Noname manuscript No. (will be inserted by the editor) Gauge Invariance for Classical Massless Particles with Spin Jacob A. Barandes Received: date / Accepted: date Abstract Wigner's quantum-mechanical classification of particle-types in terms of irreducible representations of the Poincaré group has a classical analogue, which we extend in this paper. We study the compactness properties of the resulting phase spaces at fixed energy, and show that in order for a classical massless particle to be physically sensible, its phase space must feature a classical-particle counterpart of electromagnetic gauge invariance. By examining the connection between massless and massive particles in the massless limit, we also derive a classical-particle version of the Higgs mechanism. Keywords Gauge theory * Particle physics * Spin * Classical field theory * Representation theory * Higgs mechanism 1 Introduction The ingredients of classical physics are usually simpler to visualize and understand than those of quantum theory. It is therefore worthwhile to investigate which seemingly quantum phenomena turn out to have classical realizations, if only to provide the kind of intuition that can lead to discoveries. As an important example, intrinsic spin is often regarded as fundamentally quantum in nature, but there exists a fully classical description of relativistic point particles with arbitrary masses and fixed spin. With the eventual goal of describing and extending this framework,1 we begin in Section 2 by suitably generalizing the usual Lagrangian formulation of classical physics to a more expressly Lorentz-covariant form. In Section 3, we review the classification of particle-types in terms of transitive group actions of the Poincaré group, expanding on earlier work [1,8,5] and paralleling Wigner's classification [11] of 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 comprehensive treatment of the results in this paper, see [2]. 2 Jacob A. Barandes quantum particle-types in terms of irreducible Hilbert-space representations of the Poincaré group. We will be most interested in the massless case, for which we present new results that include the emergence of a classical-particle form of electromagnetic gauge invariance. In Section 4, we revisit this appearance of gauge invariance from the perspective of the massive case in the massless limit, along the way deriving a classical-particle version of the Higgs mechanism, another novel result. 2 The Manifestly Covariant Lagrangian Formulation Consider a classical system with time parameter t, degrees of freedom qα, Lagrangian L, and action functional S[q] ≡ ∫ dtL(q, q, t), (1) where dots here denote derivatives with respect to the time t. Before we apply this framework to classical relativistic point particles, we will find it useful to recast these ingredients in a form that is more manifestly compatible with relativistic invariance. To do so, we begin by replacing t with an arbitrary smooth, monotonic parameter λ. Letting dots now denote derivatives with respect to λ, we can rewrite the action functional in the reparametrization-invariant form2 S[q, t] ≡ ∫ dλL (q, q, t, ṫ), (2) where L (q, q, t, ṫ) ≡ ṫ L(q, q/ṫ, t). (3) We introduce a raised/lowered-index notation according to qt ≡ c t, qt ≡ −c t, qα ≡ qα, pt ≡ H/c, pt ≡ −H/c, pα ≡ pα. (4) where pα are the system's usual canonical momenta, H is the system's usual Hamiltonian derived from the original Lagrangian L in (1), and c is a constant with units of energy divided by momentum. The quantities pt and pα are then expressible in terms of the function (3) as pt = ∂L ∂qt , pα = ∂L ∂qα , (5) 2 For an early example of this technique, see [4]. For a more modern, pedagogical treatment, see [3]. Gauge Invariance for Classical Massless Particles with Spin 3 and one can show that the Euler-Lagrange equations take the symmetriclooking form ṗt = ∂L ∂qt , ṗα = ∂L ∂qα . (6) Moreover, the action functional (2) now takes a form that resembles a Lorentzcovariant dot product involving a square matrix η ≡ diag(−1, 1, . . . ) that naturally generalizes the Minkowski metric tensor from special relativity, S[q] = ∫ dλ ( ptq t + ∑ α pαq α ) = ∫ dλ ( pt pα ) η ( qt qα ) , (7) despite the fact that the degrees of freedom qα are not assumed at this point to have anything to do with physical space. The action functional is then invariant under transformations( qt qα ) 7→ Λ ( qt qα ) , ( pt pα ) 7→ Λ ( pt pα ) (8) for square matrices Λ satisfying the condition ΛTηΛ = η. Thus, this reparametrization-invariant Lagrangian formulation motivates the introduction of phase-space variables qt, qα, pt, pα that transform covariantly under a generalized notion of Lorentz transformations. We therefore refer to this framework as the manifestly covariant Lagrangian formulation of our classical system's dynamics. 3 Transitive Group Actions of the Poincaré Group Wigner showed in [11] that classifying the different Hilbert spaces that provide irreducible representations of the Poincaré group yields a systematic categorization of quantum-mechanical particle-types into massive, massless, and tachyonic cases.3 As shown in various treatments, such as [1,8,5], there exists a classical analogue of this construction, one version of which we review here. Toward the end of this section and in the next section, we will present fundamental new results concerning previously unexamined features of the massless case. 3.1 Kinematics We start by laying out a formulation of the kinematics of a system that we will eventually identify as a classical relativistic particle. Given a classical system described by a manifestly covariant Lagrangian formulation, we say that its phase space provides a transitive or "irreducible" group action of the Poincaré group if we can reach every state (q, p) in the system's phase space by starting from an arbitrary choice of reference state (q0, p0) and acting with an appropriate Poincaré transformation (a, Λ) ∈ R1,3o 3 See [10] for a pedagogical review. 4 Jacob A. Barandes O(1, 3), where aμ is a four-vector that parametrizes translations in spacetime and Λμν is a Lorentz-transformation matrix. The Poincaré group singles out systems whose phase spaces consist of spacetime coordinates Xμ ≡ (c T,X)μ ≡ (c T,X, Y, Z)μ (9) and corresponding canonical four-momentum components pμ ≡ ∂L ∂Ẋμ ≡ (E/c,p)μ, (10) where we identify H ≡ E as the system's energy. We will see that such a system formalizes the notion of a classical relativistic particle. To be as general as possible, we allow the system to have an intrinsic spin represented by an antisymmetric spin tensor, Sμν = −Sνμ, (11) in terms of which we can define a proper three-vector S and a three-dimensional pseudovector S according to Sμν ≡  0 Sx Sy Sz −Sx 0 Sz −Sy −Sy −Sz 0 Sx −Sz Sy −Sx 0  μν . (12) Hence, the system's phase space consists of states that we can denote by (X, p, S) and that, by definition, behave under Poincaré transformations (a, Λ) according to (X, p, S) 7→ (ΛX + a, Λp, ΛSΛT). (13) Taking our reference state to be (0, p0, S0) (14) for convenient choices of pμ0 and S μν 0 that will be made on a case-by-case basis later, we can therefore write each state of our system as (X, p, S) ≡ (a, Λp0, ΛS0ΛT), (15) so aμ and Λμν effectively become the system's fundamental phase-space variables. To keep our notation simple, we will refer to aμ as Xμ in our work ahead, keeping in mind that these variables are independent of the Lorentz-transformation matrix Λμν . We will therefore express the functional dependence of the system's manifestly covariant action functional as S[X,Λ]. It is natural to introduce several derived tensors from the system's fundamental variablesXμ, pμ, Sμν . The system's orbital angular-momentum tensor is defined by Lμν ≡ Xμpν −Xνpμ = −Lνμ, (16) Gauge Invariance for Classical Massless Particles with Spin 5 and Lμν together with Sμν make up the system's total angular-momentum tensor: Jμν ≡ Lμν + Sμν = −Jνμ. (17) Defining the four-dimensional Levi-Civita symbol by εμνρσ ≡  +1 for μνρσ an even permutation of txyz, −1 for μνρσ an odd permutation of txyz, 0 otherwise = −εμνρσ, (18) the system's Pauli-Lubanski pseudovector is Wμ ≡ −1 2 εμνρσpνSρσ = (p * S, (E/c)S− p× S)μ. (19) The following quantities are then invariant under proper, orthochronous Poincaré transformations, and therefore represent fixed features (or Casimir invariants) of the system's phase space: −m2c2 ≡ pμpμ, (20) w2 ≡WμWμ, (21) s2 ≡ 1 2 SμνS μν = S2 − S2, (22) s2 ≡ 1 8 εμνρσS μνSρσ = S * S. (23) In the analogous quantum case, the third of these invariant quantities, the spin-squared scalar s2, would be quantized in increments of ~ (or, more precisely, ~2). In our classical context, we are essentially working in the limit of large quantum numbers, in which the correspondence principle holds and these quantities are free to take on fixed values from a continuous set of real numbers. Note, in particular, that the invariance of s2 is entirely separate from issues of quantization, just as the invariance of m2 does not require quantization. 3.2 Dynamics We now turn to the system's dynamics. In the absence of intrinsic spin, Sμν = 0, the system's manifestly covariant action functional is, from (7), given by Sno spin[X,Λ] = ∫ dλ pμẊ μ = ∫ dλ (Λp0)μẊ μ. (24) We will eventually need to establish a definite relationship between the system's four-momentum pμ and its four-velocity Ẋμ ≡ dXμ/dλ. 6 Jacob A. Barandes First, however, we will extend the action functional (24) to include intrinsic spin. We begin by introducing the standard Lorentz generators: [σμν ] α β = −iδαμηνβ + iημβδαν . (25) Using the composition property of Lorentz transformations applied to the case of infinitesimal shifts λ 7→ λ+ dλ in the parameter λ, Λ(λ+ dλ) = Λ(dλ)Λ(λ) = (1− (i/2)dθμν(λ)σμν)Λ(λ), (26) where dθμν is an antisymmetric tensor of infinitesimal Lorentz boosts and angular displacements, we have Λ(λ) ≡ Λ(λ+ dλ)− Λ(λ) dλ = − i 2 θμν(λ)σμνΛ(λ). (27) Invoking the following trace identity satisfied by antisymmetric tensors Aμν = −Aνμ, 1 2 Tr[σμνA] = iAμν , (28) we can express the rates of change θμν(λ) according to θμν(λ) = i 2 Tr[σμνΛ(λ)Λ−1(λ)]. (29) By an integration by parts, we can then recast the action functional (24) (up to an irrelevant boundary term) as Sno spin[X,Λ] = ∫ dλ 1 2 Lμν θ μν . (30) With the alternative form (30) of the action functional in hand, we can straightforwardly introduce intrinsic spin into the system's dynamics by making the replacement Lμν 7→ Jμν ≡ Lμν + Sμν . Converting the term involving Lμν back into the form (24), we thereby obtain the new action functional S[X,Λ] = ∫ dλL = ∫ dλ ( pμẊ μ + 1 2 Tr[SΛΛ−1] ) , (31) which now properly accounts for intrinsic spin. The equations of motion derived from this action functional are ṗμ = 0, (32) Jμν = 0, (33) and respectively express conservation of four-momentum and conservation of total angular momentum, in keeping with Noether's theorem and the symmetries Gauge Invariance for Classical Massless Particles with Spin 7 of the dynamics under Poincaré transformations. It follows that the PauliLubanski pseudovector (19) is conserved, Ẇμ = 0, and that the scalar quantities −m2c2 and w2 defined in (20)–(21) are guaranteed to be constant, as required. As shown in [7], constancy of the spin-squared scalar s2 defined in (22) requires the imposition of an important Poincaré-invariant condition on the system's phase space. To see why, we make use of the equation of motion (33) to compute the rate of change of s2: d dλ ( 1 2 SμνS μν ) = Sμν Ṡ μν = 2ẊνpμSμν = 0. Keep in mind that without a definite relationship between the four-momentum pμ and the four-velocity Ẋμ, this condition is nontrivial. Because it establishes a constraint on all solution trajectories in the particle's phase space, we conclude that the following Lorentz-invariant condition must hold:4 pμS μν = 0. (34) Combined with the system's equations of motion (32)–(33), this condition yields a pair of basic relationships between the system's four-momentum pμ and its otherwise-unfixed four-velocity Ẋμ: p * Ẋ = ±mc2 √ −Ẋ2/c2, (35) m √ −Ẋ2/c2 pμ = ∓m2Ẋμ. (36) The equations (32)–(36) complete our specification of the system's dynamics. 3.3 Classification of the Transitive Group Actions Specializing to the orthochronous Poincaré group, classifying the different systems whose phase spaces give transitive group actions is a straightforward exercise that parallels Wigner's approach in [11]. As derived in detail in [2], one finds that each such system can describe a massive particle m2 > 0 or a massless particle m2 = 0 with either positive energy E = ptc > 0 or negative energy E = ptc < 0, or a tachyon m2 < 0, or the vacuum pμ = 0. Furthermore, the relations (35)–(36) imply that for each of these cases, the four-momentum is parallel to the four-velocity, pμ ∝ Ẋμ. It then follows immediately from the equations of motion (32) and (33) that Lμν and Sμν are separately conserved. For a massive particle, we can take the reference state (14) to describe the particle at rest, with reference four-momentum pμ0 = (mc,0) μ. (37) 4 This condition is a classical-particle analogue of the Lorenz equation ∂μAμ = 0 that appears both in the Proca theory of a massive spin-one bosonic field and as the Lorenzgauge condition in electromagnetism. As in those field theories, the condition (34) serves to eliminate unphysical spin states. 8 Jacob A. Barandes The condition (34) then eliminates unphysical spin degrees of freedom and implies that the particle's spin tensor (12) reduces to the three-dimensional spin pseudovector S, whose possible orientations fill out a compact, fixedenergy region of the particle's phase space. On the other hand, for massless particles and tachyons, the little group of Poincaré transformations that preserve the particle's reference four-momentum pμ0 dictates that the particle's phase space at any fixed energy is seemingly noncompact, leading to infinite entropies and other thermodynamic pathologies, besides problems that arise in the corresponding quantum field theories.5 For a tachyon, the only way to eliminate this noncompactness is to require that the spin tensor vanishes, Sμν = 0, meaning that tachyons are naturally spinless. For a massless particle, by contrast, the story is more interesting. We can take the massless particle's reference four-momentum to be pμ0 = (E/c, 0, 0, E/c) μ, (38) and the condition (34), pμS μν = 0, then implies the corresponding reference spin tensor Sμν0 =  0 S0,y −S0,x 0 −S0,y 0 S0,z −S0,y S0,x −S0,z 0 S0,x 0 S0,y −S0,x 0  μν . (39) The most general little-group transformation preserving the reference fourmomentum (38) consists of a Lorentz-transformation matrix Λ of the form6 Λ(θ, α, β) = R(θ)L(α, β), (40) where R(θ) ≡  1 0 0 0 0 cos θ sin θ 0 0 − sin θ cos θ 0 0 0 0 1  (41) is a pure rotation by an angle θ around the z axis and where L(α, β) ≡  1 + ζ α β −ζ α 1 0 −α β 0 1 −β ζ α β 1− ζ  (42) is a complicated combination of Lorentz boosts and rotations. One can show that R(θ1)R(θ2) = R(θ1 + θ2), (43) L(α1, β1)L(α2, β2) = L(α1 + α2, β1 + β2), (44) 5 See, for example, , but also [6] for a more optimistic take. 6 For a derivation, see, for example, [2,10]. Gauge Invariance for Classical Massless Particles with Spin 9 so rotations R(θ) around the z axis and the Lorentz transformations L(α, β) respectively form a pair of commutative subgroups of the particle's little group. Noting that R(θ)L(α, β)R−1(θ) = L(α cos θ + β sin θ,−α sin θ + β cos θ), (45) we identify the little group as ISO(2), which is the noncompact group of rotations and translations in the two-dimensional Euclidean plane. These little-group transformation act nontrivially on the particle's reference spin tensor (39): L(α, β)S0L T(α, β) = S0 +  0 −βS0,z αS0,z 0 βS0,z 0 0 βS0,z αS0,z 0 0 −αS0,z 0 −βS0,z αS0,z 0  . (46) Hence, the only way to ensure that the massless particle has a compact phase space at fixed reference energy while still allowing for nonzero spin is to impose the following equivalence relation on the particle's phase space: (X, p, S) ∼= (X, p, S′). (47) This equivalence relation is a new result. It is a classical-particle manifestation of the gauge invariance that arises for the gauge potentialAμ in electromagnetism, and it cuts the particle's phase space at fixed energy down to a compact extent. The distinct physical states of the massless particle are then characterized by a spacetime position Xμ, a four-momentum pμ, and a helicity σ ≡ (p/|p|) *S.7 4 The Massless Limit We can better understand the origin of the novel equivalence relation (47) by starting with the massive case m > 0 and then taking the massless limit m→ 0. Our original reference state (37) degenerates for m→ 0, so we instead take the massive particle's reference four-momentum to be pμ ≡ (pt, 0, 0, pz)μ = (√ (pz)2 +m2c2, 0, 0, pz )μ . (48) This choice has the correct m→ 0 limit (38): lim m→0 pμ = (E0/c, 0, 0, E0/c) μ, E0 ≡ pzc. (49) 7 Note that if we permit parity transformations, which map σ 7→ −σ, then we must require that the equivalence relation (47) hold only for states that share the same helicity σ. 10 Jacob A. Barandes Moreover, (48) is related to our original choice (37) of reference four-momentum for the massive particle by a simple Lorentz boost Λ along the z direction, pμ = Λμνp ν 0 , (50) and the new reference value Sμν of the massive particle's spin tensor is related to its original reference value Sμν0 according to Sμν ≡ (ΛS0ΛT)μν =  0 pz mc S0,y − pz mc S0,x 0 − p z mc S0,y 0 S0,z − pt mc S0,y pz mc S0,x −S0,z 0 pt mc S0,x 0 pt mc S0,y − pt mc S0,x 0  μν . (51) For m → 0, we have pt, pz → E0/c, so the components of Sμν involving pt/mc or pz/mc diverge. Furthermore, there is a discrete mismatch in the particle's spin-squared scalar (22) between the massive case and the massless case: s2 = S20,x + S 2 0,y + S 2 0,z (massive) 6= S20,z (massless). (52) These discrepancies are hints that the massive case includes spin degrees of freedom that need to be removed before taking the massless limit. Our approach for removing these ill-behaved spin degrees of freedom is motivated by a corresponding procedure in quantum field theory that was originally developed by Stueckelberg in [9]. We start with the redefinition( Sx Sy ) 7→ mc pt ( Sx + p tφx Sy + p tφy ) = mc pt ( Sx Sy ) +mc ( φx φy ) , (53) where φx(λ) and φy(λ) are arbitrary new functions on the particle's worldline. The particle's spin tensor (51) then has the decomposition Sμν =  0 pz pt S0,y − pz pt S0,x 0 − p z pt S0,y 0 S0,z −S0,y pz pt S0,x −S0,z 0 S0,x 0 S0,y −S0,x 0  μν +  0 pzφy −pzφx 0 −pzφy 0 0 −ptφy pzφx 0 0 p tφx 0 ptφy −ptφx 0  μν , (54) Gauge Invariance for Classical Massless Particles with Spin 11 and the spin-squared scalar (22) becomes s2 = ( 1− ( pz pt )2)( (S0,x + p tφx) 2 + (S0,y + p tφy) 2 ) + S20,z. (55) The particle's spin tensor (54) is now invariant under the simultaneous transformations( Sx Sy ) 7→ ( Sx Sy ) − pt ( fx fy ) , (56)( φx φy ) 7→ ( φx φy ) + ( fx fy ) , (57) where fx(λ), fy(λ) are arbitrary functions on the particle's worldline. Our massive particle's original phase space, with states labeled as (X, p, S), is therefore equivalent to a formally enlarged phase space consisting of states (X, p, S, φ) under the equivalence relation (X, p, S, φ) ∼= (X, p, S− ptf, φ+f), suitably generalized from the reference state (X, p, S, φ) to general states (X, p, S, φ) of the system. Indeed, one can check that the specific choice (fx, fy) ≡ −(φx, φy) yields (X, p, S + ptφ, 0), which gives back the state (X, p, S) after undoing the redefinition (53) of Sμν . We can now safely take the massless limit of the system's redefined spin tensor (54): lim m→0 Sμν =  0 S0,y −S0,x 0 −S0,y 0 S0,z −S0,y S0,x −S0,z 0 S0,x 0 S0,y −S0,x 0  μν + E c  0 φy −φx 0 −φy 0 0 −φy φx 0 0 φx 0 φy −φx 0  μν , (58) and lim m→0 s2 = S20,z. (59) The degrees of freedom describing spin components perpendicular to the particle's reference three-momentum p no longer contribute to the particle's spin-squared scalar s2. If we remove these ancillary degrees of freedom by setting φx, φy equal to zero, then the particle's spin tensor (58) reduces correctly to the reference spin tensor (39) for a massless particle, and our equivalence relation (56) reduces to the gauge invariance (47). We have therefore completed our recovery of the massless case from the m→ 0 limit of a massive particle. 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