Spin Gauge Symmetry in the Action Principle for Classical Relativistic Particles

Spin gauge symmetry in the action principle for classical relativistic particles

Jan Steinhoff1, 2, ∗ 1Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany, EU 2Centro Multidisciplinar de Astrof´ısica — CENTRA, Departamento de F´ısica, Instituto Superior Técnico — IST, Universidade de Lisboa — ULisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal, EU (Dated: January 21, 2015) We suggest that the physically irrelevant choice of a representative worldline of a relativistic spinning particle should correspond to a gauge symmetry in an action approach. Using a canonical formalism in special relativity, we identify a (first-class) spin gauge constraint, which generates a shift of the worldline together with the corresponding transformation of the spin on phase space. An action principle is formulated for which a minimal coupling to fields is straightforward. The electromagnetic interaction of a monopole-dipole particle is constructed explicitly.

I. INTRODUCTION spin-tensor Sµν , where the spatial components Sij con- tain the usual spin (or flux dipole) and the components i0 Relativistic spinning particles are an important topic S give the mass dipole. The latter is non-zero if and in both classical and quantum physics. All experimen- only if the observed center of mass differs from the refer- tally verified elementary particles, except the Higgs bo- ence point. i0 son, are spinning. In the classical regime, spinning par- Actually, in special relativity, S is in one-to-one cor- ticles in relativity are a seminal topic, too, see [1,2] for respondence with the choice of a reference point. This reviews. But we are going to argue in this paper that, allows one to represent the choice of center through a µν as of now, their formulation through a classical action condition on S , called spin supplementary condition principle remains incomplete. (SSC). For instance, the physical meaning of the Fokker- ν An interesting, but sometimes problematic, feature of Tulczyjew condition Sµν p = 0 [3,4] is that one observes i0 spinning objects in relativity is that their center of mass a vanishing mass dipole S = 0 in the rest-frame of the ν depends on the observer. This is vividly illustrated in object, where its momentum p is parallel to the time di- Fig.1. On the other hand, the definition of the angular rection. In other words, the chosen center agrees with the momentum of the object, or spin, hinges on the loca- center of mass observed in the rest-frame. Because phys- tion of the center as the reference point. Hence, both ically the dynamics of the object must be independent µ the three-dimensional center and the three-dimensional of the reference worldline z (σ) we chose to represent its spin of an object depend on the observer. However, both motion, this choice is a gauge choice and the SSC can are combined to form an antisymmetric four-dimensional be understood as a gauge fixing. However, we will see that, in the context of an action formulation for spinning particles, there is a twist in this story. In general relativ- fast and heavy ity, the correspondence between the choice of center and the SSC was rigorously demonstrated only for the special ν case Sµν p = 0 [5,6], but intuitively one should expect that it holds more generally, too. vi ∆zi To date, the prototype construction for an action and canonical formalism of classical spinning particles in special relativity is given in a seminal paper of Hanson and spin Regge [2]. (For earlier work on the action see [7,8] and for the canonical formalism see [9]). However, this con- slow and light ν struction essentially covers the choice Sµν p = 0 only,

arXiv:1501.04951v1 [gr-qc] 20 Jan 2015 which has the advantage of being a covariant condition. FIG. 1. If a spinning spherical symmetric object moves with Then the covariance of the theory is manifest. But fol- i a velocity v to the left, then its upper hemisphere moves lowing the idea that the choice of a reference worldline faster with respect to the observer than its lower hemisphere. can be seen as an arbitrary gauge choice, one would in- Hence, the upper hemisphere possesses a larger relativistic mass than the lower one and the object acquires a mass dipole stead expect that the action possesses a corresponding Si0 = m∆zi. See, e.g., [1]. gauge symmetry, so that any SSC can be used equiva- lently at the level of the action. It is the main purpose of the present paper to work out a recipe for the construction of such a gauge-invariant action in special relativity, and in this sense completes the work in [2]. ∗ jan.steinhoff@aei.mpg.de; http://jan-steinhoff.de/physics/ It is important to notice that Ref. [2] is the basis of cur- 2 rent methods for the computation of general relativistic plies that particles are characterized by a mass shell and spin effects in compact binaries [10–20]. Predictions for an irreducible representation of the little group, which these spin effects are of great importance for future grav- carries a spin quantum number. An analogous group- itational wave astronomy. For instance, some black holes theoretical approach can be used in the classical context are known to spin rapidly [21, 22] and certain spin orien- [31]. tations lead to an increased gravitational wave luminosity For classical particles, we follow [2] and represent the [23]. This makes it likely that spin effects are relevant for configuration space of the particle by a position zµ and a µ the first detectable sources. If spinning compact objects Lorentz matrix ΛA , with obvious transformation prop- are modeled through an effective field theory approach erties under the Poincarégroup. Here the index A labels (EFT) [10, 16, 24], then it is vital to implement all ex- the basis of the body-fixed frame. Though this is bor- pected symmetries in the effective action. However, the rowing terminology from a rigid body, we insist that the gauge symmetry related to the choice of a representative constructed action can serve as an effective description of worldline was not considered so far. generic spinning bodies. The generators of the Poincaré It is noteworthy that classical spinning particles are group are the linear momentum pµ and the total angular based on a spin-1 representation in the present paper, momentum Jµν = 2z[µpν] + Sµν , so the Poisson brackets while Fermions transform under fractional spin represen- must read tations. But still, an action for classical spinning parti- µ µ α β α cles can also be seen as an effective theory for fermion {z , pν } = δν , {ΛA ,Sµν } = 2ΛA ηβ[µδν], (1) fields in certain limits. The clearest way to understand {S ,S } = S η − S η + S η − S η , the classical limit is through a Foldy-Wouthuysen trans- αβ µν αµ νβ αν µβ βν µα βµ να formation [25, 26]. This is a unitary transformation of all other zero, where η is the Minkowski metric. the spinors, which removes the Zitterbewegung. This µν unitary representation corresponds to the choice of the We consider the case of a massive particle here. Then we can define a standard boost which transforms the time Pryce-Newton-Wigner center [27–29] for the representa- µ tive point in the classical theory. Then the commutators, direction of the body-fixed frame Λ0 into the direction or Poisson brackets, of the three-dimensional spin and of the linear momentum pµ, position are the (rather simple) standard canonical ones. pµΛ pωµω We adopt the conventions from [2]. µ µ 0ν ν L ν = δ ν + 2 − ρ , (2) p pρω

√ µ where p := pµp and for later convenience we have II. SPINNING PARTICLES ν ν ν introduced the time-like vector ω := p /p + Λ0 . When applied to the Lorentz matrix, we obtain a new matrix µ µ ν Before we discuss classical spinning particles, it is use- Λ˜ A = L ν ΛA , or explicitly ful to recapitulate the definition of particles in quantum µ µ theory. One-particle states are defined by the property of ˜ µ p ˜ µ µ ν pν ω Λ0 = , Λi = Λi − Λi ρ . (3) transforming under an irreducible representation of the p pρω Poincarégroup [30]. That is, a particle is still the same one if it is transformed to a different position, orienta- µ We notice that Λ˜ 0 is redundant, since it is given by pµ. tion, or speed by the Poincarégroup and the irreducibil- The independent physical degrees of freedom are con- ity guarantees that it can not be separated into parts, µ tained in Λ˜ i , which carries an SO(3) index and hence so it is indeed just one particle. The linear momentum transforms under the vector (spin-1) representation of of the particle pµ follows as the eigenvalue of the trans- the little group. This is intuitively clear, since the state lation operator. Since pµ transforms as a vector under of motion is already characterized by the linear momen- Lorentz transformations, it can not label the irreducible µ 2 tum and the temporal component of the body-fixed frame representations, but the scalar pµp = M can. Now, must be redundant. The physical information of the all vectors on such a mass shell are connected by Lorentz body-fixed frame is the orientation of the object, which transformations, so one can define a standard Lorentz should be associated with three-dimensional rotations. transformation L(p) which brings p to some standard µ Because Λ µ contains irrelevant, or gauge, degrees of form. A generic Lorentz transformation of the particle A freedom, its conjugate S must be subject to a con- state can then be decomposed into the standard Lorentz µν straint. This constraint is usually also the generator of transformation L(p) and an element of the so called little the gauge symmetry [32], which must leave the physical group. The latter leaves the standard form of p invari- µ degrees of freedom Λ˜ µ invariant. From the projector-like ant. For instance, for massive particles (M2 > 0) one i structure of Eq. (3) and the fact that the spin generates usually chooses the standard form of p to point to the µ Lorentz transformations of Λ µ, we may guess that the time direction (p is boosted to the rest frame) and the i µ generator is given by S ων . Indeed, we find little group is the rotation group SO(3) (Wigner rota- µν tion), since it leaves the time direction invariant. Finally, ν ˜ ρ this simple group-theoretical definition of particles im- {Sµν ω , Λi } = 0, (4) 3

µ µ 1 µ µ µ µν ν ˙ Aµ Aµ where we made use of ω pµ/p = ωµΛ0 = 2 ωµω and u =z ˙ ,Ω = ΛA Λ , and ˙ = d/dσ. Since Λ µ µ Λ˜ i wµ = Λ˜ i pµ. We have discovered the spin gauge con- is a Lorentz matrix, it can only be varied by an in- straint finitesimal Lorentz transformation δθµν = −δθνµ, i.e., Aµ Aν µ δΛ = Λ δθν . ν p ν ν Notice that the dynamical mass M can be a general 0 ≈ Cµ := Sµν + Λ0 ≡ Sµν ω , (5) p function of the dynamical variables. Then the mass M contains all the interaction energies. It must be adapted where weak equality [33] (restriction to the constraint such that the point-particle provides an effective descrip- surface) is denoted by ≈. It is further illustrated below tion for some extended body on macroscopic scale (reduc- that this constraint physically makes sense. This spin ing the almost innumerable internal degrees of freedom gauge constraint is not an SSC (in the usual sense) and of the body to the relevant ones). The dynamics is then does not correspond to a choice for the worldline, but encoded through M and the action principle. This makes it parametrizes the possible SSCs through a gauge field the dynamical mass M analogous to a thermodynamic ν Λ0 , see below for discussion. potential, like the internal energy. It should be noted ν We notice that Ref. [2] was missing the Λ0 in this con- that EFTs are of comparable importance for both parti- ν straint. One might object that the appearance of Λ0 cle and statistical physics. The analogy of M to a ther- is breaking the SO(1,3) Lorentz invariance in its first modynamic potential suggests to apply a construction of index. However, this index belongs to the body-fixed M using symmetries and power counting arguments, as frame, whose time-like component is not an observable. usual in an EFT. For black holes, M as a function of ν Indeed, we already argued that Λ0 is a physically irrel- the dynamical variables is related to the famous laws of evant gauge degree of freedom. Furthermore, also from black hole dynamics [36], see also [37] for discussion. an EFT point of view, only SO(3) rotation invariance is We require here that the constraints in HD are related a symmetry that must be respected for the body-fixed to gauge symmetries. Then the Lagrange multipliers are frame [24, 34], instead of SO(1,3) Lorentz invariance. not fixed by requiring that the constraints are preserved We must have another constraint related to a gauge in time and represent the gauge freedom. This implies symmetry here, namely that of reparametrization invari- that the Poisson brackets between all pairs of constraints ance of the worldline parameter σ. The associated con- vanish weakly here. Constraints with this property are straint is the mass shell one [2] called first class [33]. See Ref. [32] for further discus- 0 ≈ H := p pµ − M2, (6) sion of gauge symmetry in constrained Hamiltonian dy- µ namics. In contrast to our requirement that all three which together with (5) forms the basis for an action independent components of the spin constraints are first principle. class, only one component of the spin constraint used in [2] is first class. This renders the completion of the set of constraints in [2] rather unsatisfactory, because only III. ACTION FOR SPINNING PARTICLES first class constraints require a completion through gauge fixing constraints. We are going to construct an action principle based on a Hamiltonian. The Dirac Hamiltonian HD [32, 33, 35] is the canonical Hamiltonian plus the (primary) con- IV. SPIN GAUGE CONSTRAINT straints, which are added with the help of Lagrange multipliers. However, due to reparametrization invariance, The first main objective of the present paper is to es- the canonical Hamiltonian vanishes, so that HD is just tablish Cµ as the proposed spin gauge constraint. We composed of the constraints, just argued that it should be a first class constraint. Furthermore, it should be the generator of a spin gauge λ µ HD = H + χ Cµ, (7) transformation, i.e., a shift of the representative world- 2 line ∆zµ(σ) together with the appropriate change in the where λ and χµ are the Lagrange multipliers. spin, An action principle for spinning point-particles (PP) ∆S ≈ 2p ∆z , (9) must reproduce the equations of motion of HD with Pois- µν [µ ν] son brackets from Eq. (1). The relation between action and Poisson brackets is discussed in AppendixA. The This is in agreement with the physical picture in Fig.1, µ appropriate action reads [2] which refers to the rest-frame where p = (m, 0). These are the physical requirements we have on Cµ. Z µ 1 µν These requirements are indeed met for Eq. (5). It is SPP = dσ −pµu − Sµν Ω − HD , (8) 2 easy to see that Cµ is first-class among itself, where the variables with independent variations are zµ, 2 Aµ {Cµ, Cν } = p[µCν] ≈ 0, (10) pµ,Λ , and Sµν and we introduced the abbreviations p 4 so it qualifies for a gauge constraint. Then the generator and the second one was applied more recently only [11, µ of an infinitesimal spin gauge transformation is Cµ and 13, 40, 41] in slightly different forms and contexts. The the transformations of the fundamental variables read differences arise in the way the (normalized) time vector µ δ0 is generalized to curved spacetime. However, it should µ α µ 1 µν α ∆z := { Cα, z } = P Sνα , (11) be noted that [11, 13, 41] suggest to complete the set of p µ µ constraints by Λ0 ∝ p , while here in the context of spin µ α µ ∆p := { Cα, p } = 0, (12) gauge symmetry this would lead to the first SSC, but not µ µ α µ [µ ν] to the second one. The SSC and the condition on Λ0 ∆ΛA := { Cα, ΛA } = 2 ω ΛAν , (13) α can not be chosen independently here. ∆Sµν := { Cα,Sµν } = 2p[µ∆zν] − 2[µCν], (14) Any of the above conditions can be added to the set µν µ µ where P is the projector onto the spatial hypersurface of constraints, e.g., 0 ≈ Λ0 − δ0 . This constraint then of the rest-frame, turns the spin gauge constraint into a second class constraint, which means that the set of constraints can be pµpν P µν := ηµν − . (15) eliminated using the Dirac bracket [33] (and one can solve p2 for the Lagrange multipliers). This is the usual manner in which gauge fixing is handled in the context of con- µ Notice that ∆z is a spatial vector in the rest-frame of strained Hamiltonian dynamics. µ the particle, pµ∆z = 0. This makes it obvious that the symmetry group is three-dimensional. However, one can alternatively insert the solution to Now, on the constraint surface, Eq. (14) is identical to the set of constraints into the action in a classical context. our second and final requirement in Eq. (9). That is, it is For the case of the Pryce-Newton-Wigner spin gauge, it µ µ 0 0 precisely the amount that an angular momentum changes holds Λ0 = δ0 and ΛA = δA, so the temporal compo- if the reference point is moved by ∆zµ. This shows that nents drop out of the spin kinematic term in the action, µ Cµ indeed generates a shift of the reference worldline within the object and that Cµ is the corresponding gauge 1 µν 1 ij ij ki ˙ kj constraint. Sµν Ω = SijΩ , Ω = −Λ Λ . (19) 2 2 While the physical meaning of ∆Sµν is immediately µ clear, an interpretation of ∆ΛA deserves a more detailed illustration. We recall that the physically relevant com- The kinematic term still has the same form as in Eq. (8), µ ponents of ΛA are obtained by a finite boost to the rest but the indices are 3-dimensional now and the SO(1,3) µ frame, see Eq. (2). Therefore, ∆ΛA should be given by Lorentz matrix was reduced to a SO(3) rotation ma- a standard boost to the rest-frame followed by another trix. Therefore, the standard so(1,3) Lie algebra Pois- standard boost to a frame infinitesimally close to ΛAµ. son bracket for the spin, Eq. (1), must be replaced by a It is straightforward to check using the standard boost, standard so(3) algebra for the spatial components of the Eq. (2), that this is the case. spin. However, for other gauge choices, the kinematic term will not simplify this drastically and the reduced Poisson bracket algebra will in general be more com- V. SPIN GAUGE FIXING plicated. This makes the Pryce-Newton-Wigner gauge probably the most useful one if one aims at a reduc- As usual, a gauge fixing now requires a gauge condition of variables, while the Fokker-Tulczyjew one leads µ tion, that is, a condition on the gauge field Λ0 . No- to manifestly covariant equations of motion. It should tice that the spin gauge constraint (5) does not corre- be emphasized that all gauges lead to equivalent equa- spond to a choice for a representative worldline, because tions of motion by construction here, if the mass shell µ it contains the unspecified gauge field Λ0 . The following constraint H is spin gauge invariant. choices turn the spin gauge constraint (5) into familiar Note that we obtain the same set of constraints as [2] choices for the SSC: if we choose the first gauge. However, adding a gauge µ symmetry to the theory should not be seen as adding µ p ν Λ0 ≈ ⇒ Cµ = Sµν p ≈ 0, (16) unnecessary complications. On the contrary, different p µ µ ν ν gauges are useful for different applications. For instance, Λ0 ≈ δ0 ⇒ Cµ = Sµν (p + pδ0 ) ≈ 0, (17) in Ref. [2] a transformation from Fokker-Tulczyjew to 2p0δµ − pµ Pryce-Newton-Wigner variables is considered because it Λ µ ≈ 0 ⇒ C = S ≈ 0. (18) 0 p µ µ0 simplifies the reduced Poisson brackets. Here one can directly use the second gauge fixing condition instead and The first one is due to Fokker [3], the second due to the simplification of brackets is explained by Eq. (19). Pryce, Newton, and Wigner [27–29], and the last one due This is an important consequence of our construction: to Pryce and Møller [28, 38]. In general relativity, the Different SSCs are manifestly equivalent, in particular first condition was first considered by W. M. Tulczyjew the covariant SSC is equivalent to noncovariant ones by [4], the third one by Corinaldesi and Papapetrou [39], construction. 5

VI. GAUGE INVARIANT VARIABLES straightforward to generalize the current construction to the general relativistic case, where the dynamical mass Now we turn to the second main objective of the must be coordinate invariant. present paper, which is the construction of the mass shell constraint H such that it is invariant under spin gauge However, because the mass shell constraint must be transformations. This is equivalent to written entirely in terms of the positionz ˜µ, it is sugges- tive to shift the worldine of the action to this position. {Cµ, H} ≈ 0. (20) It should be noted that one can not switch to the invariant spin and Lorentz matrix as fundamental variables, The usual way to construct invariant quantities is by because the transformation involves projections (notice µ combining objects with are invariant. That is, we aim Λ˜ i pµ = 0). This leads to an action containing a time to find a position, spin, and Lorentz matrix which have derivative of the momentum, weakly vanishing Poisson bracket with Cµ. We already µ encountered the invariant Lorentz matrix Λ˜ i given by µ Eq. (3). Recalling that Λ˜ i was obtained by a boost to the rest frame, we may guess that the following projec- Z p˙ p 1 S = dσ −p u˜µ − Sµν µ ν − S Ωµν − H . tions to the rest-frame variables, PP µ p2 2 µν D p (26) µ µ µν ν ˜ α β µ z˜ := z + S , Sµν := Pµ Pν Sαβ (21) The Poisson brackets involvingz ˜ will not be standard p2 canonical, but they can be readily obtained from Eq. (21) have the desired properties. It is indeed straightforward and the old Poisson brackets. Note that one can associate to show that these variables with a tilde have weakly Poisson brackets to an action if it contains at most first vanishing Poisson brackets with Cµ, order time derivatives (and no pathologies arise). See the AppendixA. The equations of motion are still first order, 1 which is important because otherwise more initial values {C , z˜µ} = − [δµp − δµp ]Cν ≈ 0, (22) α p2 α ν ν α would be needed. {C , S˜ } = −2P P βC ≈ 0, (23) α µν α[µ ν] β Now it is now straightforward to couple the spinning and the linear momentum is already invariant, see particle to the gravitational field, namely by replacing Eq. (12). Therefore, if the mass shell constraint depends ordinary derivatives with respect to σ in Eq. (26) by co- on these variables only, then it is invariant under spin variant ones (minimal coupling). Additionally, nonmin- gauge transformation, imal couplings can be added in the sense of an effective theory via the dynamical mass M, as long as these are µ µ µ µ M = M(˜z , p , Λ˜ i , S˜µν ) ⇒ {Cµ, H} ≈ 0. (24) evaluated at the position of the new worldlinez ˜ and µ constructed using the matter variables Λ˜ i , S˜µν , and pµ. The most simple model is given by Further development of the general relativistic case and an application to the post-Newtonian approximation is 2 2 µ 2 M = f(S˜ ) in H = pµp − M , (25) given in [42]. In the following section, we illustrate that this construction is also convenient for coupling to other ˜2 1 ˜ ˜µν and S = 2 Sµν S . Here f is an almost arbitrary func- fields, like the electromagnetic one, and see that non- tion, commonly referred to as a Regge trajectory [2]. We minimal couplings represent the multipoles of the body require it to be analytic and nonconstant (for f = const [12, 43, 44]. it follows Ωµν = 0 for any spin). This function encodes the rotational kinetic energy and the moment of inertia The time derivative of the momentum in our action is of the body. For black holes, it is related to the laws of similar to the acceleration term introduced in [45]. (It black hole mechanics. also appeared in a similar time+space decomposed form in [46, 47].) After a coupling to gravity (or the electromagnetic field), one can approximately remove this term, VII. SIMPLIFIED INVARIANT ACTION if this is desired, using a manifestly covariant shift of the worldline as introduced in [48]. This approximately cor- The construction of the last section has a problematic responds to inserting the equation of motion forp ˙µ into aspect. All fields interactions, entering through the dy- the action [49]. This transformation leads to the nonmin- namical mass M, must be taken at the positionz ˜µ, which imal coupling proportional to the Fokker-Tulczyjew SSC in general is different from the worldline coordinate zµ. used in [12]. However, in both Refs. [12, 45] this term This is particularly a problem for coupling to the grav- was introduced in order to preserve the covariant SSC, itational field, because first of all the tangent spaces at while here it arises from the requirement of spin gauge z˜µ and zµ are different and second the difference between symmetry. This distinction is significant in the context the positions is not a tangent vector. Therefore, it is not of an EFT. 6

VIII. ELECTROMAGNETIC INTERACTION from the right hand sides of the equations of motion. The equation of motion of a dynamical variable is then A minimal coupling to the electromagnetic field with expressed in terms of derivatives of HD with respect to µ other dynamical variables only, and the prefactor is the charge q can be introduced as usual by adding −qA˜µu˜ to the Lagrangian, see, e.g., Eq. (6.1) in [50, 51], mutual Poisson bracket. Explicit expressions for all Pois- son brackets are given in AppendixA. These Poisson Z µ ˜ µ brackets are similar to Eq. (5.29) in [2]. However, for SPP = dσ − pµu˜ − qAµu˜ most applications it should be sufficient to have an action principle and the equations of motion in the form p˙ p 1 µν µ ν µν given above. But it is good to know that the equations − S 2 − Sµν Ω − HD , (27) p 2 of motion follow a symplectic flow and that the Poisson brackets can be obtained explicitly if needed. An analo- where A˜ = A (˜zν ). This new term turns into a total µ µ gous calculation should be possible for the gravitational time derivative under electromagnetic gauge transforma- interaction. tions. The added term is also manifestly invariant under The finite-size and internal structure of the particle spin gauge transformations, because it only involves in- is modelled by nonminimal couplings in the dynamical variant quantities. Here it is convenient that we shifted mass M. These are composed of the (electromagnetic the worldline to the spin-gauge-invariant positionz ˜µ. gauge invariant) Faraday tensor F and its derivatives, We are going to derive the equations of motion belong- µν where an increasing number of derivatives corresponds to ing to the action (27) and explicitly construct the Poisson smaller length scales or higher multipoles. As an illus- brackets associated to it. The δp -variation leads to the µ tration for the treatment of electromagnetic multipoles, velocity-momentum relation we consider the simplest case of a dipole, which corre- p˙ p ∂H sponds to a nonminimal coupling to the Faraday tensor u˜µ = 2S˜µν ν + S˙ µν ν − D , (28) p2 p2 ∂p Fµν in the dynamical mass. For a spin-induced dipole, µ this reads and from the δS -variation it follows gq µν M2 = f(S˜2) + F˜ S˜µν , (35) 2 µν 2p[µp˙ν] ∂H µν D (29) where g is the gyromagnetic ratio. For this model, our Ω = 2 − 2 . p ∂Sµν equations of motion are in agreement with [2]. Making µ µ Finally, the δz˜µ-variation leads to the gauge choice Λ0 = p /p and requiring that it is preserved in time by above equations of motions, we find µ ν ∂HD that χ = 0 for the Lagrange multiplier of the spin gauge p˙µ = −qF˜µν u˜ + , (30) ∂zˆµ constraint in this case. A contraction of (28) with pµ then µ 2 leads to λ =u ˜ pµ/p for the Lagrange multiplier of the ˜ ˜ ˜ with the Faraday tensor Fµν := Aµ,ν − Aν,µ, and the mass shell constraint, which is related to the parametriza- δΛAµ-variation gives tion of the worldline. Then we find structural agreement with (4.29), (4.33), and (4.34) in [2] and therefore also ∂H S˙ = −2Ωρ S + 2Λ D , (31) with the Bargmann-Michel-Telegdi equations [52]. The µν [µ ν]ρ A[µ ν] ∂ΛA normalization of the dipole interaction differs compared We eliminate the time derivatives on the right hand side to [2], which is due to the fact that in [2] the dipole is proportional to the angular velocity, while here it is pro- of Eq. (28), µν portional to the spin. In [52] the SSC S uµ = 0 [53–55] q u˜µ = − S˜µν F˜ u˜α + U µ, (32) is used, which fails to uniquely define a worldline and in p2 να general leads to helical motion [54–56]. Finally, we notice that a dipole linearly induced by an external field µ ∂HD [µ ν] pν ∂HD U := − − 4δα S β 2 is modelled by couplings of the form shown in Eq. (1) ∂pµ p ∂Sαβ (33) of Ref. [57], which can be added to the dynamical mass ˜µν µ[α ν] S ∂HD η p ∂HD here. + 2 ν + 2 2 ΛAα ν , p ∂zˆ p ∂ΛA If desired, one can shift thep ˙µ-term in the action (27) µ to higher orders in the derivative of the electromagnetic which can be solved foru ˜ using Eq. (A.4) and (A.7) in field through redefinitions of the variables [49]. That Ref. [2], is, through successive redefinitions, one can turn the " # p˙µ-term into nonminimal interaction terms of increasing qS˜µν F˜ u˜µ = δµ − να U α, (34) derivative order. Consistent with neglecting finite-size α 2 1 ˜ρδ ˜ p − 2 qS Fρδ effects of a certain multipolar order in M, one can ter- minate this process at the desired order and thep ˙µ-term µν where we used S˜µν ∗S˜ = 0 (∗ denotes the Hodge dual). is effectively removed. The physical reason for the addi- Using this relation, all time derivatives can be removed tional nonminimal interactions is that a shift of position 7

−1 of a monopole q generates an infinite series of higher mul- one can directly read off Mab . For the case of electro- tipoles [48]. magnetically interacting spinning particles (27), we have µ µ ν u˜ = G ν U where q 1 IX. CONCLUSIONS Gµ = δµ − S˜µαF˜ ,P 2 = p2 − qS˜µν F˜ . (A6) ν ν P 2 αν 2 µν The spin gauge symmetry formulated in the present This leads to the Poisson brackets involvingz ˜µ paper is an important ingredient to extend the EFT for charged particles to spinning charged particles (see, e.g., S˜αµ S˜µν {zˆµ, z˜ν } = Gν = − , (A7) [58]). This should serve as a classical EFT for massive α p2 P 2 fermions. Massless particles have a different little group, ν ν {pµ, z˜ } = −G µ, (A8) so our approach need several adjustments for this case. 2 We identified spin gauge invariant variables in the present µ ν ν[α µ] {ΛA , z˜ } = 2 G p ΛAα, (A9) paper, which should be useful for a matching of the EFT. p The general relativistic case is analogous to the electro- α 4 α [β ρ] {Sµν , z˜ } = − G βδ S ν]pρ, (A10) magnetic one and is invaluable for modelling the motion p2 [µ of black holes and neutron stars. This will lead to better gravitational wave forms needed for the data analysis where we used Eq. (A.5) in [2], requirements of future gravitational wave astronomy. S˜αµ S˜µν S˜αν Gν = − = −Gµ . (A11) α p2 P 2 α p2 ACKNOWLEDGMENTS Similarly, from considering the equation of motion for pµ,

α We acknowledge inspiring discussions with Gerhard {pµ, pν } = qF˜ναG µ, (A12) Sch¨afer and Michele Levi. A part of this work 2q was supported by FCT (Portugal) through grants {Λ µ, p } = − F˜ Gβ[αpµ]Λ , (A13) A ν p2 νβ Aα SFRH/BI/52132/2013 and PCOFUND-GA-2009-246542 4q (cofunded by Marie Curie Actions). {S , p } = F˜ Gδ δ[βSρ] p , (A14) µν α p2 αδ β [µ ν] ρ and from the remaining equations Appendix A: Poisson brackets from the action 4q {Λ µ, Λ ν } = − Λ p[ρF˜ν] Gβ[αpµ]Λ , (A15) Consider an action containing at most ﬁrst order A B p4 Bρ β Aα derivatives in time. We can write it in the form α β α {Sµν , ΛA } = −2ΛA ηβ[µδν] Z 8q [β (A16) S = dt B (qb)q ˙a − H(qb) , (A1) + Λ p[δF˜α] Gσ δ Sρ] p , a p4 Aδ σ β [µ ν] ρ

{Sµν ,Sαβ} = −4S η where a, b label the dynamical variables q . The equa- α][µ ν][β a 16q tions of motion read + Sγ δδ p F˜ Gσ δ[χSρ] p . p4 [α β] [δ γ]σ χ [µ ν] ρ b (A17) Mabq˙ = ∂aH,Mab := ∂aBb − ∂bBa. (A2)

a The Poisson brackets are rather complicated. For most where ∂a = ∂/∂q , or applications, it is therefore better to work directly with a ab ab −1 the action (27) if possible. q˙ = M ∂bH,M := Mab (A3)

This can be written using Poisson brackets Appendix B: Another check against Ref. [2] q˙a = {H, qa}, (A4) The term involvingp ˙µ in Eq. (27) can be removed by if we set variable redeﬁnitions, which will then be canonical variables because the action assumes a canonical form. Here ba {X,Y } = M ∂aX∂bY. (A5) we only intend to make a connection to the results in [2]. µ µ Therefore we apply the gauge ﬁxing Λ0 = p /p, or the µ Instead of computing Mab and its inverse directly from SSC Sµν p = 0. The term containingp ˙µ can be cancelled the action, it is often easier to obtain the equations of mo- from the action by shifting back to the position zµ. Now tion and transform them to the form of Eq. (A3). Then new contributions arise from the minimal coupling term. 8

However, at the level of the action, we can neglect terms momentum µ of quadratic or higher order in the SSC Sµν p . Then one can νρ pρ can absorb all additional terms by deﬁning the canonical p = pµ + qAµ + qFµν S . (B1) µ p2

This can be solved for pµ using the identities in Appendix A of [2] and one ﬁnds agreement of (25) with (5.18) in can [2] (therein it holds πµ = pµ − qAµ).

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