1 Introduction↩︎

Astronomical magnetic fields play crucial roles in the formation and evolution of galaxies, stars, and planets. Their origin and amplification are generally explained by the dynamo mechanism (for a comprehensive review, see [1]), providing a theoretical framework for understanding the galactic magnetic fields observed today [2]. Magnetohydrodynamics (MHD) uses the electromagnetism and fluid dynamics to describe the interaction between electromagnetic fields and conductive fluids [3]. Despite its success in providing phenomenological explanations, MHD faces significant challenges. Observational evidence for the initial seed magnetic fields remains elusive, and the theoretical mechanisms proposed for their generation are not yet fully understood [4]. Furthermore, critical issues persist, such as understanding how the back-reaction of magnetic fields limits magnetic amplification in turbulent flow and the mechanisms reasonable for the generation/maintenance of large-scale galactic magnetic fields [5]. These challenges partly stem from the fact that, in MHD, electromagnetic fields are described by using particle dynamics based on field theory, in contrast to the description of fluids using hydrodynamics. As a result, different theoretical frameworks are employed, leading to a lack of a unified approach, which poses a major obstacle to achieving a consistent and comprehensive interpretation of the underlying physics.

In this paper, we propose a novel theoretical approach to address these challenges. Our research aims to develop a unified framework that integrates electromagnetism and fluid dynamics, thereby transcending their conventional separate treatment in MHD. We anticipate that this unified framework will not only provide a more fundamental understanding of the interaction between electromagnetic fields and fluids but also offer a relativistic extension necessary for explaining high-energy astrophysical phenomena [6] such as supernova explosions, gamma-ray bursts, and black hole jets in addition to nuclear fusion.

Before delving further, it is important to outline a theoretical motivation for this study. In modern physics, particles are described as fields inhabiting spacetime, endowed with specific symmetries and properties such as spin, charge, and mass. The field-theoretic approach to matter and particles originates from the electromagnetism, where electric and magnetic interactions were unified within the framework of a local \(U(1)\) gauge theory. Quantum field theory, built on this foundation, has been remarkably successful in describing fundamental interactions, including electromagnetic, weak, and strong forces. This framework has also been applied successfully to gravity, at least in classical regime. However, limitations in existing paradigms, such as the treatment of singularities, have driven the exploration of alternative frameworks like string theory [7], loop quantum gravity [8], and the emergent point of view for gravity [9]–[11].

On the other hand, the relativistic fluid and thermodynamic approach assumes that matter or particles reside in a three-dimensional Euclidean space, referred to as "matter space," with their dynamics governed by mappings between this matter space and four-dimensional spacetime, illustrated in Fig. 1, [12]–[18].

In this work, we adopt this perspective by considering matter (specifically, photons) to reside in a three-dimensional matter space, while the electromagnetic fields observed in four-dimensional spacetime represent projections of this underlying system. Our approach encompasses scenarios incorporating magnetic sources naturally while exhibiting dual symmetry. Consequently, the Aharonov-Bohm effect arises solely from the matter-space gauge potential, while simultaneously ensuring natural helicity conservation. Moreover, nonlinear corrections analogous to mass renormalization are intrinsically incorporated. From a broader conceptual standpoint, our approach establishes a unified foundation for treating electromagnetism and fluid dynamics within a single framework, thereby offering valuable insights into the fundamental interplay between matter and electromagnetic fields.

In a certain sense, our approach is analogous with the emergent concept of gravity proposed by [9]–[11], which interprets gravity as a phenomenon arising from thermodynamic principles. Similarly, our framework suggests that the dynamics of electromagnetic fields can be understood as an emergent phenomenon resulting from the pull-back mechanism depicted in Fig. 1. This approach emphasizes that describing collective phenomena through fluid dynamics provides a more natural representation of physical systems.

In Sec. 2, by using the constraints and “matter-space gauge symmetry” of form fields pulled back from a 3-dimensional matter space to 4-dimensional spacetime, we derive the electromagnetic field equations featuring a magnetic monopole-like source. In Sec. 3, we derive the remaining Maxwell equation through a relativistic fluid Lagrangian formulation that treats the constituent form fields as independent variables. Upon imposing the constraints established in the matter-space formulation, these equations reduce to a generalized form of Maxwell’s equation, which further simplifies to the standard form. In Sec. 4, we show that the imposition of duality symmetry admits the matter-space formulation to construct electromagnetism in a more systematic way in the absence of external magnetic charge carriers. In Sec. 5, we summarize our principal results and discuss various theoretical aspects of the framework.

2 Gauge symmetry and matter space formulation↩︎

In this work, we consider one-fluid system incorporating one-matter space flowing along \(m^a\) defined below. Let the three-dimensional matter space have coordinates \(-\infty < M^A < \infty\), where \(A = 1, 2, 3\), as illustrated in Fig. 1. The coordinates \(M^A(x^a)\) co-move with their respective world-lines in four-dimensional spacetime. We are interested in only form fields to be associated with matter, which are anti-symmetric tensor fields, rather than symmetric ones. Since the matter space is three-dimensional, the following form fields may exist: a 0-form field \(\phi(M^A)\), a 1-form field \(\boldsymbol{A} = A_A (M^A)\, dM^A\), a 2-form field \(\boldsymbol{F} = \frac{1}{2!} F_{AB}(M^A) \, dM^A \wedge dM^B\), and a 3-form field \(\boldsymbol{M}=\frac{1}{3!} M_{ABC}(M^A) \, dM^A \wedge dM^B \wedge dM^C\). The fundamental conjecture in this work is that these forms characterize the matter itself.

In fluid dynamics the 3-form field \(\boldsymbol{M}\) typically encodes the direction of fluid flow. Its spacetime dual, defined as \[\label{m a} m^a \equiv \frac{1}{3!} \epsilon^{abcd}M_{bcd},\tag{1}\] corresponds to the flow vector of the fluid, where the 3 form \(M_{abc}\) is the spacetime projcection of the matter space field, \(M_{abc} =(\nabla_a M^A)(\nabla_b M^B) (\nabla_c M^C)M_{ABC}\). In this sense, when we construct electromagnetism based on the dynamics of a single matter space, the results must inherently possess a directional structure aligned with \(m^a\). In ordinary fluid dynamics, the flow field \(m^a\) is often interpreted as encoding the conservation of particle number. However, for the case of the electromagnetic field, the number density may not be necessarily conserved because the photon is massless. In this sense, the conservation law \[\label{conservm} \nabla_a m^a=0 ,\tag{2}\] which comes from \(\boldsymbol{d M}=0\), does not imply particle number conservation. Because we are interested in the electromagnetism, we begins with the 1,2-form fields and later identify \(m^a\) as a kind of vector denoting helicity flux.

Reminding this point, we reconstruct electromagnetism based on the behavior of the remaining fields. The corresponding fields induced in spacetime, through the mapping in Fig. 1, are given by \[\begin{align} \label{inducedstfields} &\phi(x^a)\equiv \phi(M^A(x^a)), \nonumber \\ & A_a(x^a) \equiv (\nabla_a M^A(x^a)) A_A(M^B(x^a)), \nonumber\\ & F_{ab}(x^a) \equiv (\nabla_a M^A(x^a))(\nabla_b M^B(x^a)) F_{AB}(M^C(x^a)) , \end{align}\tag{3}\] in addition to \(M_{abc}\) mentioned in Eq. @{eq:m a} .

Before proceeding further, we clarify how the “matter-space gauge symmetry” is realized in this framework. Since our interest lies in the matter space flowing along \(m^a\), the induced fields obeys specific constraints. Because the matter space has lower dimensionality than the physical spacetime, relations that are trivial within the matter space can yield nontrivial consequences once mapped to spacetime. For instance, in the three-dimensional matter space the wedge products \(\boldsymbol{M} \wedge \boldsymbol{A}\) vanishes identically, as no 4-forms exist. However, upon projection to spacetime one finds \[\begin{align} \label{M94A2} 0= \boldsymbol{M} \wedge \boldsymbol{A} = & \frac{1}{3!} M_{ABC} A_{D} dM^A \wedge dM^B \wedge dM^C \wedge dM^D \nonumber \\ =& \frac{1}{3!} M_{abc} A_{d} dx^a \wedge dx^b \wedge dx^c \wedge dx^d \nonumber\\ =& \frac{1}{3!} M_{abc} A_{d} \epsilon^{abcd} \times (\boldsymbol{Vol}) = -m^d A_d \times (\boldsymbol{Vol}) \end{align}\tag{4}\] where \((\boldsymbol{Vol})\) denotes the 4-dimensional volume form. Here, Eq. 4 implies the non-trivial condition, \[\label{M^A} m^a A_a =0.\tag{5}\] Thus, the potential \(A_a\) is restricted to directions orthogonal to \(m^a\). The corresponding field strength \(\boldsymbol{d A}\) must remain invariant under a transformation \(A_a\to A'_a = A_a + \nabla_a \phi'\). However, if the scalar field \(\phi'\) satisfies \(m^a \nabla_a \phi' \neq 0\), the transformed potential fails to remain within the matter-space form, as then \(m^a A'_a \neq 0\). Accordingly, scalar fields naturally split into two classes: i) \(\phi'\), which displace the potential outside the matter space; ii) \(\phi\), which preserve the restriction and thus generate the “matter-space gauge symmetry”, characterized by \[\label{m^dphi} m^a \nabla_a \phi=0.\tag{6}\] This condition arises equivalently from the matter-space relation \(\boldsymbol{M} \wedge \boldsymbol{d \phi}=0\). Hence, the matter-space scalar \(\phi\) encodes the gauge degrees of freedom, which is pull-backed from the matter space. By contrast, scalars with \(m^a \nabla_a \phi' \neq 0\) belong to the first class and do not generate admissible transformations.

Therefore, within the matter space formulation, the full gauge symmetry of electromagnetism does not survive; rather, it is restricted to the matter-space symmetry obeying 6 . This restriction reflects the fact that the potential is constructed from matter-space structures transported along \(m^a\). As we show later, despite this restriction, it remains possible to construct electromagnetism consistently by invoking the duality between electric and magnetic fields. Finally, from the definition of \(m^a\), we find \[\label{orthom} m^a F_{ab} = 0,\tag{7}\] restricting the gauge field also normal to \(m^a\).

At this point, we need to examine the degrees of freedom for the matter space. Fundamentally, the matter space has three degrees of freedom consisting of \(M^A\) with \(A=1,2,3\). On the other hand, in four-dimensional spacetime, the electromagnetic potential \(A_a\) and fields \(F_{ab}\) consist of four and six-variables, respectively. The gauge potential is restricted to satisfy 5 , leaving three degrees of freedom; this restriction changes not the physics but the gauge description, since observables are represented by the field strength. The electromagnetic field, given the direction \(m^a\), also has three freedoms satisfying \(m^a F_{ab}=0\)¹. In this sense, the 6 freedoms of the electromagnetic field is redistributed to 3+3 for the direction of \(m^a\) and for the electromagnetic field from matter space, respectively. If we are to construct a general electromagnetic field based on the matter-space approach, we need to consider multi-fluid theory with many propagating directions, "\(m^a\)" vectors.

As mentioned, we interpret \(\phi\) as the scalar for gauge degrees of freedom. Let us display the consequences on the field \(\boldsymbol{F}\). The 4-form equation, \[\begin{align} 0 &= \boldsymbol{d\phi} \wedge \boldsymbol{dF}= \frac{1}{4!} (\nabla_{[a} \phi) (\nabla_b F_{cd]}) dx^a \wedge dx^b \wedge dx^c \wedge dx^d, \label{dphi:dF} \end{align}\tag{8}\] presents a nontrivial relation: \[\begin{align} \label{DphiDF} \epsilon^{abcd} (\nabla_{[a} \phi)(\nabla_b F_{cd]}) = 0. \end{align}\tag{9}\] Since \(\phi\) is considered to represent the gauge degrees of freedom, this identity must hold for any scalar \(\phi\), satisfying Eq. 6 . Rewriting the equation 9 by using the dual field \(*F^{ab} \equiv \frac{1}{2} \epsilon^{abcd}F_{cd}\), \[\begin{align} \label{DphiDF1} \epsilon^{abcd} (\nabla_{[a} \phi)(\nabla_b F_{cd]}) =(\nabla_a\phi) (\nabla_b \,*F^{ab}) = 0\qquad \forall \phisatisfyingm^a \nabla_a \phi =0 . \end{align}\tag{10}\] Because of the constraint \(m^a \nabla_a \phi =0\), this equation gives, \[\label{*dF} \nabla_{b} *F^{ab} = \alpha m^a \neq 0,\tag{11}\] where \(\alpha\) is a scalar². Notice that this equation is similar to a source equation for the dual field \(*F^{ab}\) with source, \(\alpha m^a\). When \(\alpha \neq 0\), therefore, this equation denotes the presence of a magnetic monopole-like contribution³.

For the time being, we consider \(\alpha=0\) case and discuss the \(\alpha\neq 0\) case in Sec. 4. Then equation 8 gives \[\label{Maxeq1} \nabla_{[a} F_{bc]} =0 ~,\tag{12}\] which is similar to the Bianchi identity. Therefore, the two form \(F_{bc}\) can be written as \(F_{bc} = \nabla_b X_c- \nabla_c X_b\) with a 1-form field \(\boldsymbol{X}\).

In a similar manner, the relationships \(\boldsymbol{d A} \wedge \boldsymbol{F}=0\) and \(\boldsymbol{A} \wedge \boldsymbol{dF}=0\) present \[\begin{align} \label{orthoDAF} &\epsilon^{abcd} (\nabla_{[a} A_{b]} ) F_{cd} = *F^{ab} (\nabla_{[a} A_{b]} ) =0 ,\qquad \nonumber \\ &\epsilon^{abcd} A_{a} \nabla_{[b} F_{cd]} =0 \,. \end{align}\tag{13}\] In other words, \(*F_{ab}\) is orthogonal to \(\nabla_{[a} A_{b]}\), i.e., \(*F^{ab} (\nabla_{[a} A_{b]} ) =0\). Furthermore, the constraint equation, \(\boldsymbol{F} \wedge \boldsymbol{F} = 0\), which is also a 4-form field, presents \[\begin{align} \label{tildeff} *F^{ab}F_{ab} = \epsilon^{abcd} F_{ab} F_{cd} = 0. \end{align}\tag{14}\] To summarize, the spacetime-induced fields satisfy the relations in Table 1.

Because of the Bianchi identity 12 , equation (C) in Table 1 is automatic. On the other hand, equations (B) and (D) impose constraints on the relationship between the two 1-forms, \(X_a\) and \(A_a\). In general, the 1-form field \({\boldsymbol{X}}\), which plays the role of a typical gauge field, may not be a matter-space form. Explicitly, if \(X_a\) is a matter space form, any addition of a constant vector having component along \(m^a\) direction falsify the assumption. However, one may always choose a scalar \(\phi'\) so that \(m^a X_a' =0\) where \(X_a' =X_a + \nabla_a \phi'\). Because we assume that all the properties of the matter should be described by the matter space fields, we exclude the \(X_a\) which does not satisfy \(m^a X_a =0\). With this choice, both \(X_a\) and \(A_a\) are the matter space fields and satisfy \(m^a X_a =0=m^aA_a\). Then, the two quantities have only 3 degrees of freedoms, respectively. Because one of the freedoms is fixed by the gauge choice, \(X_a = A_a + \nabla_a \phi\) for some scalar \(\phi\), the two conditions (B) and (D) are enough to relate the two up to the gauge transform. This result leads to the following expression: \[\begin{align} \label{FdA} F_{ab} = \nabla_a A_b - \nabla_b A_a ~, \end{align}\tag{15}\] where \(F_{ab}\) satisfies all the constraints listed in Table 1 and represents a subset of Maxwell’s equations.

It is essential to distinguish gauge fields that satisfy \(m^a A_a=0\) from those that do not. To clarify this point, let us consider the Aharonov-Bohm effect in the presence of a magnetic flux. Take \(m^a\) along the time direction, so that \(\vec{E}=0\) while \(\vec{B} \neq 0\). For a charge \(q\) traveling along a closed path \(P\), the quantum phase shift is given by \[\label{AB} \Delta \varphi \propto q \int_{P}A_a dx^a ,\tag{16}\] where \(A_a\) denotes the vector potential. If the path is varied by \(\delta x^a\), the effect differs depending on the direction:

• A spatial variation \(\delta x^i\) can change the enclosed magnetic flux, thereby altering the phase.

• A temporal variation \(\delta t\), however, cannot change the flux because the magnetic field is time-independent in this setup. Consequently, the scalar potential \(A_0\) does not contribute to the geometric phase.

This example shows that only the part of the gauge potential constrained by \(m^aA_a=0\), which originates from the matter space, produces observable effects. It thereby supports our conjecture that the essential physical content is encoded in the matter-space fields themselves.

3 Electromagnetism from relativistic fluid dynamics↩︎

We now turn to the Lagrangian formulation of relativistic fluid dynamics and derive the remaining Maxwell equations. Our framework focuses on fluid of matter characterized by the vector field \(A_a\) and the antisymmetric field \(F_{ab}\) without introducing additional fields. We consider terms up to quadratic order in the fields but intentionally omit the \(A^2\) term, which is typically associated with a “mass" term. Since the scalar is regarded as representing gauge degrees of freedom, it is not included explicitly. Additionally, we do not explicitly include kinetic terms arising from gradients of the field values as usual in fluid mechanics.

The relativistic Lagrangian, which is a scalar in (3+1)-dimensional spacetime, depends on the field with properties mentioned above and the spacetime metric \(g_{ab}\), is given by \[\begin{align} \label{lag} \Lambda(A_a, F_{ab}, g_{ab}) = -\frac{1}{4} F^{ab} F_{ab} - j^a A_a, \end{align}\tag{17}\] where \(j^a\) denotes an external current vector, added manually. In this formulation (17 ), \(A_a\) and \(F_{ab}\) are treated as independent fields, and the Lagrangian itself does not inherently encode their dynamics. However, as aforementioned, imposition of the constraint equations listed in Table 1 establishes the connection between \(A_a\) and \(F_{ab}\) and hence develops dynamics in the resulting fields.

Introducing a Lagrangian displacement vector \(\xi^a\), the relationship between the Lagrangian variation \(\Delta\) and the Eulerian variation \(\delta\) is given by \[\label{var} \Delta = \delta + \mathcal{L}_{\xi},\tag{18}\] where \(\mathcal{L}_{\xi}\) denotes the Lie derivative with respect to \(\xi^a\). Here, \(\Delta\) and \(\delta\) capture changes relative to a reference configuration and variations with respect to spacetime fields, respectively. Since the Lagrangian variation of the matter space coordinates \(M^A\) vanishes, i.e., \(\Delta M^A = 0\), we obtain: \[\label{eulervarMA} \delta M^A = - \mathcal{L}_{\xi} M^A.\tag{19}\] This relation allows the variation of the action to be expressed in terms of the displacement \(\xi^a\) rather than the flux.

The Eulerian variation of \(F_{ab}\) is given by \[\begin{align} \label{varFab} \delta F_{ab} &= \delta F_{AB} (\nabla_a M^A)(\nabla_b M^B) \nonumber \\ &\quad + F_{AB} \left[(\nabla_a \delta M^A)\nabla_b M^B + (\nabla_a M^A)(\nabla_b \delta M^B)\right] \nonumber \\ &= -\xi^c \nabla_c F_{ab} - (\nabla_a \xi^c)F_{cb} - (\nabla_b \xi^c)F_{ac} \nonumber \\ &= -\mathcal{L}_{\xi} F_{ab}. \end{align}\tag{20}\] Similarly, for the vector field \(A_a\), the variation takes the form \[\label{varAa} \delta A_a = -\mathcal{L}_{\xi } A_a.\tag{21}\] As noted above, Eqs. 20 and 21 are merely restatements of \(\Delta F_{ab} =0\) and \(\Delta A_a =0\).

By performing a complete variation of the Lagrangian \(\Lambda(A,F)\) with respect to both \(F_{ab}\) and \(A_a\), and utilizing the variations in Eqs. 20 and 21 , we get \[\begin{align} \label{varLag} &\frac{\delta (\sqrt{-g} \Lambda)}{\sqrt{-g}} = \delta \Lambda + \frac{1}{2} \Lambda g^{ab} \delta g_{ab} \nonumber \\ &= \Pi^{ab} \left(-\mathcal{L}_{\xi} F_{ab}\right) + j^a \left(-\mathcal{L}_{\xi} A_a \right) + \left(\frac{\partial \Lambda}{\partial g_{ab}} + \frac{1}{2} \Lambda g^{ab} \right) \delta g_{ab} \nonumber \\ &=\left[ A_e\nabla_a j^a-3\Pi^{ab}\nabla_{[e}F_{ab]} + 2F_{ea}\nabla_b \Pi^{ba} +2j^a\nabla_{[a}A_{e]} \right]\xi^e \nonumber \\ & \quad + \left[ \frac{\partial \Lambda}{\partial g_{ab}} + \frac{1}{2}\Lambda g^{ab} \right]\delta g_{ab} + \text{total derivatives}. \end{align}\tag{22}\] Here, the conjugate to \(F_{ab}\) is \[\begin{align} \label{defofmomFab} &\Pi^{ab} = \frac{\partial \Lambda}{\partial F_{ab}} = \left(\frac{\partial F}{\partial F_{ab}}\right)\left(\frac{\partial \Lambda}{\partial F}\right) = \frac{F^{ab}}{F} \Pi, \end{align}\tag{23}\] where \(\Pi \equiv \partial\Lambda/\partial F\). With respect to the Lagrangian 17 , we get \(\Pi / F =- 1/2\) and \(\Pi^{ab}=- F^{ab}/2\). Finally, requiring the charge conservation associated with the external source, \[\begin{align} \label{chaconserv} \nabla_a j^a = 0 , \end{align}\tag{24}\] we obtain field equations from (22 ), which are given by \[\begin{align} \label{eqofmotion} 3 F^{ab}\nabla_{[e}F_{ab]} -2\left(\nabla_b F^{ba}\right) F_{ea} + 4 j^a \nabla_{[a} A_{e]} = 0. \end{align}\tag{25}\]

Substituting the equation (15 ) into the field equation (25 ) gives \[\begin{align} \label{eqm2} F_{ea} \left[ \nabla_b F^{ab} - j^a \right] = 0. \end{align}\tag{26}\] The equation 26 leads to ⁴ \[\begin{align} \label{Maxeq2} \nabla_b F^{ab} = j^a . \quad \end{align}\tag{27}\] This equation is the last piece of the Maxwell equation other than Eq. 12 . Consequently, it becomes evident that equations 12 and 27 consist the Maxwell equations describing electromagnetism, where \(F_{ab}\) is interpreted as the electromagnetic field tensor and \(A^a\) as the vector potential.

The charge-conservation relation 24 imposed by hand to ensure gauge invariance, can instead emerge naturally from a matter-space construction. Introduce a 3-form field \(\boldsymbol{N} = \frac{1}{3!} N_{ABC}(N^{A}) dN^{A} \wedge dN^{B} \wedge dN^{C}\), defined on a three-dimensional charge-carrier matter space with coordinates \(\{N^A\}\) (where \(A = 1, 2, 3\)), where the charge of a carrier is \(q\). From this 3-form one constructs the number-flow 4-vector \(n^a \equiv \frac{1}{3!} \epsilon^{abcd} N_{bcd}\), which is dual to the induced field \(N_{abc} \equiv (\nabla_a N^A)(\nabla_b N^B)(\nabla_c N^C) N_{ABC}\). The charge current, \[\label{j:n} j^a \equiv q n^a ,\tag{28}\] serves as the source term in the field equations. Because the matter space is three-dimensional, its 3-form satisfies \(\boldsymbol{dN} = 0\), which projection to spacetime \(\nabla_a n^a=0\) implies the charge-conservation equation 24 (cf.[16]). Thus the charge conservation law emerges naturally as a direct consequence of the particle-number conservation in external source, which itself follows from the absence of 4-forms in matter space. This result highlights that the \(U(1)\) symmetry underlying electromagnetism is intrinsically built into the matter-space formulation.

We now revisit the ambiguity noted in Footnote 4 regarding Eq. 26 , and explore the possibility of nonlinear modifications to Maxwell’s equations. By comparing the orthogonality condition 7 with Eq. 26 , we notice that a term proportional to \(m^a\) can be added to the right-hand side. Accordingly, we express the field equation as \[\begin{align} \label{Maxeq3} \nabla_b F^{ab} = q (n^a + \beta m^a )~, \end{align}\tag{29}\] where \(j^a = q n^a\) as in Eq. 28 , and \(\beta\) is a scalar parameter⁵. When \(\beta = 0\), Eq. 29 reduces to the standard Maxwell equations (cf.Eqs. 15 , 27 ) that govern classical electromagnetism. In contrast, Eqs. 5 , 6 , and 7 imply that \(m^a\) can be expressed nonlinearly in terms of \(A^a\) and \(F_{ab}\), recalling that energy densities are typically quadratic in the fields. Therefore, for \(\beta \neq 0\), Eq. 29 describes a generalzied class of nonlinear Maxwell equations, where the additional contribution is proportional to the energy flux carried by the electromagnetic field.

4 Electromagnetic duality and its consequences↩︎

Let us now revisit Eq. 11 and analyze the case where \(\alpha \neq 0\). This equation corresponds to a modified version of Maxwell’s equations that includes a magnetic ‘source-like’ term, \(\alpha m^a\). We argue that when we accept that electromagnetism possesses the duality symmetry, \((\vec{E}, \vec{B}) \to (\vec{B}, -\vec{E})\), a particular case of the continuous dual symmetry with respect to the electric-magnetic rotation, \[\vec{E} \to \vec{E} \cos \theta + \vec{B} \sin\theta, \qquad \vec{B} \to -\vec{E} \sin \theta + \vec{B} \cos\theta,\] the Bianchi identity 12 follows naturally from the structure of the theory, provided that no external magnetic sources (monopoles) are present.

In the presence of the \(\alpha\)-term, the standard Bianchi identity in Eq. 12 no longer holds in its conventional form. Consequently, the complementary part of Maxwell’s equations is not governed by Eq. 27 , but rather by the modified dynamics of Eq. 25 . However, this conclusion is premature. Before concluding so, let us apply the duality symmetry to Eq. 29 and investigate its implications. To this end, consider an external magnetic source current, \[j_M^a = q_M n_M^a,\] where \(n_M^a\) is the conserved fluid 4-vector (\(\nabla_a n_M^a=0\)) defined on the 3-dimensional magnetic charge charrier space \(N_M\), which carries magnetic charge \(q_M\). Under the duality transform \(F^{ab} \to *F^{ab}, ~ q \to q_M, ~n^a \to n_M^a\) including the sources, Eq. 29 becomes \[\begin{align} \label{finalMaxwell} \nabla_{b} *F^{ab} &= q_M(n_M^a + \bar \beta m^a), \end{align}\tag{30}\] with \(\beta \to \bar \beta\). Comparing with Eq. 11 , we get \(\alpha = q_M \bar \beta\). This relation yields an important conclusion: even if \(\bar \beta \neq 0\), in the absence of an external magnetic charge carrier (\(q_M=0\)) there is no source for the dual field, i.e. \(\nabla_b *F^{ab}=0\). Therefore, the Bianchi identity 12 remains valid whenever external magnetic charge carriers are absent and the theory admits duality symmetry. This observation provides an alternative justification for employing Eq. 29 , which was originally derived under the assumption of the Bianchi identity.

It is important to emphasize that the above conclusion relies on two assumptions: (i) the theory respects duality symmetry, and (ii) no magnetic charge carriers are present. If either of these conditions fails, the argument does not hold.

When the theory lacks duality symmetry and \(\alpha \neq 0\), the Maxwell equations derived in sections 2 and 3 become \[\begin{align} \label{speloavw} \nabla_{b} *F^{ab} &= \alpha m^a, \nonumber\\ F_{ea} \left(\nabla_b F^{ba} - 6 j^a_M \right) &= 2 j^a \nabla_{[a} A_{e]}, \end{align}\tag{31}\] where we used \(\nabla_{[e} F_{ab]} = 2 \epsilon_{ecab} j^c_M\). In contrast to the dual-symmetric case, these equations reveals an intrinsic asymmetry. This asymmetry ariese because, in our construction, electromagnetism is not a fundamental gauge field but rather emerges dynamically from the underlying matter content of a single fluid. Importantly, the magnetic ‘current’ \(\alpha m^a_M\) is not an independent quantity but is directly tied to the underlying fluid motion.

When external magnetic charges are present and the theory preserves duality symmetry, Eq. 29 can no longer be employed. In this situation, the fully dual symmetric formulation is required. One natural extension is to introduce a second, dual matter space \(*M\), distinct from the original matter space \(M\). In such a framework, the conventional vector potential \(A^a\) would reside in \(M\), while the dual potential \(C^a\) would be associated with \(*M\), allowing for a manifestly symmetric description of the electric and magnetic sectors. The model Lagrangian 17 would then require a reformulation into a fully dual-symmetric manner [19]–[22]. We defer the detailed development of this extended framework for future work.

In conclusion, the duality symmetry is indispensable if the present theory recover the classical electrodynamics irrespective of the presence of magnetic monopole.

5 Summary and discussions↩︎

In summary, we have reformulated electromagnetism within the matter-space framework, commonly employed to describe fluid dynamics. The central assumption of this work is that all properties of the electromagnetic field are encoded in the matter-space fields themselves. As supporting evidence, we have argued that the Aharonov-Bohm effect arises solely from the matter-space gauge potential 16 . This leads to the concept of a restricted gauge symmetry, which imposes the condition \(m^a A_a=0\) on the gauge field. Although such restricted gauge symmetry disrupts the standard Bianchi identity, we have demonstrated that imposing the dual symmetry restores conventional electromagnetism.

We began with a Euclidean matter space \(\{M^A\}\) (\(A = 1, 2, 3\)), incorporating 0-, 1-, and 2-form fields. These form fields were subsequently mapped onto spacetime, yielding the orthogonality conditions in Eqs. 5 , 6 , and 7 and constraint equations summarized in Table 1. Based on these orthogonality conditions and constraints, we establish the matter-space gauge symmetry for the matter space fields and the Bianchi identity. To complete the framework, we have employed a Lagrangian formulation for relativistic fluid dynamics. The proposed model, governed by the Lagrangian 17 , inherently lacks intrinsic dynamics and is formulated in terms of the induced fields up to quadratic order. By substituting the constraint equations into the equations of motion, we derived the remaining Maxwell’s equations 27 . As a result, thanks to the constraint equations, the dynamics of fields has been generated. We then employed the electric-magnetic duality symmetry and show the legality of the Bianchi identity when there are no external magnetic charge. We also have shown that the charge conservation relation naturally appears from the presence of a charge-carrier matter space at Eq. 28 .

Now, let us discuss various aspects of the theory. We first clarify the role of the vector \(m^a\) in relation to the electromagnetic field arising from our one-fluid construction. Given a field configuration \(F_{ab}\) and a gauge potential \(A_a\) (up to gauge scalars \(\{\phi\}\)), the direction of \(m^a\) is fixed by the orthogonality conditions (such as \(m^aF_{ab}=0\)), and its overall scale is determined (up to an overall constant) by the conservation law \(\nabla_a m^a=0\). Hence \(m^a\) should be regarded as a derived quantity, determined by \(F_{ab}\) and \(A_a\), rather than as an independent field. In particular, the orthogonality condition \[\label{KelHelA} m^aF_{ab}=0 \quad\Longrightarrow\quad m^a\nabla_{[a}A_{b]}=0,\tag{32}\] is formally identical to the force-free condition \(v^a\omega_{ab}=0\) in ideal fluid dynamics, where \(v^a\) denotes the fluid 4-velocity and \(\omega_{ab}=2\nabla_{[a}v_{b]}\) its vorticity. By analogy, one may regard \(A_a\) as defining a flow potential along \(m^a\), with \(F_{ab}\) playing the role of its vorticity. Eq. 32 then implies that this vorticity is Lie-transported (frozen-in) by \(m^a\), i.e. conserved in the sense of the Kelvin-Helmholtz theorem. Thus \(F_{ab}\) can be viewed as an inviscid vorticity field generated by the gauge potential \(A_a\), and Eq. 7 encapsulates the conservation of this "electromagnetic vorticity" along the matter flow. Equivalently, when \(m_a\) is not twist (i.e. hypersurface-orthogonal flow), \(\nabla_{[a}m_{b]}=0\), \(A_a\) is Lie-conserved : \[\label{LiewrtmofA} \pounds_{m}A_b = m^a\nabla_a A_b + A_a\nabla_b m^a = 2\nabla_{[a}m_{b]}\,A^a =0.\tag{33}\] Hence the line integral \(\oint A_a\,dx^a\) around loops comoving with \(m^a\) is conserved. More generally, even if \(\nabla_{[a}m_{b]}\neq0\), \(\pounds_{m}A_b\) reduces to a gradient when combined with \(m^aA_a=0\), so the circulation of \(A_a\) around comoving loops remains conserved.

A natural question is: what quantity in electromagnetism corresponds to the fluid vorticity? Combining the two equations (B) and (D) in Table 1 and defining the helicity 4-vector, \[\begin{align} \label{defJ} \mathcal{H}^a \equiv \epsilon^{abcd} A_{[b} F_{cd]} = A_b *F^{ba}~, \end{align}\tag{34}\] we derive another nontrivial conservation equation: \[\begin{align} \label{defcurrent} \nabla_a \mathcal{H}^a = 0. \end{align}\tag{35}\] Even though this form appears gauge dependent, helicity becomes gauge-invariant and physically meaningful for a closed or periodic systems or under certain boundary conditions, e.g., \(\vec{B} \cdot \hat{n} =0\) on the boundary, where \(\hat{n}\) denotes the normal vector at the boundary. Notice that explicit calculation gives \[\label{d H} \nabla_a \mathcal{H}^a = F_{ab} *F^{ab} = - 4 \vec{E} \cdot \vec{B}.\tag{36}\] This result shows that the helicity conservation holds because this theory is the electromagnetism from one-fluid system satisfying Eq. 14 . The vector \(\mathcal{H}^a\) satisfies \[\begin{align} \label{orthoH} \mathcal{H}^a \nabla_a \phi = 0, \quad \mathcal{H}^a A_a = 0, \quad \mathcal{H}^a F_{ab} = 0. \end{align}\tag{37}\] Consequently, up to an overall constant normalization, we may identify \(m^a \propto \mathcal{H}^a\), so that the helicity current itself provides the vorticity vector in the electromagnetic context⁶.

In this framework, the flow vector \(m^a\) is expressed directly in terms of the electric and magnetic fields, \(\vec{E}\) and \(\vec{B}\). We perform a standard \(3+1\) decomposition of the field strength tensor as follows: \[\label{eq:3431decomp} F_{0i} = E_i,\quad F_{ij} = \varepsilon_{ijk} B^k\,.\tag{38}\] From this, the components of \(m^a\) are derived as \[\begin{align} \label{eq:mwrtEB1} m^0 &\propto \mathcal{H}^0 = \epsilon^{0ijk} A_i F_{jk} = A_i B^i\,, \\ m^i &\propto \mathcal{H}^i = \epsilon^{i0jk} A_0 F_{jk} + \epsilon^{ij0k} A_j F_{0k} = A_0 \epsilon^{ijk} B_k + \epsilon^{ijk} A_j E_k\,. \end{align}\tag{39}\] In the Coulomb gauge (\(A_0 = 0\)), the expressions simplify to \[\begin{align} \label{eq:mwrtEB2} m^0 \propto \vec{A} \cdot \vec{B}\,, \qquad \vec{m} \propto \vec{A} \times \vec{E} \,. \end{align}\tag{40}\] Here, \(\mathcal{H}^0 = \vec{A} \cdot \vec{B}\) corresponds to the magnetic helicity density, while \(\vec{\mathcal{H}} = \vec{A} \times \vec{E}\) represents the helicity flux. Consequengly, Eq. 35 reduces to the continuity equation \[\begin{align} \label{eq:continuity} \frac{\partial}{\partial t}(\vec{A}\!\cdot\!\vec{B}) +\nabla\!\cdot(\vec{A}\times\vec{E}) =0\,, \end{align}\tag{41}\] indicating that \(\mathcal{H}^a\) is conserved and gauge-invariant under appropriate boundary conditions. Thus, identifying \(m^a\) with a suitably scaled \(\mathcal{H}^a\) naturally interprets the fluid flow vector as a helicity current within the one-fluid model. Not all electromagnetic fields can be realized in this one-fluid framework; more general configurations require multi-fluid descriptions or different matter-space structures. In future work, one could explore relaxations of these assumptions to classify broader classes of field configurations.

This nonlinear correction in Eq. 29 is analogous to mass renormalization effects, reminiscent of the motivation behind Born-Infeld electrodynamics [23]. Because Eq. 29 contains nonlinear contributions, it is interesting to examine the electron self-energy problem. When \(\beta = 0\), the electron self energy problem appears as follows: The electric energy density of a static charge \(q\) located at \(r=0\) is \[\begin{align} \label{energydenb610} u(r) \;=\; \frac{\epsilon_0 E_r^2}{2} \;=\; \frac{q^2}{32\pi^2 \epsilon_0\,r^4}, \end{align}\tag{42}\] where \(\epsilon_0\) is the electric permittivity of free space. So integrating over volume, the total electric energy becomes \[\begin{align} \label{totenergyb610} U(\delta) = \int_{r>\delta} u\,dV \sim \frac{q^2}{8\pi\epsilon_0\,\delta}, \end{align}\tag{43}\] where \(\delta\) denotes the characteristic size of the electron. Taking the infinitesimal limit \(\delta \to 0\), the energy diverges. To compensate this divergence, one should choose the bare mass of electron to have negative infinite value.

The divergence of the self-energy disappears when \(\beta >0\). In this case, the solution to Eq. 29 is \[\begin{align} \label{Coulombfieldbneq0} E_r = \frac{q}{4\pi \epsilon_0 r^2(1+ \frac{ r_c}{ r})}, \qquad r_c \equiv \frac{\beta q^2}{8\pi} \end{align}\tag{44}\] with all other components vanish. Note that when \(r \gg r_c\), the formula reproduces the \(\beta=0\) result. The electric potential is modified around \(r \sim r_c\). In high energy \(e^+e^-\) scattering experiments, the Coulomb potential is kept to \(10^{18}\)m [24]–[26]. This result presents an upper bound on the value of \(\beta\): \[r_c = \frac{\beta \rm{e}^2}{8\pi} < 10^{-18}\rm{m}.\] Near \(r\to0\), however, the denominator modifies the divergence so that the energy density \[\begin{align} \label{energydenbneq0} u(r) = \frac{\epsilon_0 E_r^2}{2} =\frac{q^2}{32\pi^2 \epsilon_0 r^4(1+ \frac{\beta q^2}{8\pi r})^2} \end{align}\tag{45}\] over any volume be finite. Integrating over the whole space, the total energy stored in the electro-static field is \[\begin{align} \label{totenergybneq0} U_{\rm total} = \int_0^\infty 4\pi r^2 u(r)\,dr = \frac{1}{\beta \epsilon_0} = \frac{q^2}{8\pi \epsilon_0 r_c }. \end{align}\tag{46}\] If we take this value to be the electron mass, we have \(r_c \sim 1.4 \,{\rm fm}\), which is comparable to the nuclear size and a bit smaller than the so-called classical electron radius, \(2.8\, {\rm fm}\) [27]. Note that \(r_c\) does not denote the size of electron because the electric charge in this model is located at the center \(r=0\), which makes the electric field diverge there. Therefore, this result is consistent with the known upper bound of the electron size \(10^{-22}\,{\rm m}\). Even though this result is better than that of the classical electrodynamics, it fails to explain the self-energy problem in the classical level because it does not satisfy the previous bound. Therefore, this theory still requires the quantum mechanical treatment to resolve the self-energy problem.

In this new framework, the observable corresponding to the particle concept is based on the matter space, where the particle’s degrees of freedom is represented by the number of coordinates of this space. While the classical equations of motion within this framework are equivalent to those of conventional electromagnetism, this equivalence may not necessarily extend to the quantum regime. As a result, the process of quantization in matter space emerges as a significant and intriguing area for further exploration. Additionally, a key direction for future research will be to investigate how spinor fields and other symmetries can be realized within the context of matter space.

Acknowledgements↩︎

This work was supported by the National Research Foundation of Korea(NRF) grant with grant number RS-2023-00208047 (H.K.,J.H.,Y.Y.), NRF-2020R1A6A1A03047877 (J.H.), RS-2025-00553127 (J.H.), and NRF-2020R1F1A1068410 (J.L.).

References↩︎