Quantum fields in Minkowski spacetime

The theory of quantum fields in curved spacetime is a generalization of the well-established theory of quantum fields in Minkowski spacetime. To a great extent, the behavior of quantum fields in curved spacetime is a direct consequence of the corresponding flat spacetime theory. Local entities, such as the field equations and commutation relations, are to a large extent determined by the principle of general covariance and the principle of equivalence. However, global entities which are unique in Minkowski spacetime lose that uniqueness in curved spacetime. For example, the vacuum state, which in Minkowski spacetime is determined by Poincaré invariance, is not unambiguously determined in curved spacetime. This ambiguity is closely tied to the phenomenon of particle creation by certain gravitational fields, as in the expanding universe or near a black hole. It is logical, therefore, to review the relevant aspects of flat spacetime quantum field theory. This will serve to establish the necessary background, to fix our notation, and to highlight those aspects of the theory which can be carried over to curved spacetime, as well as those which lose their meaning in curved spacetime. We will often be brief, emphasizing concepts while omitting many derivations, and only touch on particular topics. Our discussion of the curved spacetime theory in later chapters will be more detailed. In this initial chapter, we discuss the canonical formulation, including the Schwinger action principle and the relation between symmetry transformations and conserved currents (Schwinger 1951b, 1953).