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 \(\mathrm{CCR}^*\)-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.