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arxiv: 2606.09538 · v1 · pith:HQ2MFRZOnew · submitted 2026-06-08 · 🧮 math-ph · math.MP

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

Yoshitsugu Sekine This is my paper

Pith reviewed 2026-06-27 14:44 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Bose-Einstein condensationNelson modelPauli-Fierz modelKMS statesfunctional integral representationsoff-diagonal long-range orderzero moderesolvent algebra
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The pith

If the test-function space distinguishes the zero mode, absence of off-diagonal long-range order is equivalent to vanishing condensate density and trivial BEC ideal in Nelson and Pauli-Fierz models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs KMS states for the Nelson model, the spinless Pauli-Fierz model, and the Pauli-Fierz model with spin by means of functional integral representations. After simultaneous removal of infrared and ultraviolet cutoffs in the point-source versions, it establishes that several criteria become equivalent when the physical test-function space can distinguish the zero mode: absence of off-diagonal long-range order, vanishing of the zero-mode form, vanishing of the condensate density, the order-parameter criterion, and triviality of the BEC directions together with the BEC ideal. This equivalence functions as a no-go result showing that Bose-Einstein condensation cannot occur under those conditions. A sympathetic reader would care because the result ties together different mathematical tests for condensation in models of bosons interacting with quantized fields.

Core claim

Under the condition that 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. The paper also supplies a uniform description of the infrared quotient and the BEC ideal in the resolvent algebra for the three models and formulates an operator-algebraic sufficient condition for separately given spatially translation-invariant KMS states.

What carries the argument

Functional integral representations of KMS states that remain valid after simultaneous infrared and ultraviolet cutoff removal, which allow proof of the listed equivalences among BEC criteria when the test-function space distinguishes the zero mode.

If this is right

  • The infrared quotient and BEC ideal admit a uniform description in the resolvent algebra across the Nelson and Pauli-Fierz models.
  • An operator-algebraic sufficient condition holds for the existence of separately given spatially translation-invariant KMS states.
  • If any one of the equivalent conditions such as absence of off-diagonal long-range order is satisfied, then the condensate density vanishes and the BEC ideal is trivial.
  • The same set of equivalences applies uniformly to the three models once the zero-mode distinction condition is met.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence may indicate that realistic physical regimes of these models, where zero modes are distinguishable, exclude condensation altogether.
  • Analogous no-go statements could be sought in other models of bosons coupled to quantized fields that admit similar functional integral constructions.
  • The requirement that the test-function space distinguish the zero mode may serve as a diagnostic for whether condensation is possible in approximate numerical treatments of the same Hamiltonians.

Load-bearing premise

The functional integral representations used to construct the KMS states for the point-source models remain valid after simultaneous removal of infrared and ultraviolet cutoffs.

What would settle it

An explicit KMS state for one of the point-source models after cutoff removal that exhibits non-vanishing condensate density while the test-function space distinguishes the zero mode.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs KMS states for the Nelson model, the spinless Pauli-Fierz model, and the Pauli-Fierz model with spin by means of functional integral representations. For the point-source versions after simultaneous removal of infrared and ultraviolet cutoffs, it proves that, when the physical test-function space distinguishes 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 all equivalent. The paper also gives a uniform description of the infrared quotient and the BEC ideal in the resolvent algebra for the three models and states an operator-algebraic sufficient condition for spatially translation-invariant KMS states.

Significance. If the functional-integral representations remain valid after cutoff removal, the equivalences supply a no-go theorem for BEC in these interacting quantum-field models by linking several independent characterizations of condensation. The uniform treatment across the Nelson and Pauli-Fierz families and the combination of functional-integral constructions with operator-algebraic arguments constitute a clear technical contribution. The derivation of the equivalences directly from the functional-integral representation and resolvent-algebra properties, without introduction of fitted parameters, is a methodological strength.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: The central claim that the listed equivalences constitute a no-go theorem for the physical (cutoff-removed) point-source models rests on the assertion that the functional-integral representations of the KMS states survive simultaneous infrared and ultraviolet cutoff removal. The manuscript states that the representations are used after removal but supplies neither an explicit limit argument nor a reference to a prior result establishing existence of the joint limit for the point-source Nelson and Pauli-Fierz Hamiltonians; this step is load-bearing for the extension from regularized to physical models.
minor comments (2)
  1. The term 'BEC ideal' is introduced in the abstract without a preceding definition or reference; a short clarifying sentence in the introduction would improve readability for readers outside the immediate subfield.
  2. Notation for the resolvent algebra and the infrared quotient is used uniformly across models but is not collected in a single preliminary section; a brief table or paragraph summarizing the common objects would aid comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the load-bearing step in extending the equivalences to the physical models. We address the single major comment below and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2: The central claim that the listed equivalences constitute a no-go theorem for the physical (cutoff-removed) point-source models rests on the assertion that the functional-integral representations of the KMS states survive simultaneous infrared and ultraviolet cutoff removal. The manuscript states that the representations are used after removal but supplies neither an explicit limit argument nor a reference to a prior result establishing existence of the joint limit for the point-source Nelson and Pauli-Fierz Hamiltonians; this step is load-bearing for the extension from regularized to physical models.

    Authors: We agree that the manuscript does not supply an explicit limit argument or citation establishing that the functional-integral representations of the KMS states remain valid after simultaneous IR and UV cutoff removal for the point-source models. This omission weakens the direct applicability of the no-go theorem to the physical Hamiltonians. In the revised manuscript we will add either a self-contained sketch of the joint limit (where feasible) or a precise reference to prior results on the existence of the cutoff-removed KMS states via functional integrals for the Nelson and Pauli-Fierz families. The equivalences themselves are derived for the regularized models and then transferred; the revision will make the transfer step explicit. revision: yes

Circularity Check

0 steps flagged

Equivalences derived from functional-integral KMS constructions without reduction to inputs

full rationale

The paper constructs KMS states via functional integral representations for the Nelson and Pauli-Fierz models, then removes IR/UV cutoffs for point-source cases and derives equivalences among ODLRO absence, zero-mode form vanishing, condensate density vanishing, order-parameter criterion, and triviality of BEC directions/ideal (conditional on test-function space distinguishing the zero mode). These steps rely on operator-algebraic properties of the resolvent algebra and the validity of the representations in the joint limit, which are external to the target equivalences rather than defined in terms of them. No fitted parameters are renamed as predictions, no self-citation chain bears the central claim, and no ansatz or uniqueness theorem is smuggled in. The derivation chain is self-contained against the stated constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of operator algebras and quantum statistical mechanics; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math KMS condition defines thermal equilibrium states
    Invoked to construct the states via functional integrals (abstract).
  • domain assumption Functional integral representations exist for the Nelson and Pauli-Fierz Hamiltonians
    Used to build the KMS states after cutoff removal.

pith-pipeline@v0.9.1-grok · 5643 in / 1303 out tokens · 23749 ms · 2026-06-27T14:44:56.757063+00:00 · methodology

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mean-Field Bose--Einstein Condensation and Condensate Ideals in the Resolvent Algebra

    math-ph 2026-07 unverdicted novelty 5.0

    In the mean-field BEC regime of the imperfect Bose gas, zero-mode covariance defines a mean-field BEC ideal in the resolvent algebra, with occupation-number and Brownian-loop formulations recovering consistent density...

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