Noname manuscript No. (will be inserted by the editor) On Magnetic Forces and Work Jacob A. Barandes Received: date / Accepted: date Abstract We address a long-standing debate over whether classical magnetic forces can do work, ultimately answering the question in the affirmative. In detail, we couple a classical particle with intrinsic spin and elementary dipole moments to the electromagnetic field, derive the appropriate generalization of the Lorentz force law, show that the particle's dipole moments must be collinear with its spin axis, and argue that the magnetic field does mechanical work on the particle's elementary magnetic dipole moment. As consistency checks, we calculate the overall system's energy-momentum and angular momentum, and show that their local conservation equations lead to the same force law and therefore the same conclusions about magnetic forces and work. We also compute the system's Belinfante-Rosenfeld energy-momentum tensor. Keywords Gauge theory * Particle physics * Spin * Electromagnetism * Relativistic multipole moments * Conservation laws 1 Introduction Textbook treatments and research articles on classical electromagnetism, such as [10,9], often suggest that magnetic fields cannot do mechanical work. On the other hand, everyday examples of bar magnets lifting other bar magnets would seem to suggest otherwise. In this paper, we show that there exists a classical way to understand how magnetic fields can indeed do work.1 We start in Section 2 with a review of the kinematics of classical relativistic point particles with intrinsic spin and permanent, elementary dipole moments. In Section 3, we couple a particle of this kind to the electromagnetic field and derive its dynamics, showing, in particular, that magnetic forces can classically J.A. Barandes (ORCID: 0000-0002-3740-4418) Jefferson Physical Laboratory, Harvard University, 17 Oxford Street, Cambridge, MA 02138 E-mail: barandes@physics.harvard.edu 1 For a more extensive treatment of the results in this paper, see [1]. 2 Jacob A. Barandes do work on the particle via its elementary magnetic dipole moment. We also show as a matter of self-consistency that the particle's elementary dipole moments must be collinear with the particle's intrinsic spin. In Section 4, we derive expressions for the overall system's energy-momentum and angular momentum, and show that their associated conservation laws lead to the same equations of motion as before, thereby providing further confirmation that magnetic fields can do work on a particle with elementary dipole moments. We conclude with one more new result by calculating the system's BelinfanteRosenfeld energy-momentum tensor. 2 The Kinematics of a Relativistic Elementary Dipole To start, we will need a relativistic description of the kinematics of a classical particle with intrinsic spin. 2.1 The Phase Space for a Relativistic Massive Particle with Spin Following [2,3,12], we model the particle's kinematics using spacetime coordinates Xμ = (c T,X)μ, energy E, four-momentum pμ = (E/c,p)μ, positive inertial mass m > 0, and antisymmetric spin tensor Sμν by identifying the particle's phase space as a transitive or "irreducible" group action of the orthochronous Poincaré group. The states in this phase space take the form (X, p, S) and are each obtained from the reference state (0, (mc,0), S0) by an appropriate Poincaré transformation (a, Λ) ∈ R4 nO(1, 3) according to (X, p, S) = (a, Λ(mc,0), ΛS0Λ T). (1) Here the coordinates Xμ = aμ and the variable Lorentz-transformation matrix Λμν are treated as the particle's fundamental phase-space variables, with the condition that ΛTηΛ = η = diag(−1,+1,+1,+1). 2.2 Charge and Elementary Dipole Moments We can couple the particle to the electromagnetic field by assigning the particle an electric-monopole charge q and an antisymmetric elementary dipole tensor mμν , so that the particle is an elementary dipole. We note that elementary dipoles of this kind are neither of the Ampère model, which consist of loops of moving electric monopoles, nor of the Gilbert model, which consist of pairs of hypothetical magnetic monopoles. In particular, the elementary dipoles that we examine here represent a classical extension of Maxwell's original theory of electromagnetism, as Maxwell's theory includes dipoles only of the Ampère type.2 2 We thank David Griffiths for pointing out this defining feature of Maxwell's original theory. On Magnetic Forces and Work 3 We let uμ ≡ dXμ/dλ denote the particle's four-velocity and γ ≡ u0/c denote the particle's associated Lorentz factor, where uμ is not generically normalized to u2 = −c2 unless the worldline parameter λ is taken to be the particle's proper time τ . The particle's four-velocity then takes the form uμ = (γc, γv)μ. (2) The particle has four-dimensional electric-monopole current density jνe (x, t) = (ρe(x, t)c, Je(x, t)) ν = quν 1 γ δ3(x−X) (3) and elementary-dipole density Mμν = mμν 1 γ δ3(x−X), (4) with overall current density jν(x, t) = jνe (x, t) + ∂μM μν(x, t), (5) where (1/γ)δ3(x −X) is the Lorentz-invariant form of the three-dimensional Dirac delta function. It follows immediately from (3) that the particle's electric-monopole density ρe = j t e/c, its electric-monopole current density Je = (j x e , j y e , j z e ), and its threevelocity v ≡ dX/dt satisfy the basic relationship Je = ρev. (6) We emphasize that no such relationship holds for the particle's elementary dipole moments, which, again, are not assumed to arise from any underlying motion of electric monopoles. As in [9], by introducing suitable four-vectors πμ and μμ and antisymmetric tensors πμν ≡ 1 mc (pμπν − pνπμ), (7) μμν ≡ 1 mc εμνρσpρμσ, (8) we can write the particle's elementary dipole tensor in terms of an electric part πμν and a magnetic part μμν as mμν = πμν + μμν , (9) or, equivalently, as mμν ≡  0 cπx cπy cπz −cπx 0 −μz μy −cπy μz 0 −μx −cπz −μy μx 0  μν . (10) 4 Jacob A. Barandes Here εμνρσ is the four-dimensional Levi-Civita symbol (with εtxyz ≡ +1), and πν(λ) and μμ(λ) are related to their reference values πμ0 ≡ (0,π0)μ and μμ0 ≡ (0,μ0)μ and the particle's variable Lorentz-transformation matrix Λμν(λ) according to πμ(λ) ≡ Λμν(λ)πν0 , (11) μμ(λ) ≡ Λμν(λ)μν0 . (12) 3 The Dynamics of a Relativistic Elementary Dipole Next, we turn to a discussion of the particle's dynamics. 3.1 The Action Functional for a Relativistic Massive Particle with Spin In the absence of external interactions, as shown in [2,3,12], we can encode the dynamics of a particle with intrinsic spin in terms of the manifestly covariant action functional Sparticle[X,Λ] = ∫ dλ 1 2 Jμν θ μν = ∫ dλ ( pμẊ μ + 1 2 Tr[SΛΛ−1] ) , (13) where λ is a smooth and monotonic but otherwise arbitrary parameter along the particle's worldline, Jμν = Lμν + Sμν is the particle's total angularmomentum tensor, Lμν ≡ Xμpν − Xνpμ is its orbital angular-momentum tensor, θμν is an antisymmetric tensor of boost and angular degrees of freedom, and we ignore irrelevant boundary terms. Consistency of the particle's dynamics with the required invariance of the quantities p2 ≡ −m2c2 and s2 ≡ (1/2)SμνSμν requires the auxiliary phase-space condition pμS μν = 0. (14) 3.2 The Particle's Equations of Motion Our next step will be to couple the particle to the electromagnetic field and obtain the particle's equations of motion, from which we will be able to infer the appropriate generalization of the Lorentz force law. Given the charge and elementary dipole moments outlined above, the overall action functional for the elementary dipole and the electromagnetic field is given by On Magnetic Forces and Work 5 S[X,Λ,A] ≡ Sparticle[X,Λ] + Sfield[A] + Sint[X,Λ,A] = ∫ dλ ( pμẊ μ + 1 2 Tr[SΛΛ−1] ) (Sparticle) + ∫ dt ∫ d3x ( − 1 4μ0 FμνFμν ) (Sfield) + ∫ dt ∫ d3x jνAν (Sint), (15) where Fμν ≡ ∂μAν − ∂νAμ is the usual Faraday tensor, jν = jνe + ∂μMμν is the particle's overall current density, and the interaction term in the final line ensures that extremizing the action functional with respect to the electromagnetic gauge field Aμ yields the Maxwell equations in their usual form. The first line in this action functional (Sparticle) is fixed by group theory, the second line (Sfield) defines the vacuum in the pure Maxwell theory, and the third line (Sint) provides the canonical coupling between the particle and the electromagnetic field in a manner consistent with the Maxwell equations and the particle's features as laid out in the previous section. After an integration by parts, we can write the interaction term in the final line as Sint[X,Λ,A] = ∫ dt ∫ d3x ( jνeAν − 1 2 MμνFμν ) . (16) Collecting together all the terms that involve the particle's degrees of freedom, we obtain Sparticle+int[X,Λ,A] = ∫ dλ ( pμẊ μ + 1 2 Tr[SΛΛ−1] ) + ∫ dt ∫ d3x jνeAν + ∫ dt ∫ d3x ( − 1 2 ) MμνFμν , (17) which we can further reduce to the form Sparticle+int[X,Λ,A] = ∫ dλLparticle+int, (18) for a manifestly covariant Lagrangian defined by Lparticle+int ≡ pμẊμ + 1 2 Tr[SΛΛ−1] + qẊνAν − 1 2c √ −Ẋ2mμνFμν . (19) It follows from a straightforward calculation that the particle's equations of motion, expressed in terms of the particle's proper time τ , are then 6 Jacob A. Barandes dp dτ μ = −quνF νμ − 1 2 mρσ∂μFρσ − 1 2c2 d dτ (uμmρσFρσ) = −quνF νμ − 1 2 mρσ(ημν + uμuν)∂νFρσ − 1 2c2 d dτ (uμmρσ)Fρσ, (20) as obtained in [6,12,8], and dSμν dτ = −(uμpν − uνpμ)− (mμρF νρ −mνρFμρ), (21) which generalizes the results of [4,6,12]. 3.3 The Non-Relativistic Limit with Time-Independent External Fields In the non-relativistic limit and ignoring self-field effects-so that we can replace the overall electric and magnetic field with the external fields Eext and Bext-the equations of motion (20)–(21) reduce to dE dt ≈ v * (qEext +∇(π *Eext + μ *Bext)), (22) dp dt ≈ q(Eext + v ×Bext) +∇(π *Eext + μ *Bext), (23) dJ dt ≈ X× dp dt + π ×Eext + μ×Bext, (24) where the particle's four-momentum in this limit is pμ = (E/c,p)μ ≈ (mc2 + (1/2)mv2,p)μ, (25) and the particle's overall angular-momentum pseudovector J is made up of orbital and spin contributions according to J ≡ L + S = (Lyz, Lzx, Lxy) + (Syz, Szx, Sxy). (26) The dynamical equation (23) tells us that the electromagnetic force on the particle is F = qEext + qv ×Bext +∇(π *Eext) +∇(μ *Bext). (27) We observe that the usual Lorentz force law, qEext +qv×Bext, is enhanced in the presence of the particle's elementary dipole moments by the appearance of two additional dipole terms, ∇(π *Eext) +∇(μ *Bext), in which the magnetic field appears on an equal footing with the electric field. Accordingly, the On Magnetic Forces and Work 7 magnetic field contributes to the work done by the external electromagnetic field, W ≡ ∫ dX * F: W = ∫ B A dt (qv *Eext) +∆(π *Eext) +∆(μ *Bext). (28) Moreover the rate at which work is done is in agreement with the dynamical equation (22). We have reached the key conclusion of this paper-namely, that magnetic forces can do work on classical particles with elementary dipole moments.3 We next turn to a detailed treatment of self-consistency conditions on the particle's dynamics, as well as obtain the necessary formulas for determining the particle's four-velocity uμ in the presence of a nonzero electromagnetic field. Later on, we will analyze electromagnetic forces and work done on the particle from the standpoint of local conservation laws. 3.4 Implications of Self-Consistency Taking a derivative of the phase-space condition pμS μν from (14) yields the self-consistency requirement dpμ dτ Sμν + pμ dSμν dτ = 0, which entails that the particle's four-momentum pμ and its four-velocity uμ = dXμ/dτ (now normalized to u2 = −c2) are related by pμ = meffu μ + bμ. (29) Here meff, which plays the role of an effective mass, is defined by meff ≡ − m2c2 p * u , (30) and the four-vector bμ, which is orthogonal to the particle's four-momentum, b * p = 0, is given by bμ ≡ 1 p * u ( dpν dτ Sνμ − pν(mνρFμρ −mμρF νρ) ) . (31) As in [12], we regard (29) as an implicit formula for the particle's four-velocity uμ. This formula ensures, in particular, that the particle's four-momentum pμ has constant norm-squared p2 = −m2c2. 3 We thank Sebastiano Covone for suggesting the relevance of these results to the Bohr-van Leeuwen theorem [5,11]. The Bohr-van Leeuwen theorem assumes the original Lorentz force law without contributions from elementary dipole moments, and asserts that a non-rotating system of particles, when treated classically, always has a vanishing average magnetization at thermal equilibrium. The theorem's implication is that phenomena like diamagnetism can only be understood in terms of quantum effects, a view challenged by our results, at least in principle. 8 Jacob A. Barandes For vanishing field, Fμν = 0, the relationship (29) reduces to the familiar equation pμ = muμ, as expected. On the other hand, when the electromagnetic field is nonzero, Fμν 6= 0, (29) has the form pμ = muμ + (terms of order 1/c2). (32) This relation ensures that there is no ambiguity over whether we should identify the particle's energy E as ptc or utmc2 for the purposes of quantifying the work done by the field on the particle in the non-relativistic regime. Invoking the spin tensor's equation of motion (21), together with the phasespace condition (14), pμS μν = 0, and the constancy of the particle's spinsquared scalar s2 ≡ (1/2)SμνSμν , we find d dτ (s2) = d dτ ( 1 2 SμνS μν ) = (Sρμm μσ − Sσμmμρ)Fρσ = 0, (33) which yields the condition Sρμm μσ = Sσμm μρ. (34) In the particle's reference state, this equality produces the relations π0 × S0 = 0, μ0 × S0 = 0, } (35) which dictate that the particle's elementary electric and magnetic dipole moments must be collinear with the particle's spin pseudovector S0: π0 = 1 c Ξ S0, μ0 = Γ S0.  (36) Here Ξ is a pseudoscalar and Γ is the particle's scalar gyromagnetic ratio. We can understand these relationships physically as telling us that if the particle's elementary-dipole vectors were not collinear with the particle's spin axis, then torques exerted on the particle by the electromagnetic field would cause the particle's overall spin to speed up or slow down, in violation of the constancy of s2. 4 Conservation Laws For completeness, we verify that the equations of motion (20)–(21) also follow from local conservation of energy-momentum and angular momentum. To On Magnetic Forces and Work 9 begin, we recall the relevant version of Noether's theorem, which states that if a system's dynamics has a continuous symmetry, qα 7→ q′α = qα + δεqα, δεqα = ∑ b gqα,bεb, (37) where the quantities εb parameterize the symmetry and the quantities gqα,b characterize its precise form, then we have the following conservation law: Qb ≡ ∑ α ∂L ∂qα gqα,b − fb, dQ dt = 0. (38) Here Qb are a set of conserved quantities, L is the system's Lagrangian, qα are its degrees of freedom, and the functions fb are related to the change in the Lagrangian according to L 7→ L+ δεL, δεL = d dt (∑ b fbεb ) = ∑ b dfb dt εb. (39) 4.1 Local Conservation of Energy-Momentum In order to employ Noether's theorem to obtain the overall system's energymomentum tensor, we examine the behavior of the system under a translation in spacetime by an infinitesimal four-vector εμ. The particle's phase-space variables transform as Xμ(λ) 7→ X ′μ(λ) ≡ Xμ(λ) + εμ, Λμν(λ) 7→ Λ ′μ ν(λ) ≡ Λμν(λ), } (40) and the electromagnetic gauge potential transforms as Aμ(x) 7→ A′μ(x) ≡ Aμ(x− ε) = Aμ(x)− ∂νAμ(x)εν . (41) By an application of Noether's theorem to the particle's manifestly covariant Lagrangian L ≡ Lparticle+int defined by (18) and the Lagrangian density L for the overall system defined in terms of the action functional S[X,Λ,A] ≡ 10 Jacob A. Barandes∫ dt ∫ d3xL from (15), one finds that the overall system's conserved fourmomentum is expressible as Pν = ∂L ∂Ẋρ gXρ,ν + ∫ d3x (−nμ) ∂L ∂(c∂μAρ) gAρ,ν − fν = pν + qAν + 1 2c2 uνm στFστ + 1 c ∫ d3x (−nμ) ( Hμρ∂νAρ − δμν ( 1 4μ0 F ρσFρσ )) = 1 c ∫ d3x (−nμ)Tμcan,ν , (42) where nμ ≡ (−1,0)μ is a unit timelike vector orthogonal to the three-dimensional spatial hypersurface of integration. In this expression, the overall system's canonical energy-momentum tensor is given by Tμνcan = T μν can,particle + T μν can,field, (43) with the contributions from the particle and the field given respectively by4 Tμνcan,particle ≡ u μpν 1 γ δ3(x−X) (44) and Tμνcan,field ≡ H μρF νρ − ημν 1 4μ0 F 2 + 1 2c2 uμuνmρσFρσ 1 γ δ3(x−X) + ∂ρ(H μρAν). (45) Here Hμν is the auxiliary Faraday tensor: Hμν ≡ 1 μ0 Fμν +Mμν = 1 μ0 Fμν +mμν 1 γ δ3(x−X). (46) The last term in (45) is a total spacetime divergence with vanishing divergence ∂μ∂ρ(H μρAν) = 0 on its μ index, and its temporal component ∂ρ(H tρAν) has vanishing integral over three-dimensional space under the assumption that the fields go to zero sufficiently rapidly at spatial infinity. We emphasize that in our approach, all the terms in the overall system's canonical energy-momentum tensor follow from the systematic application of Noether's theorem to the relevant action functionals. 4 The authors of [9] decompose the overall energy-momentum tensor by including the interaction terms with the energy-momentum tensor for the particle, an approach that obscures the work being done by the electromagnetic field on the particle. On Magnetic Forces and Work 11 We can integrate the local conservation law ∂μT μν can = 0 over three-dimensional space to compute the time derivative of the particle's four-momentum pν : dpν dt = 1 c d dt ∫ d3xT tνcan,particle = −1 c d dt ∫ d3xT tνcan,field = ∫ d3x ( − ∂μ ( HμρF νρ − ημν 1 4μ0 F 2 )) − 1 2c2 d dt (uνmρσFρσ) = −quμFμν +mρμ∂μF νρ − 1 2c2 d dτ (uνmρσFρσ). After invoking the electromagnetic Bianchi identity ∂μF νρ+∂ρFμν+∂νF ρμ = 0, we obtain the equation of motion (20). Our formulas above for the overall system's canonical energy-momentum tensor are new results. By replicating the particle's equation of motion (20), they provide further support for the key claim of this paper-that magnetic forces can classically do work on particles with elementary dipole moments. 4.2 Local Conservation of Angular Momentum Next, we use Noether's theorem to examine the overall system's angular momentum and its local conservation. Under an infinitesimal Lorentz transformation Λinf = 1 + i 2 ερσσρσ, (47) the particle's phase-space variables transform as Xμ(λ) 7→ X ′μ(λ) ≡ (ΛinfX(λ))μ = Xμ(λ) + i 2 ερσ[σρσ] μ νX ν(λ), Λμν(λ) 7→ Λ ′μ ν(λ) ≡ (ΛinfΛ(λ))μν = Λμν(λ) + i 2 ερσ[σρσ] μ λΛ λ ν(λ).  (48) The second of these two transformation laws is equivalent to the following transformation rule for the particle's Lorentz parameters θμν(λ): θμν(λ) 7→ θ′μν(λ) ≡ θμν(λ) + εμν . (49) Meanwhile, the gauge field Aμ(x) transforms as Aμ(x) 7→ A′μ(x) ≡ (A(Λ−1infx)Λ −1 inf )μ ≡ Aλ((1− (i/2)ερσσρσ)x)(δλμ − (i/2)ερσ[σρσ]λμ) = Aμ(x)− ∂νAμ(x)(i/2)ερσ[σρσ]νλxλ −Aλ(x)(i/2)ερσ[σρσ]λμ. (50) 12 Jacob A. Barandes Noether's theorem (38) then yields the system's overall angular-momentum tensor, up to an overall minus sign: − Jνρ = ∂L ∂Ẋα gXα,νρ + 1 2 ∂L ∂θαβ gθαβ ,νρ + ∫ d3x (−nμ) ∂L ∂(c∂μAα) gAα,νρ − fνρ = − ( pα + qAα − 1 2 (−uα/c2)mσλFσλ ) (Xνδ α ρ −Xρδαν ) − Sνρ − 1 c ∫ d3x (−nμ) ( Hμα − δμσ ( 1 4μ0 F 2 )) × ∂σAα(xνδσρ − xρδσν ) − 1 c ∫ d3x (−nμ)(HμνAρ −HμρAν) = − ∫ d3x (−nμ)J μcan,νρ. (51) Here we have identified the system's canonical angular-momentum flux tensor as J μνρcan = Lμνρ + Sμνρ, (52) with orbital contribution Lμνρ ≡ xν 1 c Tμρcan − xρ 1 c Tμνcan (53) and spin contribution Sμνρ = 1 c uμSνρ 1 γ δ3(x−X) + 1 c (HμνAρ −HμρAν). (54) We naturally read off the spin flux tensors for the particle and the field respectively as Sμνρparticle = 1 c uμSνρ 1 γ δ3(x−X), (55) Sμνρfield = 1 c (HμνAρ −HμρAν). (56) Integrating the local conservation law ∂μJ μνρcan = 0 over three-dimensional space and taking advantage of the local conservation ∂μT μρ can = 0 of the overall canonical energy-momentum tensor Tμρcan, we can compute the time derivative On Magnetic Forces and Work 13 of the particle's spin tensor as follows: dSνρ dt = d dt ∫ d3xStνρparticle = − d dt ∫ d3x 1 c (xνT tρcan − xρT tνcan +HtνAρ −HtρAν) = − ∫ d3x ∂μ(x νTμρcan − xρTμνcan +HμνAρ −HμρAν) = − 1 γ (uνpρ − uρpν)− 1 γ (mνσF ρσ −mρσF νσ). We therefore see that local conservation of angular momentum yields the equation of motion (21). 4.3 The Belinfante-Rosenfeld Energy-Momentum Tensor The overall system's canonical energy-momentum tensor (43) is not symmetric on its two indices, a feature that is required of the energy-momentum tensor that locally sources the gravitational field in general relativity. To conclude this paper, we follow the standard Belinfante-Rosenfeld construction5 to construct a properly symmetric energy-momentum tensor, which will likewise represent a new result. We start by introducing a new tensor Bμρν ≡ c 2 (Sμνρ + Sνμρ + Sρμν) = −HμρAν + 1 2 (uμSνρ + uνSμρ + uρSμν) 1 γ δ3(x−X). (57) We then obtain a symmetric, locally conserved energy-momentum tensor Tμν for the overall system from the relation Tμν = Tμνcan + ∂ρBμρν :6 Tμν = 1 2 (uμpν + uνpμ) 1 γ δ3(x−X) + 1 2 HμρF νρ + 1 2 HνρFμρ − ημν 1 4μ0 F ρσFρσ + 1 2c2 uμuνmρσFρσ 1 γ δ3(x−X) + 1 2 ∂ρ(Sμνρparticle + S νμρ particle). 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