One of the many outstanding problems in trying to construct a quantum
field theory of gravitation concerns the appropriate interpretation of quantum
states for configurations that make no overt reference to ``time''. (It is
difficult in general to endow the theory with a traditional Hilbert space
structure based on a hermitian inner-product and a unitary evolution.) Although
many alternative schemes have been suggested difficulties in interpretation
remain. Some of the difficulties are intrinsic to the infinite dimensional
aspect of field quantisation and in this respect one often seeks guidance by
studying truncated field configurations corresponding to situations with high
symmetry. There the many cogent arguments that urge caution in extending
deductions from such models to the full quantum field theory. However symmetric
models are a useful theoretical laboratory for testing ideas that may have more
general validity, In the context of ``mini-superspace'' models it has been
noticed that it is possible to implement the Hamiltonian constraint for
Bianchi-type models in general relativity, on a multicomponent wavefunction.
In the paper `A Spinor Equation for Quantum Cosmology' (with M Onder, T Dereli), Phys Letts B323 1994 134-140, focus is on a particular minisuperspace analysis that gives rise to a Hamiltonian constraint, classically describing the zero energy configuration of an oscillator ghost-oscillator pair. This gives rise to a Wheeler-DeWitt equation that has occurred in a number of different contexts and provides a solution to a number of puzzles.
In this particular model a symmetry vector of the DeWitt metric on superspace is chosen to construct, from a particular quantum state, a probability density that has maxima in the vicinity of classical cosmological loci. An internally consistent interpr a cosmological metric.
Our probability interpretation is reminiscent of that following from
the non-relativistic Schrodinger equation in the presence of a complex
potential. In the Schrodinger situation the use of a complex potential models
the absorptive properties of an open system. For a closed system a
non-hermitian hamiltonian is usually regarded as pathological. However in the
context of gravitation such a reaction requires caution. For example, if the
non-unitary evolution of a pure state of matter to a mixed state via the
Hawking process can be maintained when gravitational back reaction is taken
into account then probability conservation in a gravitational context may not
In such a scenario it is tempting to conjecture that it is the existence of degenerate classical geometries that are mandatory to accommodate the absorption of probability flux in the Euclidean domains. Just as the creation (and annihilation) of a classical cosmology may correspond to such domains where a classical spacetime description breaks down, the same may be true at the end points of localised gravitational collapse.