The No Boundary Proposal
The question of our most probable past formulated above is of course
only answerable within the context of some definite theory which tells you the
probability measure on past spacetimes. The usual arguments for eternal
inflation are vague on this point, and basically just assume that for some
reason a region existed in which there were the right conditions for inflation
to start. Hartle and Hawking proposed
an ansatz(which is a fancy way of saying they guessed!) the formula for relative
probabilities. One can easily imagine
an infinite number of possible ansatzes for the initial conditions of the
Universe. At the present time many of these might be perfectly consistent with
observation. But Hartle and Hawking’s proposal is appealing because it is based
on general ideas which have a rationale beyond cosmology. In a strong sense I think it is ‘the most
conservative thing you could do’. It may well fail precisely because it is too
conservative. Space and time may be emergent rather than fundamental
properties. Describing the Universe as a manifold may not be appropriate to its
early moments. Nevertheless precisely because the no boundary proposal is not just cooked up to make inflation work,
its failings and limitations may teach us something deeper about what is in
fact required.
In physics we are now used to the fact that all we can ever predict in
practice is probabilities. This was
even true in classical physics, because nothing is ever measured perfectly and
in consequence all predictions carry ‘error bars’. But in quantum physics the
issue is more fundamental, because the quantities we use to describe the world,
and which we need in order to predict its future or to extrapolate back into the
past (like position and velocity of a
particle) are in principle impossible to measure beyond a certain limiting
accuracy. Only certain questions are allowed, and these involve correlations between physical quantities.
Paradoxically, in spite of the limitations quantum mechanics imposes on our
ability to predict classical properties of the world, quantum physics is
actually a far more complete theory than classical physics because it predicts
the correlations with perfect
accuracy.
Hartle and Hawking proposed to use the formulation of quantum mechanics
defined by Dirac and Feynman, called the path integral formulation, to define
all cosmological correlators. This formulation is widely accepted and is the
basis for modern treatments of gauge field theories, well tested in the
laboratory. But it usually does not
claim to solve the initial conditions problem. Usually the path integral
formula is used to give the quantum mechanical amplitude to be in a state B at time t_{f}, given that the system was in a state A at time t_{i}.
There is one situation however, in which it becomes unnecessary to
specify exactly what state the system was in. That is when the system is in
thermal equilibrium. In this case, two
beautiful things happen to the DiracFeynman formula. First, we identify the
initial state A with the final
state B, and sum over all possible states. Second, real time is continued to imaginary time,
and the temperature of the system then replaces the role of the period of the
system in imaginary time. In
nongravitational physics, one still needs to specify the temperature in order
to specify all correlators. But in the
context of gravity, the size of the system in imaginary time becomes a
dynamical quantity to be determined by the theory itself. Thus Hartle and Hawking’s proposal can
actually become a completely selfreferential prescription for the initial
conditions which does not need any input except the dynamical laws of physics
(the Lagrangian).
It may seem paradoxical that quantum physics yields better defined
probability measures than classical physics because in quantum physics certain
quantities are inherently uncertain (Heisenberg’s ‘uncertainty principle’). But
in fact the two aspects are closely related. For example, the state of a
particle in a box is described in classical theory by its location and
velocity, which can in principle be measured with absolute certainty. If we limit the energy of the particle,
there are still an infinity of possible states for it to be in (for example
differing infinitesimally in location).
In quantum physics, however, if we limit the energy there are only a
finite number of possible states. In all of these states both the location and
velocity of the particle is to some extent uncertain.
The finiteness imposed by quantization is one aspect of obtaining a
well defined probability measure. But there is another aspect, namely the box!
If the box is infinite, there are an infinite number of possibilities for the
particle and again the probability measure is ill defined. In many situations
the precise nature of the box is irrelevant, as long as it is bigger than any
relevant scale in the problem. But in some situations, it is allimportant. For
example, a gas particle in a box performs a random walk as it collides with
other particles. For some questions,
such as the average speed of the particle, the size of the box is irrelevant.
But for others, such as ‘how far does it travel’, the walls of the box are all
important. In cosmology, I want to
argue that causality plays a role similar to the walls of the box.
Before proceeding let me list a few of the ‘technical’ difficulties to
be faced in implementing Hartle and Hawking’s idea.

Einstein gravity is nonrenormalizable.
This objection refers to the bad short distance properties of the theory. These
are certainly important but not central to the discussion here. Theories such as supergravity with improved
ultraviolet properties are not conceptually different as far as the problems
we are discussing.

The Euclidean Einstein action is not
positive definite, and therefore the Euclidean path integral is ill defined.
This is the ‘conformal factor’ problem in Euclidean quantum gravity and I shall
return to it below. Recent work has shown that at least for some choices of
physical variables, and to quadratic order, this problem is overcome.This means that at least to the extent that approximation calculations have so far
been possible, the problem is overcome.

The sum over topologies in four
dimensions is likely to diverge, as happens in string theory. Most likely one
will need some formulation in which manifolds of differing topologies are
treated together.
These problems are formidable. The main hope, it seems to me, is that
our Universe appears to be astonishingly simple, and well described by a classical
solution of great symmetry with small fluctuations present at a level of a part
in a hundred thousand. This suggests
that we may be able to accurately describe it using perturbation theory and
classical solutions.
Contributed by: Dr. Neil Turok
