Wigner-Souriau translations and Lorentz symmetry of chiral fermions

Chiral fermions can be embedded into Souriau's massless spinning particle model by"enslaving"the spin, viewed as a gauge constraint. The latter is not invariant under Lorentz boosts; spin enslavement can be restored, however, by a subsequent Wigner-Souriau (WS) translation, analogous to a compensating gauge transformation. The combined transformation is precisely the recently uncovered twisted boost, which we now extend to finite transformations. WS-translations are identified with the stability group of a motion acting on the right on the Poincare group, whereas the natural Poincare action corresponds to action on the left. The relation to non-commutative mechanics is explained.


INTRODUCTION
Semiclassical chiral fermions can be described by the phase-space action where a(p) is a vector potential for the "Berry monopole" in p-space, ∇ p × a = p 2|p| 2 , p = p/|p|, p = 0. Here A(x) and φ(x) are [static] vector and scalar potentials and e is the electric charge [1 -6]. A distinctive feature is that spin is "enslaved" to the momentum, i.e., identified with 1 2 p, see (1. 3). An intriguing aspect of the model is its lack of manifest Lorentz symmetry. Recently [5], it was shown, though, that modifying the dispersion relation in (1.1) as h = |p|+eφ(x)+ p·B 2|p| yields a theory which is covariant w.r.t. Lorentz transformations. Turning off the external field, their expression # (6) reduces to where β is an infinitesimal Lorentz boost. This formula has also been found, independently, by relating the chiral fermion to Souriau's model of a relativistic massless spinning particle through their spaces of motions [6,7]. The latter has an additional degree of freedom identified with "unchained" spin and represented by the vector s.  [12].
Section 5 clarifies the geometry hidden behind : while the natural Poincaré action corresponds the left-action of the Poincaré group on itself, WS translations correspond to the right-action of the stability group of a motion.

SPINNING MODELS
Both the chiral and the Souriau models can conveniently be described within Souriau's framework [6,7], where the classical motions are identified with curves or surfaces in some evolution space, and are tangent to the kernel of a closed two-form; see [6] for details. We limit our considerations to the free case; coupling to external fields has been discussed in the literature listed in [6,13].
We first consider the [free] chiral model (1.1). It has been shown [6] that the associated variational problem admits an alternative geometric formulation. To that end, we introduce the seven dimensional evolution space V 7 = T (R 3 \{0}) × R described by triples (x, p, t), and endow it with the two-form σ defined by ijk p i dp j ∧ dp k , h = |p|. (2.1) The two-forms ω and σ are closed, since ∇ p · b = 0. The kernel of σ defines an integrable distribution, whose leaves [integral manifolds] can be viewed as generalized solutions of the variational problem. Here, the kernel is one-dimensional and a curve (x(t), p(t), t is tangent to it iff the equations of motion, dx dt = p, dp dt = 0, (2.2) are satisfied [6]; the solution is plainly p = p 0 = const, x(t) = x 0 + p t, x 0 = const, i.e., the motion is in the p direction with the velocity of light.
The Souriau model admits a similar description [6]. Restricting ourselves again to the free case, the evolution space here is 9-dimensional and is described by The evolution space is endowed with the closed two-form

4)
A (3 + 1)-decomposition can be introduced by writing, in a Lorentz frame, R = (r, t), and P = (p, |p|) where p = 0. The components S µν of the spin tensor can in turn be split into space and space-time components, and can be deduced from Eq. (3.6) in [6]. A particular solution of (2.6) is obtained by embedding the chiral solution above into the spin-extended evolution space V 9 by identifying r with x in (2.2) and completing [trivially] with a constant spin vector, p = p 0 = const, r(t) = r 0 + p t, r 0 = const ., s(t) = s 0 = const . A remarkable feature of Eqs (2.6) is that for an arbitrary 3-vector W , the transformation takes a solution of (2.6) into another, equivalent one. This transformation is referred to as a Wigner-Souriau (WS) translation [4,[6][7][8][9]. The kernel of (2.4) is invariant under WStranslations and is in fact 3-dimensional, swept by the images of the embedded solutions.
The spin vector here, s, is not necessarily "enslaved", i.e., may not be parallel to the momentum, p. The spin constraint in (2.3) implies nevertheless that the projection of the spin onto the momentum and the perpendicular component Σ satisfy, respectively, see (2.5). From the WS action above we infer that Σ → Σ + p × (p × W ). It follows that Σ can be eliminated : choosing W = Σ/|p| carries Σ to zero.
The Poincaré group acts naturally on V 9 , namely according to [6,7] where ω, β, γ, ε are identified with infinitesimal rotations, boosts, translations and timetranslations; their action duly projects to Minkowski space-time as the natural one. In what follows, we focus our attention at boosts; WS-translations will be studied further in Sec. 5.

MODEL
Now we embed the evolution space of the chiral model, V 7 , into that of the massless spinning particle, V 9 . We note first that, by (2.8), spin enslavement, (1.3), is equivalent to which, viewed as a constraint, defines a seven dimensional submanifold of V 9 that we parametrize with r, p, t and denote (with a suggestive abuse of notation) still V 7 . Eqns (2.6) and (2.8) imply thatΣ = |p| pṫ −ṙ . Requiring Σ = 0 is therefore consistent with the dynamics : the motions of the chiral system lie in the intersection of the 3-dimensional characteristic leaves of V 9 with the surface V 7 defined by spin enslaving; they remain therefore motions also for the extended dynamics [17]. A chiral motion is embedded into V 9 by respecting the gauge condition (1.3), namely as γ(t) = r(t) = x(t), p(t), s(t) = 1 2 p(t) .
So far we studied infinitesimal actions only. But our strategy is valid also for finite transformations, as we show it now. Firstly, from Appendix C of [6] we infer the action of a finite boost with b on the evolution space V 9 , namely which is consistent with the infinitesimal action (2.9).
Then, starting with enslaved spin, s = 1 2 p, we find, consistently with (4.1), which does not vanish in general: spin is unchained.
At last, the finite WS-translation (2.7) with restores the validity of (1.3): spin is re-enslaved, s = 1 2 p . The combined transformation for finite boosts, We now turn to the space of motions [6,7], defined as the quotient of the evolution space by the characteristic foliation of σ; we denote it by M . The equations of motion of the spin-extended system, (2.6), imply that is in fact a constant of the motion, dx/dt = 0. It can be used therefore to label the motion, i.e., a characteristic leaf in V 9 . The conserved momentum, p, is another good coordinate set on M , which is 6-dimensional, and whose points can therefore be labeled byx and p = 0.
In [6], we derived the twisted boost (1.2) from the Poincaré action on the space of motions.
Conversely, the latter can be obtained from our construction here. Boosting in V 9 according to (2.9) we find, δ β r− pt = − p β·(r− pt) and In view of (4.7), the preceding terms combine to yield The Poincaré group acts on the dual Lie algebra by the co-adjoint representation, defined by Coad g µ · X = µ · g −1 Xg, for all X ∈ g. We denote here by µ = (M, P ) ∈ g * a "moment" of the Poincaré group, and by µ · X = 1 2 M µν Λ µν − P µ Γ µ its contraction with X ∈ g, see [7]. Contracting the Maurer-Cartan 1-form, g −1 dg, with and arbitrary fixed element µ 0 of the dual of Lie algebra, yields a real one-form on the group G; we denote by σ = d µ 0 · g −1 dg (5.1) its exterior derivative. As a general fact, the characteristic leaves of the two-form σ, defined by its kernel, are identified with the classical motions [11] associated with µ 0 . Indeed, the space of all motions, M , of an elementary system for the group G is interpreted by Souriau [7] as the orbit of some basepoint µ 0 under the coadjoint action, The stability Lie algebra, g 0 , is therefore 4-dimensional, parametrized by (β, ε, λ), where β ⊥ p 0 , ε ∈ R, and λ ∈ R represents an infinitesimal rotation around p 0 . We note that the evolution space V 9 is in fact the quotient of the Poincaré group by rotations around p 0 , and (2.9) above is in fact the projection of the infinitesimal left-action of the group G to V 9 ≈ G/SO(2), whereas the 2-form (5.1) projects as (2.4), as anticipated by the notation.
Remember now that the Poincaré group also acts on itself from the right, g → gh −1 , for h ∈ G; its infinitesimal right-action is therefore given by matrix multiplication, δ X g = −gX, where X = δh at h = 1. Choosing in particular X ∈ g 0 , the action on the group reads it generates its kernel [7]. Renaming the Poincaré translation C as R = (r, t), a space-time event, shows that the right-action on G of a vector X from the stability algebra g 0 yields, This transformation satisfies the condition p · δ X r = δ X t required for an infinitesimal WStranslation; conversely, any WS-translation W is of the form (5.3), with β perpendicular to p and λ arbitrary. Therefore (with a slight abuse) we will refer to g 0 acting from the right as a WS-translations.
Comparing now (5.3) with (4.2) allows us to conclude that while a boost β acting on the left unchains the spin, the latter is re-enslaved by a WS-translation with the same boost, β, acting from the right.
It is readily verified that the right-action of an X from the stability algebra g 0 acts as δ X p = 0 and δ X s = 1 2 p × ( p × β), consistently with the infinitesimal action. Thus, after rotations around p 0 are factored out, the right-action of the stability subalgebra g 0 projects to V 9 as the WS translations. to V 9 and then to M 6 and R 3,1 , respectively.

CONCLUSION
In this Letter, we re-derived the twisted Lorentz symmetry (1.2) of chiral fermions by embedding the theory of Refs. [1-5] into Souriau's massless spinning model [6,7] by spin enslavement, (1.3), viewed as a gauge fixing. The latter is not boost invariant, but enslavement can be restored by a suitable compensating WS-translation, which is analogous to a gauge transformation [12]. Our formula (4.6) extends (1.2) to finite transformations.
In other words, a free massless relativistic particle behaves as a 3-brane. Coupling to an external electromagnetic field breaks the WS "gauge" symmetry, so that spin can not be enslaved; the motions then take place along curves: the particle gets localized [6].