Microlocal Analysis of Quantum Fields on Curved Spacetimes

2000 Mathematics Subject Classification (primary): • 81T13 Yang-Mills and other gauge theories in quantum field theory • 35L10 Second-order hyperbolic equations • 35L40 First-order hyperbolic systems • 58J40 Pseudodifferential and Fourier integral operators on manifolds; (secondary): • 81T28 Thermal quantum field theory • 35L15 Initial value problems for second-order hyperbolic equations • 35L45 Initial value problems for first-order hyperbolic systems • 58J47 Propagation of singularities; initial value problems on manifold • 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

This monograph focuses on free fields and the corresponding quasi-free states, and, more precisely, on Klein Gordon fields and Dirac fields. The first chapters are devoted to preliminary material on CCR -algebras, quasi-free states, wave equations on Lorentzian manifolds, microlocal analysis, and to the important Hadamard condition characterizing physically acceptable quantum states on curved spacetimes. In the later chapters, more advanced tools of microlocal analysis, like the global pseudo-differential calculus on non-compact manifolds, are used to construct and study Hadamard states for Klein Gordon fields by various methods, in particular, by scattering theory and by Wick rotation arguments. In the last chapter, the fermionic theory of free Dirac quantum fields on Lorentzian manifolds is described in some detail. This monograph is addressed to both mathematicians and mathematical physicists. Mathematicians will find the book useful as a rigorous exposition of free quantum fields on curved spacetimes and as an introduction to some interesting and physically important problems arising in this domain. Mathematical physicists may find this text a helpful introduction to the use of more advanced tools of microlocal analysis in this area of research.