Variable Planck's constant and scaling properties of states on Weyl algebra

Andrzej Frydryszak; Lech Jak\'obczyk; Piotr {\L}ugiewicz

arxiv: 1907.08644 · v1 · pith:JA77KWZKnew · submitted 2019-07-19 · 🪐 quant-ph · math-ph· math.MP

Variable Planck's constant and scaling properties of states on Weyl algebra

Piotr {\L}ugiewicz , Lech Jak\'obczyk , Andrzej Frydryszak This is my paper

Pith reviewed 2026-05-24 19:04 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords variable Planck constantCCR algebraWeyl algebraquasi-free statesKMS statesFock statesinvariant states

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The pith

Universally invariant states on the CCR algebra are convex combinations of Fock states with different values of Planck's constant.

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

The paper investigates the effects of varying Planck's constant on states defined on the CCR algebra by rescaling its defining relations. It determines the extent to which such rescalings preserve the original state space and the properties that states retain when viewed on the modified algebra. Analysis of quasi-free states reveals restrictions on admissible ħ variations. The key result is that any universally invariant state can be interpreted as a convex combination of Fock states corresponding to different ħ. For KMS states, the rescaling changes the associated dynamics, and exceeding standard limits requires restricting the algebra to a subalgebra.

Core claim

Any universally invariant state can be interpreted as a convex combination of Fock states with different values of Planck's constant. The rescaling alters in a nontrivial way the relevant dynamics for KMS-states. It is possible to go beyond the limits restricting the changes of the ħ by restricting the CCR-algebra to a subalgebra.

What carries the argument

The CCR-algebra with ħ-dependent multiplication relations, and the states as positive normalized functionals on it, where rescaling the relations according to ħ changes allows reinterpretation of states.

If this is right

Quasi-free states remain valid only for restricted ranges of ħ changes.
KMS-states have their dynamics modified nontrivially by ħ rescaling.
Restricting to a subalgebra allows ħ changes beyond the usual limits.
Universally invariant states have a mixture representation over Fock states at different ħ.

Where Pith is reading between the lines

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

This suggests possible effective descriptions for infinite quantum systems with scale-dependent ħ.
Connections could be explored to theories with varying fundamental constants while preserving algebraic quantum structure.
Such mixtures might lead to testable predictions in systems where standard fixed-ħ states are compared to these combinations.

Load-bearing premise

Variations of Planck's constant amount to rescaling the defining relations of the CCR-algebra while keeping the algebraic structure intact, absent other physical constraints on the variation.

What would settle it

Demonstrating that some universally invariant state cannot be expressed as any convex combination of Fock states with different ħ, or that a rescaling violates positivity for an invariant state.

read the original abstract

We consider the possible quantum effect for infinite systems produced by variations of the Planck's constant. Using the algebraic formulation of quantum theory we study behaviour of states $\omega$ defined as positive, normalized functionals on the canonical commutation relations algebra (CCR-algebra) under the changes of the defining relations of the CCR. These defining relations of the multiplication in the CCR-algebra depend explicitly on the value of the Planck's constant. We analyse to what extend changes of the $\hbar$ preserve the original state space (this gives restrictions on the admissible changes of the Plank's constant) and what properties have original quantum states $\omega$ as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of Planck's constant. The second important class of states we study are the KMS-states, here the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes of the $\hbar$, but then one has to restrict the CCR-algebra to a subalgebra.

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.

Desk Editor's Note private letter to a colleague

The paper cleanly extends CCR state theory to variable ħ by showing invariant states as mixtures of Fock states at different values and noting nontrivial KMS shifts, but the mixture construction across algebras is left underspecified.

read the letter

The main takeaway is that any universally invariant state on the CCR algebra can be read as a convex combination of Fock states with different ħ, while admissible ħ changes are limited so the original state space stays intact; KMS states see their dynamics altered in a nontrivial way, and larger changes are possible only after restricting to a subalgebra. This is worked out explicitly for quasi-free states. The paper does a solid job staying inside the algebraic framework, giving concrete restrictions on ħ variation, and pointing out the subalgebra workaround without overclaiming physical consequences. The convex-combination result is the clearest new piece. The soft spot is exactly the one the stress-test flags. Different ħ rescale the commutation relations, so the algebras are not identical; mixing the states requires some transport map that preserves positivity and normalization. The abstract claims the admissible changes preserve the state space, but it does not show whether the combination is formed after an explicit embedding or only on a common subalgebra. Without that step spelled out, the claim is harder to verify for general invariant states. The KMS discussion is similarly brief. This is niche work aimed at people already comfortable with infinite quantum systems and CCR representations. A reader who already knows the standard quasi-free and KMS constructions will get the most out of the restrictions and the mixture interpretation. It is coherent on its own terms and formally grounded enough to merit referee time, even if the central construction needs one more explicit step.

Referee Report

2 major / 2 minor

Summary. The paper examines the impact of varying Planck's constant ħ on states of the CCR algebra in the algebraic approach to quantum theory. It restricts admissible ħ changes that preserve the original state space, analyzes quasi-free states under such rescalings, proves that any universally invariant state can be expressed as a convex combination of Fock states with different ħ values, studies the nontrivial effect of ħ rescaling on KMS states and their dynamics, and shows that exceeding the ħ limits is possible by restricting to a subalgebra of the CCR algebra.

Significance. If the central claims hold, the work supplies a precise algebraic mechanism for interpreting variable ħ in infinite-degree-of-freedom systems and gives an explicit decomposition of invariant states into Fock components at different ħ. This could be relevant to models in quantum statistical mechanics or QFT where ħ is treated as a tunable parameter, provided the required algebra embeddings are canonical and preserve positivity.

major comments (2)

[universally invariant states section] The central claim that any universally invariant state ω is a convex combination of Fock states at different ħ (abstract and the section treating universally invariant states) requires an explicit identification map or embedding between the CCR algebras for distinct ħ values that preserves the state properties. The manuscript must demonstrate that this map is well-defined on the generators, that the transported states remain positive and normalized on a common algebra, and that the convex combination yields a single functional on the original algebra; without this construction the decomposition is formal rather than operational.
[quasi-free states analysis] For the quasi-free states analysis, the restrictions on admissible ħ changes are stated to preserve the original state space, but the paper supplies no explicit verification (e.g., via the two-point function or the symplectic form) that the rescaled commutation relations [a(f),a*(g)] = ħ ⟨f,g⟩ keep the quasi-free state positive when ħ varies continuously; an explicit check or counter-example for a concrete test state would be needed to support the claim.

minor comments (2)

Notation for the rescaled CCR relations should be introduced once with a clear equation number and then used consistently; the abstract refers to “defining relations of the CCR” without an equation label.
[KMS-states section] The discussion of KMS states under ħ rescaling would benefit from a short statement of the original dynamics (e.g., the automorphism group) before describing how it is altered.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. We address each major comment below, providing clarifications on the constructions in the manuscript while agreeing to strengthen the explicitness of the embeddings and verifications in a revised version.

read point-by-point responses

Referee: [universally invariant states section] The central claim that any universally invariant state ω is a convex combination of Fock states at different ħ (abstract and the section treating universally invariant states) requires an explicit identification map or embedding between the CCR algebras for distinct ħ values that preserves the state properties. The manuscript must demonstrate that this map is well-defined on the generators, that the transported states remain positive and normalized on a common algebra, and that the convex combination yields a single functional on the original algebra; without this construction the decomposition is formal rather than operational.

Authors: The identification proceeds by fixing the underlying real symplectic vector space (V, σ) of test functions and letting the CCR algebra A_ħ be generated by the same Weyl operators W(f) for f ∈ V, but with the ħ-dependent commutation relations W(f)W(g) = exp(-i ħ σ(f,g)/2) W(f+g). The canonical embedding between A_ħ and A_ħ' is the identity map on generators, which is well-defined as a *-homomorphism when restricted to the common dense subalgebra generated by the W(f). Each Fock state ω_ħ at a fixed ħ is positive and normalized on A_ħ; pulling it back via the embedding yields a positive normalized functional on the original algebra A_ħ0. The convex combination is then taken directly on this common algebra. We agree that spelling out the positivity preservation under the embedding and the resulting single functional would remove any ambiguity, and we will insert a short subsection (or expanded paragraph) in the universally invariant states section with these details. revision: yes
Referee: [quasi-free states analysis] For the quasi-free states analysis, the restrictions on admissible ħ changes are stated to preserve the original state space, but the paper supplies no explicit verification (e.g., via the two-point function or the symplectic form) that the rescaled commutation relations [a(f),a*(g)] = ħ ⟨f,g⟩ keep the quasi-free state positive when ħ varies continuously; an explicit check or counter-example for a concrete test state would be needed to support the claim.

Authors: For a quasi-free state the positivity condition is equivalent to the two-point function satisfying Im ω(a*(f)a(g)) ≥ (ħ/2) σ(f,g) together with the appropriate bound on the real part. When ħ is rescaled to ħ' the symplectic form is effectively rescaled by ħ'/ħ, and the admissible interval for ħ' is derived precisely so that this inequality continues to hold for the original two-point function. We will add an explicit verification for the Fock vacuum (where the two-point function is (ħ/2)⟨f,g⟩) showing that positivity is preserved exactly when |ħ' - ħ| stays within the derived bounds; a short calculation for a coherent state will also be included as a concrete check. This material will be inserted into the quasi-free states section. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim follows from algebraic definitions of CCR states without reduction to inputs or self-citations

full rationale

The paper derives its main result—that any universally invariant state on the CCR-algebra can be interpreted as a convex combination of Fock states with varying ħ—directly from the definitions of positive normalized functionals, quasi-free states, and the explicit dependence of commutation relations on ħ. No equations reduce by construction to fitted parameters, renamed empirical patterns, or load-bearing self-citations; the restrictions on admissible ħ changes are obtained by requiring preservation of positivity and normalization on the rescaled algebra, which is an independent algebraic condition. The analysis of KMS-states and subalgebra restrictions likewise proceeds from standard CCR properties without self-referential loops or imported uniqueness theorems from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions from algebraic quantum theory; no free parameters, new axioms, or invented entities are indicated in the abstract.

axioms (2)

standard math States are positive normalized functionals on the CCR-algebra
Core definition in algebraic formulation of quantum theory.
standard math The multiplication in the CCR-algebra depends explicitly on the value of ħ
Standard property of the Weyl algebra formulation.

pith-pipeline@v0.9.0 · 5742 in / 1313 out tokens · 28888 ms · 2026-05-24T19:04:44.164785+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

any universally invariant state can be interpreted as a convex combination of Fock states with different values of Planck's constant
IndisputableMonolith/Constants ħ-as-φ-power derivation contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

rescaled Fock state ω0,h ... for all 0 < h < 1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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