CHIRAL FERMIONS & MASSLESS SPINNING PARTICLES II : TWISTED POINCARE SYMMETRY. C. Duval, M. Elbistan P. Horvathy, P.-M. Zhang

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1 CHIRAL FERMIONS & MASSLESS SPINNING PARTICLES II : TWISTED POINCARE SYMMETRY C. Duval, M. Elbistan P. Horvathy, P.-M. Zhang July 17, 2015

2 c-model Stephanov & Yin PRL 2012 (semi)classical model ( (p ) dx S = + ea dt ( p + eφ ) a dp dt ) dt (1) a(p) momentum-dependent vector potential for Berry monopole of strength s = 2 1 in p-space p a = Θ p 2 p 2. (2) Spin enslaved : conserved angular momentum J = x p p (3) No manifest Lorentz symmetry. However J-Y. Chen et al. PRL 113 (2014) 18, δx = β t + β p 2 p, δp = p β, δt = β x (4) twisted boost cf. J-W Chen et al. PRL 110, (2013). Aim: derive (4) from Souriau s model with natural Poincaré symmetry. 1

3 Souriau 1970 ( Lagrange 1810) : (i) Dynamics determined by closed two-form σ of constant rank r, defined on evolution space V. Motions [curves or surfaces] characteristic leaves, tangent to kernel of ker σ two-form σ. (ii) Factoring out char. leaves symplectic manif (M, ω) called space of motions. Can be viewed as abstract substitute for phase space. 2

4 Symplectic description of chiral model Variation of chiral action (1) yields eqns of motion for position x and momentum p 0, ( 1 + eθ B ) dx dt ( ) dp 1 + eθ B dt = p + e E Θ + (Θ p) e B = e E + e p B + e 2 (E B) Θ where E, B electric & magnetic field. (5) Evolution space V 7 = T (R 3 \{0}) R = { } (p, x, t) (6) endowed with Souriau two-form σ = ω dh dt, ω = ω 0 + e 2 ɛ ijkb i dx j dx k, (7) ω 0 = dp i dx i (1/2) 2 p 3 ɛijk p i dp j dp k (8) h = p + eφ (9) If det(ω αβ ) = (1 + e Θ B) 2 0, kernel of σ 1- dimensional; γ(τ) = (x(τ), p(τ), t(τ) ) tangent to ker σ iff eqns of motion (5) satisfied. 3

5 Free part : for E = B = 0 Θ-terms drop out dx dt = p, Eqns integrated: dp dt = 0. (10) x(t) = x + p t, p(t) = p, (11) with x, p const. Space of free motions M 6 = V 7 / ker σ described by conserved quantities x = x(t) p t p. (12) Two-form σ projects to Space of motions (M, ω), ω = ω 0 = d p i d x i (1 2 ) 2 p 3ɛijk p i d p j d p k. 4

6 Souriau s Massless Spinning Particles Souriau particle V 9 { R, P, S 1969 Free relativistic massless spinning described by 9-dim evolution space } P µ P µ = 0, S µν P ν = 0, 1 2 S µνs µν = s 2 Endowed with closed two-form σ = dp µ dr µ 1 2s 2 dsµ λ Sλ ρ ds ρ µ. (14) Dynamics given by foliation whose leaves are tangent to ker σ; world-sheet [or world-line] obtained by projecting leaf to Minkowski space, yielding corresponding spacetime track. (13) Kernel using constraints curve (R(τ), P (τ), S(τ)) tangent to ker σ iff P µ Ṙ µ = 0, Ṗ µ = 0, Ṡ µν = P µ Ṙ ν P ν Ṙ µ, (15) where dot d/dτ. Spacetime velocity Ṙ orthogonal to momentum P. 5

7 Wigner-Souriau (WS) translations Souriau 1970 (15) integrated using spacetime vectors W orthogonal to P, P µ W µ = 0, R µ R µ + W µ, P µ P µ, S µν S µν + (P µ W ν P ν W µ ). (16) Any point in leaf reached by suitable W at each point kernel is 3-dim, projects to spacetime as 3d surface in R 3,1, spanned by all vectors orthogonal to P motions of free massless spinning particle on 3-dimensional planewave tangent to light-cone at each spacetime event R. Particle not localized in spacetime. Wigner 39, Penrose 67, Souriau 69. 6

8 Leaf ( 3-brane ) invariant by WS-shift in (16) considered same motion. Each leaf defines motion. M 6 = V 9 / ker σ Space of motions is collection of leaves. R = (r, t) where r, t position & time in chosen Lorentz frame. Spin tensor S ij = ɛ ijk s k, s = spin vector S j4 Σ j = ( p s ) (17) j = 1, 2, 3 j s = 1 2 p p Σ, Σ p = 0 (18) orthogonal decomposition Important fact : p s = 1 2 (19) 7

9 N.B. 1 2 not length of spin vector s but projection onto momentum. For arbitrary W R 3 Wigner-Souriau (WS) translation W = (W, p W ). Acts on V 9 r r + W t t + p W (20) p p p 4 p 4 (21) s s + p W (22) S j4 S j4 + p j ( p W ) p W j (23) S j4 component eliminated by WS translation W = p s p = Σ p (24) enslaves spin s 1 2 p. SPIN eliminated as indept degree of freedom. 8

10 x = g p = r p t + p s p (25) itself conserved. Leaf space of motions M 6 = V 9 / ker σ T (R 3 \{0}). (26) labeled by x and p 0. Two-form σ descends to space of motions M 6 as symplectic two-form ω = dp i d x i (1 2 ) 2 p 3 ɛijk p i dp j dp k. (27) same as for c-model! c-model space of motions S-model 9

11 Poincaré symmetry I. C. Duval and P. A. Horvathy, Chiral fermions as classical massless spinning particles, Phys. Rev. D (2015) arxiv: [hep-th]. Poincaré algebra spanned by (Λ, Γ) where Λ = (Λ µν ) belongs to Lorentz algebra and Γ = (Γ µ ) is translation in Minkowski space. 10

12 Poincaré acts naturally on V 9 : δr µ = Λ µ νr ν + Γ µ, δp µ = Λ µ νp ν, δs µν = Λ µ ρs ρν Λ ν ρs ρµ. (28) Souriau form σ (14) invariant symmetry. Action descends to space of motions (M 6, ω). Noetherian conserved quantities P µ = I µ, M µν = R µ P ν R ν P µ + S µν (29) conserved linear and Lorentz momentum. In (3 + 1): parametrize Poincaré algebra by Λ ij = ɛ ijk ω k, Λ i4 = β i, Γ = (γ, ε), (30) ω, β, γ R 3, ε R are inf. rotations, boosts and space- resp. time translations. Inf. Poincaréaction on V 9 given by δr = ω r + βt + γ, δt = β r + ε, δp = ω p + β p, δs = ω s β ( p s), (31) projects as natural action on Minkowski space. 11

13 Explicit form of Poincaré momenta, present matrix M = (M µν ) as M ij = ɛ ijk l k, M j4 = g j with l, g 3-vectors. { l = r p + s, (32) g = p r pt + p s. Poincaré algebra acts on space of motions as δ p = ω p + p β, δ x = p p ω x + β ( 2 p 2) β x p + γ ε p p. (33) Duval, Ziegler parameter vectorfield (33) leaves free symplectic structure invariant symplectic Noether theorem associates 10 consts of the motion, l = x p p angular momentum g = p x boost momentum p = p linear momentum E = p energy (34) N.B. conserved quantity x = g p in (25) is (boost momentum)/energy. (35) 12

14 Poisson brackets calculated using (27), {l i, l j } = ɛ k ij l k, {l i, g j } = ɛ k ij g k, {l i, p j } = ɛ k ij p k, {l i, E} = 0, {g i, g j } = ɛ k ij l k, {g i, p j } = E δ ij, {g i, E} = p i, {p i, p j } = 0, {p i, E} = 0, (36) Poincaré algebra. Casimir invariants m 2 = p 2 + E 2 = 0, l p = 1 2, (37) N.B. infinitesimal action extends to finite action. S-model carries zero-mass & spin- 1 2 representation of Poincaré group, realized by natural action. 13

15 Twisted Poincaré symmetry of free c-model Strategy I : import natural Poincaré symmetry of S-model to chiral system through their common space of motions. x = x pt c-model #(11) r pt + p s p S-model #(25) (38) in terms of coordinates (p, x, t) on chiral evolution space V 7, Poincaré action (33) becomes twisted, δx = ω x + β t + γ + β p, 2 p (39) δp = ω p + p β, δt = β x + ε. Confirms Lorentz action Chen, Son, Stephanov... Conserved Lorentz quantities, l = x p p, g = p x pt. (40) Spin contribution to conserved angular momentum: enslaved 15

16 Identity of spaces of motion allows to export natural Poincaré symmetry of latter to become twisted ( dynamical ) symmetry of former. N.B. Twisted boost action (4) i.e. δx = β t + β p 2 p, δp = p β, δt = β x cf. (39) is not usual, natural one on ordinary spacetime. In fact, it is not action on spacetime at all, since it also involves momentum variable p; is sort of dynamical symmetry. x should not be considered as position variable, because does not transform under a boost as positions should. 16

17 Strategy II : Duval-Elbistan-PAH-Zhang, Wigner- Souriau translations and Lorentz symmetry of chiral fermions, Phys. Lett. B 742 (2015) 322, arxiv: Embedding chiral evolution space V 7 into that, V 9, of massless spinning particle of Souriau through Spin enslavement Analogy with symmetries of gauge fields Consider gauge field F with [ not unique!] (gauge potential A µ, F µν = µ A ν ν A µ. X inf. spacetime transformation. Gauge field symmetric if L X A µ = µ W. (41) Chosen gauge potential may not be invariant [ex: monopole: No rotation-invariant vector potential does exist!] Choosing vector potential gauge fixing [e.g. Coulomb gauge A = 0, or Lorentz gauge µ A µ = 0 etc]. May be violated by transf. X. However, variation of gauge potential can be compensated by gauge transf. which restores gauge fixing. 17

18 Return to chiral fermions: Condition Σ j S j4 = ( p s) j = 0 i.e. s = 1 2 p (42) [achieved by WS translation] defines submanifold Ṽ 7 of V 9 identified with V 7. Restriction of two-form σ of V 9 to Ṽ 7 is (7) (8). Motions of chiral system lie within leaves of 3-dimensional foliation of V 9 V 9 Chiral evolution space can be embedded, V 7 Ṽ 7, into that, V 9, of massless spinning particle by enslaving spin by requiring p s = 0. Dynamics consistent with embedding. 18

19 V 9 19 Lorentz symmetry inherited from embedding? no : consider chiral motion x pt embedded into V 9 s.t. s = 1 p 2 and boost. Although spin is boost-invariant, p is not δ(p s) = 1 2 β p 0 (43) However restored by compensating gauge [i.e. WS] transformation yielding twisted Lorentz symmetry of c-model

20 Coupling to electromagnetic field Conventional minimal coupling rule : 4-momentum shifted by 4-potential, p µ p µ ea µ. (44) not what is proposed by Stephanov et al in (1): rule (44) used for 4-momentum (p, h), but in Berry term Θ(p) momentum not shifted. half-way-rule used in (1) is instead consistent with Souriau s prescription, who works with same evolution space as for free particle, but adds electromagnetic field strength F to free two-form (14), σ σ + ef, (45) where e is electric charge. σ closed, dσ = 0, because F is closed 2-form of Minkowski space. Rules (44) and (45) equivalent in spinless case only. 20

21 Minimal coupling of S-model Souriau s prescription (45) applied to massless spinning model yields, on V 9, closed 2-form σ = dp µ dr µ 1 2s 2 dsµ λ Sλ ρ dsρ µ + e 2 F µν dr µ dr ν. Lengthy calculation eqns of motions Ṙ µ = P µ + Sµν F νρ P ρ 1 2 S F, Ṗ µ = ef µ νṙν, assuming Ṡ µν = P µ Ṙ ν P ν Ṙ µ (46) (47) S F S αβ F αβ 0. (48) Spin-field coupling breaks WS invariance spin degree can not be eliminated, leaving us with 9 1 = 8-dim. space of motions (phase space locally). Dim of ker σ drops from 3 1. In terms of (3 + 1)-decomposition : assuming (a) 1 2 S F 2 1S αβf αβ = s (B p E) 0, (b) p B 0, 21

22 B p E ṙ = s p s (B p E), (49) ( p B) ṫ = s p s (B p E). (50) Condition (a) henceforth assumed satisfied. Condition (b) : when not satisfied ṫ = 0, motion instantaneous. Assume regularity conds; merging eqns (49)- (50) : dr dt dp dt ds = B p E p B, = e ( E + dr dt B) = e E B p B p, = p dr dt = p B p B p ( p E) p B dt (51) p expected on r.h.s. of velocity relation cancels out; electric charge drops out; momentum dynamics decouples from spin as long as latter does not vanish; scalar spin s 0 disappears also from all eqns..

23 Eqs (51) d p/dt = 0 direction of p unchanged during motion. Spin is not independent degree of freedom, its (for spacetime dynamics irrelevant) motion entirely determined by other dynamical data. Ex.: crossed constant electric & magnetic fields (like in Hall effect), E = E ˆx, B = B ẑ. E B = 0 p const. of motion, so is angle θ between B and p (θ π/2 for p B 0). Assume init. momentum in x-z plane. Eqns of motion solved by ( r(t) = (cos θ) 1 ẑ E ) B ŷ t + r 0, p(t) = p 0, ( s(t) = p tan θ ŷ + E ( ) cos θ ˆx sin θ ẑ) t + s 0. B (52) Thus, in addition to const.-speed vertical motion, particle also drifts perpendicularly to electric field with Hall velocity E/B. Spin moves perpendicularly to p projection on p remains const, s(t) p = 1 2. Spin decoupled, but can not be enslaved : s and p do not remain parallel even for such init. cond. 22

24 Motion in constant Hall-type electric and magnetic fields. Init. position, r 0, chosen in y-z plane, init. momentum and spin parallel and in x-z plane. Spatial motion, r(t), is combination of constant-velocity Hall drift perp. to E and B, combined with const-velocity vertical drift. Momentum, p, conserved, but spin, s(t), has curious, Hall-type motion in plane perp to momentum. Velocity superluminal ( dr/dt > 1), diverges as θ π/2; for p B = 0 instantaneous motions with infinite velocity, parallel to z axis. 23

25 Anomalous coupling Minimal model curious, but not completely satisfactory. Generalize. Clue : allow masssquare to depend on coupling of spin to e.m. field, P µ P µ = eg 2 S F, (53) where used shorthand S F S αβ F αβ. constant g : gyromagnetic ratio. Real Novel evolution space { P, R, S P µ P µ = eg } 2 S F, S µνp ν = 0, 2 1S µνs µν = s 2, endowed with closed two-form, (54) σ = dp µ dr µ 1 2s 2 dsµ λ Sλ ρ dsµ ρ + 2 1eF µν dr µ dr ν. (55) NB. formally same as (46) up to different massshell constraint. 24

26 New eqns of motion on Ṽ 9. Ṙ µ = P µ 1 1 [ (g 2) S µν F (g + 1) S αβ F αβ νρ P ρ g S µν ν F ρσ S ], ρσ Ṗ µ = ef µ ν Ṙν eg 4 µ F ρσ S ρσ, Ṡ µν = P µ Ṙ ν P ν Ṙ µ + eg 2 [S µ ρ F ρν S ν ρ F ρµ ]. (56) Zero-rest-mass counterparts of Bargmann-Michel- Telegdi eqns for massive relativistic particles. Reduce to (46) for g = 0. In normal case g = 2 resulting from Dirac eqn anomalous velocity canceled, but new contribution involving derivative of em field. (3 + 1)-decomposition. R = (r, t), P = (p, E), S j4 = ( p E s ) j (57) where spin tensor still defined as in (??), but new dispersion relation, E = p 2 eg 2 S F. (58) Decomposing em field into electric and magnetic components, ( 1 2 S F = s B p ) E E. (59) 25

27 Tedious calculation yields very complicated (3+ 1)-form of eqns of motion (56). Consider g = 2, assume that fields are constant; then complicated system simplifies to one reminiscent of a massive relativistic particle, (g=2) E dr dt dp dt ds = p, = e = e ( E + dr (( p E 2 s ) ) dt B, E + s E B ) dt (60) assuming E is real. [NB: p 0 implies that E can not vanish]. In pure magnetic field momentum and spin satisfy eqns of identical form, dp dt = e E p B, ds dt = e E s B. (61) Multiplying by p, s and by B, resp. p = const 0, s = const 0, p B = const, s B = const, p z = const, s z = const, E = p 2 es B = const. 26

28 Both momentum and spin vectors precess around direction of magnetic field, B = Bẑ with common angular velocity ω = eb/e, p(t) = (p 0 e i(eb/e)t, p z ), (62) s(t) = (s 0 e i(eb/e)t, s z ), (63) where p 0 = p x + ip y, s 0 = s x + is y. Therefore r(t) = ( ip0 eb e i(eb/e)t, p ) z E t + r 0. (64) Weak pure magnetic field s = 1 2 p (consistent) E p eg 4 S F = p e p B 2 p, (65) also proposed by Chen et al, Manuel et al. N.B. similar to motion of c-model in pure, const. magnetic field: p(t) = (p 0 e i(eb/ p m)t, p z ) (66) p (1+eΘ B) E = p 2 es B p ( 1 e s B p 2 ). 27

29 Motion in pure const magnetic field B. Both momentum p(t) and spin s(t) precess with identical angular velocity eb/e around B direction. Position r(t) spirals on cylinder around B-axis, obtained by combining precession with vertical drift of supporting cone. 28