Construction of Currents in Causal Fermion Systems

Felix Finster; Patrick Fischer

REVIEW 3 major objections 6 minor 8 references

Construction of Currents in Causal Fermion Systems

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Probing the linearized field equations of a causal fermion system with wave functions recovers Maxwell's equations sourced by the Dirac current, with a coupling constant independent of the regularization scale.

desk verdict A genuinely new probing formalism for causal fermion systems, with a plausible rank-one recovery of Maxwell—but the coupling constant rests on an imported, unverified replacement of a singular function, so the derivation is not yet complete. read the letter →

arxiv 2507.09633 v1 pith:YFCO4ZHK submitted 2025-07-13 math-ph math.MP

Felix Finster , Patrick Fischer This is my paper · ORCID

classification math-phmath.MP MSC 49S0581T2083C47

keywords causalfermionsystemsactionprincipleCFScurrentlinearizedfieldequationsMaxwellDiraccontinuumlimitiε-regularization

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 close the gap between the abstract variational principle of causal fermion systems and the concrete classical equations of physics. The authors introduce a CFS current built from the linearized field equations and show that probing this current with wave functions, then Taylor-expanding in spacetime, converts the linearized equations into a hierarchy of tensorial equations of increasing rank. Carrying out the rank-one step in $i\varepsilon$-regularized Minkowski spacetime with two fermion types, they recover Maxwell's equations with the Dirac current as source, $\partial_\mu F^{\mu\nu}(x) = \alpha\, \varphi(x)\,\gamma^\nu \varphi(x)$, where the coupling constant $\alpha$ is independent of the regularization length $\varepsilon$. If correct, this is a concrete bridge from the causal action principle to electrodynamics, and the same systematic procedure is set up to obtain Einstein's equations from rank-two probing and Planck-scale corrections from higher-rank tensors.

What carries the argument

The central object is the CFS current, $J_x(v) := ((D_v Q)\Psi)(x) + (QD_v\Psi)(x) - rD_v\Psi(x)$, whose vanishing is exactly the linearized field equation for a perturbation $v$ of the causal fermion system. The method is carried by probing: multiplying $J_x(v)$ by $\Psi(y)^*$, and on a manifold by its Taylor derivatives, turns the abstract operator equation into tensorial equations of increasing rank. In the Minkowski computation the central step is the proportionality of the structure functions $f_s$ and $f_a$ for the Maxwell and Dirac perturbations, obtained after replacing the singular function $T^{(1)}_m$ by a smooth function $g(x,y)$ with $g(x,x)=c$; the single real factor $1/\alpha = 1/(196\pi^2) - (2\pi/3)\,c$ then fixes all five coefficient ratios at once and makes $\alpha$ independent of $\varepsilon$.

What would settle it

Recompute the equality of the five ratios in (5.18) using the actual singular function $T^{(1)}_m$, logarithmic pole and all, instead of the smooth replacement $g(x,x)=c$; if the ratios are not all equal to the same real constant, the rank-one recovery of Maxwell's equations is an artifact of that substitution.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the linearized field equations of a causal fermion system can be read off pointwise: the operator-valued CFS current $J_x(v)$ vanishes if and only if its products with the wave-evaluation operator $\Psi(y)^*$ vanish for all $y$, and when spacetime has a manifold structure this becomes the vanishing of the family of tensorial equations $J_x(v)\,\partial_{\mu_1}\cdots\partial_{\mu_n}\Psi(x)^* = 0$. Working out the rank-one members of this family in $i\varepsilon$-regularized Minkowski spacetime with two fermion types, the paper obtains that the Dirac current of the charged type is proportional to the divergence of the electromagnetic field tensor, $\partial_\mu F^{\mu\nu} = \alpha\, \varphi\gamma^\nu \varphi$. The proportionality constant $\alpha$ is real, and because five separate coefficient ratios coincide, it is independent of the regularization scale; the recovered Maxwell equations are therefore Lorentz invariant in the continuum limit even though the $i\varepsilon$ regularization itself breaks Lorentz invariance.

Load-bearing premise

The load-bearing premise is that the singular function $T^{(1)}_m$ appearing in the Maxwell perturbation may be replaced by a smooth function $g(x,y)$ whose coincidence value $g(x,x)=c$ is a real constant imported from earlier work; the Maxwell-Dirac proportionality, and with it the recovered Maxwell equation and the value of $\alpha$, depends on that substitution.

Editorial extensions

If this is right

Rank-one probing reproduces Maxwell's equations with the Dirac current as source; an axial version of the same argument yields the axial Maxwell-Dirac equation for a single fermion type.
The construction generalizes to non-abelian gauge fields, giving a route to the Yang-Mills equations from the same probed-current machinery.
Probing at second order is expected to yield the Einstein equations, in line with the continuum-limit analysis for metric perturbations.
Higher-rank tensorial equations supply systematic corrections, including Planck-scale terms of order $m\varepsilon$ that break Lorentz invariance in a controlled way.
Because $\alpha$ is independent of $\varepsilon$, the leading-order electrodynamics is Lorentz invariant even though the regularization itself singles out a time direction.

Reading between the lines

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

The dependence of $\alpha$ on the undetermined constant $c$ means the framework does not yet predict the fine-structure constant; fixing $c$ from the causal action itself is a concrete next step implied by the paper's logic.
The fact that one proportionality factor controls all five coefficient pairs suggests a mechanism worth testing: any perturbation or regularization preserving that ratio pattern would give the same field equation with a rescaled coupling, so checking a second regularization scheme would test the scheme-independence of $\alpha$.
For a single fermion type the formalism predicts that vectorial interactions are absent at rank one, because they preserve the twofold eigenvalue degeneracy and drop out of the Lagrangian; only axial currents survive, a structural prediction that could be probed within the theory.

Share X Bluesky LinkedIn Reddit HN

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

Bottom line: this is a solid, interesting contribution to the causal fermion systems program, and I would send it to a referee. The central claim that a rank-one probing of the linearized causal action reproduces Maxwell’s equations is plausible, but it is conditional on a step imported from earlier work rather than proved here.

What is genuinely new: the CFS current and probing formalism (Theorem 3.3), the analytic continuation argument in Theorem 4.2, and the explicit continuum-limit computation for the iε-regularized Minkowski vacuum. The linear algebra with the closed chain is clean, the formal series solution in Proposition 4.5 is proved carefully, and the paper is honest about what is still expectation: Einstein equations and Planck-scale corrections are explicitly framed as future work. The regularization breaks Lorentz invariance, and the authors show the recovered equation is Lorentz invariant in the limit, which is a nice check.

The soft spots are concentrated in Section 8.2. Equation (8.17) is a legitimate smooth approximation, but Eqs. (8.18)–(8.19) replace the singular function T_m^(1) by a smooth function g with g(x,x)=c, where c is an undetermined real constant imported from [4, Section 3.7] via the microlocal chiral transformation. The proportionality of the Maxwell and Dirac structure functions, and therefore the Maxwell equation (5.27) and the coupling constant alpha, all ride on this replacement. Since c is not fixed by the present paper, the form of Maxwell’s equations is recovered but the coupling is not predicted. The stress-test concern about this step is right. The citation to [4] is appropriate; the issue is that the transformative step is cited rather than demonstrated or verified here. A second, smaller gap: the reduction from the full first-order perturbation (7.12) to the 'relevant term' (4.25) is asserted, and I did not find a systematic estimate showing that the omitted terms do not contribute to the CFS current. This is probably fixable, but it is load-bearing enough that a referee should ask for it. Finally, the equivalence 'if and only if' for the combined perturbation is checked at zeroth and first moments; the extension to all moments is plausible but not written out.

Who is this for: people working in or seriously evaluating causal fermion systems. A general field-theory audience should read Sections 5 and 8.2 before trusting the physics conclusion. My recommendation: send to peer review with a conditional-accept stance. The framework is worth engaging, and the gaps I named are addressable. If the authors can derive c or show that the chiral transformation is fixed by the action, and if they check the omitted terms in (7.12), the paper would be much stronger.

Referee Report

3 major / 6 minor

Summary. The paper introduces the notion of a CFS current J_x(v) for a perturbation v of a causal fermion system, together with a probing procedure that converts the linearized field equations J_x(v)=0 into operator equations J_x(v)Psi(y)^*=0 and, on a manifold, into sequences of tensorial equations obtained by differentiating the wave evaluation operator. The main application is to i-epsilon-regularized Minkowski spacetime: the authors construct perturbations of the fermionic projector corresponding to Dirac particles and to (axial and vector) gauge potentials, compute the resulting CFS current in the continuum limit, and claim to recover the Maxwell equations, with the Dirac current as source, for two fermion types, and the axial Maxwell equation for one fermion type. They argue that the coupling constant is independent of the regularization length epsilon.

Significance. If the derivation were complete, the paper would provide a systematic, first-principles route from the causal action principle to classical electrodynamics, with a clear framework for higher-rank equations (Einstein equations, Planck-scale corrections). The paper contains several valuable and clean contributions: Theorem 3.3 (equivalence between vanishing of the CFS current and its probed version), Theorem 4.2 (the expansion theorem for i-epsilon-regularized Minkowski spacetime), the explicit solution of the perturbed Klein-Gordon equation (Proposition 4.5), and the detailed computation of the structure functions in Appendix 8. The claim that the coupling constant is independent of epsilon is a non-trivial and potentially important structural result. However, the central recovery of Maxwell's equations rests on an import from [4] that replaces the singular function T_m^{(1)} by an arbitrary constant c, so the coupling constant alpha is not predicted and the proportionality between the Maxwell and Dirac structure functions is not derived within the paper. This weakens the claim that classical electrodynamics is derived rather than merely recovered after an undetermined input.

major comments (3)

[Section 8.2, Eqs. (8.18)-(8.19)] The proportionality of the Maxwell structure functions f_s^(M), f_a^(M) to the Dirac structure functions is obtained by replacing the logarithmically singular function T_m^{(1)} with a smooth function g(x,y) such that g(x,x)=c, an arbitrary real constant. This replacement is not derived in the present paper, is not applied consistently to the perturbed projector or to the Dirac-side computation, and is stated to require at least three generations although the model in Section 5.3 has only two fermion types. As a consequence, the coupling constant alpha in Eq. (5.27), given by 1/alpha = 1/(196pi^2) - (2pi/3)c in Eq. (8.38), is not predicted by the theory, and the proportionality itself is conditional on the chiral transformation not introducing additional tensor structures or modifying f_s^(D), f_a^(D). This is the most load-bearing step of the derivation.
[Section 4.3, Eq. (4.25)] The reduction from the full first-order perturbation (7.12) to the single term (4.25) is asserted without detailed justification. The full expansion contains terms proportional to A_mu, to F_mu nu, and to partial_nu F^{nu mu} with different T^{(n)} prefactors; the paper does not show why all but the partial_nu F^{nu mu} term fail to contribute to the CFS current. This matters because the Maxwell current computation in Section 8.2 uses the two-term expression (5.13) involving both T^{(0)} and T^{(1)}, which is not obviously the continuum limit of (4.25).
[Sections 5.2-5.3, Eqs. (5.19) and (5.27)] The statement that the combined perturbation preserves the restricted EL equations 'if and only if' the current relation holds is verified only for the zeroth and first moments of the CFS current. Theorem 4.2 requires the vanishing of all derivatives partial_mu1 ... partial_mun Psi(x)^* for all n, and the second and higher moments are not computed; moreover, Section 6.2 explicitly anticipates nonzero second-moment content (the energy-momentum tensor). Therefore the 'if and only if' is not established, and the recovery of the Maxwell equation as a consequence of the full linearized field equations remains incomplete.

minor comments (6)

[Eq. (4.9)] The quantity r is introduced as ||y-x||^2 but is used throughout as the radial distance (e.g., in the integrals over dr r^2 in Section 8.3); please fix the definition.
[Section 5.1] The diagonalizability of A_xy for r>0 is invoked from Proposition 4.4, but the non-vanishing of 1/4 tr[(A_xy-b)^2] is not explicitly checked; please state the explicit expression for this quantity.
[Section 5.2] The constants C^(D,*) and C^(M,*) used in Eq. (5.18) are not listed explicitly; the reader is referred to Appendix 8.3 where only the combined constants are given. Please provide the individual constants or explain how their ratios are obtained.
[Eqs. (5.19) and (5.27)] Linearity of J_x and the definition 1/alpha = C^(M)/C^(D) in Eq. (5.18) imply that cancellation of the two contributions gives j = -alpha j^D (up to sign conventions for alpha). Please specify the sign convention used for alpha or state whether the constants C^(M) and C^(D) have opposite signs.
[Throughout] There are numerous typos and grammatical errors, e.g., 'archieved' (Section 4.2), 'addtion' (Section 2.2), 'Sequentially' (Section 5.1), and 'F akult¨at' in the affiliation; the manuscript would benefit from a careful proofreading.
[Section 8.3] The integrals defining the constants (e.g., C^(0) in Eq. (8.23)) are formally divergent as epsilon -> 0; the paper should comment on the convergence and on how the epsilon-independence of the ratios is justified.

Circularity Check

1 steps flagged · score 4.0 of 10

Rank-one Maxwell recovery is an emergent result, but the proportionality and coupling constant rest on an undetermined constant c imported from the authors' prior work via the microlocal chiral transformation (Eqs. 8.18-8.19).

ansatz smuggled in via citation [Section 8.2 (Maxwell Current), Eqs. (8.18)-(8.19); coupling constant in Eq. (8.38); Maxwell recovery in Eq. (5.27)]
"As shown in [4, Section 3.7], this pole can be absorbed by a so-called microlocal chiral transformation ... For this paper, we just replace T (1) m (x, y) by the smooth function g(x, y) ≈ g(x, x) =: c ∈ R. Thus, we get f (M ) s (t, r) = (1/196π2 − 2π/3 c) f (D) s (t, r), f (M ) a (t, r) = (1/196π2 − 2π/3 c) f (D) a (t, r)."

The equality of structure functions f_s^(M)=K f_s^(D) and f_a^(M)=K f_a^(D) with K=1/196π^2 - (2π/3)c is the exact input that makes all ratios in (5.18) equal and yields 1/α=K in (8.38). It is not derived from the causal action; it is obtained by replacing the singular T_m^(1) with an arbitrary smooth-function value c. The legitimacy of the replacement is attributed to the microlocal chiral transformation of [4, Sec. 3.7], a work by the same author.

full rationale

The central derivation is not circular: the CFS current (3.1) is defined from the linearized field equations (2.24), and the probing theorem (3.3) and the Paley-Wiener expansion theorem (4.2) are proved in the paper. The Maxwell equation (5.27) is not an input; it arises as the condition that the combined particle plus vector-potential perturbation solves the homogeneous linearized field equations. The Dirac and Maxwell current computations in Sections 8.1-8.2 are explicit and self-contained up to one step. That step is the replacement of T_m^(1) by a smooth function with value c at x=y, justified only by an appeal to the same author's monograph [4] (microlocal chiral transformation). This makes the proportionality (8.18)-(8.19) and therefore the coupling constant α (8.38) dependent on an undetermined constant, so the interaction strength is not predicted and the rank-one recovery is conditional. Because the Maxwell equation's form still emerges from the framework and no fitted parameter is relabeled as a prediction, the circularity score is moderate rather than severe.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The CFS current Jx(v) and the probing construction are mathematical tools, not new physical entities. No new particles, forces, or dimensions are postulated in this paper.

free parameters (3)

epsilon = artificial scale epsilon > 0
Regularization length in the i-epsilon-regularized Minkowski vacuum; the final coupling is claimed to be independent of it.
r = 0
Lagrange multiplier for the trace constraint; set to zero in the continuum limit to solve the restricted EL equations. Whether r=0 is forced by the constraints is not shown.
c = real constant
Value of the smooth replacement g(x,x) for the singular T_m^{(1)}; enters the coupling alpha through Eq. (8.38) and is not determined by the theory in this paper.

assumptions (7)

domain assumption Causal action principle with volume, trace, and boundedness constraints
Starting point of the theory; taken from prior work [6].
domain assumption Linearized field equations (2.24) characterize perturbations preserving the restricted EL equations
Taken from [3]; not reproved in this paper.
domain assumption i-epsilon-regularized Minkowski vacuum with H built from negative-energy Dirac solutions
Concrete realization used for computation; assumes a Minkowski background.
standard math Paley-Wiener theorem for holomorphic extension of Psi(z)^* in a complex neighborhood
Used in Theorem 4.2 to prove equivalence of all-order probing.
domain assumption Fermionic wavefunctions and vector potentials are almost constant on the regularization scale
See Eq. (5.9) and the approximations in Appendix 8; needed to pull currents out of integrals.
ad hoc to paper Microlocal chiral transformation replaces T_m^{(1)} by a smooth function with constant value c
Required to define the Maxwell structure functions; imported from [4] and stated to require at least three generations.
domain assumption Dropping terms of order epsilon^3 in the continuum limit and restricting to r>0
Used in Section 5.1 to simplify eigenvalues and projectors; the singularity at r=0 has measure zero.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Construction of Currents in Causal Fermion Systems." pith.science (2026). https://pith.science/paper/YFCO4ZHK

@misc{pith2026250709633,
  author       = {Pith},
  title        = {Pith review of: Construction of Currents in Causal Fermion Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFCO4ZHK}},
  note         = {Machine review of arXiv:2507.09633}
}

read the original abstract

This paper presents a novel and systematic formalism for deriving classical field equations within the framework ofcausal fermion systems, explicitly accounting for higher-order corrections such as quantum effects and those arising from spacetime discreteness. Our method, which also generalizes to non-abelian gauge fields and gravitation, gives a systematic procedure for evaluating the linearized field equations of causal fermion systems. By probing these equations with specific wave functions and employing Taylor expansions, we reformulate them as a family of tensorial equations of increasing rank. We show that, for rank one, this approach recovers the established classical dynamics corresponding to Maxwell's equations. In addition, the approach gives rise to higher-rank tensorial equations, where the second-rank equations are expected to encode the Einstein equations, and higher-rank tensors potentially reveal new physics and systematic corrections.

Figures

Figures reproduced from arXiv: 2507.09633 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages

[4]

F. Finster. The Continuum Limit of Causal Fermion Systems . Vol. 186. arXiv:1605.04742 [math-ph], Fundamental Theories of Physics. Springer, Cham, 2016, pp. xi+548

arXiv 2016
[1]

A variational principle in discrete space-time: Existence of minimiz- ers

F. Finster. “A variational principle in discrete space-time: Existence of minimiz- ers”. In: arXiv:math-ph/0503069, Calc. Var. Partial Differential Equations 29.4 (2007), pp. 431–453

arXiv 2007
[2]

Causal variational principles on measure spaces

F. Finster. “Causal variational principles on measure spaces”. In: arXiv:0811.2666 [math-ph], J. Reine Angew. Math. 646 (2010), pp. 141–194

arXiv 2010
[3]

Solving the linearized field equations of the causal action principle in Minkowski space

F. Finster. “Solving the linearized field equations of the causal action principle in Minkowski space”. In: arXiv:2304.00965 [math-ph], Adv. Theor. Math. Phys. 27.7 (2023), pp. 2087–2217

arXiv 2023
[5]

Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles

F. Finster and N. Kamran. “Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles”. In: arXiv:1808.03177 [math-ph], Pure Appl. Math. Q. 17.1 (2021), pp. 55–140. 30 REFERENCES

arXiv 2021
[6]

Finster, S

F. Finster, S. Kindermann, and J.-H. Treude. Causal Fermion Systems: An Intro- duction to Fundamental Structures, Methods and Applications . arXiv:2411.06450 [math-ph], to appear in Cambridge Monographs on Mathematical Physics. Cam- bridge University Press, 2025, pp. xii+483

arXiv 2025
[7]

Causal Fermion Systems: Spacetime as the Web of Correlations of a Many-Body Quantum System

P. Fischer and C.F. Paganini. “Causal Fermion Systems: Spacetime as the Web of Correlations of a Many-Body Quantum System”. In: arXiv:2504.19272 [math-ph], in preparation (2025)

arXiv 2025
[8]

K. Yosida. Functional Analysis. Sixth. Vol. 123. Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1980, pp. xii+501. F akult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany Email address: finster@ur.de F akult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany Email address: pa...

work page 1980

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [4]

F. Finster. The Continuum Limit of Causal Fermion Systems . Vol. 186. arXiv:1605.04742 [math-ph], Fundamental Theories of Physics. Springer, Cham, 2016, pp. xi+548

arXiv 2016

[2] [1]

A variational principle in discrete space-time: Existence of minimiz- ers

F. Finster. “A variational principle in discrete space-time: Existence of minimiz- ers”. In: arXiv:math-ph/0503069, Calc. Var. Partial Differential Equations 29.4 (2007), pp. 431–453

arXiv 2007

[3] [2]

Causal variational principles on measure spaces

F. Finster. “Causal variational principles on measure spaces”. In: arXiv:0811.2666 [math-ph], J. Reine Angew. Math. 646 (2010), pp. 141–194

arXiv 2010

[4] [3]

Solving the linearized field equations of the causal action principle in Minkowski space

F. Finster. “Solving the linearized field equations of the causal action principle in Minkowski space”. In: arXiv:2304.00965 [math-ph], Adv. Theor. Math. Phys. 27.7 (2023), pp. 2087–2217

arXiv 2023

[5] [5]

Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles

F. Finster and N. Kamran. “Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles”. In: arXiv:1808.03177 [math-ph], Pure Appl. Math. Q. 17.1 (2021), pp. 55–140. 30 REFERENCES

arXiv 2021

[6] [6]

Finster, S

F. Finster, S. Kindermann, and J.-H. Treude. Causal Fermion Systems: An Intro- duction to Fundamental Structures, Methods and Applications . arXiv:2411.06450 [math-ph], to appear in Cambridge Monographs on Mathematical Physics. Cam- bridge University Press, 2025, pp. xii+483

arXiv 2025

[7] [7]

Causal Fermion Systems: Spacetime as the Web of Correlations of a Many-Body Quantum System

P. Fischer and C.F. Paganini. “Causal Fermion Systems: Spacetime as the Web of Correlations of a Many-Body Quantum System”. In: arXiv:2504.19272 [math-ph], in preparation (2025)

arXiv 2025

[8] [8]

K. Yosida. Functional Analysis. Sixth. Vol. 123. Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1980, pp. xii+501. F akult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany Email address: finster@ur.de F akult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany Email address: pa...

work page 1980