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REVIEW 3 major objections 4 minor 48 references

Wave equations for spin 3/2 quantum fields

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that spin-3/2 fields can be described in a previously unstudied Lorentz representation, the direct sum of the single- and double-spin chiral sectors, obtained from the 20-dimensional irreducible representation of SO(1,4).

desk verdict The paper's central claim of a new spin-3/2 representation is contradicted by the Hurley-Sudarshan classification it cites; the review portions are useful but the headline result doesn't hold. read the letter →

arxiv 2506.23057 v1 pith:LKWO4Z7R submitted 2025-06-29 hep-th

J. Escamilla-Muñoz , S. Gómez-Ávila This is my paper · ORCID
classification hep-th
keywords spin-3/2fieldsLorentzgrouprepresentationsDuffin-Kemmer-PetiauformalismRarita-SchwingerequationJoos-WeinbergrepresentationcovariantkineticoperatorsSO(14)embeddingrelativisticwaveequations
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 revisits the menu of Lorentz-group representations that can support wave equations for a massive spin-3/2 particle. Working under the rule that only representations with at most two spin sectors, and with spin 3/2 as the highest spin, are admissible, it recovers the familiar Joos-Weinberg, double-spin-chiral, and Rarita-Schwinger cases and then identifies a fourth: the representation $(1,1/2)\oplus(1/2,1)\oplus(3/2,0)\oplus(0,3/2)$, which mixes the single- and double-spin chiral sectors. According to the authors, this representation has not previously been studied as a home for spin-3/2 wave equations, and its operator-space decomposition contains scalar, vector, second-rank, and third-rank covariant kinetic operators. If the claim holds, the representation is a new candidate framework of Duffin-Kemmer-Petiau type, and a possible alternative to the interacting Rarita-Schwinger theory. The paper's result is a possibility statement: it identifies the candidate, not yet a fully constructed and verified wave equation.

What carries the argument

The machinery is the operator-space decomposition of a Lorentz representation: for a field transforming in a representation $R$, the operators on $R$ decompose as $R\otimes R^*$ into Lorentz-irreducible pieces, and the covariant kinetic terms are read off from the pieces that can contract with momenta. The classification combines this decomposition with a parity ($\mathbb{Z}_2$) grading that separates block-diagonal from block-antidiagonal operators. For the new candidate, the representation $(1,1/2)\oplus(1/2,1)\oplus(3/2,0)\oplus(0,3/2)$ is the 20-dimensional irreducible representation of SO(1,4) reduced to SO(1,3), and its tensor square contains scalar, vector, symmetric-tensor, and third-rank tensor pieces, producing kinetic terms of momentum rank one, two, and three. The Duffin-Kemmer-Petiau algebra, originally constructed for spin 0 and 1, is the template: the new representation is proposed as a DKP-type generalization for spin 3/2.

What would settle it

Construct the wave equation from the new representation's first-, second-, and third-order kinetic terms, count the independent rest-frame solutions, and examine the propagator: if the theory propagates more or fewer than four degrees of freedom, or shows ghost or superluminal modes, the claim that this representation is a viable spin-3/2 framework is refuted.

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Extended reading notes

Core claim

The paper's central claim is that, within the class of Lorentz representations that contain spin 3/2 as their highest spin and at most two spin sectors, there are four possible constructions and the fourth is new. In addition to the single-spin chiral (Joos-Weinberg) representation $(3/2,0)\oplus(0,3/2)$, the double-spin chiral representation $(1,1/2)\oplus(1/2,1)$, and the standard Rarita-Schwinger representation $(1,1/2)\oplus(1/2,1)\oplus(1/2,0)\oplus(0,1/2)$, the paper identifies $(1,1/2)\oplus(1/2,1)\oplus(3/2,0)\oplus(0,3/2)$ as a previously unstudied possibility. This representation is obtained by reducing the 20-dimensional irreducible representation of SO(1,4) to Lorentz representations, and its operator space decomposes to give kinetic terms of first, second, and third rank in momenta. The authors state that wave equations built from this representation have not appeared in the literature, making it the paper's main new contribution to the search for consistent spin-3/2 field theories.

Load-bearing premise

The central claim rests on treating the existence of covariant kinetic operators in the representation's operator-space decomposition as sufficient evidence that the representation is a viable candidate for a spin-3/2 wave equation, without actually constructing the equation or checking that it propagates the right degrees of freedom.

Editorial extensions

If this is right

  • The new representation becomes a concrete candidate setting in which to write spin-3/2 wave equations of Duffin-Kemmer-Petiau type, with kinetic operators of order one, two, and three in momenta.
  • It provides an alternative frame to the Rarita-Schwinger vector-spinor formalism, whose interacting versions are known to propagate acausal modes and lose canonical commutation relations.
  • The SO(1,4) origin of the representation suggests a five-dimensional route to DKP-type spin-3/2 equations, analogous to the known embedding of spin-0 and spin-1 DKP theories.
  • The classification narrows the search space for consistent spin-3/2 constructions: within the stated assumptions, any candidate must be one of the four listed representations, or a justification for allowing more than two spin sectors.

Reading between the lines

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

  • Going beyond the paper, the natural next test is to construct a concrete Lagrangian from the new representation's kinetic terms and check that the rest-frame spectrum contains exactly four propagating degrees of freedom ($2j+1$ for massive spin 3/2), with no ghosts or superluminal modes.
  • The paper's restriction to at most two spin sectors is a design choice rather than a mathematical necessity; relaxing it could either eliminate the new candidate or open additional representations, so the novelty claim is contingent on that choice.
  • The same SO(1,4) branching logic could be applied to other higher-dimensional irreducible representations to generate DKP-type candidates for higher spins, though higher-rank kinetic terms would likely make the resulting equations higher-derivative theories.
  • If the new candidate fails consistency checks, the paper's reusable contribution would be its explicit operator-space decompositions, which remain a catalog of covariant kinetic operators for other constructions.
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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

3 major / 4 minor

Summary. This paper reviews relativistic wave equations for spin-3/2 fields in different Lorentz representations. It recovers the Joos-Weinberg single-spin chiral, double-spin chiral, and Rarita-Schwinger frameworks, analyzes their operator-space decompositions, and discusses the DKP meson algebra. The main advertised result is the identification of the representation (3/2,0) ⊕ (0,3/2) ⊕ (1,1/2) ⊕ (1/2,1) as a "new possibility" for spin-3/2 DKP-type wave equations, based on its occurrence in the SO(1,4) 20-dimensional irrep. No explicit wave equation or beta-matrix construction is provided for this representation.

Significance. If the claimed representation were genuinely new and a consistent wave equation could be built in it, the result would be a useful addition to the spin-3/2 literature, potentially relevant for BSM phenomenology and for understanding DKP-type algebras. However, the central novelty claim is not supported by the manuscript's own cited classification, and the paper explicitly defers the construction of the corresponding theory to future work. The review portions and the SO(1,4) embedding observations are useful and clearly organized, but the advertised new result needs substantial revision.

major comments (3)
  1. [Section VI; Conclusions item 4] Equation (6) is exactly the direct sum of Hurley-Sudarshan family 1 for n = 3/2, namely (3/2,0) ⊕ (1,1/2), and its parity conjugate (0,3/2) ⊕ (1/2,1), both of which appear in the list quoted by the authors. Therefore the statement "wave equations defined in this representation have not been previously studied" is contradicted by reference [25]. The authors should either show that no linear DKP-type equation exists for the direct sum despite existing for each summand, or retract the novelty claim.
  2. [Section IX; abstract] The paper calls Eq. (6) a "novel DKP-type theory" but explicitly states that its study is a future perspective. The abstract's "we find a new possibility" is therefore not backed by an explicit wave equation, beta matrices, or a degree-of-freedom count. The existence of covariant kinetic operators in the operator-space decomposition (Section VIII) is not by itself a demonstration that a consistent wave equation exists.
  3. [Section II] The classification is restricted by an ad hoc rule ("at most two spin sectors" and spin j as the highest spin present). The paper does not justify why this rule is physically necessary, and it is used to exclude representations that might also be viable. Since the conclusions present this as a classification, the rule should be argued from consistency conditions rather than convenience.
minor comments (4)
  1. [Abstract; Introduction] "Joss-Weinberg" should be "Joos-Weinberg" throughout the manuscript.
  2. [Eq. (35)] Equation (35) contains apparent index typos: the term "ηµρη0µ" should presumably be "ηµρη0ν", and one of the "M0µ" factors should be "M0ν" so that the expression is symmetric in µ and ν as expected.
  3. [Eq. (59)] In the branching rules for the 20 irrep, the first line is labeled "35′" but should be "35"; the second line is "35′". As printed, the 35 irrep is missing from the branching table.
  4. [Eq. (20)] The notation "12" and "42" in Eq. (20) is nonstandard; the authors should clarify that these denote two scalar and four vector irreps, respectively, or write the multiplicities explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No fitted-parameter circularity; the derivation is self-contained. The main caveat is a novelty overclaim undercut by the paper's own Hurley-Sudarshan citation, but that is a literature issue, not a circular step.

full rationale

The derivation chain is self-contained. The candidate representations in Eqs. (5)-(6) are obtained from textbook SO(1,3) tensor products and from the SO(1,4) branching rules in Eqs. (53), (57), and (59); the operator-space decompositions (e.g., Eqs. (30), (37), (54), (57)) are computed by explicit tensor and Young-diagram methods. No parameter is fitted and no output is imported from a fit. The only self-citation, [18], supplies the covariant-basis technique and the parity-boost interpretation; it is background and not load-bearing for the central classification. The principal caveat is a novelty/correctness concern, not circularity: Eq. (6) equals Hurley-Sudarshan family 1 with n=3/2, namely (3/2,0)⊕(1,1/2), plus its parity conjugate (0,3/2)⊕(1/2,1), as quoted in Sec. VI. Therefore the claim that wave equations in this representation have not been previously studied is questionable, but this does not make the derivation circular, so the circularity score remains low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical free parameters appear in the paper; the only adjustable coefficients, α and β in Eq. (17), are said to be fixed by spin and mass. The central claim rests on standard representation-theoretic background plus two unproven modeling assumptions: the two-spin-sector/highest-spin restriction and the inference from available kinetic operators to a viable wave equation. No new physical entities such as particles, forces, or dimensions are introduced.

assumptions (6)
  • standard math Wigner classification of irreducible Poincaré representations and the massive orbit labels (m², j)
    Used throughout Section II to define spin-3/2 fields; standard result, not derived in the paper.
  • standard math Isomorphism so(1,3) ≃ su(2) ⊕ su(2), with O(1,3) irreps (a,b) or (a,b) ⊕ (b,a)
    Basis for labeling fields by (a,b) in Section II and for all tensor-product decompositions.
  • standard math Operator-space decompositions (a,b) ⊗ (a,b) and cross products are computed by standard Young-tableau methods and, for the covariant basis, by the parity construction in [18]
    Sections III-V rely on these decompositions without re-deriving them in the present paper.
  • domain assumption A viable spin-3/2 representation must contain at most two spin sectors and have the desired spin as the highest spin present
    Explicit restriction in Section II; it selects which representations are classified and is not derived from dynamics.
  • ad hoc to paper The presence of covariant kinetic operators in the operator-space decomposition suffices to establish a 'possibility' for a wave equation
    Used in Sections VII-VIII and Conclusions to promote the new representation to candidate status without an explicit equation or consistency check.
  • domain assumption Rest-frame parity projection, boosted to arbitrary frames, produces covariant kinetic equations
    Taken from [18] and used in Sections III-IV to obtain Dirac and Joos-Weinberg type equations; background assumption.

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Cite this review

Pith. "Pith review of Wave equations for spin 3/2 quantum fields." pith.science (2026). https://pith.science/paper/LKWO4Z7R

@misc{pith2026250623057,
  author       = {Pith},
  title        = {Pith review of: Wave equations for spin 3/2 quantum fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKWO4Z7R}},
  note         = {Machine review of arXiv:2506.23057}
}
abstract

In this work, we review formulations of wave equations for spin-3/2 fields constructed from different Lorentz group representations. We analyze the Joss-Weinberg single-spin chiral representation and the double-spin chiral representation, focusing on the structure of their covariant operators. We explore the Duffin-Kemmer-Petiau (DKP) formalism and its algebraic properties, originally introduced for spin--0 and spin-1 particles, and here considered as a potential framework for spin 3/2. As a result, we recover the well-known Rarita-Schwinger representation and we find a new possibility in the $(3/2,0) \oplus (0,3/2) \oplus (1,1/2) \oplus (1/2,1)$ representation.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

48 extracted references · 39 canonical work pages

  1. [25]

    E. Wigner. On unitary representations of the inhomogeneous lorentz group. The Annals of Mathematics , 40(1):149, January 1939. ISSN 0003-486X. doi: 10.2307/1968551. URL http://dx.doi.org/10.2307/1968551

    doi:10.2307/1968551 1939

  2. [1]

    m2 > 0, with well–defined values for the Casimir operators C2 = P 2 and C4 = W 2 with W σ = 1 2 ϵσρµν PρMµν and eigenvalues w2 = −m2s(s + 1)

    work page

  3. [2]

    m = 0 with P 0 > 0, with well defined value for the helicity λ in W µ + λP µ = 0

    work page

  4. [3]

    m = 0 y P µ = 0, which uniquely corresponds to what we call the vacuum representation

    work page

  5. [4]

    a free-field equation is nothing but an invariant record of which components are superfluous

    m2 < 0, with well–defined values for the Casimir operators C2 = P 2 and C4 = W 2 with W σ = ϵ. (These are called tachyonic representations). arXiv:2506.23057v1 [hep-th] 29 Jun 2025 2 In the following we will center our attention in representations in the first orbit, that is, fields associated with massive particles. The spacetime translation comprise an ...

    arXiv 2025

  6. [5]

    Non–chiral representations j 2 , j 2 for j = 0, 1 only

    work page

  7. [6]

    Single spin chiral representations ( j, 0)χ

    work page

  8. [7]

    For j = 0, 1 2 , 1 these include all of the fields used in the Standard Model

    Double spin chiral representations j − 1 2 , 1 2 χ, and of course, direct sums of these. For j = 0, 1 2 , 1 these include all of the fields used in the Standard Model. In this note, we will review existing wave equations for spin (3 /2), the Joos-Weinberg single–spin chiral represen- tation 3 2 , 0 ⊕ 0, 3 2 , the double–spin chiral representation 1, 1 2 ⊕...

    work page

Show all 48 references
  1. [8]

    ( n, 0) ⊕ n − 1 2 1 2 ,

  2. [9]

    n + 1 2 1 2 ⊕ (n, 0) ⊕ n − 1 2 , 1 2 ,

    ( n, 0) ⊕ n + 1 2 , 1 2 , 3. n + 1 2 1 2 ⊕ (n, 0) ⊕ n − 1 2 , 1 2 ,

  3. [10]

    (1 , 0) ⊕ 1 2 1 2 ⊕ (0, 1). Coming back to the double–spin chiral representation for j = 3 2 , where a vector operator was available to construct a linear wave equation, we note that this representation is not in the list. Since this representation has a left and a right compo...

  4. [11]

    Single spin chiral representation 3 2 , 0 ⊕ 0, 3 2 , where we find a higher–derivative theory

  5. [12]

    Double spin chiral representation 1, 1 2 ⊕ 1 2 , 1 giving rise to a linear, quadratic or cubic wave equation. 10

  6. [13]

    The well-known Rarita-Schwinger representation 1, 1 2 ⊕ 1 2 , 1 ⊕ 1 2 , 0 ⊕ 0, 1 2 , where we also find a linear, quadratic or cubic wave equation

  7. [14]

    The study of the novel DKP–type theory in the 1, 1 2 ⊕ 1 2 , 1 ⊕ 3 2 , 0 ⊕ 0, 3 2 representation is an interesting perspective of this work

    The novel 1, 1 2 ⊕ 1 2 , 1 ⊕ 3 2 , 0 ⊕ 0, 3 2 representation, mixing the single and double spin chiral representations, and with first, second and third rank kinetic terms. The study of the novel DKP–type theory in the 1, 1 2 ⊕ 1 2 , 1 ⊕ 3 2 , 0 ⊕ 0, 3 2 representation is an i...

  8. [15]

    The Quantum theory of fields

    Steven Weinberg. The Quantum theory of fields. Vol. 1: Foundations . Cambridge University Press, Cambridge, UK, 1995. ISBN 0521550017

    1995

  9. [16]

    Navas et al

    S. Navas et al. Review of particle physics. Physical Review D , 110(3):030001, August 2024. ISSN 2470-0029. doi: 10.1103/physrevd.110.030001. URL http://dx.doi.org/10.1103/PhysRevD.110.030001

    2024 doi

  10. [17]

    The structure and interpretation of the standard model

    Gordon McCabe. The structure and interpretation of the standard model. In Gordon McCabe, editor, The Structure and Interpretation of the Standard Model , volume 2 of Philosophy and Foundations of Physics , pages i–252. Elsevier, Amster- dam, The Netherlands, 2007. doi: https:/...

    2007 doi

  11. [18]

    Outstanding questions: physics beyond the standard model

    John Ellis. Outstanding questions: physics beyond the standard model. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , 370(1961):818–830, February 2012. ISSN 1471-2962. doi: 10.1098/ rsta.2011.0452. URL http://dx.doi.org/10.1...

    1961

  12. [19]

    Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies

    Elcio Abdalla et al. Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies. Journal of High Energy Astrophysics , 34:49–211, 2022. ISSN 2214-

    2022

  13. [20]

    Feynman Rules for Any Spin

    Steven Weinberg. Feynman Rules for Any Spin. 2. Massless Particles. Phys.Rev., 134:B882–B896, 1964. doi: 10.1103/ PhysRev.134.B882

    1964

  14. [21]

    H. L. Anderson, E. Fermi, E. A. Long, and D. E. Nagle. Total cross sections of positive pions in hydrogen. Physical Review, 85(5):936–936, March 1952. ISSN 0031-899X. doi: 10.1103/physrev.85.936. URL http://dx.doi.org/10.1103/PhysRev. 85.936

    1952 doi

  15. [23]

    On a theory of particles with half integral spin

    William Rarita and Julian Schwinger. On a theory of particles with half integral spin. Phys.Rev., 60:61, 1941. doi: 10.1103/PhysRev.60.61

    1941 doi

  16. [24]

    Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential

    Giorgio Velo and Daniel Zwanziger. Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential. Phys.Rev., 186:1337–1341, 1969. doi: 10.1103/PhysRev.186.1337

    1969 doi

  17. [26]

    Theory of group representations and applications

    Ryszard Raczka and Asim Orhan Barut. Theory of group representations and applications . World Scientific Publishing, Singapore, Singapore, 2 edition, January 1986

    1986

  18. [27]

    Feynman Rules for Any Spin

    Steven Weinberg. Feynman Rules for Any Spin. Phys.Rev., 133:B1318–B1332, 1964. doi: 10.1103/PhysRev.133.B1318

    1964 doi

  19. [28]

    Boulware and S

    David G. Boulware and S. Deser. Can gravitation have a finite range? Physical Review D , 6(12):3368–3382, December

  20. [29]

    The explicit relation between the dkp equation and the klein-gordon equa- tion

    Djahida Bouchefra and Badredine Boudjedaa. The explicit relation between the dkp equation and the klein-gordon equa- tion. AIP conference Proceedings, 2183(1):5, 2016. doi: https://doi.org/10.1063/1.5136204. Department of Mathematics and Computer Science, University Center Abd...

    2016 doi

  21. [30]

    Sudarshan

    Kenneth Johnson and E.C.G. Sudarshan. Inconsistency of the local field theory of charged spin 3/2 particles. Annals Phys., 13:126–145, 1961. doi: 10.1016/0003-4916(61)90030-6

    1961 doi

  22. [31]

    University of paris thesis

    G Petiau. University of paris thesis. Acad´ emie Royale De Belgique. Classe Des Sciences. M´ emoires. Collection, 16(2):1, 1936

    1936

  23. [32]

    R. J. Duffin. On the characteristic matrices of covariant systems. Phys. Rev. , 54:1114–1114, Dec 1938. doi: 10.1103/ PhysRev.54.1114. URL http://link.aps.org/doi/10.1103/PhysRev.54.1114

    1938 doi

  24. [33]

    N. Kemmer. The particle aspect of meson theory. Proc. R. Soc. A , 173:91–116, Nov 1939. doi: 10.1098/rspa.1939.0131. URL https://doi.org/10.1098/rspa.1939.0131

    1939

  25. [34]

    Napsuciale

    Selim G´ omez-´Avila and M. Napsuciale. Covariant basis induced by parity for the ( j, 0) L(0, j) representation. Phys. Rev. D, 88:096012, Nov 2013. doi: 10.1103/PhysRevD.88.096012. URL https://link.aps.org/doi/10.1103/PhysRevD.88. 096012

    2013 doi

  26. [35]

    On the Representation theory of inhomogeneous Lorentz groups as the foundation of quantum mechanical kinematics

    Hans Joos. On the Representation theory of inhomogeneous Lorentz groups as the foundation of quantum mechanical kinematics. Fortsch.Phys., 10:65–146, 1962. doi: 10.1002/prop.2180100302

    1962 doi

  27. [37]

    and Ernst

    Dharam Vir Ahluwalia and D.J. and Ernst. (j,0) + (0,j) covariant spinors and causal propagators based on Weinberg formalism. Int.J.Mod.Phys., E2:397–422, 1993. doi: 10.1142/S0218301393000145. 11

    1993 doi

  28. [38]

    Ahluwalia and D.J

    D.V. Ahluwalia and D.J. Ernst. New arbitrary-spin wave equations for (j, 0)⊕(0, j) matter fields without kinematic acausal- ity and constraints. Physics Letters B , 287(1):18–22, 1992. ISSN 0370-2693. doi: https://doi.org/10.1016/0370-2693(92) 91869-B. URL https://www.scienced...

    1992

  29. [39]

    Benjam ´ ın Grinstein, Donal O’Connell, and Mark B. Wise. The lee-wick standard model. Phys. Rev. D , 77:025012, Jan

  30. [41]

    D. Anselmi. Fakeons and lee-wick models. . High Energ. Phys. , 144:141, Feb 2018. doi: 10.1007/JHEP02(2018)141

    2018 doi

  31. [42]

    Algebraic study of a class of relativistic wave equations

    William J Hurley and E.C.G Sudarshan. Algebraic study of a class of relativistic wave equations. Annals of Physics, 85(2): 546–590, 1974. ISSN 0003-4916. doi: https://doi.org/10.1016/0003-4916(74)90422-9. URL https://www.sciencedirect. com/science/article/pii/0003491674904229

    1974

  32. [43]

    B. C. Clark, S. Hama, G. R. K¨ albermann, R. L. Mercer, and L. Ray. Relativistic impulse approximation for meson-nucleus scattering in the kemmer-duffin-petiau formalism. Physical Review Letters, 55(6):592–595, August 1985. ISSN 0031-9007. doi: 10.1103/physrevlett.55.592. URL ...

    1985 doi

  33. [44]

    K¨ albermann

    G. K¨ albermann. Kemmer-duffin-petiau equation approach to pionic atoms.Physical Review C, 34(6):2240–2243, December

  34. [46]

    R. E. Kozack, B. C. Clark, S. Hama, V. K. Mishra, R. L. Mercer, and L. Ray. Spin-one kemmer-duffin-petiau equations and intermediate-energy deuteron-nucleus scattering. Physical Review C , 40(5):2181–2194, November 1989. ISSN 0556-2813. doi: 10.1103/physrevc.40.2181. URL http:...

    1989 doi

  35. [48]

    M. D. Scadron and R. L. Thews. Inconsistency of a duffin-kemmer-petiau model for meson decay rates. Physical Review D, 9(7):2180–2182, April 1974. ISSN 0556-2821. doi: 10.1103/physrevd.9.2180. URL http://dx.doi.org/10.1103/PhysRevD. 9.2180

    1974 doi

  36. [49]

    Bhabha relativistic wave equations

    R-K Loide, I Ots, and R Saar. Bhabha relativistic wave equations. Journal of Physics A: Mathematical and General , 30 (11):4005, jun 1997. doi: 10.1088/0305-4470/30/11/027. URL https://dx.doi.org/10.1088/0305-4470/30/11/027

    1997 doi

  37. [1972]

    doi: 10.1103/physrevd.6.3368

    ISSN 0556-2821. doi: 10.1103/physrevd.6.3368. URL http://dx.doi.org/10.1103/PhysRevD.6.3368

    doi

  38. [1986]

    doi: 10.1103/physrevc.34.2240

    ISSN 0556-2813. doi: 10.1103/physrevc.34.2240. URL http://dx.doi.org/10.1103/PhysRevC.34.2240

    doi

  39. [2008]

    URL http://link.aps.org/doi/10.1103/PhysRevD.77.025012

    doi: 10.1103/PhysRevD.77.025012. URL http://link.aps.org/doi/10.1103/PhysRevD.77.025012

    doi

  40. [4048]

    URL https://www.sciencedirect.com/science/article/pii/ S2214404822000179

    doi: https://doi.org/10.1016/j.jheap.2022.04.002. URL https://www.sciencedirect.com/science/article/pii/ S2214404822000179

    2022 doi

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