Wigner function under changes of reference frames

G. F. Torres del Castillo; J. Berra-Montiel

REVIEW 3 major objections 4 minor 11 references

Wigner function under changes of reference frames

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Under unitary frame changes of the special class (13), the Wigner function transforms by the integral formula (23)/(25), and in the three worked examples this reduces to $W'(x,p)=W(X,P)$.

desk verdict A clean, short derivation that gets the affine examples right, but the central formula misses a Jacobian factor and does not extend to dilations or nonlinear frames as claimed. read the letter →

arxiv 2411.09890 v1 pith:WIB4VIJH submitted 2024-11-15 math-ph math.MPquant-ph

J. Berra-Montiel , G. F. Torres del Castillo This is my paper · ORCID

classification math-phmath.MPquant-ph MSC 81S30

keywords WignerfunctionphasespacequantummechanicsreferenceframesWigner-Weylquantizationunitarytransformationsquasi-probabilitydistributioncoordinate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Wigner functions are phase-space portraits of quantum states: they contain all the information in the wavefunction, though they can take negative values and therefore are quasiprobabilities. This paper asks how that portrait changes when the experimenter moves to a different reference frame, and derives an answer for the class of frame changes implemented by a unitary operator that sends position operators to functions of position only and momentum operators to functions of momentum only. The main result is an integral formula for the transformed Wigner function, in both position and momentum representations, expressed directly through the old wavefunction evaluated at the transformed coordinates, with two phase functions constrained by a consistency relation. In the three worked examples—spatial translations, Galilean boosts, and constant acceleration—the phases cancel and the law reduces to $W'(x,p)=W(X,P)$: evaluate the old Wigner function at the new phase-space coordinates. A correct transformation law of this kind would make it straightforward to move phase-space descriptions of quantum states between frames, and the authors intend the same method to extend to nonlinear, non-inertial, and ultimately quantum reference frames.

What carries the argument

The carrying mechanism is the wavefunction transformation law $\psi'(x)=e^{-i\alpha(x,t)/\hbar}\psi(X(x,t))$, imported from Ref. [9], together with the phase-consistency condition (21) derived from the inner product $\langle x|p\rangle$. Inserting this law into the defining integral of the Wigner function produces the position-space formula (23), and the same step in the momentum representation produces (25). The phases $\alpha$ and $\beta$ do the real work: in affine displacements they cancel, which is why simple substitution works there, while for nonlinear $X$ and $P$ they remain and change the functional form of the Wigner function.

What would settle it

For a unitary dilatation $X=\lambda x$, $P=p/\lambda$ with $\lambda\neq 1$, compute the transformed Wigner function two ways: from Eq. (23) using the phase-only wavefunction rule, and from the directly defined Wigner function of the squeezed wavefunction $\psi'(x)=e^{-i\alpha(x,t)/\hbar}\psi(\lambda x)$. A mismatch by a factor $\lambda^n$, or a failure of normalization, would show that Eq. (23) is incomplete for non-volume-preserving frames; exact agreement would show the phase-only formula is more general than the derivation suggests.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Eq. (23): if $\hat U\hat x_i\hat U^{-1}=\hat X_i(\hat x,t)$ and $\hat U\hat p_i\hat U^{-1}=\hat P_i(\hat p,t)$, then the Wigner function of the transformed state is $$W'(x,p)=\frac{1}{(2\pi\hbar)^n}\int \$psi^{{*}}$(X((x+y)/2,t))\,\psi(X((x-y)/2,t))\,$e^{{-\frac{i}}${\hbar}[\$\alpha$((x-y)/2,t)-\$\alpha$((x+y)/2,t)-p\cdot y]}\,dy,$$ with the momentum-space analogue (25) obtained by Fourier transformation. The phases $\alpha$ and $\beta$ are not free: they must satisfy $p\cdot x=\beta(p,t)-\alpha(x,t)+P(p,t)\cdot X(x,t)$, the condition that makes the position and momentum descriptions of the new frame mutually consistent. For spatial translations, Galilean boosts, and constant acceleration, the phases cancel after substitution and the transformed Wigner function is simply $W'(x,p)=W(X,P)$.

Load-bearing premise

The load-bearing premise is that the frame change is a unitary transformation in which each new position operator depends only on the old position operator and each new momentum operator only on the old momentum operator, together with the imported phase relation (21); if a frame change mixes coordinates with momenta, or if no real phase functions satisfy (21), then formulas (23) and (25) do not follow.

Editorial extensions

If this is right

For spatial translations, Galilean boosts, and constant accelerations, the transformed Wigner function obeys $W'(x,p)=W(X,P)$, so the phase-space portrait moves rigidly to the new coordinates.
Because position and momentum marginals of the Wigner function are $|\psi(x)|^2$ and $|\tilde\psi(p)|^2$, formula (23) predicts exactly how both probability densities change under the frame transformation.
The phase factors $\alpha(x,t)$ and $\beta(p,t)$ are kept explicitly, so the same formulas are set up to handle nonlinear transformations where those phases do not cancel.
The momentum-representation formula (25) provides an independent route to the same result, giving a built-in consistency check for applications.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, Eq. (23) should be tested on a nonlinear coordinate map such as $X=x+\lambda x^2$; the consistency condition (21) may have no real solution for $\alpha,\beta$ in such cases, which would mean the admissible frame changes form a narrower class than the unitary form (13) alone suggests.
Beyond the paper, the phase-only wavefunction transformation (16) implicitly assumes that the coordinate map $X$ preserves the normalization of position eigenstates; for dilatations or other non-volume-preserving maps a Jacobian factor is likely needed, so the formulas as written are probably complete only for unit-Jacobian frames.
If the paper's transformation law is correct, an experimental tomography reconstruction of a displaced or accelerated state should show the original Wigner function rigidly shifted, which would be a direct phase-space test of the Galilean and acceleration examples.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is the short version: the paper is a modest, readable note that derives explicit Wigner-transformation formulas for a restricted class of frame changes and gets its three worked examples right. But the main formula as printed is not correct for transformations with nontrivial Jacobian, so the concluding claim about nonlinear and non-inertial frames is unsupported.

What is actually new are Eqs. (23) and (25), integral representations obtained by substituting the wave-function rule psi'(x) = exp(-i alpha) psi(X(x)) into the standard Wigner definition. I do not see those integrals in the cited literature, and for the spatial translation, Galilean, and constant-acceleration cases they reproduce the expected W'(x,p) = W(X,P). The calculation is transparent and easy to follow.

The soft spot is load-bearing. Eq. (16) states U^{-1}|x> = exp(i alpha)|X(x)>, but with the standard normalization = delta(x-x'), the ket |X(x)> is normalized to delta(X(x)-X(x')), not delta(x-x'). Unitarity forces an extra sqrt(|det J_X|) factor. That factor is missing from Eqs. (17) and (23). The examples all have det J_X = 1, which hides the error. Take X = 2x, P = p/2, realized by U psi(x) = sqrt(2) psi(2x); Eq. (23) gives W'_paper = W(2x,p/2)/2, which is not normalized. So the formula as printed is valid only for volume-preserving X, and the conclusion's claim about nonlinear transformations is false in that generality.

Two smaller issues. The essential rule (17) is imported from Ref. [9], by one of the authors, and not re-derived. That is acceptable if Ref. [9] is solid, but the new formula inherits its limitations. And the Galilean beta in Eq. (33) does not satisfy Eq. (21) as written; the term -mV^2 t is missing. Because only differences beta(p+u/2) - beta(p-u/2) enter Eq. (25), the final position-space Wigner results are unaffected. Minor, but worth fixing.

Who is this for? Someone collecting transformation laws for Wigner functions under affine, unit-determinant frame changes will find the examples useful. Someone wanting the nonlinear or quantum-reference-frame extension will not, at least not from this paper.

If the journal wants an incremental phase-space note, I would send it to a referee with a specific request to check the normalization and to restrict the claims to det J_X = 1. As is, I would not cite it.

Referee Report

3 major / 4 minor

Summary. The paper studies the transformation law of the Wigner function under changes of reference frames. Starting from a unitary operator U satisfying the restricted conditions U x U^{-1} = X(x,t) and U p U^{-1} = P(p,t), and using the wavefunction transformation psi'(x) = exp(-i alpha(x,t)/hbar) psi(X(x,t)) imported from Ref. [9], the authors derive an integral formula (Eq. (23)) for the Wigner function in the transformed frame, and a momentum-space analogue (Eq. (25)). They illustrate the formula with spatial translations, Galilean boosts, and constant acceleration, obtaining in each case W'(x,p) = W(X,P).

Significance. If the general formula (23) were correct, it would provide a compact tool for computing Wigner functions under reference-frame changes, including time-dependent ones, and the paper is clearly written with instructive examples. However, the generality is overstated: the derivation misses a Jacobian normalization factor, so the central formula fails for non-unit-determinant transformations, and the claimed extension to nonlinear transformations is inconsistent with the hypotheses (13). The examples themselves are all of the form X = x - f(t), P = p - g(t), for which the Jacobian is 1 and the results are known; the novelty therefore rests entirely on the flawed general claim.

major comments (3)

[Eq. (16) and Eq. (23)] Equation (16) omits the Jacobian determinant factor. For a unitary U satisfying (13), the standard delta-normalization <X(x)|X(x')> = delta(X(x)-X(x')) = delta(x-x')/|det J_X| requires U^{-1}|x> = sqrt(|det J_X(x,t)|) e^{i alpha(x,t)/hbar} |X(x,t)>. Without this factor, the right-hand side of (16) is not a unit vector. Consequently, Eq. (23) is missing the corresponding normalization factor and is incorrect for transformations with non-unit Jacobian, such as X = 2x, P = p/2 (realized by U psi(x) = sqrt(2) psi(2x)). For that example, Eq. (23) yields a Wigner function with total integral 1/2 instead of 1. All examples in Section 4.1 have X(x) = x - f(t), so det J_X = 1 and the defect is hidden. This is load-bearing because the abstract and Section 5 claim the formula applies to general changes of reference frames.
[Section 5] The claim that the method extends to nonlinear transformations and non-inertial frames is unsupported and in fact incompatible with the derivation. The hypotheses (13) require U x U^{-1} = X(x) and U p U^{-1} = P(p). Combining these with the commutation relation [X(x), P(p)] = i hbar forces X and P to be affine transformations with constant Jacobian determinants. For a nonlinear X, the unitary induced by the coordinate change would make U p U^{-1} a differential operator depending on x, violating (13). Thus the concluding statement that the approach can be applied to 'nonlinear transformations and non-inertial changes of reference frames, as well as quantum reference frames' is not supported by the paper's own framework.
[Eq. (33)] Equation (33) is inconsistent with the phase condition (21). Substituting X = x - Vt and P = p - mV into (21) gives alpha - beta = -p·Vt - mV·x + m V^2 t. The proposed alpha = -mV·x + xi(t) and beta = p·Vt + xi(t) yield alpha - beta = -p·Vt - mV·x, which is missing the m V^2 t term. This error does not affect the final Wigner expression (23) because only differences of alpha at different x enter and the missing term is independent of x, but it is still an internal inconsistency in the determination of alpha and beta and should be fixed.

minor comments (4)

[Eqs. (34) and (38)] The normalization factor (2 pi hbar)^3 should be written as (2 pi hbar)^n for consistency with the general notation, even though the examples are set in R^3.
[Eq. (30)] There is an extra closing parenthesis in the exponent: 'e^{- i/hbar (x-a)·u})' should read 'e^{- i/hbar (x-a)·u}'.
[End of Section 4.1.3] The sentence explaining that the examples take the form W'(x,p) = W(X,P) because the unitary operators are displacement operators is helpful but could be expanded: the phase factors xi(t) and the x-independent parts of alpha always cancel in the Wigner integrand, which is why only the coordinate substitution survives.
[Introduction / Section 5] The paper would benefit from acknowledging that the transformation law W'(x,p) = W(X,P) for translations and Galilean boosts is already well known in the phase-space literature; this would help calibrate the novelty claim and the review of prior work.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the only relevant issue is a minor self-citation to Ref. [9], whose content is re-derived in the text.

full rationale

The transformation law (23) is derived, not assumed: it follows by substituting the wavefunction transformation (17) into the definition (22). Equation (17) is itself derived in Eqs. (15)-(16) from the unitary ansatz (13), and the phase constraint (21) is derived from the scalar product (19)-(20), so the argument is self-contained up to the stated assumptions. Ref. [9], by one of the present authors, is cited as the starting point of Section 3, but its content is reproduced rather than imported as an unverified black box, hence the self-citation is not load-bearing. The examples are evaluations of formula (23), not fits. A mathematical caveat should be noted separately: Eq. (16) omits the Jacobian normalization |det J_X|^{1/2} required by unitarity when det J_X ≠ 1, so Eq. (23) is not valid for non-volume-preserving transformations as written; this is a correctness issue, not a circularity. The conclusion that the method extends to nonlinear frames is therefore unsupported by the printed derivation, but the derivation chain itself does not reduce to its inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard Wigner-Weyl quantization, on a restricted class of unitary transformations, and on a wave-function transformation rule taken from the authors' earlier Ref. [9]. No genuinely new entities are introduced. The only hand-chosen quantity is the gauge phase xi(t), which drops out of the final Wigner transformation.

free parameters (1)

xi(t) = unspecified (arbitrary real function of t)
Introduced in Eqs. (28), (33), and (37) to parametrize the gauge phase in alpha and beta. It cancels in the Wigner transformation (23) because only differences of alpha at shifted arguments appear, so it does not affect the final result.

assumptions (4)

standard math Standard Wigner-Weyl quantization properties, including the integral kernel (4), the inverse Wigner transform (6), and the normalization (9).
Invoked in Section 2 as background from Refs. [2-8]. No derivation of these properties is given in this paper, but they are standard phase-space quantum mechanics.
domain assumption Unitary operator U satisfying the separated transformation Eq. (13): U x U^-1 = X(x,t) and U p U^-1 = P(p,t).
This restricts the class of frame changes to those where the new position is a function of old position only and the new momentum a function of old momentum only. It excludes canonical transformations that mix x and p. Used to derive Eqs. (15)-(17).
domain assumption Wave-function transformation rule (17): psi'(x) = exp(-i alpha(x,t)/hbar) psi(X(x,t)), with phase constraint (21) p dot x = beta - alpha + P dot X.
Imported from Ref. [9] by one of the authors without proof. This is the bridge between unitary frame changes and wave-function substitution, and it is load-bearing for the Wigner transformation formulas.
standard math Standard inner product for position and momentum eigenstates: <X|P> = (2 pi hbar)^(-n/2) exp(i P dot X / hbar).
Used in Eqs. (19)-(20) to obtain the phase constraint (21).

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Pith. "Pith review of Wigner function under changes of reference frames." pith.science (2026). https://pith.science/paper/WIB4VIJH

@misc{pith2026241109890,
  author       = {Pith},
  title        = {Pith review of: Wigner function under changes of reference frames},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WIB4VIJH}},
  note         = {Machine review of arXiv:2411.09890}
}

read the original abstract

In this paper, we investigate the transformation laws of the Wigner function under changes of reference frames. By employing the coordinate transformation of the wave functions, we derive an integral representation for the transformed Wigner function in both position and momentum representations. To illustrate our results, we include some basic examples.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [9]

G. F. Torres del Castillo and B. C. N´ ajera Salazar, Transformation of a wavefunction under changes of reference frame, Rev. Mex. F ´ ıs.63, 185 (2017)

work page 2017

[2] [1]

Moshinsky and T

M. Moshinsky and T. H. Seligman, Canonical transformations to action and angle variables an d their representations in quantum mechanics , Ann. Phys. 114, 243–272 (1978)

work page 1978

[3] [2]

Bayen, M

F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. St ernheimer, Deformation theory and quantization I. Deformations of symplectic structures , Ann. Phys. 111, 61–110 (1978)

work page 1978

[4] [3]

Zachos, Deformation Quantization: Quantum Mechanics lives and wor ks in phase-space , Int

C. Zachos, Deformation Quantization: Quantum Mechanics lives and wor ks in phase-space , Int. J. Mod. Phys. A 17 297 (2002), arXiv:hep-th/0110114

arXiv 2002

[5] [4]

R. P. Rundle and M. J. Everitt, Overview of the Phase Space Formulation of Quantum Mechanic s with Application to Quantum Technologies , Adv. Quant. Tech. 4, 2100016 (2021)

work page 2021

[6] [5]

Reed and B

M. Reed and B. Simon, Methods of Modern Mathematical Physics . Volume I: Functional Analysis, 2nd edn. (Academic, San Diego, 1980)

work page 1980

[7] [6]

L. A. Takhtajan, Quantum Mechanics for Mathematicians , Graduate Studies in Mathematics Vol. 95 (American Mathematical Society, 2008). Wigner function under changes of reference frames 9

work page 2008

[8] [7]

Weinbub and D

J. Weinbub and D. K. Ferry, Recent advances in Wigner function approaches , Appl. Phys. Rev. 5, 041104 (2018)

2018