Quantum Field Theory on Curved Backgrounds, I. The Euclidean Functional Integral

Abstract

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder-Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.

33 pages

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Cited by in corpus (14)

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• Higher Derivative Scalar Quantum Field Theory in Curved Spacetime
• Reflection Positivity and Monotonicity
• Non-existence of Skyrmion-Skyrmion and Skyrmion-anti-Skyrmion static equilibria
• Reflection positivity in higher derivative scalar theories
• Reflection positivity for the circle group
• Snowmass White Paper: The Quest to Define QFT
• Lattice radial quantization by cubature
• Reflection Positive Stochastic Processes Indexed by Lie Groups
• A relationship between scalar Green functions on hyperbolic and Euclidean Rindler spaces
• KMS conditions, standard real subspaces and reflection positivity on the circle group
• Transition amplitudes and sewing properties for bosons on the Riemann sphere
• Green functions and dimensional reduction of quantum fields on product manifolds
• Euclidean scalar Green functions near the black hole and black brane horizons