aa r X i v : . [ phy s i c s . g e n - ph ] O c t Lagrangian for the Frenkel electron
Alexei A. Deriglazov ∗ Depto. de Matem´atica, ICE,Universidade Federal de Juiz de Fora, MG, BrasilandLaboratory of Mathematical Physics,Tomsk Polytechnic University, 634050 Tomsk,Lenin Ave. 30, Russian Federation
We found Lagrangian action which describes spinning particle on the base of non-Grassmannvector and involves only one auxiliary variable. It provides the right number of physical degreesof freedom and yields generalization of the Frenkel and BMT equations to the case of an arbitraryelectromagnetic field. For a particle with anomalous magnetic moment, singularity in the relativisticequations generally occurs at the speed different from the speed of light. Detailed discussion of theultra-relativistic motion is presented in the work: A. A. Deriglazov and W. G. Ramirez, World-linegeometry probed by fast spinning particle, arXiv:1409.4756.
I. INTRODUCTION
Relativistic description of rotational degrees of freedom of a body starting from the proper Lagrangian has a longhistory, see [1–4] and references therein. Since the spin operators in quantum theory satisfy the angular-momentumalgebra, closely related problem consist in establishing of variational formulation which should lead to classical equa-tions of spinning electron [6–8]. One possibility here is to assume the Frenkel spin-tensor J µν be the compositequantity J µν = 2( ω µ π ν − ω ν π µ ) formed by non-Grassmann vector ω µ and its conjugated momentum π ν [3, 5, 9, 10].Since spin should be described by two physical degrees of freedom, we need to impose some constraints on eight basicvariables ω µ and π ν . Inclusion of the constraints into a variational problem turns out to be rather nontrivial task.Though a number of vector models [5, 9, 11–13] yield Frenkel and BMT equations, they also contain extra degreesof freedom. At the classical level one can simply ignore them. However, they should be taken into account duringquantization procedure, this leads to quantum models essentially different from the Dirac electron. In the recentworks [14–17] we partially solved this task by presenting a number of equivalent Lagrangians with the right physicalsector. Free theory can be described by the Lagrangian without auxiliary variables L = − mc √− ˙ xN ˙ x + √ a √ ˙ ωN ˙ ω − g ( ω − a ) , (1)where N µν ≡ η µν − ω µ ω ν ω is projector on the plane transverse to the direction of ω µ . The free parameters a and a determine the value of spin. The corresponding relativistic quantum mechanics has been identified [15] with one-particle (positive energy) sector of the Dirac equation. The problem here is that even the minimal interaction ec A µ ˙ x µ leads to a theory with the number and algebraic structure of constraints different from those of free theory. So theinteracting theory has been constructed [16, 17] on the base of Lagrangian with four auxiliary variables g i L = 12( g g − g ) [ g ( ˙ xN ˙ x ) − g ( ˙ xN Dω ) + g ( DωN Dω )] + ec A µ ˙ x µ − g ω − a ) − g m c + g a . (2)Spin interacts with A µ through the derivative D defined in Eq. (5). This yields a generalization of Frenkel andBMT equations to the case of an arbitrary electromagnetic field [16]. In the present work we obtain more economicformulation which involves only one auxiliary variable.We work in four-dimensional Minkowski space with the metric η µν = ( − , + , + , +). For contraction of indexes weuse the notation ˙ x µ ˙ x µ = ˙ x , ˙ x µ N µν ˙ x ν = ˙ xN ˙ x , N µν ˙ x ν = ( N ˙ x ) µ , F µν J µν = ( F J ), F µα J αν = ( F J ) µν and so on. ∗ Electronic address: [email protected] The last term in (1) represents velocity-independent constraint which is well known from classical mechanics. So, we might followthe classical-mechanics prescription to exclude the auxiliary variable g from the formulation. But this would lead to lose of manifestcovariance of the formalism. II. LAGRANGIAN AND HAMILTONIAN FORMULATIONS
Consider spinning particle with mass m , electric charge e and magnetic moment µ interacting with an arbitraryelectromagnetic field F µν = ∂ µ A ν − ∂ ν A µ = ( F i = E i , F ij = ǫ ijk B k ). We study the following Poincare andreparametrization invariant Lagrangian action on configuration space with coordinates x µ ( τ ), ω µ ( τ ) and g ( τ ): S = Z dτ g (cid:20) ˙ xN ˙ x + gDωN Dω − q [ ˙ xN ˙ x + gDωN Dω ] − g ( ˙ xN Dω ) (cid:21) − g m c + α ω + ec A ˙ x. (3)This depends on one free parameter α which determines spin of the particle. We take α = ~ , this corresponds tospin one-half particle, see below. The only auxiliary variable is g , this provides the mass-shell condition (16). It hasbeen denoted N µν ≡ η µν − ω µ ω ν ω , then N µν ω ν = 0 . (4)Together with ˜ N µν ≡ ω µ ω ν ω , this forms a pair of projectors: N + ˜ N = 1, N = N , ˜ N = ˜ N , N ˜ N = 0. Thesquare-root appeared in the Lagrangian seem to be typical structure [2] for the models which imply the Frenkel-typecondition J µν P ν = 0.To introduce coupling of the position variable x with electromagnetic field we have added the minimal interaction ec A µ ˙ x µ . As for spin, it couples with A µ through the term Dω µ ≡ ˙ ω µ − g eµc F µν ω ν . (5)This is the only term we have found compatible with symmetries and constraints which should be presented in thetheory. In particular, under reparametrizations the variable g transforms as g = ∂τ ′ ∂τ g ′ . This implies homogeneoustransformation law of Dω , Dω = ∂τ ′ ∂τ D ′ ω ′ , and, at the end, reparametrization invariance of the Lagrangian. In turn,this provides the expected mass-shell condition P − eµ c ( F J ) + m c = 0, see below.Switching off the spin variables ω µ from Eq. (3), we arrive at familiar Lagrangian of spinless particle L = g ˙ x − g m c + ec A ˙ x . Integrating over the auxiliary variable g we obtain L = − mc √− ˙ x + ec A ˙ x . This is equivalent to thestandard Lagrangian of spinless particle in terms of physical variables ~x ( t ), L = − mc p c − ˙ ~x + eA + ec ~A ˙ ~x , if werestrict ourselves to the class of increasing parameterizations of the world-line. This implies positive g ( τ ). So westudy (3) under the assumptions dtdτ > g ( τ ) >
0. In the presence of spin, our Lagrangian is a complicated functionof g even in the case of free theory.Let us construct Hamiltonian formulation of the model. Conjugate momenta for x µ , ω µ and g are denoted as p µ , π µ and π g . Besides, we use the condensed notation q [ ˙ xN ˙ x + gDωN Dω ] − g ( ˙ xN Dω ) ≡ √ and P µ ≡ p µ − ec A µ .Contrary to p µ , the canonical momentum P µ is U (1) gauge-invariant quantity.Since π g = ∂L∂ ˙ g = 0, the momentum π g represents the primary constraint, π g = 0. Expressions for the remainingmomenta, p µ = ∂L∂ ˙ x µ and π µ = ∂L∂ ˙ ω µ , can be written in the form P µ = 12 g ( N ˙ x µ − K µ ) , K µ ≡ [ ˙ xN ˙ x + gDωN Dω ] ( N ˙ x ) µ − g ( ˙ xN Dω )( N Dω ) µ √ ; (6) π µ = 12 ( N Dω µ − R µ ) , R µ ≡ [ ˙ xN ˙ x + gDωN Dω ] ( N Dω ) µ −
2( ˙ xN Dω )( N ˙ x ) µ √ . (7)The functions K µ and R µ obey the following remarkable identities K = ˙ xN ˙ x, R = DωN Dω, KR = − ˙ xN Dω, ˙ xR + DωK = 0 , ˙ xK + gDωR = √ . (8)Due to Eq. (4), contractions of the momenta with ω µ vanish, that is we have the primary constraints ωπ = 0 and P ω = 0. One more primary constraint, P π = 0, is implied by (8).Hence we deal with a theory with four primary constraints. Hamiltonian has the form H = p ˙ x + π ˙ ω − L + λ i T i , (9)where λ i are the Lagrangian multipliers for the primary constraints T i . To construct manifest form of the Hamiltonian,we note the equalities P = g [ ˙ xN ˙ x − ˙ xK ] and π = [ DωN Dω − DωR ]. Then, using (8) we obtain g P + 12 π = L , (10)where L is the first line in Eq. (3). Further, using Eqs. (8) we have p ˙ x + π ˙ ω ≡ P ˙ x + ec A ˙ x + πDω + g eµc ( πF ω ) = 2 L + ec A ˙ x − g eµ c ( F J ) , (11)where the Frenkel-type spin-tensor appeared J µν = 2( ω µ π ν − ω ν π µ ) . (12)Using (11) and (10) in (9) we arrive at the Hamiltonian H = g (cid:16) P − eµ c ( F J ) + m c (cid:17) + 12 (cid:16) π − αω (cid:17) + λ ( ωπ ) + λ ( P ω ) + λ ( P π ) + λ g π g . (13)The fundamental Poisson brackets { x µ , p ν } = η µν and { ω µ , π ν } = η µν imply { x µ , P ν } = η µν , {P µ , P ν } = ec F µν , (14) { J µν , J αβ } = 2( η µα J νβ − η µβ J να − η να J µβ + η νβ J µα ) . (15)According to Eq. (15) the spin-tensor is generator of Lorentz algebra SO (1 , ωπ , ω and π are Lorentz-invariants, they have vanishing Poisson brackets with J µν . To reveal the higher-stage constraints we write theequations ˙ T i = { T i , H } = 0. The Dirac procedure stops on third stage with the following equations π g = 0 ⇒ P − eµ c ( F J ) + m c = 0 ⇒ λ C + λ D = 0 , (16)( ωπ ) = 0 ⇒ π − αω = 0 , (17)( P ω ) = 0 ⇒ λ = gCM c , (18)( P π ) = 0 ⇒ λ = − gDM c . (19)We have denoted M = m − e (2 µ + 1)4 c ( F J ) , C = − e ( µ − c ( ωF P ) + eµ c ( ω∂ )( F J ) ,D = − e ( µ − c ( πF P ) + eµ c ( π∂ )( F J ) . (20)The last equation from (16) turns out to be a consequence of (18) and (19) and can be omitted. Hence the Diracprocedure revealed two secondary constraints written in Eqs. (16) and (17), and fixed the Lagrangian multipliers λ and λ . The multipliers λ g and λ and the auxiliary variable g have not been determined. H vanishes on the completeconstraint surface, as it should be in a reparametrization-invariant theory.We summarized the algebra of Poisson brackets between constraints in the Table I. The constraints π g , T , T and T form the first-class subset, while T and T represent a pair of second class. The presence of two primary first-classconstraints π g and T is in correspondence with the fact that two lagrangian multipliers remain undetermined withinthe Dirac procedure. This also indicates on two local symmetries which must be presented in the theory. One of themis the standard reparametrization invariance. Another is the spin-plane symmetry discussed in the next section. T T T T T T = P − µe c ( F J ) + m c T = π − αω − T ≈ − T ≈ α ( ω ) T ≈ T = ωπ T ≈ − T ≈ T ≈ T = P ω C T ≈ T ≈ T − M c ≈ − M c T = P π D − α ( ω ) T ≈ − T ≈ − T + M c ≈ M c TABLE I: Algebra of constraints.
Hamiltonian (13) determines evolution of the basic variables according the following equations˙ x µ = g ( T µν P ν + Y µ ) , ˙ P µ = ec ( F ˙ x ) µ + g µe c ∂ µ ( F J ) , (21)˙ ω µ = g eµc ( F ω ) µ + g CM c P µ + π µ + λ ω µ , ˙ π µ = g eµc ( F π ) µ + g DM c P µ − α ( ω ) ω µ − λ π µ , (22)We have denoted T µν = η µν − ( µ − a ( JF ) µν , Y µ = µa J µα ∂ α ( JF ) ,a = e M c = − e m c − e (2 µ + 1)( JF ) . (23)The interaction leads to modification of the Lorentz-force equation. Only for the “classical” value of magneticmoment, µ = 1, and constant electromagnetic field the constraints (18) and (19) would be the same as in the freetheory, λ = λ = 0. Then T µν = η µν , Y µ = 0, and four-velocity becomes proportional to P µ , see (21). Contributionof anomalous magnetic moment µ = 1 to the difference between ˙ x µ and P µ is proportional to Jc ∼ ~ c , while the termwith a gradient of field is proportional to J c ∼ ~ c .All the basic variables have ambiguous evolution. x µ and P µ have one-parametric ambiguity due to g while ω and π have two-parametric ambiguity due to g and λ . The quantities x µ , P µ and the spin-tensor J µν turn out to beinvariant under spin-plane symmetry. So they can be observable quantities. Equations (21) together with˙ J µν = g eµc ( F J ) [ µν ] + 2 P [ µ ˙ x ν ] , (24)form a closed system. The remaining ambiguity due to g presented in these equations reflects the reparametrizationsymmetry of the theory.The term α ω in the Lagrangian (3) provides the constraint ω π = α = ~ . Together with ωπ = 0, this impliesfixed value of spin J µν J µν = 8( ω π − ( ωπ ) ) = 6 ~ , (25)for any solution to the equations of motion. Besides, the constraints ω P = π P = 0 imply the Pirani condition [18–20] J µν P ν = 0 . (26)The variables x , P and J have vanishing Poisson brackets with the second and third terms in (13). Hence these termsdo not contribute into equations (21) and (24), and can be omitted from Hamiltonian. Further, we can construct theDirac bracket for the second-class pair T and T , after that they also can be omitted from (13). Then the relativisticHamiltonian acquires an expected form H = g (cid:16) P − eµ c ( F J ) + m c (cid:17) . (27)The equations (21) and (24) follow from this H with use of Dirac bracket, ˙ z = { z, H } DB .The first equation from (21) together with T -constraint can be used to exclude the variables P µ and g . For g weobtain √ − g µν ˙ x µ ˙ x ν m r c , where the effective metric g µν is given by (33). So, the presence of g in Eq. (5) implies highlynon-linear interaction of spinning particle with electromagnetic field. Excluding P µ and g , we obtain closed systemof Lagrangian equations for the set x, JD g h m r ( ˜ T D g x ) µ i = ec ( F D g x ) µ + µe m r c ∂ µ ( JF ) + D g ( bac Y µ ) , (28)˙ J µν = eµ √− ˙ xg ˙ xm r c ( F J ) [ µν ] − b ( µ − m r c √− ˙ xg ˙ x ˙ x [ µ ( JF ˙ x ) ν ] + 2 ba ˙ x [ µ Y ν ] . (29)Besides, all solutions satisfy the Lagrangian analog of Pirani condition J µν [( ˜ T ˙ x ) ν − bam r c p − ˙ xg ˙ xY ν ] = 0 , (30)as well as to the value-of-spin condition J µν J µν = 6 ~ . We have denoted by˜ T µν = η µν + ( µ − b ( JF ) µν , b = 2 a µ − a ( JF ) ≡ − e m c − eµ ( JF ) , (31)the inverse matrix for T , Eq. (23). Interaction of spin with the external field yields the radiation mass m r m r = m − µe c ( JF ) − ( Y gY ) c , (32)as well as the effective metric g µν = ( ˜ T T ˜ T ) µν = [ η + b ( µ − JF + F J ) + b ( µ − F JJF ] µν ,D g = 1 √− ˙ xg ˙ x ddτ . (33)The equations (28)-(30) coincide with those obtained in [16] from the Lagrangian with four auxiliary variables. Inthe approximation O ( J, F, ∂F ) and when µ = 1 they coincide with Frenkel equations.Let us specify the equation for spin precession to the case of uniform and stationary field, supposing also µ = 1and taking physical time as the parameter, τ = t . Then (30) reduces to the Frenkel condition, J µν ˙ x ν = 0, while (29)reads ˙ J µν = e √− ˙ x m r c ( F J ) [ µν ] . We decompose spin-tensor on electric dipole moment ~D and Frenkel spin-vector ~S asfollows: J µν = ( J i = D i , J ij = 2 ǫ ijk S k ) . (34)Then ~D = − c ~S × ~v , while precession of ~S is given by d~Sdt = e √ c − ~v m r c h − ~E × ( ~v × ~S ) + c~S × ~B i . (35) III. SPIN SURFACE AND ASSOCIATED SPIN FIBER BUNDLE T . While spin-sector of our model consists of the basic variables ω µ and π µ , quantum mechanics obtained in terms ofspin-tensor J µν . The passage from ω and π to J is not a change of variables, and acquires a natural interpretation inthe geometric construction described below. Generalization of this construction on the case of SO ( k, n ) Lie-Poissonmanifold can be found in [21].In the previous section we have obtained the following constraints in spin-sector: P ω = 0 , P π = 0 , (36) ωπ = 0 , π − αω = 0 , (37)It should be noticed that the Lagrangian (2) implies ωπ = 0, π − a = 0 and ω − a = 0 instead of (37). So, theLagrangian (3) does not appear from (2) by removing the auxiliary variables g , g and g .To see the meaning of Lorentz-invariant constraints (36) and (37), we consider this surface in Lorentz frame whichimplies P µ = ( P ,~ ω = π = 0. Taking this into account, the constraints (37) determinesthe following surface in R ( ~ω, ~π ) T = { ~ω~π = 0 , ~π − α~ω = 0 } , (38)that is ~ω and ~π represent a pair of orthogonal vectors with ends attached to the hyperbole y = αx . The constraints(36) imply J µν P ν = 0. In the rest frame this gives J i = 0, that is the spin-tensor has only three components whichwe identify with non-relativistic spin-vector, J ij = 2 ǫ ijk S k . Due to the constraints (38) the spin-vector belong totwo-dimensional sphere of radius √ αJ ij J ij = 8 α, or ~S = α, so we assume α = 3 ~ . (39)We call this the spin surface. The chosen value of parameter corresponds to spin one-half particle.Hence, to describe spin in the rest frame, we have six-dimensional space of basic variables R ( ~ω, ~π ), the spin-tensorspace R ( J ij ∼ ~S ) and the map f : R → R , f : ( ~ω, ~π ) → ~S = ~ω × ~π, rank ∂ ( S i ) ∂ ( ω j , π k ) = 3 . (40)According to previous section, all trajectories ~ω ( τ ) , ~π ( τ ) lie in the manifold (38) of R . f maps the manifold T ontospin surface, f ( T ) = S .Denote F S ∈ T preimage of a point ~S ∈ S , F S = f − ( ~S ). Let ( ~ω, ~π ) ∈ F S . Then the two-dimensional manifold F S consist of the pairs ( k~ω, k ~π ), k ∈ R + , as well as those obtained by rotation of ( k~ω, k ~π ) in the plane of vectors ~ω and ~π . So elements of F S are related by two-parametric transformations ~ω ′ = ~ωk cos β + ~π k | ~ω || ~π | sin β, ~π ′ = − ~ω | ~π | k | ~ω | sin β + ~π k cos β. (41)In the result, the manifold T acquires natural structure of fiber bundle T = ( S , F , f ) with base S , standard fiber F , projection map f and structure group given by transformations (41). As local coordinates of T adjusted with thestructure of fiber bundle we can take k, β , and two coordinates of the vector ~S . By construction, the structure-grouptransformations leave inert points of base, δS i = 0.The Lorentz-invariant equations (36), (37) together with the map J µν = 2 ω [ µ π ν ] represent this construction in anarbitrary Lorentz frame. In the dynamical realization given in previous section, structure group acts independentlyat each instance of time and turn into the local symmetry. k -transformations provide reparametrization invarianceof the action (3). The spin-plane rotations β are associated with the first-class constraints T and T and selects J asthe physical (observable) variable. IV. DISCUSSION
We obtained the generalization (28) and (29) of Frenkel and BMT equations to the case of an arbitrary electro-magnetic field. They follow from the Lagrangian (3) which also yields the constraints (16), (25) and (30), providingthe right number of physical degrees of freedom. Some relevant comments are in order.The relativistic equation (35) automatically incorporates the Thomas precession [4, 23–25]. Indeed, let in instan-taneous rest frame of the particle we have F ′ µν = ( ~E ′ = const , ~B ′ = 0). Then Eq. (35) tell us that spin does notexperience a torque in the rest frame, d~S ′ dt ′ = 0. Consider a frame where the particle has velocity ~v . In this frame thefield is F µν = ( ~E, ~B = c ~v × ~E ), where ~E is determined by Lorentz boost of ~E ′ [24]. An observer in the laboratoryframe detects the Thomas precession (35). Expressing ~B through ~E , the equation (35) can be written as follows: d~Sdt = e √ c − ~v m r c ~v × ( ~S × ~E ).Classical analog of the Pauli Hamiltonian [22] contains the term ~S · ~E × ~v + c~S · ~B , while the relativistic theory(27) implies c J µν F µν = ~S · ~E × ~v + c~S · ~B . Both Hamiltonians are written in a laboratory system. The difference isthe famous one-half factor. Our analysis clearly shows the origin of this discrepancy on the classical level: we dealwith two different sets of variables. Our variables obey noncommutative Dirac brackets while in the Pauli theorythe brackets supposed to be canonical. To compare the Hamiltonians, we need manifest form of (time-dependent)canonical transformation among the two formulations. Probably, the projection operator method for diagonalizationof Dirac brackets [26–28] could be used to this aim.Even for uniform fields, behavior of our spinning particle with anomalous magnetic moment ( µ = 1) differs fromthat of Frenkel and BMT. This is due to two structural modifications implied by the Lagrangian which provides thenecessary constraints . First, velocity is not proportional to the canonical momentum, see Eq. (21). Second, ininteracting theory we necessarily have the Pirani condition J µν P ν = 0 on the place of Frenkel condition J µν ˙ x ν = 0.In the Lagrangian formulation this leads to the equation h ˜ T ˙ x √− ˙ xg ˙ x i ˙ = f , which has the structure different from that ofFrenkel and BMT, h ˙ x √− ˙ x ˙ x i ˙ = f . This results in extra contribution to the standard expression for the Lorentz force,¨ x ∼ F ˙ x + O ( J ). So the complete theory implies an extra spin-orbit interaction as compared with the approximateFrenkel and BMT equations. For instance, BMT electron in a constant magnetic field moves around a circle on theplane orthogonal to the field. For our particle, the circular motion is perturbed by slow oscillations along the magneticfield [16].Frenkel condition implies ~D = 0 in the rest frame, that is zero electric dipole moment. In contrast, the Piranicondition (30) predicts small non-vanishing electric dipole moment ~D ∼ ~S × ( ~S × ~E ).As it should be in a Lorentz-invariant theory, the speed of light c represents the invariant scale in our model: ifone observer concludes that a particle has the speed c , all other inertial observers will make the same conclusion. Atthe same time, when µ = 1 our equations of motion necessarily involve the factor √− ˙ xg ˙ x instead of the standardrelativistic-contraction factor √− ˙ x . Computing the acceleration implied by (28)-(30), we obtain ~a ∼ √− ˙ xg ˙ x ~f with ~f being non-singular function as ˙ xg ˙ x →
0. So the factor determines critical speed ~v cr which the spinning particle cannot overcome during its evolution in external field. The critical speed is determined as a solution to ˙ xg ˙ x = 0. Thissurface is slightly different from the sphere c − ~v = 0. Indeed, we compute − ( ˙ xg ˙ x ) = c − ~v + 4 b ( µ − (cid:2) π ( ωF ˙ x ) + ω ( πF ˙ x ) (cid:3) . (42)As π and ω are space-like vectors, the last term is non-negative, so | ~v cr | ≥ c . Let us confirm that this term not alwaysvanishes as | ~v | = c , that is critical velocity could be different from c . Assume the contrary, that the last term in (42)vanishes, then ωF ˙ x = − ω ( ~E~v ) + ( ~ω, c ~E + ~v × ~B ) = 0 ,πF ˙ x = − π ( ~E~v ) + ( ~π, c ~E + ~v × ~B ) = 0 . (43)This implies (see the notation (12) and (34)) c ( ~D ~E ) + ( ~D, ~v × ~B ) = 0. Consider the case ~B = 0, then it shouldbe ( ~D ~E ) = 0. On other hand, for the homogeneous field the quantity J µν F µν = 2 h ( ~D ~E ) + 2( ~S ~B ) i = 2( ~D ~E ) is aconstant of motion [16]. Let us take the initial conditions for spin such, that ( ~D ~E ) = 0. Then critical speed of ourparticle in this field will be different from the speed of light. Similar conclusion has been made by Hanson and Reggewith respect to their relativistic spherical top [2].Detailed discussion of the ultra-relativistic motion is presented in [29] Comparing with Frenkel, our formulation fixes the value of spin.
V. ACKNOWLEDGMENTS
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