Quantum Gravity

While a satisfactory theory of full quantum gravity continues to elude us, the attempt to anticipate some of the properties of such a theory has led to many interesting developments. Especially, Hawking's 1974 prediction of black hole evaporation, which was based on consideration of quantum field theory in curved spacetime, suggests that there must be yet-to-be-discovered deep interconnections between quantum theory, gravity and thermodynamics. More generally, the very existence of the problem of quantum gravity has changed our perspective on each of the separate theories of classical general relativity and quantum field theory and focussed attention on issues (e.g. the problem of singularities in classical general relativity or the problem of locality in quantum field theory) which might be expected to be of relevance for the unification problem. Further, both at the theoretical and experimental/observational level, the two subject areas have now essentially merged, with very-high-energy phenomena believed to have dominated the era just after the big bang and hence to have determined the present structure of the universe.

Recent research by the York group concerns the following issues: 

1. Quantum field theory in curved spacetime.

(a) Quantum energy inequalities on the renormalised stress-energy tensor.

Most forms of classical matter have a positive energy density, a fact closely related to our experience of gravity as an attractive force. However, the situation is very different in quantum field theory, where the energy density can typically be made as negative as we like. Recent work by Fewster has been directed towards developing "quantum energy inequalities" for quantum fields which show (roughly speaking) that the magnitude and duration of negative energy densities are constrained: the energy density cannot be too negative for too long. Further details (including references) can be found here.

(b) Infrared properties of the gravitational field in the Early Universe.

Higuchi, with students Kouris and Weeks, has shown that the growth at long distance of the graviton two-point function in an inflationary spacetime is not reflected in physical two-point functions. He is currently examining claims in the literature that the cosmological constant would be suppressed in such spacetimes due to the above-mentioned growth of the graviton two-point function. See gr-qc/0212031 and references therein.

(c) The Hadamard condition.

In Minkowski space there is an obvious natural vacuum state for quantum field theory picked out by Poincare invariance. General spacetimes do not have any symmetries and therefore no single natural vacuum state. Instead, a class of physical states is indentified (the Hadamard class). The first fully precise definition of the Hadamard condition was given by Kay and Wald in 1991; more recently Radzikowski showed that this condition could be reformulated in terms of microlocal analysis. Work by the York group has extended and refined this formulation; in particular, it plays an important role in Fewster's work on quantum inequalities and was used by Kay in work (with Radzikowski and Wald) on time machine spacetimes (see below).

2. Dirac quantisation of general relativity.

General relativity may be regarded as a constrained dynamical system. Although there is a standard method for quantizing constrained systems, due to Dirac, there are obstacles to applying this method to general relativity. There has been a claim that these obstacles can be overcome, and Higuchi is currently trying to determine whether or not this claim is justified.

3. Classical limit of radiation reaction in quantum field theory

Higuchi has studied a model in quantum field theory in which one can consider the radiation reaction to a charged particle and then take the classical limit. He found that the result agrees with the standard Lorentz-Dirac theory (quant-ph/0208017). He is currently attempting to compare the radiation reaction in the Lorentz-Dirac theory and that in the quantum theory of a more realistic model.

4. Gravitationally induced decoherence.

Kay is presently continuing to work, partly together with Varqa Abyaneh, on a theory (see hep-th/9810077) according to which it is the entanglement between the quantum gravitational field and the quantum state of macroscopic quantum systems which is ultimately responsible for those systems being (with some probability distribution) in one or other of a number of definite spatial configurations. In particular, in the case of Schrodinger's cat this theory offers an explanation as to why the cat has to be either dead or alive.

Over the past decade or so, various other issues have been investigated by the York group. These include:

The theory of the Hawking effect.

In the early 1990s, Kay was involved in proving mathematical theorems which improved our understanding of why it is that - as discovered by Stephen Hawking in 1974 - black holes have to be hot. 

The question of whether it is possible in principle to manufacture a "time machine".

Work by the York group has contributed to the present understanding that time machines cannot be manufactured. In particular, Kay has been involved in proving rigorous mathematical theorems (see, e.g., gr-qc/9603012 and gr-qc/9708028) which may be interpreted as telling us that it is impossible to manufacture a time machine which would allow travel to the past by warping spacetime in a way which is describable in terms of the traditional notions of spacetime geometry familiar from Einstein's classical theory of general relativity. These theorems thus tend to lend support to Stephen Hawking's Chronology Protection Conjecture although they don't exclude the possibility of time machines which are so exotic that they warp spacetime in ways which are not describable in classical terms. For a recent overview of the work by Kay mentioned above see gr-qc/0103056.

Other work by the York group has studied the properties of quantum systems on spacetimes containing time machines (setting aside the question of whether such a thing could be manufactured). Kay proposed a weakening of the usual formulation of quantum field theory for such situations; this was further investigated by Fewster and Higuchi (gr-qc/9508051) and then Fewster (gr-qc/9804012). This led to the conclusion that - at least in two dimensions - the proposed framework could not be implemented on generic time machine spacetimes. Other work conducted by Fewster, Higuchi and Wells (gr-qc/9603045) also supports the view that quantum theory (at least as presently understood) cannot generally be formulated in the presence of time machines.

The interaction of black holes with classical & quantum fields.

Higuchi showed that the absorption cross section by a black hole (of arbitrary spacetime dimension) of any kind of massless scalar particle is exactly equal to the horizon area (hep-th/0108144). He also investigated how a charge just outside a black hole responds to Hawking radiation (see gr-qc/0011070 and references therein).

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