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Thermal equilibrium states in perturbative Algebraic Quantum Field Theory in relation to Thermal Field Theory

Joao Braga De Goes E Vasconcellos

Published Dec 10, 2019• 0 citations
Publisher

Abstract

In the first part, we analyse the properties of an interacting, massive scalar field in an equilibrium state over Minkowski spacetime. We compare the known real- and imaginary-time formalisms of Thermal Field Theory with the recent construction by Fredenhagen and Lindner of a KMS state for perturbative interacting theories in the context of perturbative Algebraic Quantum Field Theory, in the adiabatic limit. In particular, we show that the construction of Fredenhagen and Lindner reduces to the real-time formalism only if the cocycle which intertwines between the free and interacting dynamics can be neglected. Furthermore, the Fredenhagen and Lindner construction reduces to the ordinary imaginary-time formalism if one considers the expectation value of translation invariant observables. We thus conclude that a complete description of thermal equilibrium for interacting scalar fields is generally obtained only by means of the state constructed by Fredenhagen and Lindner, which combines both formalisms of Thermal Field Theory. We also discuss the properties of the expansion of the Fredenhagen and Lindner construction in terms of Feynman diagrams in the adiabatic limit. We finally provide examples showing that the real- and the imaginary-time formalisms fail to describe thermal equilibrium already at first or second order in perturbation theory. The results presented in this part are summarized in (BDP19). In the second part, we discuss the so-called secular effects, characterized by the appearance of polynomial divergences in the large time limit of truncated perturbative expansions of expectation values in Quantum Field Theory. We show that, although such effect is an artifact of perturbation theory, and thus may not be obtained via exactly solving the dynamical equation if possible, they do not represent the breakdown of perturbation theory itself. Instead, we show that the polynomial divergences follow from a bad choice of state, and we present examples of states which produce expectation values whose perturbative expansion does not present secular effects. In particular, we point that it is possible to obtain non time-divergent perturbative expressions from thermal equilibrium states for the interacting theory. This last part is based on a research project which, by the time this thesis was written, had not been concluded yet.

DOI
10.15167/braga-de-goes-e-vasconcellos-joao_phd2019-12-10

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In the first part, we analyse the properties of an interacting, massive scalar field in an equilibrium state over Minkowski spacetime. We compare the known real- and imaginary-time formalisms of Thermal Field Theory with the recent construction by Fredenhagen and Lindner of a KMS state for perturbative interacting theories in the context of perturbative Algebraic Quantum Field Theory, in the adiabatic limit. In particular, we show that the construction of Fredenhagen and Lindner reduces to the real-time formalism only if the cocycle which intertwines between the free and interacting dynamics can be neglected. Furthermore, the Fredenhagen and Lindner construction reduces to the ordinary imaginary-time formalism if one considers the expectation value of translation invariant observables. We thus conclude that a complete description of thermal equilibrium for interacting scalar fields is generally obtained only by means of the state constructed by Fredenhagen and Lindner, which combines both formalisms of Thermal Field Theory. We also discuss the properties of the expansion of the Fredenhagen and Lindner construction in terms of Feynman diagrams in the adiabatic limit. We finally provide examples showing that the real- and the imaginary-time formalisms fail to describe thermal equilibrium already at first or second order in perturbation theory. The results presented in this part are summarized in (BDP19). In the second part, we discuss the so-called secular effects, characterized by the appearance of polynomial divergences in the large time limit of truncated perturbative expansions of expectation values in Quantum Field Theory. We show that, although such effect is an artifact of perturbation theory, and thus may not be obtained via exactly solving the dynamical equation if possible, they do not represent the breakdown of perturbation theory itself. Instead, we show that the polynomial divergences follow from a bad choice of state, and we present examples of states which produce expectation values whose perturbative expansion does not present secular effects. In particular, we point that it is possible to obtain non time-divergent perturbative expressions from thermal equilibrium states for the interacting theory. This last part is based on a research project which, by the time this thesis was written, had not been concluded yet.
In this thesis interacting relativistic and non relativistic Quantum Field Theories (QFT) are studied, investigating their equilibrium configurations and their long time evolution in the framework of perturbative Algebraic Quantum Field Theory, studying also convergence in a specific model. In the first part, concerning the relativistic case, we study, in formal perturbation theory, interacting KMS states generalising results by Fredenhagen and Lindner to fermionic field theories. As an application their stability is investigated together with the characterisation of the Debye screening effect in plasmas via the explicit solution of the Semiclassical Maxwell equations. Connected to these results, we also initiate a systematic study of secular growths appearing in the truncated perturbative series and affecting the validity of perturbation theory in its long time predicting capability. We restrict to theories on Minkowski spacetime and provide, for interactions depending on time via a switching process, a sufficient condition preventing secular growths in the truncated perturbative series. In the second part, we focus on a model of non relativistic quantum field theory. We construct equilibrium states at positive temperature in the presence of a condensation phase for a gas of non relativistic Bose particles on an infinite space interacting through a localised two body interaction. The convergence is proven, both for the interacting KMS state and for the entropy of the interacting KMS state relative to the free KMS state. The strategy of our proof combines ideas of Algebraic QFT, Constructive QFT and methods of statistical mechanics in the operator algebraic setting. Limits where the localisation of the two-body interaction is removed are eventually discussed, in combination with limits of the other physical parameters of the theory such as temperature, strength of the interaction and chemical potential. Finally, in the appendix, we present results on the Araki-Uhlmann relative entropy in fermionic quantum field theories, as well as an entropy-area relation, obtained via semiclassical methods, for the cosmological horizon in the static patch of de Sitter spacetime sourced by a coherent excitation of a free quantum scalar field. This completes the collection of original results obtained during the PhD.
In this thesis we study some non-equilibrium aspects of an interacting, massive scalar field theory treated perturbatively. This is done analysing some properties of the interacting KMS state constructed by Fredenhagen and Lindner [FL14] in the framework of perturbative Algebraic Quantum Field Theory. In the first part we treat the stability of KMS states, namely we check whether the free state evolved with the interacting dynamics converges to the interacting state. In the meantime we also analyse the return to equilibrium, that is the analogous property with the role of the free and interacting quantities exchanged. We prove that those two properties hold if the perturbation potential is of spatial compact support and that they fail otherwise, even if an adiabatic mean is considered. While the stability leads to non-curable divergencies, the analysis of return to equilibrium gives something finite, which is interpreted as a non-equilibrium steady state. The novelty of this non-equilibrium state drove us to try to characterise it in more details. To do so, in the second part we introduce relative entropy and entropy production for perturbative quantum field theory, justifying those definitions by proving their main properties. Furthermore, we showed that they are well-defined in the adiabatic limit if we consider densities. These two definitions allowed to prove that the non-equilibrium steady state is thermodynamically trivial, namely it has zero entropy production. The present thesis is based on [DFP18a, DFP18b].
Methods of thermal field theory can broadly be put into two categories, according to their use of imaginary time or real time in the ensemble averages. The method of imaginary time was first proposed by Matsubara [1]. The real time method was introduced later, originally by Schwinger [2] and Keldysh [3] and followed by others [4], who applied it to study non-equilibrium processes. It was also applied to equilibrium situations by Mills [5]. The real time method was reformulated in the so-called thermofield dynamics by Umezawa and collaborators [6], who suggested a doubling of field variables in the statistical context. Niemi and Semenoff [7] clarified this apparent doubling in terms of the time contour needed in the real time formulation. A survey of early literature is available in [8]. Physical applications including later works are presented also in books [9, 10]. For systems in thermal equilibrium, one can develop perturbative expansion in both methods in parallel to the conventional method in vacuum. The difference shows up in the form of the propagators. Because of the finite interval of imaginary time, the frequencies are discrete, giving the propagator as a sum over these frequencies. On the other hand, the real time method has two infinite intervals of time, giving rise to a 2 × 2 matrix for the propagator. The majority of calculations reported in the literature are done in the imaginary time method, avoiding matrix propagators in the real time method. For systems out of thermal equilibrium, the real time method suggests itself as the appropriate framework to calculate physical quantities. However, for quantities such as the response functions (we address this problem in the last chapter) the end results relate to thermal equilibrium and so can be evaluated by either method. But, to follow the time dependence of quantities during non-equilibrium phases, such as in the early universe [11–13] – a topic too distant to discuss in this book – the real time method is the only option. In this book we adopt the real time method to calculate thermal quantities in equilibrium. This chapter is devoted to deriving the matrix form of propagators arising in this method.
This is the first of a series of papers which describe the functional-integral approach to the study of the statistical and kinetic properties of nonequilibrium quantum fields in flat and curved spacetimes. In this paper we treat a system of self-interacting bosons described by \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ scalar fields in flat space. We adopt the closed-time-path (CTP or ``in-in'') functional formalism and use a two-particle irreducible (2PI) representation for the effective action. These formalisms allow for a full account of the dynamics of quantum fields, and put the correlation functions on an equal footing with the mean fields. By assuming a thermal distribution we recover the real-time finite-temperature theory as a special case. By requiring the CTP effective action to be stationary with respect to variations of the correlation functions we obtain an infinite set of coupled equations which is the quantum-field-theoretical generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Truncation of this series leads to dissipative characteristics in the subsystem. In this context we discuss the nature of dissipation in interacting quantum fields.To one-loop order in a perturbative expansion of the CTP effective action, the 2PI formalism yields results equivalent to the leading 1/N expansion for an O(N)-symmetric scalar field. To higher-loop order we introduce a two-time approximation to separate the quantum-field effects of radiative correction and renormalization from the statistical-kinetic effects of collisions and relaxation. In the weak-coupling quasiuniform limit, the system of nonequilibrium quantum fields can subscribe to a kinetic theory description wherein the propagators are represented in terms of relativistic Wigner distribution functions. From a two-loop calculation we derive the Boltzmann equation for the distribution function and the gap equation for the effective mass of the quasiparticles. One can define an entropy function for the quantum gas of quasiparticles which satisfies the H theorem. We also calculate the limits to the validity of the binary collision approximation from a three-loop analysis. The theoretical framework established here can be generalized to nonconstant background fields and for curved spacetimes.
Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.