Hadamard states for quantized Dirac fields on Lorentzian manifolds of bounded geometry

We consider Dirac equations on even dimensional Lorentzian manifolds of bounded geometry with a spin structure. For the associated free quantum field theory, we construct pure Hadamard states using global pseudodifferential calculus on a Cauchy surface. We also give two constructions of Hadamard states for Dirac fields for arbitrary spacetimes with a spin structure.

[1]  N. Dencker.  On the propagation of polarization sets for systems of real principal type , 1982 .

[2]  M. A. Shubin,et al.  Spectral theory of elliptic operators on non-compact manifolds: qualitative results , 1992 .

[3]  Ko Sanders.  The locally covariant Dirac field , 2009, 0911.1304.

[4]  Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime , 2000, math-ph/0003015.

[5]  K. Nomizu,et al.  Foundations of Differential Geometry, Volume I. , 1965 .

[6]  A. Bernal,et al.  On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem , 2003, gr-qc/0306108.

[7]  C. Gérard,et al.  Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry , 2016, 1602.00930.

[8]  J. Bourguignon,et al.  A Spinorial Approach to Riemannian and Conformal Geometry , 2015 .

[9]  R. Verch,et al.  A local-to-global singularity theorem for quantum field theory on curved space-time , 1996 .

[10]  Oussama Hijazi,et al.  A conformal lower bound for the smallest eigenvalue of the Dirac operator and killing spinors , 1986 .

[11]  F. Finster,et al.  A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds II - Space-Times of Infinite Lifetime , 2013, 1312.7209.

[12]  A. Trautman.  Connections and the Dirac operator on spinor bundles , 2008 .

[13]  Bernd Ammann,et al.  Complex Powers and Non-compact Manifolds , 2002 .

[14]  J. Sjöstrand.  Projecteurs adiabatiques du point de vue pseudodifférentiel , 1993 .

[15]  Passivity and Microlocal Spectrum Condition , 2000, math-ph/0002021.

[16]  On microlocalization and the construction of Feynman propagators for normally hyperbolic operators , 2020, 2012.09767.

[17]  C. Gérard,et al.  Communications in Mathematical Physics Construction of Hadamard States by Pseudo-Differential Calculus , 2022 .

[18]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , 1983 .

[19]  J. Dimock.  Dirac quantum fields on a manifold , 1982 .

[20]  Mark Sweeny,et al.  Singularity structure of the two-point function in quantum field theory in curved spacetime , 1978 .

[21]  J. Eldering.  Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry , 2012, 1205.2753.

[22]  L. Fatibene,et al.  Spin structures on manifolds , 1998 .

[23]  C. G'erard.  Microlocal Analysis of Quantum Fields on Curved Spacetimes , 2019, 1901.10175.

[24]  P. Gauduchon,et al.  Generalized cylinders in semi-Riemannian and spin geometry , 2003, math/0303095.

[25]  Marek J. Radzikowski.  Micro-local approach to the Hadamard condition in quantum field theory on curved space-time , 1996 .

[26]  R. Verch,et al.  Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime , 2000, math-ph/0008029.

[28]  Y. Kordyukov,et al.  Lp-Theory of elliptic differential operators on manifolds of bounded geometry , 1991, Acta Applicandae Mathematicae.

[29]  The stress energy tensor of a locally supersymmetric quantum field on a curved spacetime , 1995, gr-qc/9505014.

[30]  Some counterexamples about globally hyperbolic spacetimes , 2021, 2110.13672.

[31]  Yvette Kosmann.  Dérivées de Lie des spineurs , 1971 .

[32]  Y. Egorov,et al.  Fourier Integral Operators , 1994 .

[33]  Antonio N. Bernal,et al.  Communications in Mathematical Physics Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes , 2005 .

[34]  Nuclearity, Local Quasiequivalence and Split Property for Dirac Quantum Fields in Curved Spacetime , 2001, math-ph/0106028.

[35]  C. Gérard,et al.  Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes , 2016, 1609.00190.

[36]  R. Geroch.  Spinor structure of space-times in general relativity. i , 1968 .

[37]  The Hadamard Condition for Dirac Fields and Adiabatic States on Robertson–Walker Spacetimes , 1999, gr-qc/9906076.