Microlocal Analysis of Quantum Fields on Curved Spacetimes

Abstract

Contents 1. Introduction 1 2. Free Klein-Gordon fields on Minkowski spacetime 5 3. Fock quantization on Minkowski space 11 4. CCR algebras and quasi-free states 14 5. Free Klein-Gordon fields on curved spacetimes 30 6. Quasi-free states on curved spacetimes 44 7. Microlocal analysis of Klein-Gordon equations 48 8. Hadamard states 55 9. Vacuum and thermal states on stationary spacetimes 63 10. Pseudodifferential calculus on manifolds 70 11. Construction of Hadamard states by pseudodifferential calculus 79 12. Analytic Hadamard states and Wick rotation 89 13. Hadamard states and characteristic Cauchy problem 98 14. Klein-Gordon fields on spacetimes with Killing horizons 109 15. Hadamard states and scattering theory 115 16. Feynman propagator on asymptotically Minkowski spacetimes 120 17. Dirac fields on curved spacetimes 126 References 139

The fermionic signature operator in de Sitter spacetime

“…Among the plethora of all possible states, one wants to select those which are physically sensible. Indeed, these are characterized mathematically by the so-called Hadamard condition, a constraint on the wavefront set of the underlying two point distribution, see [24,31] for recent surveys on this topic.…”

“…Yet the attention has been focused mainly on bosonic scalar free field theories. On the contrary, if one considers spinor fields, which will be at the heart of this paper, only few techniques have been thoroughly analysed, ranging from a positive frequency splitting on static backgrounds, to pseudodifferential calculus [24], to more ad hoc methods e.g. [9,25,26] in which cosmological spacetimes were considered.…”

Partial Differential Equations and Quantum States in Curved Spacetimes

“…Hadamard states play a distinguished role in the algebraic formulation of quantum field theory, see, e.g., [26][27][28] for a recent review.…”

“…One can show [1,27] that singularities of solutions of the wave equation on globally hyperbolic spacetimes propagate along light-like geodesics. Now, the four-dimensional version of our Hamiltonian (7) vanishes identically on light-like covectors.…”

Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

“…This quantization scheme goes under the name of algebraic quantum field theory, and it is especially well-suited for formulating quantum theories also on manifoldsee e.g., [18,36] for textbook, to [4,5,10,35,45] for recent reviews, [13][14][15][16] for homotopical approaches and [19-21, 24-29, 32-34] for some applications.…”