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%----------------------------TITLE-------------------------------------

\rightline {SU-GP-93-12-2 (DRAFT)}

\bigskip

\centerline{{\bf FORKS IN THE ROAD, ON THE WAY TO QUANTUM GRAVITY}\footnote{*}
{{\noindent\parindent=0pt\baselineskip=12pt
This paper is a slightly expanded version of a talk given at the
symposium on {\it Directions in General Relativity}, held at the University
of Maryland, College Park in May, 1993, in honor of Dieter Brill and
Charles Misner.  Aside from the addition of selected references, the main
changes are a more precise wording of some of the remarks near the end of
section IV, and a footnote updating the results reported in Section VI.}}}
%-------------------AUTHOR--and--FIRST ADDRESS------------------------------
\bigskip
\baselineskip=12pt  % (just for address)
\centerline {\it Rafael D. Sorkin}
\smallskip
\centerline {\it Department of Physics}
\centerline {\it Syracuse University}
\centerline {\it Syracuse, NY 13244-1130}
\smallskip
\centerline {\it internet address: rdsorkin@mailbox.syr.edu}
%--------------------------ABSTRACT---------------------------------------
\bigskip
\centerline{\it Abstract}
\smallskip
\leftskip=1.5cm\rightskip=1.5cm \singlespace
In seeking to arrive at a theory of ``quantum gravity'', one faces
several choices among alternative approaches.  I list some of these
``forks in the road'' and offer reasons for taking one alternative
over the other.  In particular, I advocate the following: the
sum-over-histories framework for quantum dynamics over the
``observable and state-vector'' framework; relative probabilities over
absolute ones; spacetime over space as the gravitational ``substance''
(4 over 3+1); a Lorentzian metric over a Riemannian
(``Euclidean'') one; a dynamical topology over an absolute one;
degenerate metrics over closed timelike curves to mediate
topology-change; ``unimodular gravity'' over the unrestricted
functional integral; and taking a discrete underlying structure (the
causal set) rather than the differentiable manifold as the basis of
the theory.

In connection with these choices, I also mention some
results from unimodular quantum cosmology,
sketch an account of the origin of black hole entropy,
summarize an argument that the quantum mechanical measurement scheme
breaks down for quantum field theory,
and offer a reason why the cosmological constant of the
present epoch might have a magnitude of around $10^{-120}$ in natural
units.

\bigskip
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%------------------------------MAIN TEXT------------------------------------
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\bigskip\medskip

\noindent{\bf I. A laundry list of alternatives concerning Quantum Gravity}
\medskip\nobreak
The organizers have named this conference ``Directions in General
Relativity'', and in that spirit I want to talk in some generality about
directions in Quantum Gravity.  On the way to a theory of quantum gravity
there are many forks in the road, or in other words alternatives one must
choose among, or questions one must answer, before proceeding farther.  I
will begin by listing some of those alternatives and questions which seem
to me the most important, and then will try to advocate answers to several
of them in a manner that places them in an overall context.  I hope that
along the way, a coherent approach to quantum gravity will be seen to
emerge, even if I will not be able to give a definitive answer to every
question raised.

Also along the way, I will also mention a few new or lesser-known results
relevant to the alternatives we will be considering, including an
interpretation of black hole entropy and a possible ``non-unitarity''
associated with ``unimodular'' quantum cosmology.  But first the laundry
list itself (including a personal choice of one reference representing each
alternative\footnote{*}
%
{The references here and in the rest of this paper are meant to be
indicative only, they are not intended to be comprehensive or historically
representative in any manner.} )
%

%% Would it be better to underline rather than italicize the
% alternatives?  Also should we hang indent rather than use
% ``item''?

%% references to the listed alternatives are numbered [??3]-[??23]

\item{} Is the signature of the spacetime metric {\it Lorentzian} or {\it
Euclidean}?

\item{} Should we allow {\it degeneracies in the metric}  or {\it closed
timelike curves}  (or possibly both)?

\item{} Should we {\it fix the 4-volume} in the gravitational
Sum-over-histories or extend the  {\it sum over all 4-volumes},
as is normally assumed?

\item{} Is the deep structure of spacetime {\it discrete} or {\it continuous}?

\item{} Which feature of spacetime is most basic, its {\it causal
order}, its {\it metric}, or its {\it topology}?

\item{} Is topology {\it dynamical} or is it {\it absolute}?

\item{} Is the entropy of a black hole {\it outside} or {\it inside} its
horizon?

\item{} What  really exists, the {\it history} or the {\it wave-function}?

\item{}    Should we approach the ``quantization'' of gravity via the {\it
sum-over-histories} or via {\it canonical quantization}?

\item{}  Is probability  {\it absolute/unconditional} or
        {\it relative/conditional}?  (do quantum probabilities make sense?)

\item{} Is the cosmological constant, $\Lambda$, {\it approximately} or {\it
        exactly} zero?


\bigskip
{\noindent\bf II. What is {\underbar{Classical}} Gravity?}
\smallskip\nobreak

In order to begin placing these alternatives in context, it is useful to go
back to the classical theory we are trying to ``quantize''.  Like the
majority of physical theories, General Relativity has a threefold structure
comprising a ``kinematical (or substantial) part'', answering the question
``{\it What} is there---what `substance' are we dealing with?''; a
``dynamical part'', answering the question ``{\it How} does this substance
behave?'';  and a ``phenomenological part'', answering the question ``How
does this substance which is there {\it manifest itself} in a way
accessible to us?''.

In the case of General Relativity, the Kinematics comprises a
{\it differentiable manifold} $M$ of dimension four, a {\it Lorentzian
metric} $g_{ab}$ on $M$, and a structure which, although it is closely
intertwined with the metric, I want to regard as distinct, namely the
{\it causal order-relation} $\prec$.  The Dynamics is then simply the
Einstein equation $G_{ab} = T_{ab}$, or in case non-gravitational
matter is absent, the purely geometrical statement that the metric is
Ricci-flat.  Finally, the Kinematics and Dynamics manifest themselves
as the familiar Phenomena of length, time, inertia, gravity (in
the narrow sense of the word), causality (for example the impossibility
of signaling faster than light), etc.

% summary:
%     (Substance)
%      Kinematics               Dynamics            Phenomenology
%    --------------          -------------          -------------
%    $M$, $g_ab$, <          $G_ab$ =$T_ab$         lengths, times
%                                                   inertia, gravity,
%                                                   causality,...

 \bigskip
{\noindent\bf III. Why quantize? (And what does this mean?)}
\medskip\nobreak

According to classical General Relativity, the metric behaves
deterministically, but of course this is inconsistent with the stochastic,
quantum behavior of the matter to which the metric couples via the Einstein
equation.  Thus, a theory of quantum gravity in the broadest sense of those
words would just be some theory having both classical gravity and quantum
field theory in flat spacetime as limits (the latter being our best theory
of non-gravitational matter to date).  However, most of us who speak of
{\it quantum} gravity, I think mean something more specific than just this;
and so a major question whose answer defines one's approach to quantum
gravity is:
\item{$\bullet$}{ In what sense do we expect this theory to be {\it quantum}?}

\noindent
To this I would add two further basic questions:
\item{$\bullet$}{Do we need a new kinematics}
 (as well as the new dynamics which ``quantization'' entails)?; and
\item{$\bullet$}{What is the phenomenology of quantum gravity?}

\noindent
The rest of this talk is essentially an essay in answering these three
questions, beginning with the first of them.

\bigskip
{\noindent\bf IV. What is a quantum theory?  (two views)}
\medskip\nobreak

There are of course many different viewpoints on how quantum mechanics is
to be interpreted, but I will concentrate here on two broadly opposed
attitudes, which I will call the $\Psi$-framework and the
Sum-over-histories framework.

According to the former view the essence of quantum mechanics resides in
its mathematical structure: a {\it Hilbert space}, an {\it algebra of
operators} to be interpreted physically in terms of {\it measurements}; and
a ``{\it projection postulate}'', which tells us how to take the results of
measurements into account in predicting probabilities for future
measurements.  In this framework, the central object is the state-vector
$\Psi$ (which is why I am calling it the $\Psi$-framework), and the
physical interpretation is made in terms of observables.  (See almost any
textbook, for example [??24].)

Closely allied with the $\Psi$-framework is the {\it canonical
quantization} approach to quantum gravity.  Although different variants of
this approach may employ different combinations of the basic dynamical
variables, they all work solely with {\it space} (in the sense of a
spacelike hypersurface), as opposed to spacetime.  (For a review see
[??25], for alternative choices of canonical variables see [??26].)

>From the Sum-over-histories point of view, quantum mechanics is understood
quite differently, namely as {\it a modified stochastic dynamics
characterized by a non-classical probability-calculus in which alternatives
interfere.} To see the essence of quantum mechanics in this way goes back
at least to Heisenberg's Chicago lectures [??27], and of course is
associated most closely with the name of Feynman [??28].  Within this
framework, the spacetime {\it history} itself is the central object.  It
exists in the same sense in which a history is taken to exist in classical
physics, and the physical interpretation can thus be made directly in terms
of properties of this history---what John Bell [??29] called `beables' (a
word that I always thought was some kind of joke until I realized that he
meant it to be pronounced ``be-ables'')---rather than indirectly in terms
of ``observables''.  (See also [sukanya] for a statement of this view.)

Since the sum-over-histories is by nature a ``spacetime approach'', it
naturally leads to a version of quantum gravity which works with {\it
spacetime} as opposed to data on a hypersurface. [??31]

%% [[discuss their relation as in following?]]
%
%     States in Sum/hist arise as summaries of the past and  refer
%     to hypersurfaces.  Ndiyo maana Canonical Quantization goes with
%     Psi-framework, Spacetime approach goes with Sum/hist.


In comparing these two attitudes, I think it is fair to say that the
$\Psi$-framework is mathematically better developed (although this applies
less to quantum field theory than it does to quantum mechanics in the
narrow sense), whereas the sum-over-histories framework is, to my mind,
more satisfactory philosophically, because it avoids the positivistic
refusal to contemplate anything besides our generalized sense perceptions.

Now why, aside from its philosophical advantages, do I favor the
sum-over-histories/spacetime approach to quantum gravity over the
$\Psi$-framework/canonical quantization approach?  An important part of the
answer has to do with what has been called ``the problem of time'' [??32],
though the plural `problems' would probably be a more appropriate word in
this connection.

One such problem which affects the $\Psi$-framework concerns the temporal
meaning of the ``logical ordering'' required by the projection postulate.
In employing that postulate, one writes the projections in a definite
sequence determined by the order of the observations {\it in time}; but how
can such a rule avoid leading to a vicious circle in a theory in which time
itself is one of the things being ``observed''?  A second, closely related
difficulty concerns the ``frozen formalism'' which results when one applies
the formal rules of canonical quantization to General Relativity (or to any
generally covariant Lagrangian theory).  In consequence of the Hamiltonian
constraints, the ``physical observables'' are necessarily all
time-independent (they are what Karel Kucha{\v r} [??33]
calls `perennials'), and
one seems forced into an attempt to ``fix the time-gauge'' in order to
recover a semblance of spacetime from the disembodied spacelike
hypersurface to which the formalism directly refers.  This is not only
technically questionable, but it can easily smuggle arbitrary answers to
important physical questions (like whether collapse to a singularity is
avoidable) into the theory in the guise of a gauge-choice.  Finally, in a
framework based on ``observables'' rather than ``beables'', how are we to
speak about (say) the early universe, if there were no observers then and
none in the offing for a long time to come?  Since some of the most
important applications of quantum gravity are likely to be precisely to the
early universe, this also appears to present a serious difficulty.

None of these ``problems of time'' would seem to exist for the
sum-over-histo\-ries/space\-time approach.  Time itself doesn't need
to be recovered, because it is there from the very beginning as an
aspect of the spacetime metric.  The projection postulate is
irrelevant because there is no state-vector to be ``reduced''.  And
the early universe existed just as much as we ourselves do here and
now, even if from our vantage point it is relatively remote and
inaccessible.

There is another point I want to mention here, which is presented more
fully in my contribution to Dieter Brill's Festschrift [??34], and that is
that---independently of any problems related to general covariance---the
$\Psi$-framework starts to break down, in a certain sense, already for
quantum field theory in flat spacetime.  One of the seeming advantages of
the $\Psi$-framework vis-as-vis the sum-over-histories is that it appears
to possess a more comprehensive measurement scheme, telling us what in
principle can be measured (every selfadjoint operator) and prescribing (at
least formally) how to design an interaction-Hamiltonian to effect the
corresponding measurement (see [??35]).  In contrast, there exists (so far)
no equally comprehensive theory of measurements within the
sum-over-histories framework.  Now this lack is not the great disadvantage
for the sum-over-histories which it would be for the $Psi$-framework,
because measurement is not a fundamental notion for the
sum-over-histories.  Nonetheless, the question of what can and cannot be
measured is clearly an important one for any theory.  However, it turns out
that the simple measurement scheme which the $\Psi$-framework appears to
possess is physically viable only for quantum mechanics in the narrow sense
of non-relativistic point-particle mechanics.

The point is that in Quantum Field Theory, it proves inconsistent with
causality to assume that every observable constructed from the field
operators within a given spacetime region $R$ can be measured by operations
confined entirely to $R$; to be able to do so would lead to the possibility
of superluminal signaling.  There is no time here to repeat the argument
[??34] in detail, but it considers three spacetime regions $A$, $B$, and $C$
arrayed so that communication is possible from $A$ to $B$ and from $B$ to
$C$, but not from $A$ to $C$.  Specifically one can choose $B$ to be a
thickened spacelike hyperplane, with $A$ and $B$ being spacelike separated
points which are respectively to the past and future of $B$.  Assuming that
arbitrary localized ideal measurements were possible in these regions, the
argument concludes that an experimenter stationed at $A$ could transmit
information to a colleague at $C$ by deciding whether or not to perform a
certain observation, given that both know that a certain other observation
{\it will} be performed in the intervening region $B$.

Thus, one must reject the assumption of arbitrary localized measurements,
and it becomes a priori unclear, for quantum field theory, which
observables can be measured consistently with causality and which can't.
This would seem to deprive the $\Psi$-framework for quantum field theory of
any definite measurement theory, leaving the issue of what can actually be
measured to (at best) a case-by-case analysis, just as it remains (so far)
within the sum-over-histories framework.  (Notice as well that most of the
hypersurface observables with which a canonical formulation of gravity
would presumably deal, are likely to run into locality troubles of this
same sort.)  As pointed out above, this actually puts the $Psi$-framework
at a disadvantage, because for it, the notion of measurement {\it is}
fundamental.

Finally, I want to return to the more general comparison of the two opposed
frameworks in order to stress what seems to me to be the great practical
advantage of a spacetime approach vis a vis a purely spatial one.  In fact
the questions one wants to ask of a quantum gravity theory are---due
ultimately to the diffeomorphism invariance of General Relativity---all of
an unavoidably spacetime character.  For example, one may want to study how
the horizon area of a black hole responds to the emission of Hawking
radiation.  Or one may want to ask whether the cosmological expansion we
are now experiencing was actually preceded by (say) nine previous cycles of
expansion and recontraction.  Both of these questions make perfect sense if
one has access to an entire 4-geometry, but would seem to be virtually
impossible to formulate in terms of the kind of hypersurface data with
which the canonical approach works.

Well, if we do go in the direction indicated by the signpost reading
``sum-over-histories'', we come a little way down the road to a
secondary fork concerning the proper interpretation of the ``quantum
probabilities'' which that formalism yields.  In fact, the
sum-over-histories will furnish a probabilistic answer to almost any
question about the history you care to ask (more precisely, it will
furnish relative probabilities for the elements of any partition of
the set of all histories into mutually exclusive and exhaustive
subsets [logan]), but it is easy to see on physical grounds that most of these
``quantum probabilities'' cannot be very meaningful.  A central question
for this approach is therefore, under what circumstances such
probabilities do acquire meaning (an issue represented in my initial
laundry list by the question whether probability is relative or
absolute).

Here there are two alternative points of view that I know of.  According to
the first view [??37] probabilities are {\it absolute} and {\it
unconditional} (in the sense that their application rests on no assumption
about the history other than a choice of a cosmological initial condition)
but have meaning
only with respect to a fixed partition which obeys the condition that Jim
Hartle and Murray Gell-Mann call `decoherence'.  However there are in
general very many decohering partitions, not all of whose
probability-assignments will be compatible with each other.\footnote{*}
%
{For this reason, it could be misleading to describe such probabilities as
absolute: they are relative to a choice of decohering partition.}
%
(An interesting example of such an incompatibility is that a given
partition can have its decoherence destroyed by subsequent
``observational activity'' which has the effect of making the original
alternatives interfere\footnote{$^\dagger$}
%
{Or stated more generally and precisely: there can exist pairs of
partitions $P'$ and $P''$, based respectively on earlier and later
properties of the history, such that $P'$ and $P''$ each decohere, but
their ``union'' $P'\vee P''$ does not.  Here $P'\vee P''$ is the partition
which asks about both the earlier and the later properties together; its
elements are the atoms of the lattice of sets generated by the elements of
$P'$ and $P''$ via intersection and complementation.}.)
%
%% [[Is this last a correct way to situate this problem?]]
%% example: beam splitter followed by diaphragm whose ``diffractive
%% shadows'' overlap.
%
According to the second view [??38], probabilities are {\it relative} to a
split of the universe into subsystems, and {\it conditional} on possible
assumptions about the behavior (or even existence) of one of the subsystems
(cf. [??39]).  (For
example one subsystem could be an electron and the other a collection of
molecules in a cloud chamber.)  The criterion for the ``quantum probabilities''
to be meaningful is then not that they necessarily decohere (and hence obey
the sum-rules proper to classical probabilities), but that a sufficiently
perfect {\it correlation} obtain between the two subsystems (for example the
correlation by which the track in the cloud chamber reflects the path of
the electron) [??38].

Incidentally, in arguing for the sum-over-histories/spacetime approach in
preference to the $\Psi$-framework/canonical quantization one, I would not
want to give the impression that I think that mathematical studies of the
operator constraints are necessarily irrelevant.  Indeed, some formulations
of the sum-over-histories effectively employ something akin to a Hilbert
space norm on wave-functions as a mathematical intermediary in computing
quantum probabilities, and making such a formulation mathematically
well-defined might still require a Hermitian inner product on the space of
solutions $\psi$ to the operator-constraints.  On the other hand, a new
kinematics in general {\it would} render the constraints irrelevant (except
possibly in some approximate effective theory), and this brings us to the
second basic question, the one about changing the kinematics.


\bigskip
{\noindent\bf V. A Modified Kinematics?}
\medskip\noindent
\nobreak
{(a)} {\it Lorentzian or Euclidean Signature?}
\smallskip\nobreak

People have suggested several possible modifications of the kinematics
of classical General Relativity, and the one I want to discuss first
is perhaps the least radical---though it's radical enough.  It
proposes [??40]
 that the sum-over-geometries be conducted with
positive-definite (Riemannian) metrics instead of  Lorentzian ones.
Such a replacement carries less fundamental import if one interprets
this sum within the $\Psi$-framework, where the histories have meaning
only as intermediaries used to find a wave-function $\Psi$; but even
so, it represents a significant modification.

The main motivation for altering the signature in this way seems to be
that tunneling phenomena appear, in an ``instanton'' approximation, to
occur via Riemannian solutions of the field equations.  But something
like this is already true in quantum {\it mechanics}, where one can
use an imaginary-time path to compute the WKB approximation to barrier
penetration, as in the classic problem of alpha-decay.  Such a
calculational technique can be interpreted as an infinite-dimensional
saddle point approximation to the path-integral, and from this point
of view the complex-time path, or saddle point, merely {\it summarizes}
(to leading order in $\hbar$) the contribution of a large number of
real-time paths.  Certainly its use would not normally be
taken to imply that
physical time turns imaginary while the alpha particle is ``under the
barrier''.  In the same way, a gravitational instanton should
presumably be understood as summarizing the contribution of a large
number of Lorentzian histories.

On this view, the notorious ambiguities [??41] in the choice of saddle
point and contour which affect the so-called Euclidean functional integral
will only be resolved (to the extent that this can be done at all without
recourse to an underlying discrete theory) when one has succeeded in
deriving the Euclidean-signature expression by analytic continuation from
a Lorentzian starting point.  Carefully observing the (appropriate infinite
dimensional generalizations of the) rules for deformation of complex
contours ought then to answer such questions as which saddle points
contribute, and what are the relative signs of their contributions.  For my
own enlightenment, I actually went through the corresponding analysis
in detail in the much simpler case of one-dimensional barrier
penetration, and I can affirm that everything works out just as it should,
including the fact that passage through the barrier results in
damping rather than amplification.

%% Here we could include the detailed formulas, if desirable

%% could mention also someone's question why not include ALL signatures, in
% line with the Principle of Relativity which told us to include all
% topologies. I think this was Dieter's question.

\medskip\noindent
{(b)} {\it Should We Sum over Different Topologies?}
\smallskip\nobreak

A kinematic question of another type is whether one should include more
than one topology in the sum-over-histories.  (Here, incidentally, I have
posed the question in terms of the sum-over-histories, not only because
that was the ``fork'' we followed earlier, but mainly because the
$3+1$-framework doesn't lend itself to a dynamical topology in any known
manner.)
To this question, my answer would be: ``yes, and again yes''.  Yes first of
all, because it seems contrary to the spirit of Relativity to make of the
topology the only absolute element, which ``affects without being
affected''.

And yes secondly, because of a further reason which is perhaps more
substantive, though not as widely appreciated.  Namely, I want to
claim that the study of topological geons (e.g. [??42])
leads to the conclusion that
{\it a dynamical metric requires a dynamical topology}.  The basis of
this claim is the observation that, for a type of particle such as a
geon, whose existence is {\it defined} in terms of the spatial
topology, any procedure of ``second quantization'' by definition
forces the topology to change because the number of geons cannot
vary if the topology remains constant.  But experience shows
that a relativistic particle which is only ``first quantized'' is not
physically consistent (among the well-known difficulties, such a
particle moves faster than light, as mentioned by Bob Wald
yesterday [??43]), and there is no reason to expect geons to be immune from
this imperative.  In this sense, I believe one can say without too
much exaggeration that quantum gravity without topology change is
simply inconsistent physically [??44].\footnote{*}
%
{This reasoning cannot be airtight, of course, because there might in
theory be an absolute, ``god-given'' topology of spacetime for which no
geons at all can exist, for example $S^3 \times R$.}

A further consideration is that, without topology change, it is possible
to (formally) quantize General Relativity so that
certain geons violate the normal connection between spin and
statistics [??44]
(for a not-only-formal treatment in $2+1$-dimensions
see [??45]).  Considering that some process of
pair-creation/annihi\-la\-tion seems always to underly the known
proofs of spin-statistics theorems, it is thus a natural guess that
the correct incorporation of topology-change into quantum gravity
would automatically set up a correlation between exchange and rotation
which would exclude the spin-statistics violating possibilities [??46].
Some
recent evidence in favor of this idea comes from the finding by Fay
Dowker that precisely such a correlation occurs for geons of certain
types which have been pair created via ``U-tube'' cobordisms (this
sort of cobordism being a universal mechanism of pair
creation) [??47].
If it is true that a correct incorporation of topology-change must
reinstate the spin-statistics correlation for topological geons, then
conversely, the requirement that this in fact occur will
serve as an important test of any theory of quantum gravity, or to put
it another way, as an important clue in the formulation of such a
theory.


{\underbar {\it Remark}} Topology-change not only speaks against
canonical quantization, but it also excludes what is sometimes called
the ``covariant'' approach to quantum gravity, which works with an
operator-valued metric field on a {\it fixed} spacetime manifold [??48].

 If we do accept that topology-change must be provided for, then we
come further down that road upon a fork corresponding to my earlier
question whether it is the regularity of the metric or causality which
should be sacrificed, the choice being forced upon us by a theorem of
Geroch [??49]
according to which topology change entails either singularities
or closed timelike curves.

The choice of allowing closed timelike curves in order to preserve
regularity of the metric
can be opposed on the grounds that
it conflicts
with the causal set idea, which I will come back to in a moment.  More to
the point, recent evidence is that closed timelike curves (``time
loops'') lead to trouble with the quantum fields living in such a (background)
spacetime, specifically divergences in the stress-energy [??50]
and a
breakdown of unitarity [??51].  An older, if lesser known difficulty concerns
the pair-creation of monopoles in (5-dimensional) Kaluza-Klein theory.
It turns out that such pair creation via a globally regular Lorentzian
metric is impossible for topological reasons {\it even if} one allows
time loops to occur (as long as time-{\it orientability} is maintained) [??52].
To me, this strongly indicates that the
imposition of strict
global regularity on the metric is inappropriate, a view
which is reinforced by considering how rough the metric is likely to
be in any case, given the indications from quantum mechanics and
quantum field theory.

In fact there does exist a mechanism of topology-change which
preserves causality, at the cost of allowing the metric to degenerate
to zero at isolated spacetime points [??53].  For any compact cobordism
(interpolating manifold between two spacelike hypersurfaces of
possibly different topology) one can find a metric which degenerates
to zero at a finite number of points, in one of a  finite
number of predetermined
ways (the number depending on the spacetime dimension), and for which no
time-loops are present.  With respect to such a metric, one can view
the topology change as ``happening'' at the points of degeneration,
and one might accordingly hope for simple dynamical rules to describe
what occurs at such points.  In this direction, there is an intriguing
result in the two-dimensional case, where there are only two
possible types of ``elementary cobordism'', corresponding to the
``crotch-point'' in a ``trousers'' spacetime, and the ``crown-point'' in a
``yarmulke'' or ``big bang'' spacetime.  At the corresponding points
of degeneration, the scalar-curvature Action becomes complex in such a
way as to suppress the former type of topology change and enhance
the latter [??54].


\medskip\noindent
{(c)} {\it Is the Metric Fundamental, or Only an Effective
           Description of Something Deeper? (Causal Sets)}
\smallskip\nobreak

A much more basic kinematical question than those dealt with so far is
whether the spacetime metric should be replaced by some ``deeper''
structure of which it is only an approximate description.  Now, String
Theory [??55]
 has one answer to this, but the answer I want to discuss
follows from still another question: is spacetime ultimately
continuous or discrete?

Here, I cannot resist quoting Einstein, who wrote in 1954,
\smallskip

{\narrower\noindent
        ``The alternative continuum-discontinuum seems
         to me to be a real alternative; i.e., there is no compromise\dots\dots
         In a [discontinuum] theory space
         and time cannot occur\dots\dots  It will be especially difficult
         to derive something like a spatio-temporal quasi-order (!)
         from such a schema\dots\dots   But I hold it entirely possible
         that the development will lead there\dots\dots'' [??56]
\smallskip}
\noindent
(In this quotation the `(!)' is mine, put there because the words
`spatio-temporal quasi-order' seem so obviously to be calling for
a theory based on causal sets!)
He goes on to say that ``{\dots\dots} This objection [against the continuum] is
not decisive only because one doesn't know, in the contemporary state
of mathematics, in what way the demand for freedom from singularity
(in the continuum theory) limits the manifold of solutions.''  The
objection in question  was that quantum mechanics teaches that
a bounded system can be described by a finite set of ``quantum
numbers'', and such a description is incompatible with the infinite
number of degrees of freedom posited by a continuum theory.  (The
loophole preventing the objection from being ``decisive'' was that an
appropriate classical field theory might be able to replace quantum
theory, and still reproduce all its characteristic effects.)

In addition to this argument for a fundamental discreteness there are
several contradictions in existing theories which speak powerfully for
the same conclusion.  These contradictions, which I call ``the three
infinities'' (or perhaps four depending on how you count them), include
the divergences of Quantum Field Theory, the singularities of
classical General Relativity, the apparent non-renormalizability of
naively quantized gravity [??57], and the apparently infinite value of the
black hole entropy if no cutoff is present.  The final item in this
list rests on an interpretation of horizon entropy to which I will
return below; to my mind it is the least adequately appreciated of the
common arguments for discreteness.

If we accept all these indications then we come immediately upon a
subsequent multiple fork in the road corresponding the question of
what the discrete substratum actually is.  To attempt an answer
at this point would seem to be hopeless if there is not at least
what I would call some sort of ``structural bridge'' between the
continuum and the underlying discontinuum, i.e. some structural
analogy which would allow one to understand how the former can
``emerge'' from the latter in appropriate circumstances.  People have
sought the source of such an analogy in at least three properties of
the continuum, its {\it topology} [??58], its {\it metric} [??59], and its
causal
{\it order} [??60]; and I personally have at one time or another been drawn
to all of them before deciding, in connection with the causal set
hypothesis, that it is the {\it order} or ``causal structure'' of the
spacetime continuum which, together with one component of the metric
(its determinant), should be viewed as being its most fundmental
property.

The causal set hypothesis which I have just alluded to, posits that
the structure of the discrete substratum is that of a locally-finite
partial-ordering (= causal set), and establishes the correspondence
between this underlying structure and the overlying, ``emergent''
Lorentzian manifold by making the causal order and volume-measure of
the latter correspond to the intrinsic order and ``counting measure''
of the former (i.e. spacetime volume = number of elements).

For a general introduction to causal sets and a partial review of work
on that idea see [??61].
I will not discuss them further here, except to point to one last fork
in the ensuing road which raises the possibility of allowing the
analog of closed timelike curves to occur in the underlying discrete
ordered set.  Such a generalized structure (a ``directed graph'')
would be a possible alternative to the causal set as presently
defined, but it seems to me to be unnatural, because {\it even if} one
does let such ``cycles'' occur in the substratum, it still seems
impossible to broaden the rules of correspondence with the continuum
in such a way as to allow a Lorentzian manifold having time loops to
emerge as a valid approximation to a discrete directed graph.  In this
sense, one can {\it predict} that time loops must be absent from the
continuum, whether or not their analog is admitted into the underlying
ordered set.

%% [[ cite Finkelstein]]

%% [[ use following? ]]
%     Remark: An underlying discreteness doesn't necessarily mean work on
%             a continuum theory is useless, since it might apply in some
%             approximation.  If so, one expects that the discreteness
%             provides a regularizer and resolves ambiguities (e.g.
%             top-change, is T^3 a boson or fermion?).

\bigskip
\vbox{\noindent\raggedright
{\bf VI. A final dynamical fork: should we constrain the four-volume in
the sum over geometries (``unimodular gravity'')?}}
\nobreak

The alternative ``discrete versus continuous'' was my last one
concerning kinematics; but before turning to phenomenology, I want to
raise one further dynamical question, which is more naturally discussed
here than earlier.  (I say ``further'' because the whole discussion of
section IV was, of course, about dynamics.)
Specifically, the question is whether one should hold the spacetime
volume fixed in the gravitational sum-over-histories [logan=36].  Unlike with
topology-change, the phrasing of this question in sum-over-histories
form is not a matter of principle; there is an equivalent formulation
in canonical terms [??62].

Now classically, fixing the volume before varying the Action makes
essentially no difference; its only effect is to convert the
cosmological constant from a free parameter in the Lagrangian into a
free constant of integration of the resulting field equations.  The
physical significance of this ``unimodular'' constraint is therefore
solely quantum mechanical.  The motivation for adopting it comes, in
my mind, first of all from causal set theory, where a constraint on
the total number of elements seems necessary for the sum over causal
sets to converge, and this constraint translates in the continuum into
specifying the total spacetime volume.  Also, independently of any
 discreteness, a direct constraint on the spacetime volume seems
to ameliorate convergence problems with the continuum functional
integral, as is especially noticeable in the limit
corresponding to quantum
field theory in curved spacetime.  Most intriguingly, the manner in which
the cosmological constant ``becomes dynamical'' in unimodular quantum
gravity offers a new ``mechanism'' for producing a small or zero value
for it.

It is therefore interesting to ask what difference the unimodular
constraint would make in rudimentary models like the homogeneous universes
of ``mini-super\-space quantum cosmology''.  Recently Jorma Louko and I
have studied a couple of the simplest of such models with results that are
peculiar enough to be interesting, but not so crazy as to become boring
[??63].  We find, specifically for the Friedmann universe $S^3\times
\Reals$, that adopting the analog for the unimodular theory of the
``no-boundary boundary-condition'', and computing the crudest saddle point
approximation to the wave-function $\psi$ (with the most obvious saddle
point), that $\psi$ remains regular as a function of $a$, the radius of the
universe, in both the limits $a \to 0$, and $a \to \infty$.  (This is an
example of the improved convergence I spoke of, since in ``standard quantum
cosmology'', $\psi$ diverges exponentially as $a\to\infty$).  On
the other hand, $\psi$ is now a function of the 4-volume, which serves as a
kind of parameter-time $T$, and its ``evolution'' with $T$ is non-unitary
due to a flux of probability coming in from $a=0$.  One might interpret
this effect as a ``continuous creation of universes'', or perhaps better,
as an ``induced emission of new branches of the universe, all stemming from
a common root''.\footnote{*}
%
{Or maybe this is just the wrong saddle point.  In an example of the
ambiguity referred to earlier, Jorma has recently found a less obvious
saddle point whose contribution leads to a $\psi$ which is even better
behaved as $a \to \infty$ but which dies out with $T$ instead of blowing
up, suggesting either unitary evolution or a flow of probability {\it out}
through $a=0$!}

%           quantum mechanically it seems to help with convergence of
%           the functional integral (resp. normalizability of psi),
%           and with causality.
%

\bigskip
{\noindent\bf VII. What is the phenomenology of Quantum Gravity?}
\smallskip\nobreak

The third and last set of ``forks'' I want to discuss concerns the
phenomenology of quantum gravity, or more prosaically, the question of
what observable consequences we might expect a theory of quantum
gravity to possess.  If in fact a new substance underlying the
metrical field is the proper basis for such a theory, it becomes
especially important to try to foresee how this new form of matter
will manifest itself (or has already done so!); but even if only the
dynamics of gravity is to be changed, one would expect some dramatic
consequences to appear.  In this connection, let me present a second
laundry list of ``phenomenological'' questions whose answers some
people have hoped would emerge from quantum gravity.

% [[refs for all these??]]

\item{}  Why is there a metric?  And why is it Lorentzian?

\item{}  Why is Minkowski space a solution of the theory (with d=4)?
Notice that this question includes also the question of why the
cosmological constant is so small, since for $\Lambda\ne 0$, Minkowski
space would not be a solution.

\item{} Why is the gravitational Lagrangian what it is?

\item{} Why does non-gravitational ``matter'' exist (fields and/or particles)?

\item{} What is origin of black hole entropy?

\item{}    Why is the universe expanding?

\item{}    Is CPT broken?

\item{}  What are the rules for topology change?


Unlike for my first laundry list, there will be time to address only a
minority of these topics here, and I want to concentrate on two of
them which also occurred in the earlier list, namely those concerning
black hole entropy and the cosmological constant.  Before getting to
them, however, let me allude to part of the answer that causal set
theory would give to the first question of why the spacetime metric is
Lorentzian.  The point is that no other metric signature, Riemannian
or $(++--)$ or whatever, can lend a partial ordering to the events of
spacetime, because only in the Lorentzian case do the light-cones
provide a well-defined local distinction between before and after.

\nobreak
(a) {\it Is a Black Hole's Entropy Inside its Horizon?}
\smallskip\nobreak

     Although one might initially think that the entropy of a black
hole must represent the number of its interior states, such a view is
difficult to reconcile with the fact that the second law of
thermodynamics pertains effectively to processes which proceed in
ignorance of whatever is happening inside the horizon.  Since it is by
definition the autonomously developing ensemble of such exterior
processes which are responsible for the entropy increase, it would
seem most natural that the entropy itself be a property of the
exterior region.  In fact, if one adopts this view, and more
particularly if one identifies the exterior entropy with
$S = - {\rm Tr} \, \rho \, \log \rho$, where $\rho$ is the effective
Schr\"odinger-picture density-operator of a spacelike hypersurface in
the exterior region, then there exists a schematic explanation of why
$S$ as so defined necessarily increases as the hypersurface to
which $\rho$ refers advances in time.\footnote{*}
%
{In using this language I am presupposing a sufficiently classical
 approximate spacetime with respect to which a given hypersurface can be
 meaningfully located.}
%
This explanation [??64] rests on a certain theorem [??65] concerning
density-matrix evolution, and on the crucial fact (emphasized yesterday by
Jimmy York [??66]) that the the total energy not only is conserved, but is
meaningful as a property of the exterior region, since it can be read off
from the behavior of the metric in the asymptotic region or on some
suitable boundary surface.

What is more, one can estimate the contribution to the above $S$ from the
zero-point fluctuations of a free scalar field in a black hole background,
and one obtains a value which is proportional to the horizon area measured
in units of the cutoff [??67].  It is essentially this result to which I
referred earlier in adducing the black-hole entropy as one of the ``three
infinities''.  Although the story is really more complicated than this
(because the free-field approximation is probably wrong, and it is most
likely the degrees of freedom of the horizon itself which account for its
entropy) I believe that the conclusion that finite entropy requires a
cutoff is correct (cf.  Einstein's objection against the continuum quoted
earlier), and that the sketch of an explanation cited above for the
increase of the total entropy is fundamentally correct also.  If so, then
filling in the sketch so as to obtain a complete derivation of the
increasing character of a well-defined total entropy which includes a
horizon contribution of the correct magnitude will be a decisive test for
any theory of quantum gravity.

\nobreak
{(b)} {\it Is the Cosmological Constant Exactly Zero?}
\smallskip\nobreak

I want to conclude with a ``prediction'' about the cosmological
constant, $\Lambda$, which draws together a few of the ideas advanced
so far.  From unimodular gravity let us take the idea that $\Lambda$
is in some sense conjugate to the spacetime volume $V$ (earlier called
`$T$'), and from causal set theory the idea that $V$ is a measure of the
number of elements $N$.  From the the former idea, we can write in the
sense of the Uncertainty Principle,
$$
     \Delta V \, \Delta \Lambda \sim 1.
$$
But since the correspondence between $V$ and $N$ has a probabilistic
character, Poisson fluctuations in $N$ of order of magnitude $\sqrt N$
translate into an uncertainty in $V$ of
$$
   \Delta V \sim \Delta N \sim \sqrt N \sim \sqrt V.
$$
Putting these two relationships together yields a minimum uncertainty
in $\Lambda$ of
$$
     \Delta  \Lambda  \sim  {1 \over  \sqrt V},
$$
which, for the visible universe to date, is in order of magnitude,
$10^{-120}$ in natural units.  The prediction [??68]
 is thus that whatever
mechanism drives $\Lambda$ to vanish\footnote{*}
%
{One possible mechanism is that only $\Lambda=0$ is stable against
non-manifold fluctuations of the causal set.},
%
will probably leave it with a small but non-zero value of this
magnitude, which, intriguingly is just barely large enough to be
accessible to observation [??69].


If this is correct then to test a prediction of Quantum Gravity, we
might want to look outward rather than inward, and we might even have
the experimental answers in time for them to be presented at the next
birthday celebration for Dieter and Charlie.

Finally, I would like to thank several colleagues who attended this talk
for their stimulating questions and comments.

\bigskip\noindent
This research was partly supported by NSF grant PHY 9307570.


%----------------------------REFERENCES------------------------------
\vskip 0.5truein
\centerline {\bf References}
\nobreak
\medskip
\noindent
\parindent=0pt
\parskip=10pt
\baselineskip=12pt   % for single spacing

% FIRST REFERENCE

% SECOND REFERENCE


%a different style of refs:
% \item{2.} Freedman, M., `` There is no room to spare in four
%         dimensional space'', {\it AMS Notices}, {\bf 31}, 3 (1984).


\end

