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arXiv:2604.27904

No-Go Theorem for Quasiparticle BEC in the Spin-Boson Model

Yoshitsugu Sekine

Published Apr 30, 2026• 0 citations
arXivHTMLPDF

Abstract

We analyze the possibility of Bose-Einstein condensation (BEC) at finite temperature in the spin-boson model within the frameworks of functional integral representations and the resolvent algebra. Because a sesquilinear form arising from the zero mode appears, analogous to the case of the free Bose gas, a BEC-type component is also formally present in the spin-boson model. However, according to arxiv:1207.4621, quasiparticles do not undergo BEC, so we have to exclude this possibility. A no-go theorem for BEC is obtained, but we find that the spin-boson model is not a model that follows the criterion in arxiv:1207.4621.

DOI
10.48550/arxiv.2604.27904

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We analyze the possibility of Bose-Einstein condensation (BEC) at finite temperature in the spin-boson model within the frameworks of functional integral representations and the resolvent algebra. Because a sesquilinear form arising from the zero mode appears, analogous to the case of the free Bose gas, a BEC-type component is also formally present in the spin-boson model. However, according to arxiv:1207.4621, quasiparticles do not undergo BEC, so we have to exclude this possibility. A no-go theorem for BEC is obtained, but we find that the spin-boson model is not a model that follows the criterion in arxiv:1207.4621.
We discuss a no-go theorem for Bose-Einstein condensation (BEC) of quasiparticles (phonons) from the viewpoint of operator algebras, using the van Hove model. The βββ-KMS states of the van Hove model satisfy the self-consistency condition of arXiv:1207.4621. However, the self-consistency condition is a constraint concerning the definition of the field, and is insufficient to establish the no-go theorem for BEC. In this paper, we prove the no-go theorem for BEC via two routes. First, imposing time cluster properties on the βββ-KMS states precludes BEC. Second, under nonlinear dispersion with s>2s > 2s>2, the treatment of infrared divergences automatically reduces the algebra of physical observables, and BEC is mathematically excluded on the reduced algebra. In particular, the latter property admits an interpretation in terms of the ideal theory of the resolvent algebra.
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∗C^*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.
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.
Abstract The original Bose–Einstein condensation (BEC) was theoretically established in the 1920s, for the case of an ideal gas of boson particles (a consequence of the symmetry of the wave function). In the 1990s, it was realized in systems of atomic alkali gases. At the beginning of the 21st century, investigators considered its emergence in solid‐state matter, and a BEC‐like phenomenon was observed in the case of bosonic quasiparticles (elementary excitations in solids). Both types of BEC can occur in a condensation in conditions of thermodynamic equilibrium. A third type of what has been dubbed a BEC in non‐equilibrium conditions can occur in systems of bosonic quasiparticles (phonons, magnons, excitons); this phenomenon (which can be considered as cases of systems with complex behavior) follows after a certain threshold of intensity of an external pumping source is achieved. In this review, after brief comments on the two types of BEC in equilibrium, a description is presented on how a BEC‐like phenomenon arises in systems of quasiparticles that are driven away from thermodynamic equilibrium, and several examples are described. This description is given within a unique theoretical framework, and a comparison with experimental results is performed; in addition, the possible technological applications are outlined.
We construct KMS states for the Nelson model, the spinless Pauli--Fierz model, and the Pauli--Fierz model with spin by functional integral representations, and study point-source models after removal of the infrared and ultraviolet cutoffs. If the physical test-function space can distinguish the zero mode, the absence of off-diagonal long-range order, the vanishing of the zero-mode form, the vanishing of the condensate density, the order-parameter criterion, and the triviality of the BEC directions and the BEC ideal are equivalent. We also describe the infrared quotient and the BEC ideal in the resolvent algebra uniformly for the three models, and formulate an operator-algebraic sufficient condition for separately given spatially translation-invariant KMS states.
This paper investigates Bose-Einstein condensation of an ideal gas of finite-spin bosons in an external magnetic field. We generalize Bogolyubov’s theory of a weakly nonideal Bose gas to the case where the gas of finite-spin bosons is located in an external magnetic field. We find the corresponding quasiparticle spectrum and formulate the superfluidity criterion for the boson gas. The magnetization of the weakly nonideal Bose gas is also determined. Finally, we specify a method of studying kinetic processes that take place in a weakly nonideal Bose gas.