ``short
range'' profile-function for the Skyrmion, led to a remarkably good fit
(with no adjustable parameters) to the masses of three $N$ and three
$\Delta$ states.  This indicates that it may in fact be ``quantum
stabilization'' which prevents the Skyrmion from collapsing, rather than
any supplementary term in the chiral model Lagrangian.  (Papers 53, 59;
related: 46)

The issues of spin and statistics have seemed important to me for other
reasons as well.  If the sum-over-histories formulation of quantum theory
is the most satisfactory philosophically (and also the only one capable of
dealing with the key process of topology-change), it nevertheless has
trouble in handling spinorial or fermionic fields.  It therefore becomes
important to understand whether such non-geometrical fields need to be
fundamental inputs to the theory (in which case their existence would
belong in the category of clues to the deeper theory), or whether they can
realistically be derived from strictly geometrical fields.  This was
another motivation for the work recounted in the previous few paragraphs.

If the research discussed so far represents a ``top down'', approach which
proceeds gradually toward the deeper theory from what is already known,
then the complementary approach (followed by String Theory, for example) is
to ``dive downward'', guess the deeper theory, and proceed to work back
upward, trying to make contact with the clues established by the top-down
work.

This complementary approach characterizes the final set of my researches,
almost all of which have relied on a hypothesis of an (operational or
actual) spacetime ``{\it atomicity}'' as a way to overcome the infinities
referred to above.  The chosen structure evolved from the simplicial
complex of ``Regge calculus'' (Papers 1, 2, 3, 17, 81), to the finit(ary)
topological space (Papers 22, 62, 96), to the causal set (Papers 45, 48,
58, 64, 65, 77, 87, 92, 94), on which my present work centers.
[Although, for my own part, I have abandoned the simplicial and topological
models as fundamental discrete structures, they remain useful as
approximations in continuum theories, and my work on them is proving
valuable for current research on such approximation methods.]

According to the causal set hypothesis --- that a discrete set of elements
carrying an essentially causal order-relation is the microscopic
substructure underlying continuous spacetime --- the basic object of study
is a locally finite partially ordered set, and the usual inputs to General
Relativity are all derived structures, including the spacetime manifold,
its resident metric field, and perhaps all other physical fields as well.

A program to turn this idea into a theory of quantum gravity falls
naturally into two stages, a kinematic stage in which the intrinsic
properties of causal sets are understood better and their relation to
macroscopic spacetime properties clarified; and a dynamic stage in which a
``law of motion'' for causal sets is formulated and its consequences
deduced.  The bulk of the work done so far has belonged to the kinematic
stage, and has yielded strong evidence that the program is viable in the
sense that a discrete order does in fact suffice to reproduce the geometric
structures of general relativity.  In fact there are specific ways to
``read out'' information on dimensionality and on proper-time directly from
the order, and there is a growing body of computer simulations to confirm
that these techniques work.  A general insight which has emerged from this
work, is that the order-geometry correspondence is necessarily
probabilistic in nature (even at the kinematic level).

In the direction of dynamics, the most exciting finding at the moment
concerns the issue of locality, which is necessarily problematic in any
theory which incorporates both discreteness and local Lorentz invariance,
as causal set theory does.  Despite this, computer simulations now indicate
that it is possible to define (at least in 2- and 4-dimensional settings)
an Action for a scalar field on a background causal set which correctly
reproduces the continuum value, and which a fortiori is behaving locally.
Among other things, the resulting scheme offers an entirely new, and
Lorentz invariant type of ``spacetime lattice'' which could be useful for a
variety of purposes.  Most importantly, it provides an example which one
can hope to generalize to achieve an effectively local amplitude for the
causal set itself.  This would remove one of the two main stumbling blocks
in the way of defining a causal set dynamics which will correctly reproduce
general relativity in the classical limit.

Finally let me mention that the ``kinematical randomness'' alluded to above
seems to imply that the cosmological constant cannot be exactly zero.
Instead it must fluctuate in magnitude within a range which at present has
the value of around $10^{-120}$ in natural units, or in other words
precisely the magnitude necessary for $\Lambda$ to exert a cosmological
effect comparable to that of ordinary matter.


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