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A gauge invariant path integral for electrodynamics with magnetic monopoles in the Hestenes-Haddamard-Rodrigues formalism

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A Dirac-matrix field, not the vector potential, is proposed as the fundamental variable in a Euclidean path integral that puts magnetic monopoles into QED.

desk verdict A stringless monopole path integral that fails at its core variable change—32 integration variables cannot map to 6 as a Jacobian. read the letter →

arxiv 1909.01119 v1 pith:TUYEZOHP submitted 2019-08-21 physics.gen-ph

Luiz C L Botelho (LCLBotelho) This is my paper · ORCID
classification physics.gen-ph
keywords magneticmonopolesquantumelectrodynamicsEuclideanpathintegralgaugeinvarianceDiracmatricesgeometricalgebramonopoleMaxwellequationsfieldstrengthvariables
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 proposes a Euclidean, gauge-invariant path integral for quantum electrodynamics that includes magnetic monopoles — something the author says has not been available as a genuine second quantization. The field that is integrated over is not the electromagnetic potential but a complex Dirac-matrix field $[F](x)$ built from the field strengths, and the proposal is that the generating functional written over this matrix field, Eq. (11), can be changed into one written directly over the gauge-invariant tensor $F_{\mu\nu}(x)$, Eq. (18). The author's stated reason for caring is that a field-strength-based path integral for monopole electrodynamics would avoid both Dirac strings and the confining behavior of vector-potential formulations. The paper is explicit that the theory that results is non-renormalizable and not unitary, so this is a formal construction rather than a finished quantum field theory.

What carries the argument

The engine is the complex Dirac-matrix field $[F](x)$, a bivector-valued matrix written as $\frac{1}{4}(F_{\mu\nu}-\varepsilon_{\mu\nu\rho\tau}B^{\rho\tau})[\gamma^\mu,\gamma^\nu]_-$, with the auxiliary field $B_{\rho\tau}$ solving the magnetic part of the Maxwell equations in terms of the currents. It obeys the matrix 'Dirac-like' wave equation (9), and the whole construction rests on the observation that the quadratic action (10) is minimized exactly on that equation. The load-bearing identity is the measure change $D[F](x) = \det^{1/2}(L)\,D F_{\mu\nu}(x)$, where $L^{\nu\rho}{}_{\alpha\zeta}$ is the bivector pairing and $\det^{1/2}(L) = \mathrm{Tr}(\gamma_\rho\gamma_\nu\gamma_\alpha\gamma_\zeta)$. That identity converts the generating functional over matrices into the field-strength action of Eq. (18), with the free part given by the Cramer-Julia tensor action.

What would settle it

Evaluate the map $[F]\mapsto F_{\mu\nu}=\frac{1}{4}\mathrm{Tr}(\gamma_\mu[F]\gamma_\nu)+\varepsilon_{\mu\nu\sigma\rho}B^{\sigma\rho}$ on a regulated space, for instance on a lattice or with a momentum cutoff, and compute the actual Jacobian determinant of the measure change. If the determinant is not the constant $\mathrm{Tr}(\gamma_\rho\gamma_\nu\gamma_\alpha\gamma_\zeta)$, or if the map fails to be one-to-one on square-integrable matrix fields, then the field-strength path integral (18) is not equivalent to the matrix-field path integral (11).

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Extended reading notes

Core claim

The central claim is that Eqs. (11) and (18) define a second-quantized generating functional for the monopole Maxwell equations in Euclidean space-time. The basic variable is the complex Dirac-matrix field $[F](x)$, whose classical equation of motion is the matrix wave equation $(\gamma^\mu\partial_\mu)[F] = j_\nu\gamma^\nu - i\gamma^5\gamma_\alpha k^\alpha$; the Gaussian action proposed in Eq. (10) has its unique minimum on that equation, so its extremum reproduces the Maxwell equations with electric and magnetic sources. The second stage is a classical change of variables, justified 'under the hypothesis of its validity at the quantum level', with the Jacobian $\det^{1/2}(L) = \mathrm{Tr}(\gamma_\rho\gamma_\nu\gamma_\alpha\gamma_\zeta)$, which rewrites the path integral as one over the antisymmetric field strengths $F_{\mu\nu}$. With sources turned off the action collapses to the Cramer-Julia action $\frac{1}{6}\int(\partial_\mu F_{\nu\rho}+\partial_\nu F_{\mu\rho}+\partial_\rho F_{\mu\nu})^2\,d^4x$, and the sources enter through current-dependent terms in the weight. The paper presents this as a Dirac-string-free, gauge-invariant route to the second quantization of monopole electrodynamics.

Load-bearing premise

The load-bearing premise is that the change of variables from the matrix field $[F](x)$ to the tensor field $F_{\mu\nu}(x)$, with Jacobian $\det^{1/2}(L)=\mathrm{Tr}(\gamma_\rho\gamma_\nu\gamma_\alpha\gamma_\zeta)$, is valid inside the path integral; if that transformation is not a genuine functional determinant, Eq. (18) does not follow from Eq. (11).

Editorial extensions

If this is right

  • If the construction is correct, monopole electrodynamics has a formal quantum generating functional in which the integration variable is gauge invariant by construction, so no Dirac string or patched vector potential is needed.
  • At vanishing electric and magnetic sources the theory is a free Gaussian in $F_{\mu\nu}$ with the Cramer-Julia action, so all $n$-point functions of the field strength are formally computable by Wick contraction.
  • With dynamical matter, the current-squared terms (19a)-(19b) produce quartic current-current self-interactions, which is the paper's stated reason that the resulting QED with monopoles is non-renormalizable and violates unitarity rather than a conventional renormalizable theory.
  • The same field-strength variable would be the natural starting point for coupling monopoles to other dual gauge fields or to gravity, because no gauge-fixing of a potential is ever required.

Reading between the lines

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

  • A direct test of the construction would be to check whether the Jacobian identity survives a regulator: if $\det^{1/2}(L)$ acquires cutoff dependence or depends on the currents after regularization, the equivalence of Eqs. (11) and (18) is only formal and the field-strength path integral would need correction terms.
  • The same variable change could be applied to other theories whose fundamental degrees of freedom are antisymmetric tensors, such as duality-symmetric formulations; in each case the hidden assumption is that the map between the matrix field and the tensor field is one-to-one on the relevant function space.
  • Because the author presents the non-unitary, confining features as consequences of the field-strength formulation, a natural follow-up would be to search for a modified action whose extremum still gives the monopole Maxwell equations but whose Hessian is positive on the monopole sector; the path integral (18) is the template for such a search.
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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. The paper proposes an Euclidean path integral for quantum electrodynamics in the presence of magnetic monopoles, using a complex 4×4 Dirac-matrix field [F](x) as the fundamental variable. It postulates the matrix wave equation (9), defines a quadratic action (10) whose extremum is that equation, writes the path integral (11) over [F], and then claims a change of variables to the antisymmetric field-strength tensor Fμν with Jacobian det^{1/2}(L) = Tr(γργνγαγζ), leading to the generating functional (18). The paper asserts that the resulting theory is non-renormalizable, non-unitary, and confining of electric charge, and it closes with an appendix sketching a 'fermionization' of the Maxwell equations.

Significance. If the construction worked, it would provide a string-free, gauge-invariant second quantization of electrodynamics with magnetic monopoles directly in terms of field strengths, which is a conceptually interesting goal. The manuscript makes the algebraic structure explicit and states its measure and action unambiguously, which aids inspection. However, the pivotal change-of-variables step is asserted rather than derived and appears dimensionally impossible; the action is chosen ad hoc to reproduce a posted wave equation; and the advertised physics claims are not derived. The paper does not offer machine-checked proofs, numerical checks, or falsifiable predictions, so its significance depends entirely on the validity of the central formal manipulation, which I find unsupported.

major comments (3)
  1. [Sec. 1, Eqs. (14)-(16)] The change of variables from [F](x) to Fμν(x) is introduced only 'under the hypothesis of its validity at the quantum level' and is not derived. As written it cannot be valid: the measure (12) integrates over all 16 complex entries of [F], i.e. 32 real degrees of freedom per spacetime point, whereas Fμν is an antisymmetric rank-two tensor with 6 real components, and Bσρ in Eq. (8) is fixed by the sources through Eq. (7) and adds no independent integration variables. Moreover Eq. (16) identifies the claimed Jacobian with a single Dirac trace, not with the determinant of the 6×6 'tensorial matrix' L. A determinant is not a single trace, and no well-defined functional Jacobian is exhibited. Consequently the generating functional (18) is not shown to follow from the path integral (11); the central construction fails at its pivotal step.
  2. [Sec. 1, Eq. (10)] The action (10) is not derived from a variational principle; it is chosen so that its unique extremum is the posted Dirac-like wave equation (9). This makes the path integral (11) a quantization of an assumed input equation rather than a derived quantum theory of magnetic monopoles. The stated uniqueness of the minimum only shows that the classical limit reproduces the input equation, which is true by construction. Any physical conclusions drawn from this action therefore inherit the status of the postulated equation (9), and the paper does not explain why that equation is the correct starting point.
  3. [Final paragraph, footnote (*)] The claims that the resulting theory is non-renormalizable, violates unitarity, and confines electric charge are made without derivation. The appearance of current-current terms (19-a)-(19-b) suggests power-counting non-renormalizability in four dimensions, but no explicit analysis is given; non-unitarity and charge confinement are stated in the final paragraph and footnote (*) with reference to the author's earlier work, not derived within the present formalism. These are load-bearing conclusions for the paper's stated significance and need either a derivation or an explicit marking as conjectures.
minor comments (4)
  1. [Throughout] The manuscript contains many typographical and copy-editing errors, including 'matrixes', 'Rinocherus', 'condidion', and stray overbrace fragments in Eqs. (5) and (11), which impede readability.
  2. [Eq. (7)] The displayed formula for Bσρ has an ambiguous index structure: the bracket '(∂σ(kρ−jρ)−∂ρ(kσ−jσ))' is missing an explicit contraction and an overall factor, so the equation cannot be checked as written.
  3. [Appendix 1] The appendix states a converse fermionization result but does not prove the derivation of Eq. (25) from Eqs. (23)-(24), and its connection to the main path integral (18) is not explained.
  4. [Eqs. (12)-(13)] The path-integral measure is written as an infinite product over every x in R4; without a lattice or a regularization prescription the later claims about renormalizability and the formal variable change lack a well-defined mathematical basis.

Circularity Check

1 steps flagged · score 2.0 of 10

Central path integral is an explicit proposal, not a derived prediction; the only circularity-adjacent step is a load-bearing self-citation in footnote (*) for the field-strength formulation and confinement.

  1. self citation load bearing [Footnote (*), page 6, after Appendix 1]
    "On refs ([4]) we have proved that electromagnetic path integrals, when formally written in terms of vector potentials are of fourth-order, thus leading to confinement of electrical charge. This result constrains us to claim that only a new electrodynamics written directly in terms of the Electromagnetic Field strenght has chances to be consistent at a second quantized level when in presence of magnetic monopoles."

    The paper justifies the choice of a field-strength formulation and the physical conclusion of charge confinement solely by citing refs [4], which are all works by the same author. No independent proof, external benchmark, or derivation within this manuscript is supplied for the claim that vector-potential path integrals are fourth-order and confining, nor for the claimed uniqueness of the field-strength approach. The argument is load-bearing for the paper's motivation and for its closing physical claims, and it reduces to a self-citation chain rather than to evidence presented here.

full rationale

The central construction of the paper is an explicit proposal rather than a derived prediction. The action (10) is deliberately chosen so that its extremum is the already-postulated matrix wave equation (9), as the paper itself states: 'It is obvious that the (unique!) minimum of the quadratic functional eq(10) is achieved on the classical motion eq(9)'. This is a standard action-principle construction, not a circular reduction of a prediction to an input. Likewise, Eq. (18) is obtained from Eq. (11) by the variable change (14), which the paper explicitly flags as conditional: 'under the hypothesis of its validity at the quantum level'. Whether that variable change is mathematically legitimate is a correctness and rigor concern—especially since Eq. (16) identifies a determinant with a single trace—but it is not circularity. No parameter is fitted to data, and no external quantity is 'predicted' from a subset. The only circularity-adjacent element is footnote (*), where the field-strength formulation and the confinement conclusion rest exclusively on the author's own prior refs [4] without independent support. Since this self-citation does not enter the formal derivation of Z[J] in Eqs. (11) or (18), the overall circularity is minor, corresponding to score 2.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

The central construction rests on the choice of a new field [F] and an action deliberately built to reproduce a posted wave equation. The only free parameters are the coupling constants and a normalization, and the key quantum steps are assumed or drawn from the author's own prior papers.

free parameters (3)
  • electric coupling constant e
    Introduced in the electron current jμ = e ψ̄ γμ ψ (Eq. 20); treated as a free parameter, not determined by the theory.
  • magnetic coupling constant g
    Introduced in the monopole current kμ = g Ω̄ γμ Ω (Eq. 21); free parameter with no independent estimate.
  • free action normalization coefficient = 1/6
    The factor 1/6 in Eq. (17) is chosen to match the Cramer-Julia action; it is a hand-picked normalization with no derivation.
assumptions (6)
  • domain assumption Maxwell equations with both electric and magnetic source currents (Eqs. 1-2)
    The theory assumes divergenceless magnetic source kν and electric source jν, and that the Maxwell equations hold with both types of charges.
  • ad hoc to paper The Dirac-matrix representation of the field strength (Eq. 8) is the fundamental variable
    The paper postulates [F](x) = 1/4(Fμν - εμνρτ Bρτ)[γμ,γν]_- as the basic field, without deriving it from a variational principle.
  • ad hoc to paper The 'Dirac-like' matrix wave equation (Eq. 9) is posted
    The author states it is 'postulated' and that no Lagrangian is known for it, then constructs an action to reproduce it.
  • ad hoc to paper The quadratic action (Eq. 10) is a valid bosonic action for the theory
    The action is chosen solely because its minimum gives Eq. (9), with no independent physical motivation.
  • ad hoc to paper The quantum-level change of variables with determinant (Eqs. 14-16) is valid
    The text says this is 'under the hypothesis of its validity at the quantum level'; no proof is provided.
  • ad hoc to paper Non-renormalizability, non-unitarity, and confinement of electric charge (footnote *)
    These conclusions are not derived in this paper; they are cited from the author's own prior work (refs [4]).
invented entities (2)
  • Complex Dirac-matrix field [F](x)
    purpose: Fundamental field to be second-quantized instead of the electromagnetic potential
    An unobserved mathematical object with no independent experimental handle; its dynamics are chosen ad hoc.
  • Second-quantized monopole field Ω_M(x)
    purpose: Provides the magnetic monopole source current kμ through kμ = g Ω̄ γμ Ω
    The monopole field is posted without mass, coupling, or any observable prediction; no independent evidence is provided.

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Cite this review

Pith. "Pith review of A gauge invariant path integral for electrodynamics with magnetic monopoles in the Hestenes-Haddamard-Rodrigues formalism." pith.science (2026). https://pith.science/paper/TUYEZOHP

@misc{pith2026190901119,
  author       = {Pith},
  title        = {Pith review of: A gauge invariant path integral for electrodynamics with magnetic monopoles in the Hestenes-Haddamard-Rodrigues formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUYEZOHP}},
  note         = {Machine review of arXiv:1909.01119}
}
read the original abstract

We propose a new path integral for QED in the presence of magnetic monopoles on the formalism of Geometric Algebra of Hestenes-Haddamard-Rodrigues written in terms of Dirac Matrices

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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    - Space-Time Algebra, New York, Gordan & Breach - 1966

    Hestenes, D. - Space-Time Algebra, New York, Gordan & Breach - 1966

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    Botelho - Random Operators and Stochastic Equations - V

    Luiz C.L. Botelho - Random Operators and Stochastic Equations - V. 24, pp 79-92, 2016. - Luiz C.L. Botelho - Int. Journal of Theor. Physics, V. 48, pp 1554 -1558, 2009

    work page 2016

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    Botelho - International Journal of Modern Physics A, V

    Luiz C.L. Botelho - International Journal of Modern Physics A, V . 15, no. 5, 755- 770; 2000 7 - Luiz C.L. Botelho - IJTP, v. 48, 1701-1711; 2009. - Luiz C.L. Botelho - Physical Review 70D 045010-1-045010-7; 2004 . 8

    work page 2000

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Reviewed August 14, 2026 · model on record in the stance chip above.