The Generalised Principle of Perturbative Agreement and the Thermal Mass

Abstract

Abstract. The Principle of Perturbative Agreement, as introduced by Hollands & Wald, is a renormalisation condition in quantum field theory on curved spacetimes. This principle states that the perturbative and exact constructions of a field theoretic model given by the sum of a free and an exactly tractable interaction Lagrangean should agree. We develop a proof of the validity of this principle in the case of scalar fields and quadratic interactions without derivatives which differs in strategy from the one given by Hollands & Wald for the case of quadratic interactions encoding a change of metric. Thereby we profit from the observation that, in the case of quadratic interactions, the composition of the inverse classical Møller map and the quantum Møller map is a contraction exponential of a particular type. Afterwards, we prove a generalisation of the Principle of Perturbative Agreement and show that considering an arbitrary quadratic contribution of a general interaction either as part of the free theory or as part of the perturbation gives equivalent results. Motivated by the thermal mass idea, we use our findings in order to extend the construction of massive interacting thermal equilibrium states in Minkowski spacetime

Equilibrium States in Thermal Field Theory and in Algebraic Quantum Field Theory

“…Furthermore, the expectation values obtained in that limit do not depend on the form of the function χ used to smoothly switch on the interactioncf. [FL14,DHP16]. In this way an equilibrium state at inverse temperature β for a self-interacting scalar field is obtained.…”

“…We are considering first of all the limit h → 1 of the potential in the Bogoliubov map and then the limit h → 1 of every L h . Further details on this steps can be found in [FL14] and in the appendix of [DHP16].…”

“…Further details about this construction will be given in section 2. Additional properties of the state constructed in [FL14] have been discussed in [DHP16,DFP18,DFP19]. Notice that, in general, the state ω β,V satisfies the KMS condition with respect to the interacting time evolution α V t only if the correction to ω β described by the elements of the sum present in (4) for l > 1 are taken into account.…”

“…where w m = |p| 2 + m 2 . Notice that, thanks to the validity of the principle of perturbative agreement [HW05,DHP16], we can treat changes of the mass m tom = √ m 2 + λ in two ways. Either exactly substituting the mass m withm in the various propagators appearing in the theory or with perturbation theory, namely using an interaction Lagrangian of the form λφ 2 .…”

Weak Adiabatic Limit in Quantum Field Theories with Massless Particles

“…Adiabatic limit of the inclusive cross-sections in low orders of the perturbation theory of QED was considered in [43] (see also [20,21]). Finally, the existence of the expectation values of the products of the interacting fields in thermal states has been recently proved by Fredenhagen and Lindner in the EG framework with the use of the time-slice axiom and the time-averaged Hamiltonian [25] (see also [12,39]). Let us also note that the method of Epstein and Glaser has been reformulated and generalized in a number of ways.…”