Abstract We review two recent applications of operator-algebraic methods to the analysis of homogeneous Bose gases at finite temperature. The first part concerns the semiclassical description of Bose–Einstein condensation in the framework of Weyl $$C^*$$ C ∗ -algebras. In this setting, the semiclassical parameter is naturally linked to the density of the system, and the classical weak KMS condition emerges as the limit of quantum equilibrium states. We show in particular that the condensate structure is preserved in a suitable large-density regime. The second part discusses a joint work with Prof. Jan Dereziński which the Araki–Woods representation is used to formulate a systematic perturbative theory for interacting Bose gases at positive temperature. Within this representation, thermal effects are encoded directly in the field operators, allowing for a transparent implementation of Wick’s theorem and the computation of damping coefficients via the Fermi Golden Rule. The review synthesizes results obtained in two recent works presented at IQSA2025, emphasizing the conceptual role of algebraic methods in connecting semiclassical analysis, equilibrium states, and finite-temperature perturbation theory.
Abstract We review two recent applications of operator-algebraic methods to the analysis of homogeneous Bose gases at finite temperature. The first part concerns the semiclassical description of Bose–Einstein condensation in the framework of Weyl C∗ C ∗ -algebras. In this setting, the semiclassical parameter is naturally linked to the density of the system, and the classical weak KMS condition emerges as the limit of quantum equilibrium states. We show in particular that the condensate structure is preserved in a suitable large-density regime. The second part discusses a joint work with Prof. Jan Dereziński which the Araki–Woods representation is used to formulate a systematic perturbative theory for interacting Bose gases at positive temperature. Within this representation, thermal effects are encoded directly in the field operators, allowing for a transparent implementation of Wick’s theorem and the computation of damping coefficients via the Fermi Golden Rule. The review synthesizes results obtained in two recent works presented at IQSA2025, emphasizing the conceptual role of algebraic methods in connecting semiclassical analysis, equilibrium states, and finite-temperature perturbation theory.
Abstract Berezin and Weyl quantization are renowned procedures for mapping commutative Poisson algebras of observables to their non-commutative, quantum counterparts. The latter is famous for its use on Weyl algebras, while the former is more appropriate for continuous functions decaying at infinity. In this work, we define a variant of the Berezin quantization map, which acts on the classical Weyl algebra W(E,0) W ( E , 0 ) and constitutes a positive strict deformation quantization . This construction provides a natural framework to compare classical and quantum thermal equilibrium states of a Bose gas through the computation of their semi-classical limit. To this end, we first introduce a purely algebraic notion of KMS states for the classical Weyl algebra and establish that, in finite volume, there exists a unique such state, which can be interpreted as the Fourier transform of a Gibbs measure on a Hilbert space. We then construct a new class of classical KMS states that realize representations of the canonical commutation relations with infinite local density. These states arise as the semi-classical high-density limit of the quantum equilibrium states originally studied by Araki and Woods [5]. A key feature of our approach is that it preserves the macroscopic ground-state occupation of the Bose gas in the classical regime. Finally, we demonstrate that the infinite-volume classical states can be obtained as thermodynamic limits of finite-volume Gibbs states.
This thesis can be regarded as a light, but thorough, introduction to the algebraic approach to quantum statistical mechanics and a subsequent test of this framework in the form of an application to Bose-Einstein condensates. The success of the algebraic approach to quantum statistical mechanics hinges upon the remarkable properties of special operator algebras known as C^*-algebras. These algebras have unique characterization properties which allows one to readily identify the mathematical counterparts of concepts in physics while at the same time maintaining mathematical rigour and clarity. In the first half of this thesis, we focus on abstract C^*-algebras known as the canonical commutation relation algebras (CCR algebras) which are generated by elements satisfying specific commutation relations. The main result in this section is the proof of a certain kind of algebraic uniqueness of these algebras. The main idea of the proof is to utilise the underlying common structure of any of the CCR algebras and explicitly construct an isomorphism between the generators of these algebras. The construction of this isomorphism involves the use of abstract Fourier analysis on groups and various arguments concerning bounded operators. The second half of the thesis concerns the rigorous set-up of the formation of Bose-Einstein condensation. First, one defines the Gibbs grand canonical equilibrium states, and then we specialize to studying the taking of the thermodynamic limit of these systems in various contexts. The main result of this section involves two main elements. The first is that by fixing the temperature and density of the system while varying its activity and volume, there exists a limiting state corresponding to the taking of the thermodynamic limit. The second element concerns the existence of a critical density after which the limiting state begins to show the physical characteristics of Bose-Einstein condensation. The mathematical issues one faces with Bose-Einstein condensation are mainly related to the unboundedness of the creation and annihilation operators and the definition of the algebra that we are working on. The first issue is relevant to all areas of mathematical physics, and one deals with it in the standard ways. The second issue is more nuanced and is a direct result of the first issue we mentioned. In particular, we would like to define the states on an algebra which contains the operators that we are interested in. The problem is that these operators are unbounded, and, as a result, one must instead use the CCR algebra and show by extension that we can, in fact, also use the unbounded operators in this state.
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.
We present a systematic description of the structure of Bose-Einstein condensation (BEC) in the free Bose gas from the viewpoint of the correspondence between the operator-algebraic formulation based on the resolvent algebra and the functional integral representation. By clarifying the representation-theoretic structure of finite-temperature BEC states and rigorously analyzing the correspondence between their direct integral decomposition and the ergodic decomposition of the associated probability measures, we provide a framework in which general features of phase transitions-such as the emergence of order parameters, the decomposition of states, and clustering properties-are explicitly described using BEC in the free Bose gas as a concrete example. Furthermore, we construct in detail the correspondence between the decomposition of measures in the functional integral approach and that of operator-algebraic representations, thereby establishing the equivalence between the probabilistic and algebraic aspects, and providing a guiding principle for isolating the essential structures by disentangling the additional mathematical complications arising from the treatment of infrared singularities in interacting systems. These results lay a foundation for the rigorous analysis of phase transitions in non-relativistic constructive quantum field theory and quantum statistical mechanics, and serve as a starting point for extensions to interacting models.
The thermodynamic approach to density-functional theory (DFT) is used to derive a versatile theoretical framework for the treatment of finite-temperature (and in the limit, zero-temperature) Bose-Einstein condensates (BECs). The simplest application of this framework, using the overall density of bosons alone, would yield the DFT of Nunes [J. Phys. B 32, 4293 (1999)]. It is argued that a significant improvement in accuracy may be obtained by using additional density fields: the condensate amplitude and the anomalous density. Thus, two advanced schemes are suggested, one corresponding to a generalized two-fluid model of condensate systems, and another scheme, which explicitly accounts for anomalous density contributions and anomalous effective potentials. The latter reduces to the Hartree-Fock-Bogoliubov approach in the limit of weak interactions. For stronger interactions, a local-density approximation is suggested, but its implementation requires accurate data for the thermodynamic properties of uniform interacting BEC systems, including fictitious perturbed states of such systems. Provided that such data become available (e.g., from quantum Monte Carlo computation), DFT can be used to obtain high-accuracy theoretical results for the equilibrium states of BECs of various geometries and external potentials.
In this thesis we develop a gapless theory of BEC which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first and second order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infra-red divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultra-violet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T-matrix leads naturally to a high-energy renormalization. The theory developed in this thesis is therefore well-defined at both low and high energy and provides a systematic description of Bose-Einstein condensation in dilute gases. It can be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition. We present analytical results for these quantities in the homogeneous limit and describe how they can be calculated numerically for trapped gases. (author)