Pith. sign in

REVIEW 3 major objections 5 minor 22 references

The Feynman-Kac formula in deformation quantization

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A phase-space integral of the Wick-rotated star exponential of the Hamiltonian determines the ground-state energy of a quantum system.

desk verdict A clean restatement of a known trace formula with an overbroad claim and a couple of concrete errors in the examples; not novel enough to warrant refereeing as is. read the letter →

arxiv 2502.03624 v1 pith:KEWWFHM7 submitted 2025-02-05 math-ph math.MPquant-ph

Jasel Berra-Montiel , Hugo Garcia-Compean , Alberto Molgado This is my paper · ORCID
classification math-phmath.MPquant-ph MSC 81S3046F1053D5581S40
keywords deformationquantizationstarproductexponentialMoyalFeynman-KacformulaWignerfunctionsgroundstateenergyWickrotation
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

The paper argues that the Feynman-Kac formula of path-integral quantum mechanics has a direct analogue in deformation quantization: after a Wick rotation $\tau=it$, the ground-state energy $E_0$ of a Hamiltonian is recovered from the large-$\tau$ logarithm of the phase-space integral of the star exponential $\operatorname{Exp}_\star(-\tau H/\hbar)$, Eq. (19). The claim is that no operators or propagators are needed; only the star exponential of the Hamiltonian as a function on phase space is required. The authors demonstrate the formula on the free particle, the harmonic oscillator, general quadratic Hamiltonians, and the damped harmonic oscillator, obtaining the expected energies, and they point out that the linear potential example signals a spectrum unbounded from below. If correct, the result gives a purely phase-space route to ground-state energies for any system whose star exponential is known and integrable.

What carries the argument

The machinery is the star exponential and its spectral expansion. The star exponential $\operatorname{Exp}_\star(-itH/\hbar)=\sum_{n=0}^\infty \frac{1}{n!}(-it/\hbar)^n H^{\star n}$ is a phase-space function built from the Moyal star product. Its Fourier-Dirichlet expansion $\operatorname{Exp}_\star(-itH/\hbar)=2\pi\hbar\sum_n e^{-itE_n/\hbar}\rho_n$, with $\rho_n$ the normalized Wigner functions of the energy eigenstates, turns the phase-space integral into a sum over eigenvalues; after the Wick rotation $\tau=it$, the $\tau\to\infty$ limit is dominated by the smallest eigenvalue. This expansion is the bridge from the operator spectrum to a purely phase-space formula.

What would settle it

Take a one-dimensional finite square well and compute the phase-space integral of its Wick-rotated star exponential from the known propagator. If the logarithmic large-$\tau$ limit does not reproduce the lowest bound-state energy, then Eq. (19) does not hold for systems with a continuous spectrum, contrary to the paper's 'any prescribed physical system' wording.

Watch

Extended reading notes

Core claim

The central claim is that for a Hamiltonian with discrete spectrum bounded below, the ground-state energy is $$E_0 = -\lim_{\tau\to\infty} \frac{\hbar}{\tau}\ln\left[\frac{1}{2\pi\hbar}\int_{\mathbb{R}^2}\operatorname{Exp}_\star\left(-\frac{\tau}{\hbar}H\right)\,dx\,dp\right],$$ obtained from the Fourier-Dirichlet expansion (15), term-by-term phase-space integration using the normalization of the Wigner functions (16), and the Wick rotation $\tau=it$. The paper presents this as the deformation-quantization counterpart of the Feynman-Kac formula: instead of a stochastic expectation value, one needs only the star exponential of the Hamiltonian. The examples reproduce the known ground-state energies, while the linear potential illustrates what happens when the spectrum is not bounded below.

Load-bearing premise

The whole derivation rests on assuming that the spectral expansion of the star exponential can be integrated term by term and then Wick-rotated, with the long-time limit controlled by a discrete lowest energy level; the paper gives no general conditions under which this is true.

Editorial extensions

If this is right

  • For any system whose Wick-rotated star exponential can be computed and integrated, Eq. (19) yields the ground-state energy without operator diagonalization and without propagators.
  • The same phase-space integral encodes the full sum $\sum_n e^{-\tau E_n/\hbar}$, so the large-$\tau$ behavior carries spectral information beyond just $E_0$.
  • When the ground state is degenerate, the limiting integral equals the degeneracy, so the formula also detects ground-state multiplicity.
  • The damped-oscillator example shows the formula can be applied to deformed, non-Hermitian star products, producing complex ground-state energies.
  • The conclusions state that the construction is expected to generalize to quantum field theory and string theory, although no field-theoretic example is worked out.

Reading between the lines

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

  • The paper leaves implicit that, for continuous spectra, the spectral sum in Eq. (15) becomes an integral, so Eq. (19) as written needs a regularization or subtraction before the logarithmic limit is meaningful.
  • A stress test the paper does not perform is an anharmonic oscillator: one would compute $\operatorname{Exp}_\star$ order by order in the Moyal product and check that the $\tau\to\infty$ limit selects the exact anharmonic ground state rather than the harmonic-oscillator value.
  • Equation (17) is formally the analytic continuation of a phase-space partition function, so the formula can be read as the zero-temperature limit of a deformation-quantization thermal state, a direction the paper touches through the KMS condition in the quadratic example.
  • For deformed star products the same logarithmic limit returns complex energies, suggesting the formula could serve as a diagnostic for non-Hermitian or dissipative quantum systems beyond the damped oscillator.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a deformation-quantization analogue of the Feynman-Kac formula. Starting from the Fourier-Dirichlet expansion of the star exponential, Exp⋆(-itH/ℏ) = 2πℏ Σ e^{-itE_n/ℏ} ρ_n, the authors integrate over phase space, perform a Wick rotation τ = it, and take τ → ∞ to extract the ground-state energy as E0 = -lim_{τ→∞} (ℏ/τ) ln[(1/2πℏ)∫ Exp⋆(-τH/ℏ) dxdp] (Eq. (19)). Examples for the free particle, harmonic oscillator, linear potential, general quadratic Hamiltonians, and a damped oscillator are presented.

Significance. If Eq. (19) is restricted to Hamiltonians with discrete spectrum and trace-class heat semigroup, the derivation is straightforward and the harmonic-oscillator and quadratic examples check out. The paper is commendable for making the phase-space trace explicit and for identifying the linear-potential case as problematic. However, the abstract's 'any prescribed physical system' claim is not supported, and one printed example contains a sign error. With appropriate hypotheses and corrections, the paper would be a useful contribution to the deformation-quantization literature.

major comments (3)
  1. [§3, Eq. (19); §3.1.1; §3.1.3] The central claim that Eq. (19) applies to 'any prescribed physical system' is not supported. The derivation from (15) to (17) requires the Fourier-Dirichlet expansion to hold with integrable, normalized Wigner functions and allows term-by-term integration; the Wick-rotated limit in (18) additionally requires Σ e^{-τE_n/ℏ} to converge (i.e., e^{-τH} trace class) and a real, discrete spectrum bounded below. The paper's own free-particle example, §3.1.1, violates these hypotheses: Exp⋆(-τH_free/ℏ) = e^{-τp²/(2mℏ)} is independent of x, so the phase-space integral in (19) diverges and the logarithm is undefined; the quoted E0 = 0 requires an implicit volume regularization. Similarly, the linear-potential example, §3.1.3, has a divergent integral for τ → ∞; stating that 'the energy is not bounded from below' is not a consequence of (19) as written. The theorem should be stated with explicit hypotheses (discrete spectrum, trace-class semigroup, integrable star exponential) and the examples reclassified accordingly.
  2. [§3.1.4, Eq. (27)] In the Wick-rotated star exponential for the general quadratic Hamiltonian, Eq. (27) retains a factor i in the exponent: e^{-i H_q/(ℏ√(ab-c²)) tanh(√(ab-c²)τ)}. The Gaussian integration leading to Eq. (28) is valid only for the real exponent -H_q/(ℏ√(ab-c²)) tanh(...); with the printed i, the integral yields a coth-type factor, not the quoted csch. This appears to be a sign typo (the factor should be -1, not -i), but as printed the example does not follow from (19).
  3. [§3.1.5, Eqs. (34)-(35)] The damped-oscillator example extends Eq. (19) to a non-Hermitian star product and produces a complex ground-state energy E0 = ℏ(ω/2 + iγ/2). The derivation of (19) in Section 3 assumes real eigenvalues E_n and normalized Wigner functions; for the complex case one must justify the analytic continuation of the logarithm and the validity of the Tauberian limit for a non-self-adjoint generator. As printed, this example goes beyond the hypotheses under which (19) was derived.
minor comments (5)
  1. [Abstract] 'Constructing on previous work' should read 'Building on previous work'.
  2. [§3.1.2, Eq. (21)] The prefactor (cos(-iωτ/2))^{-1} equals sech(ωτ/2); writing it as such would make the Wick rotation more transparent.
  3. [§3.1.3, Eq. (25)] The sentence that the Feynman-Kac formula 'reveals' that the energy is not bounded from below is misleading, since the integral in (19) does not converge; the example should be labeled as outside the domain of applicability.
  4. [§3.1.5, Eq. (34)] The right-hand side contains e^{-iγτ/2}, making the phase-space integral complex; please specify the chosen branch of the logarithm or discuss how the real part is isolated in the limit.
  5. [Section 2, Eq. (4)] The Hilbert-Schmidt relation is stated for f ∈ L²(R^{2n}), but the star exponentials used in Section 3 are often distributions; a brief comment on the generalized-function extension would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (19) follows from the independent Fourier-Dirichlet expansion (15); self-citations only supply illustrative examples.

full rationale

The central formula (19) is derived, not assumed: the paper starts from the Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer Fourier-Dirichlet expansion (15), an independent spectral result, integrates it over phase space using the normalization (16) of the diagonal Wigner functions, obtains the partition-function identity (17), and then takes the Wick-rotated large-tau limit in which the E0 term dominates (18). At no point is E0 inserted or fitted; it emerges as the slowest-decaying eigenvalue. The self-cited paper [6] is used only to supply the explicit star exponentials in the examples of Section 3.1, and identical or overlapping formulas are also attributed to [19] and [20]; moreover the derivation of (19) itself does not invoke [6]. The example star exponentials therefore do not constitute load-bearing circular support for the main claim. The mathematical gaps noted by a critical reader (the free-particle star exponential is not integrable over R^2 without regularization, continuous spectra require replacing (15) by an integral, and Eq. (27) as printed contains a residual factor i after Wick rotation) are correctness and hypothesis concerns, not instances of the result being equivalent to its inputs by construction. Accordingly no circular step can be exhibited, and the appropriate score is 0.

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

The paper introduces no new free parameters or entities. Its central claim rests on the assumed validity of the Fourier-Dirichlet expansion and the Wick rotation for arbitrary systems, plus the imported deformed star product for the damped oscillator.

assumptions (4)
  • domain assumption The Fourier-Dirichlet expansion of the star exponential, Eq. (15), with normalized Wigner functions ρn, holds for the Hamiltonian under consideration.
    The central derivation integrates this expansion term by term to obtain Eq. (17). No proof is given; it is cited from [1] and assumed for arbitrary systems.
  • domain assumption The star exponential over phase space is integrable and the infinite sum in Eq. (17) can be analytically continued to imaginary time τ=it so that the τ→∞ limit isolates the ground state.
    The Wick rotation from Eq. (17) to Eq. (18) requires convergence and dominance by the lowest eigenvalue; this is not justified for general Hamiltonians.
  • domain assumption For the damped oscillator, the deformed star product ⋆γ still admits the Fourier-Dirichlet expansion and normalized Wigner functions, leading to Eq. (34).
    The non-Hermitian star product example is imported from [22] without derivation of the phase-space integral.
  • standard math Standard results of Weyl quantization and the Moyal product, including the trace of Weyl operators as phase space integration.
    Background used in Section 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Feynman-Kac formula in deformation quantization." pith.science (2026). https://pith.science/paper/KEWWFHM7

@misc{pith2026250203624,
  author       = {Pith},
  title        = {Pith review of: The Feynman-Kac formula in deformation quantization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEWWFHM7}},
  note         = {Machine review of arXiv:2502.03624}
}
read the original abstract

We introduce the Feynman-Kac formula within the deformation quantization program. Constructing on previous work it is shown that, upon a Wick rotation, the ground state energy of any prescribed physical system can be obtained from the asymptotic limit of the phase space integration of the star exponential of the Hamiltonian operator. Some examples of this correspondence are provided showing a novel and efficient way of computing the ground state energy for some physical models.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 17 canonical work pages

  1. [1]

    Bayen, M

    F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. St ernheimer, Deformation theory and quantization I-II Ann. Phys. 111 61–110 (1978); ibid. 111–151

    work page 1978

  2. [2]

    Inonu and E

    E. Inonu and E. P. Wigner, On the contraction of groups and their representations , Poc. Nat. Acad. Sci. USA 39 510 (1953)

    work page 1953

  3. [3]

    Waldmann, Convergence of Star Product: From Examples to a General Fram ework, EMS Surv

    S. Waldmann, Convergence of Star Product: From Examples to a General Fram ework, EMS Surv. Math. Sci. 6 1–31 (2019), arXiv:1901.11327 [math.QA]

    arXiv 2019

  4. [4]

    Generating All Wigner Functions

    T. Curtright, T. Uematsu and C. Zachos, Generating all Wigner functions , J. Math. Phys. 42, 2396 (2001), arXiv:hep-th/0011137

    work page Pith review arXiv 2001

  5. [5]

    Glimm and A

    J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View , (Springer-Verlag, New York, 1987)

    work page 1987

  6. [6]

    Berra-Montiel, H

    J. Berra-Montiel, H. Garc ´ ıa-Compe´ an and A. Molgado,Star exponentials from propagators and path integrals, Ann. Phys. 468, 169744 (2024). arXiv:2404.08815 [math-ph]

    arXiv 2024

  7. [7]

    Sharan, Star-product representation of path integrals , Phys

    P. Sharan, Star-product representation of path integrals , Phys. Rev. D20, 2 (1979)

    work page 1979

  8. [8]

    Wigner, On the Quantum Correction For Thermodynamic Equilibrium , Phys

    E. Wigner, On the Quantum Correction For Thermodynamic Equilibrium , Phys. Rev. 40, 749 (1932). The Feynman-Kac formula in deformation quantization 10

    work page 1932

Show all 22 references
  1. [9]

    L. A. Takhtajan, Quantum Mechanics for Mathematicians (Graduate Studies in Mathematics Vol. 95) (American Mathematical Society, Providence, Rhode Island , 2008)

    2008

  2. [10]

    N. C. Dias, M. de Gosson, F. Luef and J. N. Prata, Quantum mechanics in phase space: the Schr¨ odinger and the Moyal representations , J. Pseudo-Differ. Oper. Appl. 3, 367 (2012). arXiv:1209.1850 [math-ph]

    2012 arXiv

  3. [11]

    Reed and B

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

    1980

  4. [12]

    T. L. Curtright, D. B. Fairlie and C. K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space (World Scientific, 2014)

    2014

  5. [13]

    G. B. Folland, Harmonic Analysis in Phase Space , Annals of Mathematics Studies, vol. 122 (Princeton University Press, Princeton, 1989)

    1989

  6. [14]

    Kontsevich, Deformation Quantization of Poisson Manifolds , Lett

    M. Kontsevich, Deformation Quantization of Poisson Manifolds , Lett. Math. Phys. 66, 157 (2003). arXiv:q-alg/9709040

    2003 arXiv

  7. [15]

    Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer P hysics, and Financial Markets, (World Scientific, Singapore, 2006)

    H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer P hysics, and Financial Markets, (World Scientific, Singapore, 2006)

    2006

  8. [16]

    Dito, Star-product approach to quantum field theory: The free scal ar field , Lett

    J. Dito, Star-product approach to quantum field theory: The free scal ar field , Lett. Math. Phys. 20, 125 (1990)

    1990

  9. [17]

    Berra-Montiel, Star product representation of coherent state path integra ls, Eur

    J. Berra-Montiel, Star product representation of coherent state path integra ls, Eur. Phys. J. Plus 135, 906 (2020). arXiv:2007.02483 [quant-ph]

    2020 arXiv

  10. [18]

    N. C. Dias and J. N. Prata, Deformation quantization and Wigner functions , Mod. Phys. Lett. A 20, 1371 (2005). arXiv:hep-th/0504166

    2005 arXiv

  11. [19]

    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. A17 297 (2002), arXiv:hep-th/0110114

    2002 arXiv

  12. [20]

    Omori, Y

    H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Star exponential functions for quadratic forms and polar elements , Contemp. Math. 315, 38 (2002)

    2002

  13. [21]

    Basart and A

    H. Basart and A. Lichnerowicz, Conformal symplectic geometry, deformations, rigidity an d geometrical KMS conditions , Lett. Math. Phys. 10, 167 (1985)

    1985

  14. [22]

    Ditto and F

    G. Ditto and F. J. Turrubiates, The damped harmonic oscillator in deformation quantizatio n, Phys. Lett. A 352, 4 (2006), arXiv:quant-ph/0510150

    2006 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.