The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on ad hoc prescriptions for the test of pertinent features, such as the appearance of Bose–Einstein condensates. The method is illustrated in the case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet, the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.
The virtues of resolvent algebras, compared to other approaches for the treatment of canonical quantum systems, are exemplified by infinite systems of non-relativistic bosons. Within this framework, equilibrium states of trapped and untrapped bosons are defined on a fixed C*-algebra for all physically meaningful values of the temperature and chemical potential. Moreover, the algebra provides the tools for their analysis without having to rely on ad hoc prescriptions for the test of pertinent features, such as the appearance of Bose–Einstein condensates. The method is illustrated in the case of non-interacting systems in any number of spatial dimensions and sheds new light on the appearance of condensates. Yet, the framework also covers interactions and thus provides a universal basis for the analysis of bosonic systems.
The limit states of canonical ensembles of a large number of interacting bosons at a given temperature, which are confined by harmonic forces, are studied in the framework of the resolvent algebra. It is shown that the limits satisfy the KMS condition or are ground states, regardless of the type of interaction. In case of attractive forces, where the ensembles collapse, observables that become meaningless in the limit disappear from the limit representations. For repulsive forces, this can also happen if condensates with an infinite number of particles in the same state (proper condensates) appear in the limit. The resulting structures and their interpretation are illustrated by a simple model. The study of vanishing harmonic forces (thermodynamic limit) involves changes of the dynamics. It is conveniently based on derivations acting on the algebra. They are given by the commutator of the Hamiltonians with the elements of the algebra. To ensure that the images remain in the algebra, the interaction must be regularized. This is accomplished in a manner that has only a minor impact on the dynamics and may be of broader interest. With this input a relation between the strength of the confining harmonic forces and the number of particles in the ensembles is derived from the condition that the limit states are to be stationary (invariant) under the adjoint action of the unconfined, spatially homogeneous limit dynamics. This relation encompasses the conditions that are frequently used in studies of Bose-Einstein condensates.
Some advantages of the algebraic approach to many body physics, based on resolvent algebras, are illustrated by the simple example of non-interacting bosons that are confined in compact regions with soft boundaries. It is shown that the dynamics of these systems converges to the spatially homogeneous dynamics for increasing regions and particle numbers and a variety of boundary forces. The corresponding correlation functions of thermal equilibrium states also converge in this limit. Adding to the regions further particles, the limits are steady states, including Bose–Einstein condensates. They can either be spatially homogeneous, or they are inhomogeneous with varying, but finite local particle densities. In case of this spontaneous breakdown of the spatial symmetry, the presence of condensates can be established by exhibiting temporal correlations over large temporal distances (memory effects).
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.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.
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.
We present a comprehensive analysis of the ideal structure of the resolvent algebra for interacting bosonic quantum field theories. The resolvent algebra, introduced by Buchholz and Grundling as a C∗-algebraic alternative to the Weyl algebra, possesses a rich ideal structure that faithfully reflects the quantum-mechanical properties of bosonic many-body systems. Our main results establish a complete characterization of the closed two-sided ideals for the van Hove model and the generalized spin-boson model, proving that the singular subspace associated with infrared divergences corresponds to a unique maximal proper ideal. We show that the quotient by this ideal cleanly separates regular and singular contributions, and that Bose-Einstein condensation (BEC) can be characterized entirely in terms of the emergence of a nontrivial center in the von Neumann algebra obtained from the GNS representation of equilibrium states. Furthermore, we prove a no-go theorem establishing that quasiparticle excitations in the spin-boson model do not undergo Bose-Einstein condensation at finite temperature, resolving a long-standing question in the rigorous formulation of quantum statistical mechanics. The proofs draw on spectral analysis of the van Hove and spin-boson Hamiltonians, Tomita-Takesaki modular theory, the Araki-Woods direct integral decomposition, and functional integra representations of thermal equilibrium states.