including at the begining of the universe.
One can regard this as a triumph for the
principles of democracy: Why should the begining of the universe be exempt from the
laws that apply to other points. If all points are equal one can’t allow some to be more
equal than others.
To implement the idea that the laws of physics hold everywhere, one should take the
path integral only over non-singular metrics. One knows in the ordinary path integral case
40
that the measure is concentrated on non-differentiable paths. But these are the completion
in some suitable topology of the set of smooth paths with well defined action. Similarly,
one would expect that the path integral for quantum gravity should be taken over the
completion of the space of smooth metrics. What the path integral can’t include is metrics
with singularities whose action is not defined.
In the case of black holes we saw that the path integral should be taken over Euclidean,
that is, positive definite metrics. This meant that the singularities of black holes, like the Schwarzschild solution, did not appear on the Euclidean metrics which did not go inside
the horizon. Instead the horizon was like the origin of polar coordinates. The action of the
Euclidean metric was therefore well defined. One could regard this as a quantum version
of Cosmic Censorship: the break down of the structure at a singularity should not affect
any physical measurement.
It seems, therefore, that the path integral for quantum gravity should be taken over
non-singular Euclidean metrics. But what should the boundary conditions be on these
metrics. There are two, and only two, natural choices. The first is metrics that approach
the flat Euclidean metric outside a compact set. The second possibility is metrics on
manifolds that are compact and without boundary.
Natural choices for path integral for quantum gravity
1. Asymptotically Euclidean metrics.
2. Compact metrics without boundary.
The first class of asymptotically Euclidean metrics is obviously appropriate for scat-
tering calculations.
particles going
out to infinity
interaction
region
particles coming
in from infinity
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In these one sends particles in from infinity and observes what comes out again to infinity.
All measurements are made at infinity where one has a flat background metric and one
can interpret small fluctuations in the fields as particles in the usual way. One doesn’t
ask what happens in the interaction region in the middle. That is why one does a path
integral over all possible histories for the interaction region, that is, over all asymptotically Euclidean metrics.
However, in cosmology one is interested in measurements that are made in a finite
region rather than at infinity. We are on the inside of the universe not looking in from the
outside. To see what difference this makes let us first suppose that the path integral for
cosmology is to be taken over all asymptotically Euclidean metrics.
region of
measurement
asymptotically
Euclidean metric
Connected asymptotically Euclidean metric
region of
measurement
compact
metric
asymptotically
Euclidean metric
Disconnected asymptotically Euclidean metric
Then there would be two contributions to probabilities for measurements in a finite region.
The first would be from connected asymptotically Euclidean metrics. The second would
be from disconnected metrics that consisted of a compact spacetime containing the region
of measurements and a separate asymptotically Euclidean metric. One can not exclude
disconnected metrics from the path integral because they can be approximated by con-
42
nected metrics in which the different components are joined by thin tubes or wormholes
of neglible action.
Disconnected compact regions of spacetime won’t affect scattering calculations be-
cause they aren’t connected to infinity, where all measurements are made. But they will
affect measurements in cosmology that are made in a finite region. Indeed, the contribu-
tions from such disconnected metrics will dominate over the contributions from connected
asymptotically Euclidean metrics. Thus, even if one took the path integral for cosmology
to be over all asymptotically Euclidean metrics, the effect would be almost the same as if
the path integral had been over all compact metrics. It therefore seems more natural to
take the path integral for cosmology to be over all compact metrics without boundary, as
Jim Hartle and I proposed in 1983.