Abstract
We construct the Wightman and Green functions in a large class of models of perturbative QFT in the four-dimensional Minkowski space in the Epstein–Glaser framework. To this end we prove the existence of the weak adiabatic limit, generalizing the results due to Blanchard and Seneor. Our proof is valid under the assumption that the time-ordered products satisfy certain normalization condition. We show that this normalization condition may be imposed in all models with interaction vertices of canonical dimension 4 as well as in all models with interaction vertices of canonical dimension 3 provided each of them contains at least one massive field. Moreover, we prove that it is compatible with all the standard normalization conditions which are usually imposed on the time-ordered products. The result applies, for example, to quantum electrodynamics and non-abelian Yang–Mills theories.
Highlights
Covariant perturbative quantum field theory (QFT) in the Minkowski space is one of the most successful modern physical theories
The main result of the paper is the proof of the existence of the weak adiabatic limit in the four-dimensional Minkowski space in models with the interaction vertices L1, . . . , Lq satisfying one of the following conditions: (1) ∀l∈{1,...,q} dim(Ll) = 4 or (2) ∀l∈{1,...,q} dim(Ll) = 3 and Ll contains at least one massive field, where dim(B) is the canonical dimension of the polynomial B. (The case of models with the interaction vertices of mixed dimensions is briefly discussed in “Appendix B.”) It is a generalization of the results due to Blanchard and Seneor [1] mentioned in the previous paragraph
We proved the existence of the weak adiabatic limit in the Epstein–Glaser approach to perturbative QFT in a large class of models
Summary
Covariant perturbative quantum field theory (QFT) in the Minkowski space is one of the most successful modern physical theories. The physical Wightman and Green functions are defined as the limits of the vacuum expectation values of the products or the time-ordered products of the interacting fields with IR regularization. We prove that the following condition has to be satisfied for the weak adiabatic limit to exist: The self-energies of the massless fields (which are used in the definition of a given model) have to be normalized such that the physical masses of these fields vanish. The Wightman and Green functions cannot be defined by means of the weak adiabatic limit in models in which the correct mass normalization of massless fields is not possible. An example of such model is the massless φ3 theory.
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