Abstract
To make sense of quantum field theory in an arbitrary (globally hyperbolic) curved spacetime, the theory must be formulated in a local and covariant manner in terms of locally measurable field observables. Since a generic curved spacetime does not possess symmetries or a unique notion of a vacuum state, the theory also must be formulated in a manner that does not require symmetries or a preferred notion of a “vacuum state” and “particles”. We propose such a formulation of quantum field theory, wherein the operator product expansion (OPE) of the quantum fields is elevated to a fundamental status, and the quantum field theory is viewed as being defined by its OPE. Since the OPE coefficients may be better behaved than any quantities having to do with states, we suggest that it may be possible to perturbatively construct the OPE coefficients—and, thus, the quantum field theory. By contrast, ground/vacuum states—in spacetimes, such as Minkowski spacetime, where they may be defined—cannot vary analytically with the parameters of the theory. We argue that this implies that composite fields may acquire nonvanishing vacuum state expectation values due to nonperturbative effects. We speculate that this could account for the existence of a nonvanishing vacuum expectation value of the stress-energy tensor of a quantum field occurring at a scale much smaller than the natural scales of the theory.
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References
Unruh W.G.: Notes on black hole evaporation. Phys. Rev. D 14, 870–892 (1976)
Unruh W.G., Wald R.M.: What happens when an accelerating observer detects a rindler particle?. Phys. Rev. D 29, 1047–1056 (1984)
Hollands S., Wald R.M.: Local wick polynomials and time ordered products of quantum fields in curved space. Commun. Math. Phys. 223, 289–326 (2001) [gr-qc/0103074]
Hollands S., Wald R.M.: Existence of local covariant time-ordered-products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002) [gr-qc/0111108]
Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: a new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003) [math-ph/0112041]. see also K. Fredenhagen, “Locally covariant quantum field theory,” [arXiv:hep-th/0403007]
Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and Wick polynomials on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996)
Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)
Hollands, S., Wald, R.M.: Axiomatic quantum field theory in curved spacetime. arXiv:0803.2003
Hollands S.: The operator product expansion for perturbative quantum field theory in curved spacetime. Commun. Math. Phys. 273, 1 (2007) [arXiv:gr-qc/0605072]
Bernard C., Duncan A., LoSecco J., Weinberg S.: Exact spectral-function sum rules. Phys. Rev. D 12, 792–804 (1975) (see appendix)
Fulling S.A., Narcowich F.J., Wald R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime, II. Ann. Phys. 136, 243 (1981)
Hollands, S., Kopper, C.: in preparation
Rivasseau, V.: From perturbative to constructive renormalization, p. 336. University of Princeton, Princeton (1991) (Princeton series in physics)
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Fourth Award in the 2008 Essay Competition of the Gravity Research Foundation.
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Hollands, S., Wald, R.M. Quantum field theory in curved spacetime, the operator product expansion, and dark energy. Gen Relativ Gravit 40, 2051–2059 (2008). https://doi.org/10.1007/s10714-008-0672-y
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DOI: https://doi.org/10.1007/s10714-008-0672-y