Torsion, Magnetic Monopoles and Faraday's Law via a Variational Principle
aa r X i v : . [ h e p - t h ] J un Torsion, Magnetic Monopoles and Faraday’s Law via a Variational Principle
Philip D. Mannheim
Department of Physics, University of Connecticut, Storrs,CT 06269, USA. email: [email protected] (Dated: June 25, 2014)Even though Faraday’s Law is a dynamical law that describes how changing E and B fieldsinfluence each other, by introducing a vector potential A µ according to F µν = ∂ µ A ν − ∂ ν A µ Faraday’sLaw is satisfied kinematically, with the relation ( − g ) − / ǫ µνστ ∇ ν F στ = 0 holding on every pathin a variational procedure or path integral. In a space with torsion Q αβγ the axial vector S µ =( − g ) / ǫ µαβγ Q αβγ serves as a chiral analog of A µ , and via variation with respect to S µ one canderive Faraday’s Law dynamically as a stationarity condition. With S µ serving as an axial potentialone is able to introduce magnetic monopoles without S µ needing to be singular or have a non-trivialtopology. Our analysis permits torsion and magnetic monopoles to be intrinsically Grassmann, whichcould explain why they have never been detected. Our procedure permits us to both construct aWeyl geometry in which A µ is metricated and then convert it into a standard Riemannian geometry. I. INTRODUCTION
The Maxwell equations of electromagnetism in flatspace break up into two groups, the Maxwell-Ampereand Electric Gauss Laws ∇ × B − ∂ E ∂t = J e , ∇ · E = ρ e , (1)and the Faraday and Magnetic Gauss Laws ∇ × E + ∂ B ∂t = 0 , ∇ · B = 0 . (2)Since can one derive second-order wave equations for thepropagation of the E and B field strengths in a source-free region only when all of the above equations are takenin conjunction, as such all of these equations should beregarded as being on an equal dynamical footing. And ifthey are to be on an equal dynamical footing, then eachone of these equations should, like all dynamical equa-tions, be derivable via stationary variation of an action.However, the standard treatment of electrodynamics isnot formulated in this way, as it does not in fact derive allof these equations from a variational procedure. Rather,in order to develop the variational procedure that it doesuse, it relies on the fact that the Faraday-Magnetic Gaussequations immediately admit of an exact solution E = − ∂ A ∂t − ∇ φ, B = ∇ × A , (3)a solution that is unique up to gauge transformations ofthe form A → A + ∇ χ , φ → φ − ∂χ/∂t . The introductionof A and φ serves two purposes. When inserted into Eq.(1) they enable one to solve for the E and B fields once J e and ρ e are specified. And in addition they allow oneto develop a variational procedure.To discus the variational procedure it is more conve-nient to first write the Maxwell equations covariantly ina curved space where they generalize to ∇ ν F νµ = J µ , (4) ( − g ) − / ǫ µνστ ∇ ν F στ = 0 . (5)In Eq. (5) the antisymmetric rank two tensor F µν is thefield strength with components F = − E x , F = − B z etc., and J µ = ( ρ e , J e ). Using − ( − g ) − ǫ µνστ ǫ µαβγ = δ αν δ βσ δ γτ + δ ατ δ βν δ γσ + δ ασ δ βτ δ γν − δ αν δ βτ δ γσ − δ ασ δ βν δ γτ − δ ατ δ βσ δ γν ,Eq. (5) can also be written in the form ∇ ν F στ + ∇ τ F νσ + ∇ σ F τν = 0 . (6)For brevity we shall refer to Eq. (5) as Faraday’s Laweven as it encompass Gauss’ Law of Magnetism as well.With Eq. (5) possessing an exact solution of the form F µν = ∇ µ A ν − ∇ ν A µ , one introduces the Maxwell action I MAX = Z d x ( − g ) / (cid:20) − F µν F µν − A µ J µ (cid:21) , (7)with its stationary variation with respect to A µ immedi-ately leading to Eq. (4).Since this variation is a variation in which Eq. (5) isnot varied, Eq. (5) is required to hold on every varia-tional path. Thus even though Faraday’s Law is a dy-namical equation, the variation that is done is a con-strained one in which Faraday’s Law is imposed on everyvariational path, even on those that are not stationary,with the quantum path integral R DA µ exp( iI MAX ) thenbeing constrained this way as well. We shall thus seekto construct a variational procedure in which Faraday’sLaw is to only hold at the stationary minimum.
II. SETTING UP THE VARIATIONALPROCEDURE
If we do not want Faraday’s Law to hold on non-stationary paths, we cannot set F µν = ∇ µ A ν − ∇ ν A µ ,since this would immediately cause ( − g ) − / ǫ µνστ ∇ ν F στ to vanish [1]. If however, we wish to recover Faraday’sLaw at the stationary minimum, then with 8 equationsbeing embodied in Eqs. (4) and (5), we need not one buttwo 4-vector potentials, one of course being the standardvector potential A µ and the other needing to be some asyet to be identified axial vector S µ . Moreover, withoutregard to variational issues, we note that in the eventof magnetic monopoles one would ordinarily (though notquite in fact as we show below) modify Eq. (5) to( − g ) − / ǫ µνστ ∇ ν F στ = K µ , (8)with there then being both vector and axial vector cur-rent sources, for a total of 8 components. In the sameway as we couple A µ to J µ via A µ J µ we should equallyanticipate a coupling S µ K µ in the axial current sector, acoupling that is parity conserving if S µ is an axial vec-tor. The issue of constructing a variational principle forFaraday’s Law is thus related to the coupling of electro-magnetism to magnetic currents, and our objective willbe to set up a variational principle with respect to A µ and S µ that would recover Eqs. (4) and (8) at the stationaryminimum, with Eq. (5) then following in the limit inwhich we could set the monopole current to zero.Recalling the two-potential study [2, 3] of themonopole problem [4], it is very convenient to introduce X µν = ∇ µ A ν − ∇ ν A µ −
12 ( − g ) − / ǫ µνστ ( ∇ σ S τ − ∇ τ S σ ) (9)as a generalized F µν . On setting S µν = ∇ µ S ν − ∇ ν S µ ,we can rewrite X µν in terms of F µν and the dual ˆ S µν =(1 / − g ) − / ǫ µνστ S στ of S µν according to: X µν = F µν − ˆ S µν , ˆ X µν = ˆ F µν + S µν . (10)(If ǫ = +1, ǫ = − X µν , Eqs. (4)and (8) are to be replaced by ∇ ν X νµ = ∇ ν F νµ = J µ , ∇ ν ˆ X νµ = ∇ ν S νµ = K µ , ∇ ν ˆ F νµ = 0 , ∇ ν ˆ S νµ = 0 , (11)with it now being ∇ ν ˆ X νµ = K µ and not in fact ∇ ν ˆ F νµ = K µ that is to describe the monopole. If we introduce asecond set of field strengths S = − B ′ x , S = + E ′ z ,ˆ S = E ′ x , ˆ S = B ′ z , on setting K µ = ( ρ m , − J m ), wefind that in flat space Eq. (11) breaks up into two sectors,namely Eqs. (1) and (2) and the analog ∇ × B ′ − ∂ E ′ ∂t = 0 , ∇ · E ′ = 0 , ∇ × E ′ + ∂ B ′ ∂t = J m , ∇ · B ′ = ρ m . (12)Moreover, if we define E TOT = E + E ′ , B TOT = B + B ′ ,we can combine Eqs. (1), (2), and (12) into ∇ × B TOT − ∂ E TOT ∂t = J e , ∇ · E TOT = ρ e , ∇ × E TOT + ∂ B TOT ∂t = J m , ∇ · B TOT = ρ m . (13)Thus even if J m and ρ m can be neglected, it is E TOT and B TOT that are measured in electromagnetic experiments. On introducing the action I = Z d x ( − g ) / (cid:20) − X µν X µν − A µ J µ − S µ K µ (cid:21) , (14)we find that stationary variation with respect to A µ and S µ then immediately leads to Eq. (11), just as we want.Moreover, up to surface terms this action decomposesinto two sectors according to I = Z d x ( − g ) / (cid:20) − F µν F µν − A µ J µ − S µν S µν − S µ K µ (cid:21) . (15)Thus with the introduction of a magnetic current sectorwe can formulate a variational principle for Faraday’sLaw and for theories that involve magnetic monopoles,and can do so without the use of singular potentials ornon-trivial topologies [5]. However, we still need to as-cribe a physical meaning to S µ , and to this end we turnto torsion. This will lead us directly to the action givenin Eq. (15), and suggest a rationale for why the S µν sector has escaped detection and why a purely A µ -basedquantum electrodynamics works as well as it does. III. TORSION
To construct covariant derivatives in a metric theoryone introduces a connection Γ λµν . For a torsionless Rie-mann space one uses the Levi-Civita and spin connectionsΛ λµν = 12 g λα ( ∂ µ g να + ∂ ν g µα − ∂ α g νµ ) = Λ λνµ , − ω abµ = V bν ∂ µ V aν + V bλ Λ λνµ V aν = ω baµ , (16)to construct covariant derivatives such as ∇ µ g λν = ∂ µ g λν + Λ λαµ g αν + Λ ν αµ g λα and D µ V aλ = ∂ µ V aλ +Λ λνµ V aν + ω abµ V λb that transform as tensors under localtranslations and local Lorentz transformations. In Eq.(16) we have introduced vierbeins V aµ that carry an index a associated with a fixed special-relativistic reference sys-tem, with the metric being writable as g µν = η ab V aµ V bν .The covariant derivatives of g µν and V µa constructedwith Λ λµν obey the metricity conditions ∇ µ g λν = 0, D µ V aλ = 0. If one generalizes Λ λµν to ˜Γ λµν by addinga rank-3 tensor to it, covariant derivatives constructedwith ˜Γ λµν will still transform as true tensors. How-ever, they may not necessarily obey metricity conditions˜ ∇ µ g λν = 0, ˜ D µ V aλ = 0 with respect to ˜Γ λµν .To extend the geometry to include torsion one takesthe connection to no longer be symmetric on its two lowerindices, and defines the Cartan torsion tensor Q λµν Q λµν = Γ λµν − Γ λνµ . (17)To implement metricity one defines a contorsion tensor K λµν = 12 g λα ( Q µνα + Q νµα − Q ανµ ) , (18)and with K λµν one constructs connections of the form˜Γ λµν = Λ λµν + K λµν , − ˜ ω abµ = − ω abµ + V bλ K λνµ V aν = ˜ ω baµ . (19)To couple spinors to gravity in a Riemannian spacewithout torsion one uses the covariantized Dirac action I D = (1 / R d x ( − g ) / i ¯ ψγ a V µa ( ∂ µ + Σ bc ω bcµ ) ψ + H.c. ,where Σ ab = (1 / γ a γ b − γ b γ a ). To generalize this actionto include torsion one replaces ω bcµ by ˜ ω bcµ and obtains˜ I D = 12 Z d x ( − g ) / i ¯ ψγ a V µa ( ∂ µ + Σ bc ˜ ω bcµ ) ψ + H.c. (20)Integration parts, use of properties of the Dirac gammamatrices, and introduction of a coupling to A µ yields [6]˜ I D = Z d x ( − g ) / i ¯ ψγ a V µa ( ∂ µ + Σ bc ω bcµ − iA µ − iγ S µ ) ψ, (21)where S µ = 18 ( − g ) − / ǫ µαβγ Q αβγ , − ( − g ) − / ǫ µαβγ S µ = 14 [ Q αβγ + Q γαβ + Q βγα ] . (22)In the action ˜ I D we note that even though the torsion isonly antisymmetric on two of its indices, the only compo-nents of the torsion that appear in its torsion-dependent S µ term are the four that constitute that part of the tor-sion that is antisymmetric on all three of its indices. Aswell as being locally gauge invariant under ψ → e iα ( x ) ψ , A µ → A µ + ∂ µ α ( x ), ˜ I D is also locally chiral invariant [6]under ψ → e iγ β ( x ) ψ , S µ → S µ + ∂ µ β ( x ). Additionally,as noted in [7], ˜ I D is locally conformal invariant under V aµ ( x ) → Ω( x ) V aµ ( x ), ψ ( x ) → Ω − / ( x ) ψ ( x ) since, justlike the vector potential A µ , the equally minimally cou-pled S µ also has zero conformal weight [8]. The ˜ I D actionthus has a remarkably rich local invariance structure, as itis invariant under local translations, local Lorentz trans-formations, local gauge transformations, local axial gaugetransformations, and local conformal transformations.With S µ having a structure identical to the FaradayLaw structure given in Eqs. (5) and (6), and with S µ precisely being an axial 4-vector, S µ is thus the naturalquantity to act as the second potential that appears in X µν [9], and thus the natural axial vector needed to setup a variational procedure for Faraday’s Law of electro-magnetism [10]. However, in order to set up a variationalprocedure we will need to construct a kinetic energy termfor it. To generate such a kinetic energy term we appealto the Dirac action. Specifically, we recall [11], [6] thatwhen one does a path integration R D ¯ ψDψ exp( i ˜ I D ) overthe fermions (equivalent to a one fermion loop Feynman graph) one generates an effective action of the form [12] I EFF = Z d x ( − g ) / C (cid:20) (cid:20) R µν R µν −
13 ( R αα ) (cid:21) + 13 F µν F µν + 13 S µν S µν (cid:21) , (23)where C is a log divergent constant and R µν is the stan-dard (torsionless) Ricci tensor. The action I EFF possessesall the local symmetries possessed by ˜ I D , with the appear-ance of the R µν R µν − (1 / R αα ) term being characteris-tic of a gravity theory that is locally conformal invariant(see e.g. [13, 14]). Also, we take note of the fact thatpath integration over the fermions has converted termsthat are linear in A µ and S µ in ˜ I D into terms that arequadratic in A µ and S µ in I EFF . Comparing now withEq. (15), we see that the action I EFF contains preciselythe kinetic energy term we seek. Thus not only doestorsion provide a natural origin for the second potentialneeded for X µν , up to renormalization constants it alsoprovides precisely the correct action whose variation, onadding appropriately coupled sources, leads to Eq. (11)and a derivation of Faraday’s Law via a variational prin-ciple. S µ thus serves as an analog of the electromagnetic A µ , an analog that is purely geometrical.Given the geometrical structure of S µ , we note that itis also possible to give A µ an analogous such structure.Specifically, we recall that Weyl had suggested that onecould metricate electromagnetism by introducing a B µ -dependent connection for a real field B µ of the form W λµν = − g λα ( g να B µ + g µα B ν − g νµ B α ) = W λνµ , (24)as written here with a convenient charge 2 / λµν + K λµν + W λµν inthe spin connection, as noted in [15] the B µ term dropsout of the Dirac action identically, with Weyl’s B µ notcoupling to the Dirac spinor at all. The reason for thisis that the Weyl connection generates individual non-Hermitian terms of the generic form i ( ∂ µ + B µ ) ψ , and inthe full Hermitian ˜ I D such terms must cancel identically.However, given this, suppose we instead take B µ to beanti-Hermitian and set B µ = iA µ where A µ is Hermitian.Now, not only is there now no cancellation, use of thisanti-Hermitian connection is found to precisely lead tonone other than the above ˜ I D as given in Eq. (21). Thusstarting from the torsionless I D we can derive Eq. (21)in two distinct ways. If we demand local invariance ofthe action under ψ → e iα ( x ) ψ and ψ → e iγ β ( x ) ψ , we canintroduce A µ and S µ by minimal coupling or by changingthe geometry. The two potentials needed for electromag-netism can thus be put on a completely equal footing.Now a drawback in using a B µ -dependent W λµν is thatwith it parallel transport is path dependent, with the ge-ometry being a Weyl geometry rather than a Riemannianone. However, with iA µ the geometry associated with I EFF is a regular Riemannian one that uses only the con-nections given in Eq. (16). Thus by using iA µ instead of B µ we convert a Weyl geometry into a Riemannian one. IV. THE NATURE OF TORSION
While we have seen that the axial 4-vector S µ giveselectromagnetism a chiral structure, we need to commenton the fact that experimentally there is no apparent signof S µ . Moreover, since S µ is associated with torsion itis not simply a typical spacetime axial vector field. Tounderscore the special nature of torsion, we note thateven if the standard (torsionless) Riemann tensor is zero,torsion is not obliged to vanish. Torsion could thus existin a spacetime with no Riemann curvature at all. In aspace that is flat as far as the geometry of its four space-time x µ coordinates is concerned, we note that since theMinkowski metric is independent of the x µ , a non-zerotorsion might not depend on the x µ coordinates either.Given the antisymmetry of Q λµν , we can thus envisagethat S µ , and thus concomitantly its monopole source K µ as well, might depend instead on a set of Grassmann co-ordinates, coordinates that anticommute with each other.To realize this possibility, on comparing Eqs. (5) and(6) with Eq. (22), we can consider the possibility thatthe torsion can written as Q λµν = ∇ λ A µν , where A µν is an antisymmetric rank two tensor. In [10] it was sug-gested that this A µν could be the antisymmetric part ofa 16-component metric tensor. To this end, we now notethat if we introduce a set of Grassmann vierbeins ξ aµ ,then the quantity A µν = η ab ξ aµ ξ bν will be antisymmetricsince ξ aµ ξ bν + ξ bµ ξ aν = 0. Thus we can envisage space-time being enlarged to encompass both ordinary coor- dinates and Grassmann coordinates, and spaces of thistype were constructed in e.g. [16], where it was shownthat a canonical quantization in which vanishing anti-commutators were replaced by non-vanishing ones led tothe Dirac equation. As also noted in [16], because ofthe Pauli principle finite degree of freedom Grassmanncoordinates ξ bµ (as opposed to infinite degree of freedomGrassmann fields ψ ( x )) could not be macroscopically oc-cupied. Consequently, a Grassmann torsion could onlybe microscopic, with only the sector of electromagnetismthat is based on A µ ordinarily being observable in macro-scopic systems.Now we had found in Eq. (23) that at the classicallevel the A µ and S µ sectors were decoupled from eachother. However, according to the Dirac action given inEq. (21) both sectors couple to the fermions. Thus quan-tum mechanically one could have transitions between thetwo sectors mediated by fermion loops with both vectorand axial vector insertions (axial analog of light on lightscattering). This would be a small effect, and would alsobe microscopic, with a quantized Grassmann torsion notmaking any substantial modifications to QED. Thus tor-sion, and equally magnetic monopoles, might only bemanifest microscopically, where they could potentiallycontribute to physics beyond the standard model [17].Finally, if the torsion/monopole sector is only manifestmicroscopically, then macroscopically we can set E ′ , B ′ , ρ m , and J m to zero, with the standard sourceless FaradayLaw then holding for macroscopic electrodynamics. [1] If A µ is singular (Dirac string) or has a non-trivial topol-ogy (grand unified monopoles) one could evade the van-ishing of ( − g ) − / ǫ µνστ ∇ ν ( ∇ σ A τ − ∇ τ A σ ) – see e.g.J. Preskill, Ann. Rev. Nucl. Part. Sci. , 461 (1984);K. A. Milton, Rep. Prog. Phys. , 1637 (2006). How-ever, the intent of this paper is to obtain a non-vanishing( − g ) − / ǫ µνστ F στ without singularities or non-trivialtopologies.[2] S. Shanmugadhasan, Can. Jour. Phys. , 218 (1952).[3] N. Cabibbo and E. Ferrari, Nuovo Cim. , 1147 (1962).[4] While our work follows these earlier studies, it differsin three key aspects. First, our motivation is to addressFaraday’s Law and not magnetic monopoles per se, withit being Faraday’s Law that forces us to two potentialsand not monopoles. Second, we provide an explicit ori-gin for the potential S µ as being due torsion. And third,we consider a coupling of S µ to fermions, and with it weare able to derive an action for its dynamics rather thanhaving to postulate one. Some earlier work on relatingtorsion and magnetic monopoles may be found in M. Is-raelit, Found. Phys. , 205 (1998), and R. T. Hammond,Nuovo Cim. B , 725 (1993).[5] This of course does not preclude their possible existence.[6] I. L. Shapiro, Phys. Rept. , 113 (2002).[7] L. Fabbri and P. D. Mannheim, arXiv:1405.1248 [gr-qc],May, 2014. Phys. Rev. D, in press.[8] Under a local conformal transformation the torsion trans-forms as [6] Q λµν → Q λµν + q Ω − ( x )( δ λµ ∂ ν − δ λν ∂ µ )Ω( x ) where q is its conformal weight. No matter what the valueof q , under a conformal transformation S µ transformsinto itself with all derivatives of Ω( x ) dropping out.[9] Since X µν is invariant under A µ → A µ + ∂ µ α ( x ) and S µ → S µ + ∂ µ β ( x ), X µν is recognized as the chiral gen-eralization of F µν . On allowing for these two transforma-tions, A µ and S µ together contain 6 degrees of freedom,exactly the same number as E TOT and B TOT combined.[10] Using torsion one can also develop a gravitationalanalog of Faraday’s Law. See P. D. Mannheim andJ. J. Poveromo, arXiv:1406.1470 [gr-qc], June, 2014.[11] G. ’t Hooft, arXiv:1009.0669 [gr-qc], September, 2010.[12] The Riemann and Maxwell sector terms are given in [11]and references therein. The S µν S µν term is given in [6].[13] P. D. Mannheim, Prog. Part. Nucl. Phys. , 340 (2006).[14] P. D. Mannheim, Found. Phys. , 388 (2012).[15] K. Hayashi, M. Kasuya, T. Shirafuji, Prog. Theor. Phys. , 431 (1977).[16] P. D. Mannheim, Phys. Rev. D , 898 (1985).[17] As an alternative to Grassmann coordinates one couldconstruct the antisymmetric A µν via A µν = C ab V aµ V bν ,where C ab is the charge conjugation matrix. Torsionwould then be macroscopic, but could still have escapeddetection if the bridge between the A µ and S µ sectorsis purely quantum-mechanical. Another possibility is for S µµ