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arXiv:2606.09538

No-Go Theorem for BEC in the Nelson and Pauli--Fierz Models

Yoshitsugu Sekine

Published Jun 8, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2606.09538

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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.
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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.
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