These notes merely intend to memorialize Souriau's overlooked achievements
regarding geo\-metric quantization of Poincar\'e-elementary symplectic systems.
Restricting attention to his model of massless, spin-$\half$, particles, we
faithfully rephrase and expound here Sections (18.82)--(18.96) &
(19.122)--(19.134) of his book \cite{SSD} edited in 1969. Missing details about
the use of a preferred Poincar\'e-invariant polarizer are provided for
completeness

Duval, Christian

English

arXiv

These notes merely intend to memorialize Souriau's overlooked achievements regarding geo\-metric quantization of Poincar\'e-elementary symplectic systems. Restricting  attention to his model of massless, spin-$\half$, particles, we faithfully rephrase and expound here Sections (18.82)--(18.96) \& (19.122)--(19.134) of his book \cite{SSD} edited in 1969. Missing details about the use of a preferred Poincar\'e-invariant polarizer are provided for completeness

HAL AMU

A recollection of Souriau’s derivation of the Weylequation via geometric quantizationChristian DuvalTo cite this version:Christian Duval. A recollection of Souriau’s derivation of the Weyl equation via geometricquantization. 2016. <hal-01266487>HAL Id: hal-01266487https://hal.archives-ouvertes.fr/hal-01266487Submitted on 2 Feb 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A recollection of Souriau’s derivation of theWeyl equation via geometric quantizationC. Duval∗Centre de Physique The´orique,Aix Marseille Universite´ & Universite´ de Toulon & CNRS UMR 7332,Case 907, 13288 Marseille, France.February 2, 2016AbstractThese notes merely intend to memorialize Souriau’s overlookedachievements regarding geometric quantization of Poincare´-elementarysymplectic systems. Restricting attention to his model of massless,spin-12, particles, we faithfully rephrase and expound here Sections(18.82)–(18.96) & (19.122)–(19.134) of his book [10] edited in 1969.Missing details about the use of a preferred Poincare´-invariant polar-izer are provided for completeness.1 The classical model1.1 Minkowski spinors: a quick reviewWe start with some prerequisite. Denote by W 10 = SL(2,C)⋉ R3,1 the uni-versal covering of the restricted Poincare´ group, i.e., the neutral componentof the group, E(3, 1), of isometries of Minkowski spacetime, R3,1 = (R4, g),where g = −|dr|2 + dt2 (we put c = 1).∗mailto: duval-at-cpt.univ-mrs.frLet us recall how the well-known spin group, Spin(3, 1) ∼= SL(2,C), of R3,1arises. To that end, we choose the following representation of the Diracmatrices acting on the Minkowskian spinor space, C2,2, namelyγj =(0 σj−σj 0), γ4 =(0 11 0), γ5 =( −i 00 i)(1.1)where the σj , with j = 1, 2, 3, denote the Pauli matrices.All matrix representations use a tacitly chosen spinor frame with GrammatrixG =12(0 11 0).The Hermitian conjugate of ζ ∈ C2,2 reads therefore ζ = C(ζ)TG whereC : z 7→ z stands for complex conjugation, and T for transposition. TheDirac matrices are traceless and Hermitian-symmetric, i.e., γµ = γµ for allµ = 1, . . . , 4, with respect to G.1 They furthermore satisfy the Cliffordrelationsγµγν + γνγµ = 2gµνfor all µ, ν = 1, . . . 4; also γ5 = γ1γ2γ3γ4 is the chirality operator whichanticommutes with all γµ.• The spin group in 3 + 1 spacetime-dimension may be introduced asthe group of quaternionic G-unitary matrices commuting with the chirality.Thus A ∈ Spin(3, 1) iff A ∈ U(2, 2) and SH = HS where the quaternionicstructure of C4 is defined byH =(0 JCJC 0)as well as by Aγ5 = γ5A; here J = −iσ2 denotes the standard complexstructure of R2 ∼= C. As a consequence, we have Aγ(δR)A−1 = γ(̺(A)δR)where ̺(A) ∈ O(3, 1)0.2 The homomorphism ̺ : Spin(3, 1) → O(3, 1)0 haskernel Z2. In the above representation (1.1), the spin group is thereforegenerated by the matrices [10]A =(a 00 a−1)∈ Spin(3, 1) where a ∈ SL(2,C).where the bar denotes now Hermitian conjugation of 2×2 complex matrices.1We have γµ = G−1C(γµ)TG.2We use the shorthand notations γ(δR) = γµδRµ for all δR ∈ R3,1.2Introducing the two supplementary projectorsΠ =1 + iγ52& Π =1− iγ52we may alternatively view the spin group as the group of those matricesA = (A1A2A3A4) ∈ Spin(3, 1) with column vectors [1]A1 = Πζ, A2 = ΠHζ, A3 = Πζ, A4 = ΠHζwhere the ζ ∈ C2,2 satisfy the fundamental relations [10]ζζ = 1 & ζγ5ζ = 0. (1.2)Clearly, Spin(3, 1) is parametrized by the above-mentioned spinors ζ ; it isthus diffeomorphic to the manifoldΣ6 ∼= S3 × R3of those ζ ∈ C2,2 satisfying (1.2).3If we writeζ =(ζ ′ζ ′′)∈ Σ6 (1.3)with ζ ′, ζ ′′ ∈ C2 \ {0} as half-spinors, then (1.2) translates as ζ ′ζ ′′ = 1(ordinary Hermitian scalar product of C2).• The universal covering of the neutral Poincare´ group is thus diffeo-morphic to W 10 = Σ6 × E4 described by the pairs (ζ, R) with ζ as in (1.2),and R ∈ E4 = R3,1, a spacetime event.• With this preparation, we verify that the Minkowski vector P definedfor all δR ∈ R3,1 by the scalar productP · δR = ζγ(δR)Πζ√2 (1.4)is null and future-pointing.43The diffeomorphism is thus Spin(3, 1)→ Σ6 : S 7→ ζ = A1 +A3.4Since P · δR = PµδRµ with Pµ = gµνP ν , Equation (1.4) may alternatively be read asPµ = ζγµ [1 + iγ5] ζ/√2, for all µ = 1, . . . , 4.3Indeed, easy computation shows that P ·δR = ζ ′(−σ(δr)+δt)ζ ′/√2, withthe help of (1.1). We then get P · δR = E(−u · δr + δt) where u ∈ S2 ⊂ R3,and E = ‖ζ ′‖2/√2 with ‖ · ‖ the Hermitian norm of C2. Thus, the vectorP = E(u, 1) is clearly as announced.Likewise, the vector Q defined by Q · δR = −ζγ(δR)Πζ√2 is null andverifies P ·Q = −1.We have a projection πWV : W10 → V 9 defined by πWV (ζ, R) = (P,Q,R).The manifold V 9 has already been interpreted as the evolution space of themodel [10, 4], and P as the momentum of our particle with positive energy.We note, in view of the previous definitions of P and Q that V 9 = W 10/U(1)where the action of U(1) is given by eiθ(ζ, R) = (eiθζ, R).1.2 Classical motions of chiral particlesHaving in mind to describe spinning particles of half-integral spin, we findit convenient to use the spinor representation of the Poincare´ group tostart with, even at the classical level. To describe the classical dynamicsof relativistic massless particles with spin s = 12χ~ (where χ = ±1 is thehelicity), we will hence introduce a certain 1-form ̟ of the Lie group W 10designed to give rise to the space of (free) motions of such particles — aswell as its prequantization (see Section 2.1).• We recall that, given a Lie group G with Lie algebra g, the canonicalsymplectic structure of the coadjoint orbit, X = G/Gµ0 , passing throughµ0 ∈ g∗ is constructed as follows [2].5 If θ is the left-invariant Maurer-Cartan1-form of G, the exterior derivative, σ = d̟, of the 1-form̟ = 〈µ0, θ〉 (1.5)descends as the canonical symplectic 2-form, ω, of X , namelyσ = π∗GX ω.• Here, the group G = W 10 is, as above-mentioned, the direct productof Σ6 and spacetime E4. The leaves of the characteristic distribution of σconstitute the space of motions of the particle, namely X = W 10/ ker σ.As to the spacetime projections of these leaves via πWE : W10 → E4, theyare interpreted as the world-sheets of the particle.5The local notation G should not be confused with that of the above spinorial metric!4Denote by e(3, 1) the Lie algebra of the Poincare´ group, and choose nowthe origin µ0 = (M0, P0) ∈ e(3, 1)∗ of X as [10, 1]M0 = χ~γ1γ2 & γ(P0) = γ3 + γ4.Tedious calculation yields the 1-form (1.5), namely6̟ = −P · dR + χ~ζdζi(1.6)where P is as in (1.4).We then find that ker σ is 4-dimensional, that isδ(ζ, R) ∈ ker σ ⇐⇒ δζ =iχ~γ(δR)Πζ + iλζδR ∈ P⊥(1.7)with λ ∈ R.This implies, via (1.4), that δP = 0, and shows that the motions takeplace on null affine hyperplanes P⊥ in R3,1; the particle having a constantmomentum, P , it is strikingly delocalized on the world-sheets P⊥ [10, 4, 5].7• It hence turns out that the space of motions, X , is 6-dimensional, viz.,X6 ∼= T ∗C+ (1.8)where C+ ∼= R3 \ {0} is the punctured future light-cone described by themomentum P .What about the explicit form of symplectic 2-form, ω, of X6? Its expres-sion is computed in a Lorentz frame where P = (p, |p|) 6= 0, and R = (r, t).The would-be 3-position (spanning the fibres of X6 → C+) is then x = g/|p|where g is the boost-momentum [4]. The sought expression is [10]ω = dp ∧ dx− s surf & s = 12χ~ (1.9)where surf is the surface 2-form of S2 described by the direction u = p/|p|.86One can write, equivalently, ̟ = −PµdRµ− iχ~ ζadζa where ζa = Gabζb for all spinorindices a = 1, . . . , 4.7The projection to the evolution space V 9 of the leaves of kerσ are the Wigner-Souriautranslations of Ref. [4]; see Footnote 11.8The coordinates x = (x1, x2, x3) do not Poisson-commute; they are sometimes calledthe Pryce coordinates.5The projection πWX : W10 → X6 is given by πWX(ζ, R) = (p,x) wherep =1√2ζ ′σζ ′ & x = r (t = 0). (1.10)We see that the energy E = |p| is now given by|p| = 1√2‖ζ ′‖2 (1.11)in spinorial terms.2 Geometric quantization of the modelWe propose now to quantize, following [10], the above classical model usingthe technique of geometric quantization.2.1 PrequantizationLet us recall that a prequantum manifold above a symplectic manifold(X,ω) is a principal circle-bundle πY X : Y → X with connection α/~ whosecurvature descends to X as ω/~, i.e., such that dα = π∗Y Xω [10].9 See also[8, 9, 11] for an equivalent definition in terms of line-bundles.10• This (not so well-known) intermediate geometric structure, (Y, α), in-between the Poincare´ group, (W 10, ̟), and the space of classical motions,(X6, ω), will actually prove crucial in the derivation of the massless Diracequations describing the quantum chiral particles in Minkowski spacetime.This remark applies, of course, to other prequantizable homogeneous sym-plectic manifold physically relevant to describe elementary particles.• Let us rewrite, for convenience, the 1-form (1.6) ofW 10 in the new guise̟ = R · dP + χ~ZdZiwhere Z = e−i~P ·Rζ. (2.1)9We will denote by z → zY the U(1)-action on Y .10Alternatively, prequantization can be understood in terms of the inequivalent formsof the “Dirac-Feynman factor” exp(iS/~), where S is the classical action [6, 7].6We now claim that Y 7 = W 10/(ker̟∩ker d̟) is the sought prequantumbundle over (X6, ω), endowed with the prequantum 1-form α such that̟ = π∗WY α.11 As in [10], the prequantum bundle, Y 7, is readily identifiedwith the section of W 10 described by the pairs (Z ′,x) whereZ ′′ =Z ′‖Z ′‖2 & x = r (t = 0) (2.2)with Z ′ ∈ C2 \ {0}, and r ∈ R3. In view of (2.1) and (2.2), this immediatelyyieldsα = −x · dp− χ~ d|p|2i|p| + χ~Z ′′dZ ′i(2.3)where p ∈ R3 \ {0} (see (1.10)) reads nowp =1√2Z ′σZ ′. (2.4)The 1-form α/~ is clearly a U(1)-connection form; indeed dα = π∗Y Xωwhere ω is as in (1.9), and α(δ(Z ′,x)) = ~ where δ(Z ′,x) = (iχZ ′, 0) is thefundamental vector field of the U(1)-action on Y 7. The projection πY X :C2 \ {0} × R3 → R3 \ {0} × R3 is therefore given by πY X(Z ′,x) = (p,x).This is summarized by the following diagramsW 10πWY−−−→ Y 7πWVy yπYXV 9πV X−−−→ X6πV EyE4(ζ, R)πWY−−−→ (Z ′,x)πWVy yπY X(P,Q,R)πVX−−−→ (p,x)πV EyR11Let us mention that the characteristic distribution ker̟∩ker d̟ of (W 10, ̟) is givenby (1.7) with λ = 0. It is therefore 3-dimensional; its foliation corresponds to Wigner-Souriau translations [4] along P⊥, so that Y 7 ∼= W 10/R3. By construction, the 1-form ̟of W 10 descends to Y 7 as the 1-form α given by (2.3). N.B. It is easily verified that ζ′ (aswell as Z ′!) is a first-integral of the characteristic distribution of ̟; it hence passes to Y 7.72.2 Geometric quantization and the Weyl equationAs every quantization procedure, geometric quantization needs the intro-duction of wave functions serving to provide on the one hand solutions of thesought wave equations, and, on the other hand, unitary irreducible represen-tations of the classical symmetry group at hand, e.g., the Poincare´ group.• Geometric quantization primarily views wave functions as smoothU(1)-equivariant complex-valued functions, namely such thatΨ : Y → C & z∗YΨ = zΨ (2.5)for all z ∈ U(1); see Footnote 9.12• The next fundamental ingredient needed in geometric quantization isa polarization.13 Here, we will merely consider a real polarization of asymplectic manifold (X2n, ω), i.e., a (fiberwise) n-dimensional isotropic sub-bundle F ⊂ TX , i.e., verifying ω(F, F ) = 0.This choice generalizes that of the “position” or “momentum” represen-tation in quantum mechanics. The F -polarized wave functions of (Y, α)over (X,ω) are the F ♯-constant wave function (2.5), viz.,F ♯Ψ = 0 (2.6)where F ♯ is the horizontal lift of F to (Y, α); see also [8, 9, 11].• Returning to our model (X6, ω), we choose the most natural polariza-tion, i.e., the vertical polarization, F , whose leaves, p = const ., are thefibres of the bundle X6 → C+; see (1.8).14 It turns out that F = ker φ, whereφ =dp1 ∧ dp2 ∧ dp3|p| (2.7)is the polarizer [2, 3] defined by the (pull-back of the) canonical Poincare´-invariant 3-form of the punctured future light-cone C+. Note that F is indeeda polarization iffω ∧ φ = 0 & rankφ = 3.12This definition legitimizes the overall phase factor affecting quantum wave functionsas coming from the consideration of a circle bundle above the space of classical motions.13This is a geometrical object of symplectic geometry; no confusion with the opticalnotion of polarization of light!14In view of (1.9), we clearly have ω(F, F ) = 0.8An equivalent approach has been devised in [9, 11] to geometrically quantize,e.g., relativistic massless particle models.• We are ready to express the F -polarized wave functions of the modelusing this formalism. It has been shown [3] in full generality that those arethe solutions Ψ ∈ C∞(Y,C) of the differential equationφ ∧DΨ = 0 where DΨ = dΨ− i~αΨ (2.8)is the covariant derivative of the wave function, Ψ, of (Y, α). These wavefunctions, Ψ, are thus “covariantly constant” along the polarizer, φ.• Let us first examine the case, χ = −1, of negative helicity in the abovemodel. We find, with the help of (2.3) and (2.7), that the solutions, Ψ, ofEquation (2.8) must satisfy15dp1 ∧ dp2 ∧ dp3 ∧[ ∂Ψ∂xdx+∂Ψ∂pdp+∂Ψ∂Z ′dZ ′ +∂Ψ∂Z ′′dZ ′′+(i~x · dp− d|p|2|p| − dZ′′Z ′)Ψ]= 0which gives ∂Ψ/∂x = 0, ∂Ψ/∂Z ′ = 0, and ∂Ψ/∂Z ′′ = Z ′Ψ. This entails thatZ ′′ ∂Ψ/∂Z ′′ = Z ′′Z ′Ψ = Ψ; hence Ψ is homogeneous of degree 1 in Z ′′, viz.,Ψ(Z ′,x) = Z ′′ψ′(P ) for some ψ′ ∈ C∞(C+,C2). It thus follows thatΨ(Z ′,x) = Z ψ(P ) where ψ(P ) =(2Z ′0)Ψ(Z ′,x). (2.9)Using (1.1), (1.11), and (2.4), one furthermore shows that16γ(P ) =1√2(0 Z ′Z ′‖Z ′‖2 − Z ′Z ′ 0).We have just proven, if χ = −1, that the space of so-polarized wavefunctions (2.9) is isomorphic to the space of spinors ψ ∈ C∞(C+,C2,2)satisfyingγ(P )ψ(P ) = 0 & γ5ψ(P ) = iχψ(P ). (2.10)• The fundamental equations in (2.10) hold for χ = ±1 [10].15Easy calculation leads to ZdZ = −dZ ′′Z ′ − d|p|/(2|p|).16Indeed, if ξ = Z ′/‖Z ′‖ ∈ S3, one has σ(u) = 2ξξ − 1.9• To finish, let us introduce the partial Fourier transformψ˜(R) =∫C+ψ(P ) ei~P ·R φ (2.11)where ψ is now supposed to be compactly supported on the forward nulllight-cone, C+.Routine computation shows, with recourse to (2.10) and (2.11), that ψ˜ ∈C∞(E4,C2,2) is indeed the general solution of theWeyl equation of helicityχ = ±1, namelyγµ∂µψ˜ = 0 & γ5ψ˜ = iχψ˜. (2.12)The genesis of the Weyl equations by geometric quantization is originallydue to Souriau [10]. It has later been revisited by Simms & Woodhouse[9, 11] using the spinorial/twistorial formalism of Penrose; see also [3].• The universal covering, G, of the Poincare´ group consists of the matricesg =(Ad(A) γ(C)0 1)where A ∈ Spin(3, 1) & C ∈ R3,1.Since G ∼= W 10, its left-action reads g · (ζ, R) = (Aζ, ̺(A)R+C). The latterinduces, via pull-back, a natural (right-)action on the wave function (2.9) sothat g ·ψ(P ) = A−1ψ(̺(A)P ). Taking into account the Lorentz-invariance ofthe polarizer (2.7) in the integral (2.11), immediately leads to the following(anti-)representation of the universal covering of the Poincare´ group onthe solutions (2.11) of the Weyl equation (2.12), viz.,g · ψ˜(R) = A−1ψ˜(̺(A)R + C) (2.13)for all g = (A,C) ∈ Spin(3, 1)⋉ R3,1.• Let us express our Weyl system (2.12) in terms of 2-component spinors,ψ˜χ ∈ C∞(E4,C2) corresponding to helicity χ = ±1. Using the representa-tion (1.1) of the gamma matrices, those show up as followsψ˜ =(ψ˜−ψ˜+).10The Dirac operator, D = γµ∂µ, then readsD =(0 −σj∂j + ∂tσj∂j + ∂t 0).so that Dψ˜ = 0 in (2.12) yields the equivalent form of theWeyl equation,namelyσj∂jψ˜χ = χ ∂tψ˜χ (2.14)with ∂j = ∂/∂rj for all j = 1, 2, 3, where r = (r1, r2, r3) stand for theposition coordinates, i.e., the bona fide spatial translation coordinates ofthe Poincare´ group.Acknowledgements: It is a pleasure to thank P. Horva´thy for his helpand encouragement.References[1] C. Duval, “On the prequantum description of spinning particles in anexternal gauge field”, in Proc. Coll. Int. CNRS “Differential GeometricalMethods in Mathematical Physics”, Aix en Provence-Salamanca 1979,(P.L. Garcia, A. Pe´rez-Rendon, J.-M. Souriau Eds), pp 49–66, LectureNotes in Mathematics 836, Springer-Verlag (1980).[2] C. Duval, “On the polarizers of compact semi-simple Lie groups. Appli-cations”, Ann. Inst. H. Poincare´ A 34 (1981) 95–115.[3] C. Duval and J. Elhadad, “Geometric Quantization and Localizationof Relativistic Spin Systems”, in Proc. AMS of the 1991 “Joint Sum-mer Research Conference on Mathematical Aspects of Classical FieldTheory”, Seattle 1991, (M.J. Gotay, J.E. Marsden, V.E. Moncrief Eds),Contemporary Mathematics 132 (1992), 317–330.[4] C. Duval and P. A. Horvathy, “Chiral fermions as classical masslessspinning particles,” Phys. Rev. D91 045013 (2015).[5] C. Duval, M. Elbistan, P. A. Horvathy and P.-M. Zhang, “Wigner-Souriau translations and Lorentz symmetry of chiral fermions,” Phys.Lett. B 742 (2015) 32211[6] P. A. Horva´thy, “Classical action, the Wu-Yang phase factor and pre-quantization,” Proc. Int. Coll. on Diff. Geom. Meths. in Math. Phys.,Aix-en Provence ’79. Ed. Souriau. Springer Lecture Notes in Mathemat-ics 836, 67 (1980).[7] P. A. Horva´thy, “Prequantization From Path Integral Viewpoint,” Lec-ture Notes in Mathematics 905 (1982) 197.[8] B. Kostant, “On certain unitary representations which arise from aquantization theory,” in Group representations in mathematics andphysics (V. Bargmann, Ed.), pp. 237–253, Proceedings Battelle Seat-tle 1969 Rencontres; LNP 6, Springer-Verlag, Berlin, 1970.[9] D. J. Simms and N. M. J. Woodhouse, Lectures on Geometric Quanti-zation, Lecture Notes in Physics 53, Springer-Verlag Berlin-Heidelberg-New York, 1976.[10] J.-M. Souriau, Structure des syste`mes dynamiques, Dunod (1970),c©1969). Structure of Dynamical Systems. A Symplectic View of Physics,Birkha¨user, Boston (1997).[11] N. M. J. Woodhouse, Geometric quantization. Second edition, OxfordUniversity Press, New York, 1992.12

A recollection of Souriau's derivation of the Weyl equation via geometric quantization

Abstract

Similar works

Full text

Available Versions

HAL AMU