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REVIEW 2 major objections 4 minor 55 references

First Principles Quantization of a Non-Conservative Scalar Field

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Damped scalar field quantized from a doubled-variable action

desk verdict The path-integral half works, the canonical quantization doesn't: the mode normalization is off by a factor of √2 and the claimed Fock space is internally inconsistent. read the letter →

arxiv 2509.01403 v1 pith:VLJLVZFL submitted 2025-09-01 hep-th

Kauship Saha , Sandeep Aashish This is my paper
classification hep-th
keywords dissipative scalar fielddoubled variablesnon-conservative actionin-in formalismcanonical quantizationretarded and advanced Green's functionsspectral functionquasi-scalar bosons
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

This paper tries to establish that a linearly damped scalar field can be quantized from an action principle, without introducing ad hoc non-unitary terms. The construction doubles the field into a forward-evolving physical copy and a backward-evolving auxiliary copy, coupled by a non-conservative term that produces the damping. From the doubled action the authors derive retarded and advanced Green's functions and the spectral function, and report that these agree with the known results of the in-in formalism. They then construct a Fock space in which the physical quanta are decaying quasi-scalar bosons with lifetime of order 1/γ, generated by a Hermitian double Hamiltonian. If the construction is sound, it gives a first-principles route from classical dissipation to quantum field theory, and a template for quantizing other non-conservative systems.

What carries the argument

The load-bearing object is the doubled-variable action for the non-conservative scalar, S = ∫d⁴x (∂μφ− ∂μφ+ − m²φ+φ− − γφ−∂₀φ+), in which φ₊ is the physical forward-evolving field and φ₋ is an auxiliary backward-evolving field. The γ term couples the two and breaks time-reversal symmetry, encoding linear damping. This action simultaneously does three jobs: its physical limit gives the damped Klein-Gordon equation; its quadratic kernel inverts to the retarded/advanced Green's function matrix in the path integral; and its plane-wave mode expansions with e^{∓γt/2} factors, together with the cross-commutators [a, b†] and [b, a†], define the Fock space. The Hermitian 'double Hamiltonian' of Eq. (

What would settle it

Compute the equal-time commutator [φ₊(x,t), Π₋(y,t)] directly from the mode expansions (30)-(31) using only the commutators (34)-(35). If the result is not iδ³(x−y), the canonical quantization is internally inconsistent. This is a finite algebraic check, requiring no approximation.

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Extended reading notes

Core claim

The central claim is that the doubled-variable action S = ∫d⁴x (∂μφ− ∂μφ+ − m² φ+φ− − γ φ− ∂₀φ+) is a complete quantum theory of a damped scalar field. In the path-integral picture, the quadratic kernel inverts to retarded and advanced propagators whose poles are shifted by the damping γ, and the spectral function is a Lorentzian of width γ that reduces to the free-field signature as γ → 0. In the canonical picture, mode expansions with e^{∓γt/2} envelopes and cross-commutators between the physical and auxiliary sectors produce a normal-ordered double Hamiltonian that is Hermitian. The physical one-particle states evolve with a decaying factor e^{−γt/2}, so the excitations are quasi-scalar b

Load-bearing premise

The whole Fock-space construction depends on the assumption that the mode expansions (30)–(31) with the cross-commutators (34)–(35) satisfy the equal-time canonical commutation relations (32)–(33); if they do not, the vacuum and particle states are not a consistent Hilbert space.

Editorial extensions

If this is right

  • The causal propagators of a damped scalar carry pole shifts ±iγk⁰, so dissipation is imprinted directly in the two-point functions and the spectral function has width γ.
  • In the γ → 0 limit the theory reduces continuously to the free Klein-Gordon field: the spectral function returns to sign(k⁰) δ(k²−m²) and the propagators to the free Feynman form.
  • The physical excitations are quasi-scalar bosons with lifetime τ ∼ 1/γ, while the Hamiltonian remains Hermitian after normal ordering, so unitarity is preserved inside the doubled theory.
  • The same doubled-variable scheme can be applied to other non-conservative field theories, providing a quantization route that bypasses phenomenological or effective-theory treatments.

Reading between the lines

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

  • The construction suggests a general recipe: for any dissipative equation that can be written as a linear coupling between + and − fields, one can invert the doubled kernel to get causal propagators and then build a Fock space from the cross-modes.
  • Because the same γ controls damping and spectral width, any finite-temperature or interacting extension of the model would need a fluctuation-dissipation-type relation tying the noise kernel to this width.
  • Treating the γ term as a vertex ∂₀φ+ φ−, the Gaussian path integral in Eq. (21) can be used to generate Feynman rules for a dissipative interacting scalar theory, which the current paper does not work out.
  • If the canonical quantization holds, the doubled action could serve as a classical starting point for open-quantum-system calculations, offering an alternative to master-equation approaches that are usually taken as phenomenological.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper studies a real scalar field with linear damping using Galley's doubled-variable action principle. The authors introduce a non-conservative interaction K that yields the damped Klein-Gordon equation in the physical limit, derive the retarded/advanced Green's functions and spectral function from the path integral, and attempt a canonical quantization with mode expansions, a Hermitian 'double Hamiltonian,' and decaying quasi-scalar boson states. I find the path-integral derivation correct, but the canonical quantization fails: the proposed mode algebra violates the equal-time commutation relations, and the physical states have zero norm.

Significance. The path-integral part of the paper (Sec. III) is a clean and correct derivation: the quadratic action (15) leads to the matrix M(k), and its inversion gives the retarded and advanced Green's functions (24)-(25) and spectral function (26), matching the in-in/Schwinger-Keldysh results. The construction of K in Eq. (11) as a model for damping is acknowledged by the authors and is a standard model-building step, not a technical defect. However, the canonical quantization (Sec. IV) is internally inconsistent: the mode expansions and commutators do not satisfy the equal-time commutation relations, and the proposed physical Fock states have zero norm. Since canonical quantization is advertised as part of the first-principles quantization, the paper's central claim is not established. The correct path-integral results do not compensate for the invalid canonical sector.

major comments (2)
  1. [Sec. IV, Eqs. (30)-(35)] The proposed mode expansions do not satisfy the equal-time canonical commutation relations (32)-(33). Using (30), (31), and the cross-commutators (34)-(35), a direct computation with Π_- = ∂0φ_- − γφ_- gives [φ_+(x),Π_-(y)] = ∫ d^3k/(2π)^3 [2i cos(k·(x−y)) − i(γ/ω_k) sin(k·(x−y))] = 2iδ^3(x−y), since the odd sin term integrates to zero. The same factor of 2 appears in [φ_-,Π_+]. This contradicts (32)-(33) and invalidates the Fock-space construction of Sec. IV, including the Hamiltonian (40) and the decay law (49). The path-integral derivation in Sec. III is a separate, correct Gaussian inversion and is not affected.
  2. [Sec. IV.B, Eqs. (36), (42)-(44)] The physical one-particle states are null. Equation (36) sets [b(k),b†(p)]=0, so ⟨0|b(p)b†(q)|0⟩ = 0. Hence |p⟩_+ = b†(p)|0⟩ has zero norm. Equation (44) is only a cross-normalization between the physical and auxiliary sectors; it does not define a positive-definite inner product on the physical sector. Consequently the interpretation of b† as creating observable quasi-scalar bosons, the probability-conservation statement (51), and the claim that the auxiliary states 'normalize' the physical states are not supported. Correcting this requires more than a change of normalization in (30)-(31); the operator algebra itself must be rethought.
minor comments (4)
  1. [Sec. II.A and Introduction] The phrase 'without ad hoc tweaks' in the Introduction is overstated, since the interaction K in Eq. (11) is explicitly constructed so that Eq. (10) results. This is a legitimate model-building step, but it should be described as such rather than as a first-principles derivation of the damping term.
  2. [Sec. IV, Eq. (29)] The convention for k·x and the meaning of ω_k should be stated explicitly. The mode expansions (30)-(31) use both e^{-ik·x} and e^{ik·x}, and the reality/Hermiticity conditions on the fields should be spelled out.
  3. [Sec. IV, Eq. (44)] The notation −⟨k|p⟩_+ is nonstandard and confusing. Please explain which inner product is being used and how positive-definiteness is supposed to be obtained.
  4. [Sec. IV, after Eq. (40)] The term 'double hermitian' is unusual; the Hamiltonian in (40) is simply Hermitian if the underlying algebra is valid. Consider using standard terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-conservative action is openly constructed to reproduce the damped Klein–Gordon equation, and the derived Green's functions and decay follow from that action by standard Gaussian inversion and mode expansion; the one self-citation is non-load-bearing.

full rationale

The paper's action (Eq. 11) is deliberately built so that the physical-limit equation of motion is the damped Klein–Gordon equation (Eq. 10). This is stated explicitly ('By design, the dynamics of action (1) reduces to the known equation of motion'), so it is a transparent model assumption rather than a hidden circular step. The path-integral derivation of the retarded/advanced Green's functions (Eqs. 22–25) and the spectral function (Eq. 26) is a direct Gaussian inversion of the quadratic kernel M(k) obtained from that action; it is a mathematical consequence of the assumed Lagrangian, not an independently fitted prediction. Likewise, the canonical quantization introduces mode expansions that solve the damped field equations, and the resulting double Hamiltonian (Eq. 40) and the exponential decay of the physical states (Eqs. 47–49) follow from those equations of motion and the standard Heisenberg evolution. No parameter is fitted to a subset of data and then renamed a prediction. The only self-citation is Ref. [34] (Aashish & Haque), cited in the introduction as one example of prior work adopting Galley's method; it plays no role in any derivation and is therefore not load-bearing. A separate, non-circularity issue is that the equal-time commutator computed from the proposed mode expansions (Eqs. 30–31) and cross-commutators (Eqs. 34–35) yields [φ₊, Π₋] = 2i δ³(x−y), not i δ³(x−y), so the canonical quantization as written is internally inconsistent; however, this is a correctness flaw, not a circularity, and does not affect the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central claim rests on Galley's action principle, the imposed canonical commutation relations, and a mode-expansion ansatz that is not self-consistent. The damping coefficient γ is a free parameter chosen to reproduce the damped Klein-Gordon equation. The quasi-scalar boson and its partner are invented entities without independent evidence.

free parameters (1)
  • damping coefficient γ = not fitted; chosen by hand
    Introduced in the interaction K in Eq. (11) to reproduce the damped Klein-Gordon equation (10); all derived results (poles, spectral width, lifetime) depend on γ.
assumptions (4)
  • domain assumption Equal-time canonical commutation relations [ϕ₊,Π₋]=iδ and [ϕ₋,Π₊]=iδ
    Imposed in Eqs. (32)-(33) to quantize the doubled theory.
  • domain assumption Galley's principle of stationary non-conservative action yields correct classical dynamics in the physical limit
    Used to construct the action (11); relies on Ref. [28] and its extension [41].
  • ad hoc to paper The damped/anti-damped plane-wave mode expansion with normalization 1/√ω_k is a complete basis for the field operators
    Eqs. (29)-(31); not verified and inconsistent with the CCRs (see weakest_assumption).
  • domain assumption Real mode frequency ω_k² = k²+m²-γ²/4 > 0 (underdamped regime)
    The canonical quantization uses √ω_k, requiring γ<2m; the restriction is never stated.
invented entities (2)
  • Quasi-scalar boson
    purpose: Physical excitation of the damped scalar field with finite lifetime τ~1/γ; interprets the decaying modes.
    Defined via the mode expansion (30); it is a model-dependent quasi-particle with no predicted observable signature beyond the model's own Green's functions.
  • Auxiliary (partner) excitation
    purpose: Counterpart created by a† that grows as e^{γt/2} and participates in state normalization to conserve probability.
    Introduced in Eqs. (31) and (43); purely a bookkeeping device within the doubled formalism, no independent handle.

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Pith. "Pith review of First Principles Quantization of a Non-Conservative Scalar Field." pith.science (2026). https://pith.science/paper/VLJLVZFL

@misc{pith2026250901403,
  author       = {Pith},
  title        = {Pith review of: First Principles Quantization of a Non-Conservative Scalar Field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLJLVZFL}},
  note         = {Machine review of arXiv:2509.01403}
}
read the original abstract

We present the first-principles quantization of a damped scalar field within the framework of classical action principle of non-conservative systems using doubled dynamical variables. We consider a non-conservative potential term constructed to describe a linear damping of the scalar field for quantization using canonical and path-integral formalisms, and derive the two-point Green's function along with the spectral function, which are consistent with known results from the well-known in-in formalism.

Figures

Figures reproduced from arXiv: 2509.01403 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Pole of the retarded Green’s function( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

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Works this paper leans on

55 extracted references · 41 canonical work pages

  1. [1]

    Goldstein, C

    H. Goldstein, C. Poole, J. Safko, and S. R. Addison, American Journal of Physics70, 782–783 (2002)

    work page 2002

  2. [2]

    P. S. Bauer, Proceedings of the National Academy of Sciences of the United States of America 17, 311 (1931)

    work page 1931

  3. [3]

    Bateman, Physical Review38, 815–819 (1931)

    H. Bateman, Physical Review38, 815–819 (1931)

    work page 1931

  4. [4]

    Staruszkiewicz, Annalen der Physik480, 362–367 (1970)

    A. Staruszkiewicz, Annalen der Physik480, 362–367 (1970)

    work page 1970

  5. [5]

    Jaranowski and G

    P. Jaranowski and G. Schäfer, Physical Review D55, 4712–4722 (1997)

    work page 1997

  6. [6]

    Caldirola, Il Nuovo Cimento18, 393–400 (1941)

    P. Caldirola, Il Nuovo Cimento18, 393–400 (1941)

    work page 1941

  7. [7]

    Kanai, Progress of Theoretical Physics3, 440–442 (1948)

    E. Kanai, Progress of Theoretical Physics3, 440–442 (1948). 14

    work page 1948

  8. [8]

    Um, K.-H

    C.-I. Um, K.-H. Yeon, and T. F. George, Phys. Rept.362, 63 (2002)

    work page 2002

Show all 55 references
  1. [9]

    Dekker, Physics Reports80, 1–110 (1981)

    H. Dekker, Physics Reports80, 1–110 (1981)

    1981

  2. [10]

    Banerjee and P

    R. Banerjee and P. Mukherjee, J. Phys. A35, 5591 (2002), arXiv:quant-ph/0108055

    2002 arXiv

  3. [11]

    M. J. Blacker and D. L. Tilbrook, Phys. Rev. A104, 032211 (2021), arXiv:2107.05827 [quant- ph]

    2021 arXiv

  4. [12]

    Deguchi and Y

    S. Deguchi and Y. Fujiwara, Phys. Rev. A101, 022105 (2020), arXiv:2002.05435 [quant-ph]

    2020 arXiv

  5. [13]

    D. M. Gitman and V. G. Kupriyanov, Eur. Phys. J. C50, 691 (2007), arXiv:hep-th/0605025

    2007 arXiv

  6. [14]

    P. S. Riseborough, P. Hanggi, and U. Weiss, Phys. Rev. A31, 471 (1985)

    1985

  7. [15]

    Dito and F

    G. Dito and F. J. Turrubiates, Phys. Lett. A352, 309 (2006), arXiv:quant-ph/0510150

    2006 arXiv

  8. [16]

    Manzano, AIP Adv.10, 025106 (2020)

    D. Manzano, AIP Adv.10, 025106 (2020)

    2020

  9. [17]

    Nathan and M

    F. Nathan and M. S. Rudner, Physical Review B102, 10.1103/physrevb.102.115109 (2020)

    2020 doi

  10. [18]

    R. A. Bertlmann, W. Grimus, and B. C. Hiesmayr, Phys. Rev. A 73, 054101 (2006), arXiv:quant-ph/0602116

    2006 arXiv

  11. [19]

    McDonald and A

    A. McDonald and A. A. Clerk, Phys. Rev. Res.5, 033107 (2023), arXiv:2302.14047 [quant-ph]

    2023 arXiv

  12. [20]

    G. Park, Z. Huang, Y. Zhu, C. Yang, G. Kin-Lic Chan, and L. Lin, Phys. Rev. B110, 195148 (2024), arXiv:2408.15529 [quant-ph]

    2024 arXiv

  13. [21]

    T. Guff, C. U. Shastry, and A. Rocco, Sci. Rep.15, 3658 (2025), arXiv:2311.08486 [quant-ph]

    2025 arXiv

  14. [22]

    Gaset, X

    J. Gaset, X. Gràcia, M. C. Muñoz-Lecanda, X. Rivas, and N. Román-Roy, Reports on Math- ematical Physics 87, 347–368 (2021)

    2021

  15. [23]

    Boyanovsky, H

    D. Boyanovsky, H. J. de Vega, R. Holman, D.-S. Lee, and A. Singh, Physical Review D51, 4419–4444 (1995)

    1995

  16. [24]

    Crossley, P

    M. Crossley, P. Glorioso, and H. Liu, JHEP09, 095, arXiv:1511.03646 [hep-th]

    arXiv

  17. [25]

    Boyanovsky, New J

    D. Boyanovsky, New J. Phys.17, 063017 (2015), arXiv:1503.00156 [hep-ph]

    2015 arXiv

  18. [26]

    C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev. D 76, 085007 (2007), arXiv:0705.2960 [hep-th]

    2007 arXiv

  19. [27]

    McDonald and A

    A. McDonald and A. A. Clerk, Physical Review Research5, 10.1103/physrevresearch.5.033107 (2023)

    2023 doi

  20. [28]

    C. R. Galley, Physical Review Letters110, 10.1103/physrevlett.110.174301 (2013)

    2013 doi

  21. [29]

    Schwinger, Journal of Mathematical Physics2, 407–432 (1961)

    J. Schwinger, Journal of Mathematical Physics2, 407–432 (1961)

    1961

  22. [30]

    L. V. Keldysh, Sov. Phys. JETP20, 1018 (1965)

    1965

  23. [31]

    L. M. Sieberer, M. Buchhold, and S. Diehl, Reports on Progress in Physics79, 096001 (2016). 15

    2016

  24. [32]

    Feynman and F

    R. Feynman and F. Vernon, Annals of Physics24, 118 (1963)

    1963

  25. [33]

    Kälin, J

    G. Kälin, J. Neef, and R. A. Porto, JHEP01, 140, arXiv:2207.00580 [hep-th]

    arXiv

  26. [34]

    Aashish and A

    S. Aashish and A. Haque, Eur. J. Phys.37, 055202 (2016), arXiv:1601.02102 [physics.class-ph]

    2016 arXiv

  27. [35]

    N. E. Martínez-Pérez and C. Ramírez, J. Math. Phys.59, 032904 (2018), arXiv:1708.03419 [math-ph]

    2018 arXiv

  28. [36]

    N. E. Martínez-Pérez and C. Ramírez, arXiv preprint (2015), arXiv:1501.05018 [hep-th]

    2015 arXiv

  29. [37]

    Mahapatra and A

    R. Mahapatra and A. Haque, Phys. Lett. A494, 129289 (2024)

    2024

  30. [38]

    Mahapatra and S

    R. Mahapatra and S. Yadav, Eur. Phys. J. Plus140, 552 (2025)

    2025

  31. [39]

    Aykroyd, A

    C. Aykroyd, A. Bourgoin, and C. L. Poncin-Lafitte, Hamiltonian treatment of non-conservative systems (2025), arXiv:2507.18658 [physics.class-ph]

    2025 arXiv

  32. [40]

    Baggioli, M

    M. Baggioli, M. Vasin, V. Brazhkin, and K. Trachenko, Physical Review D102, 10.1103/phys- revd.102.025012 (2020)

    2020 doi

  33. [41]

    C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for classical mechanics and field theories (2014), arXiv:1412.3082 [math-ph]

    2014 arXiv

  34. [42]

    Cartas-Fuentevilla and A

    R. Cartas-Fuentevilla and A. J. C. Juarez-Dominguez, The European Physical Journal Plus 138, 10.1140/epjp/s13360-023-03839-z (2023)

    2023 doi

  35. [43]

    Carrillo-Ibarra, H

    I. Carrillo-Ibarra, H. Garcia-Compean, and F. J. Turrubiates, Int. J. Mod. Phys. A28, 1350068 (2013)

    2013

  36. [44]

    F. A. P. Alves-Júnior, A. S. Ribeiro, G. B. Souza, and J. A. Helayël-Neto, Annals Phys.481, 170140 (2025), arXiv:2311.00468 [hep-th]

    2025 arXiv

  37. [45]

    Vitiello, in24th International Workshop on Fundamental Problems of High Energy Physics and Field Theory(2001) pp

    G. Vitiello, in24th International Workshop on Fundamental Problems of High Energy Physics and Field Theory(2001) pp. 221–234, arXiv:hep-th/0110182

    2001 arXiv

  38. [46]

    Jafari and F

    M. Jafari and F. Kheirandish, Int. J. Theor. Phys.56, 1249 (2017), arXiv:1701.01750 [quant- ph]

    2017 arXiv

  39. [47]

    Jafari and F

    M. Jafari and F. Kheirandish, Int. J. Theor. Phys.57, 1224 (2018)

    2018

  40. [48]

    Bödeker and J

    D. Bödeker and J. Nienaber, Phys. Rev. D106, 056016 (2022), arXiv:2205.14166 [hep-ph]

    2022 arXiv

  41. [49]

    Refaei and F

    A. Refaei and F. Kheirandish, Int. J. Theor. Phys.55, 432 (2016), arXiv:1501.06594 [quant-ph]

    2016 arXiv

  42. [50]

    Ipek and A

    S. Ipek and A. Caticha, AIP Conf. Proc.1641, 345 (2015), arXiv:1412.5637 [quant-ph]

    2015 arXiv

  43. [51]

    Berera, M

    A. Berera, M. Gleiser, and R. O. Ramos, Phys. Rev. D 58, 123508 (1998), arXiv:hep- ph/9803394

    1998

  44. [52]

    Jacob and R

    R. Jacob and R. G. Sachs, Phys. Rev.121, 350 (1961). 16

    1961

  45. [53]

    V. I. Kuksa, Adv. High Energy Phys.2015, 490238 (2015), arXiv:1509.08686 [hep-ph]

    2015 arXiv

  46. [54]

    Giacosa, A

    F. Giacosa, A. Okopińska, and V. Shastry, Eur. Phys. J. A57, 336 (2021), arXiv:2106.03749 [hep-ph]

    2021 arXiv

  47. [55]

    M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

    1995

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