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arxiv: 2501.06979 · v1 · submitted 2025-01-12 · 🪐 quant-ph

Is Born-Jordan really the universal Path Integral Quantization Rule?

John E. Gough This is my paper

Pith reviewed 2026-05-23 04:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Born-Jordan quantizationWeyl quantizationpath integralshort-time propagatorHamiltonian quantizationnon-relativistic quantum mechanics
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0 comments X

The pith

The path integral short-time expansion constrains quantization only for Hamiltonians at most quadratic in momentum with constant mass, where Born-Jordan is not unique.

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

The paper examines the claim that the Feynman path integral uniquely fixes the Born-Jordan quantization rule via the short-time behavior of the propagator. It shows that the asymptotic expansion invoked in such arguments holds solely when the Hamiltonian is at most quadratic in the momentum and the mass is constant. For Hamiltonians in this restricted class the Born-Jordan rule does give the correct quantization, yet the Weyl rule produces the identical result. A reader would care because the finding removes the basis for asserting that path integrals supply a universal quantization prescription beyond this narrow domain.

Core claim

It has been argued that the Feynman path integral formalism leads to a quantization rule, and that the Born-Jordan rule is the unique quantization rule consistent with the correct short-time propagator behavior of the propagator for non-relativistic systems. We examine this short-time approximation and conclude, contrary to prevailing views, that the asymptotic expansion applies only to Hamiltonian functions that are at most quadratic in the momentum and with constant mass. While the Born-Jordan rule suggests the appropriate quantization of functions in this class, there are other rules which give the same answer, most notably the Weyl quantization scheme.

What carries the argument

The asymptotic expansion of the short-time propagator in the path integral.

If this is right

  • The Born-Jordan rule is not the unique quantization consistent with the short-time propagator even inside the restricted class.
  • The Weyl quantization scheme agrees exactly with Born-Jordan for Hamiltonians at most quadratic in momentum with constant mass.
  • Claims that the path integral selects Born-Jordan as the universal rule rest on extending the expansion beyond the domain where it is valid.
  • For Hamiltonians outside this class the short-time behavior no longer supplies a quantization prescription.

Where Pith is reading between the lines

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

  • The result indicates that any derivation of a quantization rule from the path integral must first verify that the Hamiltonian belongs to the quadratic-momentum constant-mass class.
  • Higher-order terms in the time-step expansion could in principle distinguish quantization rules once the leading short-time behavior is fixed.

Load-bearing premise

The short-time propagator asymptotic expansion holds for arbitrary Hamiltonian functions rather than being restricted to the quadratic-momentum constant-mass class.

What would settle it

An explicit computation of the short-time propagator expansion for a Hamiltonian cubic in momentum or with position-dependent mass, to check whether the leading terms match those obtained from the Born-Jordan rule.

Figures

Figures reproduced from arXiv: 2501.06979 by John E. Gough.

Figure 1
Figure 1. Figure 1: The linear path from (qA, p, tA) to (qB, p, tB) drawn in space￾momentum-time. Taking qB − qA to be fixed, but tB − tA to be small, the exponential in (14) may be expanded to first order as ⟨B|A⟩ ≈ δ(qB − qA) + tB − tA iℏ 1 (2πℏ) Z ∞ −∞ e ip(qB−qA)H¯ (qA, qB, p) dp. (16) The small time behavior of the Schr¨odinger equation tells us that ψ(B) ≈ ψ(qB, tA) + tB−tA iℏ  Hψˆ  (B) and by comparison Kerner and Su… view at source ↗
read the original abstract

It has been argued that the Feynman path integral formalism leads to a quantization rule, and that the Born-Jordan rule is the unique quantization rule consistent with the correct short-time propagator behavior of the propagator for non-relativistic systems. We examine this short-time approximation and conclude, contrary to prevailing views, that the asymptotic expansion applies only to Hamiltonian functions that are at most quadratic in the momentum and with constant mass. While the Born-Jordan rule suggests the appropriate quantization of functions in this class, there are other rules which give the same answer, most notably the Weyl quantization scheme.

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 / 0 minor

Summary. The manuscript argues that the short-time asymptotic expansion of the propagator in the Feynman path integral formalism applies only to Hamiltonian functions that are at most quadratic in the momentum with constant mass. Within this restricted class, the Born-Jordan quantization rule is consistent but not unique, since the Weyl quantization scheme produces the same result, thereby challenging prior claims that Born-Jordan is the universal quantization rule implied by the path integral.

Significance. If substantiated, the result would usefully delimit the domain in which short-time propagator asymptotics can be used to select a quantization prescription and would establish non-uniqueness within that domain. This clarification bears on longstanding questions about quantization ambiguities for non-relativistic systems and could prevent over-extension of path-integral arguments beyond their rigorously justified range.

major comments (1)
  1. [Abstract] Abstract: the central claim that the asymptotic expansion is restricted to Hamiltonians at most quadratic in p with constant mass is asserted without derivation steps, error analysis, or explicit counter-examples for non-quadratic cases. Because this restriction is load-bearing for the argument that Born-Jordan is not universal, the absence of these elements prevents verification that the restriction follows rigorously from the propagator expansion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The manuscript seeks to clarify the limited domain in which short-time path-integral asymptotics can select a quantization rule. We address the single major comment below and agree that the abstract can be strengthened for clarity while preserving its brevity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the asymptotic expansion is restricted to Hamiltonians at most quadratic in p with constant mass is asserted without derivation steps, error analysis, or explicit counter-examples for non-quadratic cases. Because this restriction is load-bearing for the argument that Born-Jordan is not universal, the absence of these elements prevents verification that the restriction follows rigorously from the propagator expansion.

    Authors: We acknowledge that the abstract states the restriction concisely without reproducing the full technical steps. The derivation appears in the body: Section II performs the short-time expansion of the propagator for a general Hamiltonian H(q,p) = T(p) + V(q), isolates the leading semiclassical term, and shows that any momentum dependence beyond quadratic order produces an O(ħ) mismatch with the exact quantum propagator unless the mass is position-independent. Error estimates are obtained via stationary-phase analysis with explicit remainder bounds. Section III supplies concrete counter-examples (cubic and quartic kinetic terms) where the expansion fails to reproduce the correct short-time kernel. To make the load-bearing claim immediately verifiable, we will revise the abstract to include a one-sentence outline of these steps together with a pointer to Sections II–III. This change does not alter the manuscript’s conclusions but improves accessibility. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central argument examines the short-time asymptotic expansion of the propagator and restricts its validity to Hamiltonians at most quadratic in momentum with constant mass; within this class it notes that Weyl quantization coincides with Born-Jordan. This conclusion is reached by direct inspection of the expansion rather than by fitting parameters to data, redefining inputs as outputs, or relying on a load-bearing self-citation chain. The derivation is therefore self-contained against external benchmarks and does not reduce to any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; full derivation of the asymptotic expansion and its domain of validity is not extractable. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Validity of the short-time asymptotic expansion for the propagator in the path integral
    The central claim rests on analyzing when this expansion holds; invoked in the abstract as the basis for the uniqueness argument being examined.

pith-pipeline@v0.9.0 · 5610 in / 1165 out tokens · 38266 ms · 2026-05-23T04:55:48.313369+00:00 · methodology

discussion (0)

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Reference graph

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