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Charge quantisation without magnetic poles: a topological approach to electromagnetism

The present work provides a theoretical explanation for the quantisation of electric charges, an open problem since Millikan's oil drop experiment in 1909. This explanation is based solely on Maxwell's theory, it recasts Electromagnetic theory under the language of complex line bundles; therefore, neither magnetic poles nor quantum mechanics are invoked. The existence of magnetic poles was essentially the only theoretical explanation for charge quantisation (e.g. Dirac's magnetic pole), and there is no experimental data supporting their existence.


Introduction
Essentially, the existence of magnetic monopoles was the only theoretical explanation for charge quantisation (e.g. Dirac's monopole [1]), and there is no experimental data supporting their existence -on the contrary, they have nerver been observed.
My intention with this note is to clarify some features of the Electromagnetic theory and to champion the idea that Maxwell's theory is not only intrinsically relativistic, but it naturally accommodates phenomena that are usually associated with its quantum aspects: charge quantisation, the Aharonov-Bohm effect, Dirac's equation, and photons wave functions. All of this become clear by defining the electromagnetic field (over a spacetime region) as an equivalence class of hermitian line bundles with hermitian connexion (considering the spacetime region of interest as the base manifold).
I shall justify my approach by quoting Dirac [1]: "The most powerful method of advance that can be suggested at present ...", still is, I think, "... to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities".
An appendix to this note is included. It supplements the text with definitions and theorems (together with sketch of their proofs) that are known in the literature, providing a place to fix notation and a guide for the reader not familiar with differential topology.
Throughout this article and otherwise stated, all the objects considered will be C ∞ ; manifolds are real, Hausdorff, paracompact, connected, and orientable; and physical quantities are measured using the International System of Units.

A précis of the Electromagnetic Theory
Einstein's field equations [2] provide the spacetime (M, g): a lorentzian manifold. Classical electromagnetism over a spacetime (M, g) is described by a differential 2-form, the electromagnetic field tensor ω ∈ Ω 2 (M; R), solving Maxwell's equations, where ⋆ g is the Hodge star operator of the lorentzian metric g, and the differential 1-form α ∈ Ω 1 (M; R) encodes the information about charges and currents distributions. On the Minkowski spacetime, these equations are equivalent to 1 : with ρ and the vectors J, B, and E defined by and The classical dynamics of light and charged test particles are both described by hamiltonian flows. Light trajectories are null geodesics of (M, g), whilst the trajectories of a test particle, having mass m and charge q, are integral curves of the hamiltonian flow defined by the function with v * (·) := mg x (v, ·) ∈ T * x M, on the cotangent bundle T * M endowed with the symplectic form dλ − q ǫ 0 c Pr * (ω), where dλ is the cannonical symplectic form of T * M induced by the projection Pr : T * M → M. The Lorentz force law, can be deduced from this symplectic picture [3].
3 The topological approach I shall consider only globally hyperbolic 4-dimensional lorentzian manifolds with finite first and second Betti numbers to model spacetimes; in particular, a spacetime M is diffeomorphic to R × M, and there exists a vector field X ∈ X(M; R) which is the gradient (with respect to the lorentzian metric) of a global time function on M (vide [4] and [2]).
It is not possible to probe inside regions occupied by charges or currents; thus, outside these regions the electromagnetic field tensor satisfies the source free Maxwell's equations, One first eliminates worldlines corresponding to particle (charge) distributions and the worldsheets corresponding to wire (current) distributions from the spacetime region of interest; then, one topologically introduces the source term back on Maxwell's equation. Let J ⊂ M stand for the disjoint 2 union of worldlines and worldsheets corresponding to particle and wire distributions, and M J := M − J the complement of J. If the class [N] is a generator of the second singular homology group H 2 (M J ; Z), N ω is the charge inside a spacetime region bounded by any smooth representative N; likewise, if the class [γ] is a generator of the first singular homology group H 1 (M J ; Z), the current passing through a spacetime region bounded by a smooth representative γ is c γ ı X ω. It is important to mention that ı X ω is not closed unless £ X (ω) = 0 (static fields), and this means that: if γ′ ∈ [γ], then c γ ı X ω = c γ′ ı X ω + c N £ X (ω) with ∂N = γ − γ′, and c N £ X (ω) is related to displacement currents.
I am now in a position to provide a precise definition for the electromagnetic field over a spacetime region that is compatible with Maxwell's theory, free of magnetic monopoles, and able to explain charge quantisation.
Given charge and current distributions J ⊂ M, the electromagnetic field over a spacetime (M, g) is an equivalence class of hermitian line bundles with hermitian connexion over M J , [(L, ·, · , ∇ ω )], whose hermitian connexion of any representative satisfies curv(∇ ω ) = −i e ω ⋆ g d ⋆ g ω = 0 .
The Chern class c 1 ([L]) ∈Ȟ 2 (M J ; Z) of [(L, ·, · , ∇ ω )] is fixed by knowing the charges and quantal phases 3 (cf. theorem A.3). The periods of the curvature provide, via de Rham's theorem (cf. theorem A.5), an element of the second de Rham cohomology group [ω] ∈ H 2 dR (M J ; R) lying in the image of the homomorphism betweeň H 2 (M J ; Z), the second Čech cohomology group with coefficients in the constant sheaf Z, and H 2 dR (M J ; R), induced by the inclusion Z ∋ k → 2πk ∈ R. Quantal phases provide a homomorphism between the fundamental group π 1 (M J ) and the unitary circle S 1 , which defines an element on the kernel of the aforementioned homomorphism H 2 (M J ; Z) → H 2 dR (M J ; R) (cf. lemma A.4 and theorem A.2). The curvature is fixed by selecting a representative of [ω] ∈ H 2 dR (M J ; R) which satisfies ⋆ g d ⋆ g ω = 0 and matches the currents, and possible initial (e.g. background electromagnetic radiation) or boundary (e.g. the vanishing of the electromagnetic field tensor at some regions) conditions.
One can conclude from the above that magnetic effects are a manifestation of flat hermitian line bundles with nonflat connexions, whilst the Aharonov-Bohm effect is due to flat connexions; ergo, topologically, it is impossible to choose a different reference frame where a trully electric effect -which is a manifestation of nonflat hermitian line bundles-is seen as a purely magnetic effect. Furthermore, what is usually associated with the quantum aspects of electromagnetism is encoded inȞ 2 (M J ; Z) (Chern classes, quantisation of charge) and π 1 (M J ) (holonomy of flat connexions, Aharonov-Bohm effect), classical properties are due to Ω 2 (M J ; R) (curvature, electromagnetic field tensor).

Topological charges and currents
In order to demonstrate that this definition implies quantisation of charges (cf. proposition A.4) and does not support magnetic monopoles, I shall investigate the sources of electromagnetic fields on Minkowski spacetime.
A point charge has its worldline removed from Minkowski spacetime; therefore, . If the electromagnetic field has its Chern class determined by the integer k ∈ Z, then the electric charge contained in a smooth representative of the generator of H 2 (M J ; Z) determined by the integer 1 is exactly ke, and there is no other source for this electromagnetic field.
An open wire carrying current has its worldsheet removed from Minkowski space- . A closed wire carrying current has its worldsheet removed from Minkowski spacetime; consequently, In both situations the electromagnetic field is represented by flat hermitian line bundles with trivial holonomy group -for π 1 (M) = {1}, there cannot be Aharonov-Bohm effects-, and a smooth representative of the generator of H 1 (M J ) determined by the integer 1 is exactly a closed curve around the wire: there is no other source for the electromagnetic field tensor but the current passing through the wire.

Local features
. Both π 1 (M) andȞ 2 (M; Z) are trivial; ergo, the trivial complex line bundle L = C×M is a representative for the source free electromagnetic field in vacuum.
This section is devoted to unravel information contained in the sections of L.

A wave function of a photon
The trivial complex line bundle admits a flat connexion ∇ 0 satisfying for a unitary section of the complex line bundle s ∈ Γ(L) and any complex-valued function f ∈ C ∞ (M; C).
There are two linear maps, defined via the flat connexion, from , the first map is given by: The second map is defined by Let L −1 denote the dual bundle of L. A hermitian structure ·, · provides an equivalence between the dual bundle and the complex conjugate bundleL (defined by the complex conjugate of the transition functions of L). In the particular case of

Wave functions for photons are sections of
and together with a compatibility condition, considering L ∼ = L −1 , Using the unitary section s, sections of the trivial complex line bundle can be identified with complex-valued functions, ω + j = 1 µ 0 c B j − iǫ 0 E j s, and one recovers a wave function for the photon (having a particular helicity) as in references [5] and [6].

Dirac's equation
Considering now a connexion ∇ ω in L whose curvature is −i e ω, one has a linear map from X(M; R) to the endomorphisms of Wave functions describing electrons and positrons of mass m subjected to an elctromagnetic field [(L, ·, · , ∇ ω )] are eigenvectors of the endomorphism induced by X + 3 j=1 Y j whose eigenvalue is mc: Again, by using the unitary section s, for each Ψ j there exists a complex-valued function ψ j ∈ C ∞ (M; C) such that Ψ j = ψ j s, and one recovers Dirac's equation for a wave function for the electron and positron [7].

Appendix: Hermitian line bundles
This is a summary of relevant results and definitions needed to understand this note -technicalities will be avoided here, though-, the reader can find them in references [8,9,10].  Proof: Let A j be a trivialising neighbourhood of a hermitian line bundle (L, ·, · ). There exists a diffeomorphism ϕ j : L A j → C×A j , and if z ∈ C belongs to the unitary circle, then one can define a section of C × A j via A j ∋ p → (z, p). A unitary section s j ∈ Γ(L A j ) is, then, obtained by s j (p) = ϕ −1 j (z, p). Conversely, let s j ∈ Γ(L A j ) be a section satisfying s j , s j (p) = 1 for all p ∈ A j . When there is an identification between a section s of the line bundle and a complex-valued function f , the diffeomorphism might be omitted, for the sake of simplicity, and the equality s = f will be used.

The map given by
Two hermitian line bundles (L, ·, · ) and (L ′ , ·, · ′ ) over the same manifold M are equivalent if there exists a diffeomorphism φ : L → L ′ such that, when restricted to a fibre L p , φ Lp : L p ⊂ L → L ′ is an isometry between L p and L ′ φ(p) and, when restricted to the zero section M ⊂ L, φ M : M ⊂ L → φ M (M) ⊂ L ′ is the identity. Such a diffeomorphism will be called a hermitian bundle diffeomorphism.
Definition A.5. The set of equivalence classes of hermitian line bundles endowed with a group structure given by tensor products of hermitian line bundles will be denoted by L h (M).
Let C ∞ R and C ∞ S 1 denote the sheaves of R-valued and S 1 -valued functions over a manifold M: Given a contractible open cover A = {A j } j∈I of M (which can always be obtained, e.g. using a convenient cover made of balls with respect to a Riemannian metric) and an element ϕ ofȞ 1 (M; C ∞ S 1 ), one can define a hermitian line bundle L such that each A j is a trivialising neighbourhood of it with transition functions given by ϕ(A j ∩ A k ).
The construction goes as follows: one can define L A j := C×A j with the standard hermitian structure (or any other one) on the fibre C and consider the equivalence has both a natural smooth structure, making it a complex line bundle over M, and a natural hermitian structure ·, · . Now, supposing that the hermitian line bundle (L, ·, · ) is given together with a contractible open cover A = {A j } j∈I of M where each A j is a trivialising neighbourhood, i.e. there exist bundle morphisms respecting the hermitian structure ϕ j : L A j → C × A j , one can define an element ofȞ 1 (M; C ∞ S 1 ): the diffeomorphisms satisfying the cocycle conditions.
is a short exact sequence of sheaves. Indeed, Thus, there is a long exact sequence of cohomology groups, and the fine property of the sheaf C ∞ R provides the result.
satisfying the cocycle conditions.
Under pertinent identifications, hermitian line bundles defined by elements of H 1 (M; S 1 ) are called flat.
Theorem A.2. The kernel of the homomorphismȞ 2 (M; Z) →Ȟ 2 (M; R) induced by the inclusion Z ֒→ R sits in the following short exact sequence of groups: is a short exact sequence of sheaves: The long exact sequence of cohomology groups associated with it is and the first isomorphism theorem implies that Using an isomorphism between Čech and de Rham cohomology, a class on the image of the homomorphismȞ 2 (M; Z) → H 2 dR (M; R) ∼ =Ȟ 2 (M; R) induced by the inclusion Z ֒→ R are said to be integral. satisfying: with [∇s](X) denoted by ∇ X s, and X( r, s ) = ∇ X r, s + r, ∇ X s , for any r, s ∈ Γ(L), f ∈ C ∞ (M; C) and X ∈ X(M; R).
With respect to a unitary section s j , defined over a trivialising neighbourhood A j ⊂ M, the connexion ∇ can be represented by a potential 1-form Θ j ∈ Ω 1 (A j ; R): Indeed, since s j is unitary, s j , s j = 1, for any X ∈ X(A j ; R): Lemma A.5. The potential 1-forms, Θ j and Θ k , of ∇ for each unitary section, s j and s k , defined over an intersection A j ∩ A k are cohomologous.
Proof: Since L is a hermitian line bundle, there exists a transition function e if jk ∈ C ∞ (A j ∩ A k ; S 1 ) relating the two unitary sections s j = e if jk s k ; as a result, however and, therefore, Definition A.7. The curvature of a hermitian connexion ∇ on a hermitian line bundle (L, ·, · ) over a manifold M is the imaginary 2-form, curv(∇) ∈ Ω 2 (M; iR), for any s ∈ Γ(L) and X, Y ∈ X(M; R).
The next lemma implies, in particular, that the curvature is closed.
Lemma A.6. If Θ is a potential 1-form for the connexion ∇, over a trivialising neighbourhood A ⊂ M, then the curvature is given by curv(∇) = −idΘ, and it is independent of the choice of trivialisation.
Proof: Assuming that s is the unitary section associated with Θ, for any X, Y ∈ X(A; R). Independence of the choice of trivialisation is provided by lemma A.5.
Proof: Locally, the condition implies that the potential 1-forms are equal. The 1-forms θ k − θ j are closed over A j ∩ A k ; consequently, 1 e θ k − 1 e θ j = dg jk , the maps A j ∩ A k ∋ p → e ig jk (p) ∈ S 1 give an element ofȞ 1 (M; S 1 ) (since [ 1 e ω] is integral), and the potential 1-forms 1 e θ j provide the hermitian connexion. There is a converse of the previous statement.
Proposition A.4. If the hermitian connexion ∇ ω of a complex line bundle (L, ·, · ) satisfies curv(∇ ω ) = −i e ω, then the de Rham class [ 1 e ω] is integral. Proof: Let A = {A j } j∈I be a contractible open cover of M such that each A j is a trivialising neighbourhood of L with unitary section s j . On each A j the formula ω = dΘ j holds: 1 e Θ j is the potential 1-form with respect to s j (lemma A.6). As a consequence, over each intersection 1 e Θ k − 1 e Θ j = df jk , where e if jk ∈ C ∞ (A j ∩ A k ; S 1 ) are the transition functions (lemma A.5). Now, d(f jk + f kl − f jl ) = 0 implies that f jk + f kl − f jl = a jkl ∈ R on A j ∩ A k ∩ A l . The cocycle conditions of the line bundle, e if jk e if kl = e if jl , imply that e ia jkl = 1; thus, a jkl ∈ 2πZ and [ i e ω] is integral. As a consequence of theorem A.3 and proposition A.4, flat hermitian line bundles are equivalent to hermitian line bundles admitting hermitian connexions whose curvatures are identically zero (flat connexions).