Vector Model of Relativistic Spin. (HANNOVER, SIS 16, December-2016)

Transcription

1 Vector Model of Relativistic Spin. (HANNOVER, SIS 16, December-2016) Alexei A. Deriglazov Universidade Federal de Juiz de Fora, Brasil. Financial support from SNPq and FAPEMIG, Brasil 26 de dezembro de 2016

2 (1) PLAN 1. Variational problem for vector model. Applications: 2. One-particle RQM with positive energies and Dirac equation. 3. Vector model prohibites Zitterbewegung. 4. Spin-induced non-commutativity of position variables and the problem of covariant formalism. 5. Minimal interaction with gravity and a rotating body in general relativity (Mathisson-Papapetrou- Tulczyjew-Dixon (MPTD) equations). 6. Paradoxical behavior of MPTD-particle in ultra relativistic limit. 7. Rotating body with unit gravimagnetic moment and improved MPTD-equations.

3 (2) NON RELATIVISTIC VECTOR MODEL Spin-sector: Basic variables Spin Properties Constraints Q. Mech. Ŝ i = 2 σ i [Ŝi, Ŝj ] = i ɛ ijk Ŝ k S 2 = Class. ω π = 0 Mech. {ω i, π j } = δ i j S = ω π {S i, S j } = ɛ ijk S k π 2 ω 2 = α Position sector: {x i, p j } = δ i j. Local symmetry related with first-class constraints implies: ω and π do not represent observable quantities, while S does. It obeys S 2 = α = Interaction of vector model with magnetic field yields the Pauli Hamiltonian [ m S = dt 2 ẋ2 + e ] α c Aẋ + DωNDω, i Ψ ( 1 ω 2 t = 2m (ˆp e c A)2 eg ) 2mc BŜ Ψ. x where g 2 (µ = g 2 ) is giromagnetic ratio, A = 1 2 [B r], N ij = δ ij ωiωj ω 2 the projector on the plane orthogonal to ω: N ij ω j = 0, and Dω i = ω i ge 2mc ɛ ijkω j B k. is

4 (3) RELATIVISTIC VECTOR MODEL Spin-sector: Basic variables Spin Properties Constraints Q. Mech. Ŝ i = 2 σ i [Ŝi, Ŝj ] = i ɛ ijk Ŝ k S 2 = Class. {ω i, π j } = δ ij S = ω π {S i, S j } = ɛ ijk S k ω π = 0, π 2 ω 2 = α Mech. Relat. {S µν, S αβ } = SO(1, 3) ωπ = 0, π 2 ω 2 = α Mech. {ω µ, π ν } = δ µ ν S µν = ω [µ π ν] S µν S µν = 6 2 ωp = 0, πp = 0 Frenkel spin-tensor contains non-relativistic spin: S µν = (S 0i, S ij = 2ɛ ijk S k ). Position sector: {x µ, p ν } = δ µ ν, with mass-shell constraint p 2 + m 2 c 2 = 0. Second-class constraints, which supply relativistic covariance, are taken into account with help of Dirac bracket, {A, B} D = {A, B} + 1 (mc) ({A, ωp}{πp, B} + {A, πp}{ωp, B}. Since x does not 2 commutes with the constraints, spin induces the non commutative brackets: {x i, x j } = 1 2mcp S ij. 0 Besides, they imply the Frenkel condition: S µν p ν = 0.

5 (4) LAGRANGIAN OF VECTOR MODEL Hamiltonian variational problem is unique: S H = dτ p µ ẋ µ + π µ ω µ [H 0 + λ a T a ], with H 0 = 0. Lagrangian can be obtained excluding momenta with help of equations of motion. It is difficult to produce the constraint pπ = 0: L = 1 2 [ẋ2 + ω 2 ] H = 1 2 [p2 + π 2 ]. L = 1 2 [ẋ2 + ω 2 (ẋ 2 + ω 2 ) 2 4(ẋ ω) 2 ] H = 1 2 [p2 + π 2 ], where pπ = 0. Lagrangian is (N is the projector on the plane orthogonal to ω µ : N µν η µν ωµων ω, N 2 µν ω ν = 0 ωπ = ωp = 0) L(x, ω, ẋ, ω) = 1 m2 2 c 2 α ω ẋnẋ ωn ω + [ẋnẋ + ωn ω] 2 4(ẋN ω) 2 = [ 2 ] 1 4λ ẋnẋ + ωn ω [ẋnẋ + ωn ω] 2 4(ẋN ω) 2 λ 2 (m2 c 2 α ω ) = 2 (mc) 2 α ω 2 (1 λ 2 ) 1 [ ẋnẋ ωn ω + 2 λẋn ω ] ω 0 = mc η µν ẋ µ ẋ ν - point particle. Lagrangian without auxiliary variables is presented in the first line. There are equivalent Lagrangians with one or another square root removed with help of an auxiliary variable. Lagrangian with λ is convenient to introduce spin-em interaction through magnetic moment, while the Lagrangian with λ is used to introduce spin-gravity interaction through gravimagnetic moment. In the spinless limit this gives the standard lagrangian of point particle.

6 (5) INTERACTION e Minimal interaction with EM field: c A µẋ µ ; Minimal interaction with gravity (covariantization): η µν g µν, ω µ ω µ = ω µ + Γ µ αβẋα ω β ; Spin-EM field interaction through magnetic moment g: ω µ Dω µ = ω µ λ eg 2c F µν ω ν. This yields generalization of Frenkel-BMT equations to the case of arbitrary EM field as well as MPTD equations for rotating body in general relativity. Spin-gravity interaction through gravimagnetic moment µ. Minimally interacting Hamiltonian equations has the same structure if we identify R µναβ S αβ F µν. Addition of spin-em field interaction leads only to deformation of mass-shell constraint: P 2 +(mc) 2 P 2 eµ 2c (F S)+(mc)2. So we expect that spin-gravity interaction could lead to P 2 + (mc) 2 P 2 + µ 16 (SRS) + (mc)2, and write Hamiltonian variational problem with this modification S H = dτ p µ ẋ µ + π µ ω µ λ 1 [ P 2 + µr αβµν ω α π β ω µ π ν + (mc) 2 + π 2 α ω 2 ] λa T a. Excluding momenta and Lagrangian multipliers (except λ in front of πp = 0), we obtain the Lagrangian L = (mc) 2 α ω 2 ẋnkσnẋ ωnkn ω + 2 λẋnkn ω, where the effective metric is σ µν = g µν + µr α µ β ν ω α ω β, and K = (σ λ 2 g) 1.

7 (6) ONE-PARTICLE RELATIVISTIC QM WITH POSITIVE ENERGY STATES Canonical formalism. Parametrization of physical time: τ = x0 c = t. The variables xi, p i and S µν - commute with first-class constraints, so they represent observable quantities. Working with Dirac bracket for second-class constraints, {x i, x j } D = Sij 2mcp 0, {x i, p j } D = δ ij, {S µν, S αβ } D = 2 ( g α[µ S ν]β g β[µ S ν]α),... where g µν η µν pµ p ν p 2, physical Hamiltonian acquires the expected form, H ph = cp 0 = c p 2 + (mc) 2. Quantization leads to the Schrödinger equation for two-component wave function Ψ a (t, x), a = 1, 2 i dψ dt = c ˆp 2 + (mc) 2 Ψ, Ψ, Φ = d 3 xψ Φ. Due to deformed classical brackets, the naive expressions: x i and σ i, do not represent position and spin, instead we have [positive-energy parts of Pryce (d)] operators x i ˆX i = x i + 2mc(ˆp 0 +mc) ɛijk σ j ˆp k, Ŝ i = 2mc ( ˆp 0 σ i 1 (ˆp 0 +mc) (ˆpσ)ˆpi ). Are the scalar product and probability the Lorentz-invariant quantities? Ψ, ˆX i Φ and Ψ, Ŝi Φ the Lorentz-covariant quantities? Are the mean values

8 (7) PROOF OF RELATIVISTIC COVARIANCE With the QM obtained above we associate the auxiliary construction: a space of states ψ which carries a representation of Poincaré group and admits conserved four-vector µ J µ (ψ) = 0 with positive null-component. Then we define the invariant integral over space-like surface Ω P Ω = (ψ, ψ) = 1 6 Ω ɛ µαβγj µ dx α dx β dx γ, then Ω 1 = Ω 2, and P t=const = J 0 d 3 x can be identified with probability. Then we look for a map between the canonical and covariant pictures which respect their scalar products, this will prove the covariance of Ψ, Φ. Our QM is, in fact, square root of KG equation, so the natural covariant picture is the KG equation for two-component wave function (equivalent to the Dirac equation): H + can = { Ψ(t, x); Ψ, Φ = d 3 xψ Φ } H KG = { ψ KG (x µ ), J µ = 1 m 2 c 2 ( σˆpψ) σ µ σˆpφ ψ σ µ φ } H D = { ψ D (x µ ), J µ D = ψ D γ µ φ D } There are the maps V = 1 mc σˆp+i, Ψ = V ψ KG, and W, ψ KG = W ψ D between canonical and manifestly covariant formalisms, which respect the scalar products: Ψ, Φ = (ψ KG, φ KG ) = (ψ D, φ D ), and thus shows covariance of the scalar product Ψ, Φ. (AAD, A. Pupasov-Maksimov, 2014)

9 (8) Vector model of spinning particle yields an example of one-particle relativistic quantum mechanics with positive-energy states. Dirac equation was used here as an auxiliary tool which allows to confirm relativistic covariance of the obtained QM. So we do not need to search for interpretation of negative-energy states (and so on) of the Dirac equation.

10 (9) THE PROBLEM OF COVARIANT FORMALISM - Goudsmit, Uhlenbeck 1926, Frenkel 1926, Thomas 1927,..., Khriplovich Khriplovich-Pomeransky for the case of gravity. ( ) 1 Electron in EM field: H phen = 2m (ˆp e c A)2 ea 0 + e(g 1) eg 2m 2 c2 Ŝ[ˆp E] 2mc BŜ. The fine structure of hydrogen energy levels fixes the factor g 1 in the third term, while Zeeman effect requires the factor g in the last term. For g = 1 the difference between phenomenological and covariant formalisms is 1 2 factor H phen e mc H rel e 4mc F µνs µν = e mc [ 1 2mc S[p E] BS], and standard Poisson brackets. [ 1 mc S[p E] BS], {x i, x j } D = 1 2m 2 c S ij + O ( ) 1 2 c. 3 The origin of the descripancy became more clear if we observe that H phen is accompanied by standard brackets, while vector model leads to covariant Hamilltonian with complicated brackets. So, suggestion: we work with the same theory using two different parametrizations. In leading approximation on 1 c the only non canonical bracket is {x, x} D. If in the covariant formalism we construct the variables with canonical brackets and substitute them into H rel, it turns into H phen. So, vector model solves the problem on a pure classical ground, without appeal to the Thomas precession, Dirac equation or Foldy-Wouthuysen transformation. The right Hamiltonian appears due to spin-induced non commutativity of position variables.

11 (10) MINIMAL INTERACTION WITH GRAVITY g µν and MPTD EQUATIONS OF ROTATING BODY (Mathisson (1937), Papapetrou (1951), Tulczyjew (1959), Dixon (1964)) L = 1 2 m2 c 2 α ω 2 ẋnẋ ωn ω + [ẋnẋ + ωn ω] 2 4(ẋN ω) 2, this implies ẋ µ = ẋgẋ (mc) 2 T 1µ νp ν, P 2 + (mc) 2 = 0, S 2 = 8α. P µ = 1 4 R µναβẋ ν S αβ, S µν = 2P [µ ẋ ν], S µν P ν = 0, MPTD equations of rotating body So, minimal interaction of vector model implies MPTD-equations. We estimate three-dimensional acceleration of the body as v 2 c 2. Excluding momenta P µ, we obtain second-order equation for trajectory. Denoting θ µν = R µναβ S αβ, we have [ ] T µ νẋ ν = 1 ẋ(g + h)ẋ 4mc (θẋ)µ, compare with geodesic: ẋ = 0. ẋgẋ Interaction of spin with metric produces tetrad field T µ ν = δ µ ν 1 8m 2 c S µσ R 2 σναβ S αβ and effective metric g µν G µν : G µν T α µg αβ T β ν = g µν + h µν (S). If G is the metric seen by the particle, the relativistic contraction factor ẋ(g + h)ẋ c 2 v 2 determines the behavior of the particle in ultra-relativistic limit. In the manifestly covariant equations it accompanies the 1 derivative: d c 2 v 2 dτ D, and can be easily controlled: DDx = f[dx, (Dx) 2, (Dx) 3,...] a (c 2 v 2 )f[dx, (Dx) 2, (Dx) 3,...], a (c 2 v 2 ) 2 f.

12 (11) PARADOXAL BEHAVIOR OF MPTD-PARTICLE AS v 2 c 2. Using the definition of three-dimensional acceleration compatible with coordinate-independence of speed of light, we have Point particle (geodesic in g): DDx f[(dx) 2 ] a finite, a 0. MPTD-particle (in G): DDx f[(dx) 3 ] a, a 0. Improved MPTD-particle (in g): DDx f[(dx) 2 ] a finite, a 0. Vector model allows to construct equations with improved behavior in ultra-relativistic limit. They arise if we introduce non minimal interaction of spin with gravity through gravimagnetic moment µ: L = (mc) 2 α ω 2 ẋnkσnẋ ωnkn ω + 2 λẋnkn ω, where the effective metric is σ µν = g µν + µr α µ β ν ω α ω β, and K = (σ λ 2 g) 1. The model is consistent for any value of µ and implies equations of the following remarkable structure: DDx = f[dx, (Dx) 2, ( µ 1)(Dx) 3 ] where D = 1 ẋ(g+( µ 1)h)ẋ d dτ. If we quantize the gravimagnetic moment: µ = 1, the spinning particle has an expected behavior as v 2 c 2 with respect to original metric g.

13 (12) COMPARE THE HAMILTONIAN EQUATIONS: MPTD eqs.: P µ = 1 4 (θẋ)µ, S µν = 2(P µ ẋ ν P ν ẋ µ ), S µν P ν = 0; Improved eqs.: P µ = 1 4 (θẋ)µ, They have different dynamics for small velocities. S µν = ẋgẋ 4m rc (θs)[µν], S µν P ν = 0; CONCLUSION. Vector model (VM) of spinning particle clarifies a number of issues presented in theoretical description of relativistic spin: 1. VM yields one particle relativistic quantum mechanics and clarifies its relation with the Dirac equation. 2. VM solves the problem of covariant formalism. 3. VM admits non minimal interaction of spin with gravity through gravimagnatic moment and predicts its quantization: µ = 1. As compare with MPTD equations, the resulting equations seem to be more promising candidate for description of relativistic rotating body. THANKS!