Quantum field theory on curved spacetime

at the Erwin Schrödinger Institute

Robert Wald, University of Chicago rmwa@midway.uchicago.edu

A program on ``Quantum field theory on curved space times'' was held at the Erwin Schroedinger Institute in Vienna, Austria from July 1 through August 31, 2002. The main goal of this program was to bring together researchers with expertise in general relativity and researchers with expertise in mathematical aspects of quantum field theory, in order to address some problems of mutual interest in quantum field theory in curved spacetime. Approximately 25 researchers in quantum field theory in curved spacetime and related areas participated in the program. The following is a brief summary of some of the main topics and results discussed during the program.

A great deal of progress has been made in recent years in characterizing the ``ultraviolet divergences'' of quantum fields in curved spacetime and developing renormalization theory for interacting quantum fields. Seminars by S. Hollands, K. Fredenhagen, R. Verch, and R. Wald reported on this recent progress. The difficulties resulting from the lack of a preferred vacuum state and a preferred Hilbert space representation of the canonical commutation relations for the free field have been overcome by formulating the theory within the algebraic approach. The difficulties associated with the lack of a global notion of a Fourier transform (so that the usual momentum space methods for renormalization cannot be used) have been overcome by the use of the methods of ``microlocal analysis''. Finally, the difficulties associated with the absence of a notion of ``Poincare invariance'' (or any other symmetries) in general curved spacetime have been overcome by imposing the condition that the quantum fields of interest be constructed locally and covariantly out of the spacetime metric. The upshot is that perturbative renormalization theory for quantum fields in curved spacetime is now on as sound a footing as in Minkowski spacetime. Furthermore, theories that are renormalizable in Minkowski spacetime will also be renormalizable in curved spacetime, although additional ``counterterms'' corresponding to couplings of the quantum field to curvature will arise.

Although the Hawking effect was derived more than 25 years ago, there remains a difficulty with the derivation in that it relies on the properties of quantum fields in a regime where one has no right to expect quantum field theory in curved spacetime to be a good approximation. Specifically, consider the modes of the quantum field that correspond to ``particles'' that are seen by observers near infinity to emerge from the black hole at late times. When traced backward in time, these modes become highly blueshifted and correspond to ``trans-Planckian'' frequencies and wavelengths at early times. Thus, the Hawking effect appears to rely on assumptions concerning the initial state and behavior of degrees of freedom in the trans-Planckian regime. Similar issues also arise in cosmology when considering the ``quantum fluctuations'' responsible for the formation of large scale structure at late times. Seminars by Jacobson and Unruh explained the nature of the trans-Planckian issues and described some simple models where the effects of modifying dynamical laws in the trans-Planckian regime can be analyzed. These models support the view that the Hawking effect is robust with respect to changes in physical laws in the trans-Planckian regime.

It is well known that in quantum field theory in flat or curved spacetime, the expected energy density at a point can be made arbitrarily negative. However, during the past ten years, some global restrictions on negative energy have been derived. In particular, ``quantum inequalities'' have been derived, which put a lower bound on the energy density measured along the worldline of an observer with a (smooth, compact support) ``sampling function'' $f(\tau)$. Originally, such bounds were derived by non-rigorous methods in certain special cases, but recently a rigorous and completely general derivation of quantum inequalities has been given using the methods of microlocal analysis. Many issues remain open, however, such as the derivation of optimal bounds and whether some version may hold of the average null energy condition (which asserts the non-negativity of the integral over a complete null geodesic of the stress energy tensor contracted twice with the tangent to the null geodesic). These issues were explored in seminars by Ford, Fewster, Roman, Flanagan, and Pfenning. In research arising directly from discussions occurring during the program, progress also was made toward deriving quantum inequalities for quantities other than the stress-energy tensor.

The Bisognano-Wichmann theorem states that in Minkowski spacetime, the restriction of the vacuum state to a wedge region is a KMS state with respect to a 1-parameter subgroup of the Poincare group. (This result can be viewed as a mathematically rigorous version of the ``Unruh effect''.) The mathematical theory underlying this result is the modular theory of Tomita and Takesaki. The computation of modular transformations was discussed in a seminar by Ynvason, and the determination of analogous wedge regions and states in deSitter spacetime was discussed by Guido. In Anti-de Sitter spacetime the wedges are in one to one correspondence with double cones in the Minkowski spacetime at spacelike infinity. One thus obtains an algebraic version of AdS-CFT correspondence which was discovered by Rehren. A seminar by Rehren discussed this correspondence in more detail by considering the relation between limits of fields at the boundary and the partition function for specified boundary values.

In the loop variables/quantum geometry approach to quantum gravity, one first defines a ``kinematical Hilbert space'' and then tries to define the action of the Hamiltonian constraint operator on these ``kinematical states''. In this approach, the Hamiltonian constraint operator is not intrinsically well defined (i.e., ``regularization'' is needed), but the nature of this regularization appears to be very different from the usual regularization of ``ultraviolet divergences'' occurring in quantum field theory. One of the goals of our program was to explore the nature of renormalization in the loop variables/quantum geometry approach to quantum gravity and to understand its relationship to renormalization in ordinary quantum field theory. Seminars by Lewandowski, Perez, Ashtekar, Thiemann, Bojowald, Fairhurst, and Sahlmann described in detail various aspects of the loop variables/quantum geometry approach. The extended interactions between the researchers in the loop variables/quantum geometry approach and researchers in quantum field theory resulting from these seminars as well as from numerous private discussions were very fruitful. In particular, simple quantum field theory analogs of some of the constructions used in the loop variables/quantum geometry approach were obtained and explored.

Overall, the program appears to have been very successful in promoting considerable productive interaction between groups of researchers who generally have had only limited interaction with each other. One may hope that the ``cross-fertilization'' and new collaborations initiated by these interactions will bear fruit for many years to come.