Relativistic corrections to the algebra of position variables and spin-orbital interaction

In the framework of vector model of spin, we discuss the problem of a covariant formalism \cite{Pomeranskii1998} concerning the discrepancy between relativistic and Pauli Hamiltonians. We show how the spin-induced non commutativity of a position accounts the discrepancy on the classical level, without appeal to the Dirac equation and Foldy-Wouthuysen transformation.


I. INTRODUCTION
In series of previous works [1][2][3][4] we developed a Poincare-invariant variational formulation describing particle with spin. This classical model provides a unified description of both Frenkel and BMT equations [5]. The latter are considered as a basic tool in the analysis of the polarization precession measurements [6]. In [7] we extended the variational formulation to the general relativity, where the classical models of a spinning particle are widely used to describe a rotating body in pole-dipole approximation [8][9][10][11][12][13][14][15][16]. Another possible application can be related with the kinetic theory of chiral medium, where, in the regime of weak external fields and weak interactions between spinning (quasi)-particles, each particle can be considered as moving along a classical trajectory [17].
For variational formulations provide a starting point to the canonical quantization [18], they have incredible theoretical importance connecting classical and quantum descriptions of nature. Canonical quantization of the free spinning particle (within our variational formulation, [19]) leads to the one-particle relativistic quantum mechanics with positive-energy states. It also identifies the non-commutative Pryce's d-type center of mass operator [19] 1 as the quantum observable which corresponds to the classical position variable. The noncommutative position variables were constructed already by Pryce [20]. He shown that coordinates of the relativistic center-of-mass have to obey non-trivial Poisson brackets. As a result, the corresponding quantum observables do not commute. Therefore a physically meaningful position operators of a spin-1/2 particle should be non-commutative.
Recent theoretical studies revive Snyder's attempts [21] to solve fundamental physical problems by introducing non-commutativity of the space [22]. It is believed that this fundamental non-commutativity may be important at Plank length scale λ P . Extensive studies of non-commutativity cover both classical and quantum theories, as well relativistic and non-relativistic situations. Postulating non-commutative deformation of position operators [31] one can study physical consequences and estimate possible effects. Calculations of the hydrogen spectrum corrections strongly limit possible non-commutativity of coordinate parameters in the Dirac equation [26][27][28][29][30].
In the present work we study effects of a non-commutativity of Pryce's d-type center of mass (at both classical and quantum levels) in the description of electron interacting with an electromagnetic background. Our considerations extend results of [19] towards a quantization of interacting spinning particle.
In the free theory, different candidates for the position operator are almost indistinguishable. All these operators obey the same Heisenberg equations (uniform rectilinear motion), and the difference in their expectation values is of Compton wave-length order, λ C . In the interacting case, the problem of the identification of quantum position observables becomes more complicated. 2 Fleming [25] noted: "The simplest form of interaction is that due to a static potential which may be expressed in terms of the position operator of the particle. For a relativistic particle, however, the important question arises of which position operator should be used. The conventional approach, in which the position operator is assumed to be local, forces the choice of the center of spin." 3 He also observed, that a formal substitution of Pryce d-type operator into the potential leads to some reasonable corrections: "The first correction term to a spherically symmetric local potential will be recognized as the spin-orbit coupling that Thomas derived many years ago as a consequence of classical relativity and which appears in the nonrelativistic limit of the Dirac equation for spin particles." Analogous situation was observed in general relativity, [35,[37][38][39] where a formal substitution of a non-local position variable into potential results in correct equations of motion for the spinning particle. Restricting ourselves to the case of special relativity, in the present work we provide some theoretical grounds for such substitution.
The paper is organized as follows. In Sect. 2 we formulate the problem of covariant formalism [35] concerning the discrepancy between relativistic and Pauli Hamiltonians. In Sect. 3 we give a brief description of the vector model for the classical description of a relativistic spinning particle. In Sect. 4 we consider canonical quantization of the model in the physical-time parametrization and realize classical algebra of Dirac brackets by quantum operators in the case of a stationary electro-magnetic background. This realization deforms the covariant Hamiltonian and at low energies gives Pauli Hamiltonian with correct spin-orbital interaction. In the conclusion we discuss the obtained results.

II. MODEL INDEPENDENT DISCUSSION OF THE QUANTUM AND CLASSICAL HAMILTONIANS OF A SPINNING PARTICLE
From quantum point of view, at low energies an electron interacting with a background electromagnetic field is described by the two-component Schrödinder equation. Pauli Hamiltonian 4 includes spin-orbital and Zeeman interactionsĤ where the spin operator is proportional to σ-matrices of PauliŜ i = 2 σ i . Gyromagnetic ratio g is a coupling constant of spin with an electromagnetic field. In principle, in non-relativistic theory one can expect different coupling constants for the third and the fourth terms of the Hamiltonian. Experimental observations of the hydrogen spectrum lead to the factor g − 1 in the third term and to the factor g in the last term. Thus, the Hamiltonian explains Zeeman effect and reproduces fine structure of the energy levels of the hydrogen atom. This Hamiltonian follows also from the non-relativistic limit of the Dirac equation in the Foldy-Wouthuysen representation [18,36]. From classical point of view, models of spinning particles are based on a Lagrangian or Hamiltonian mechanics, both in the relativistic and non-relativistic regime [23]. In a covariant formulation, the spin part of the Hamiltonian describing an interaction between spin S and electromagnetic field reads We emphasize that the expression (2) follows from the analysis of all possible terms in covariant equations of motion and thus is a model-independent [35]. It can also be predicted from symmetry considerations on the level of a Hamiltonian. For instance, if we take the Frenkel spin-tensor S µν with the covariant condition S µν P ν = 0, the only Lorentz-invariant combination that could give the desired terms written in (2) is For the classical gyromagnetic ratio g = 2, the classical spin-orbital interaction in (2) differs by the famous and troublesome factor 5 of 1 2 from its quantum counterpart in (1). It seems that quantization of H spin−em−cov will not reproduce quantum behavior given byĤ spin−em . The issue about this difference was raised already in 1926 [34] and still remains under discussion [35].
pares Pauli, Foldy-Wouthuysen, Czachor, Frenkel, Chakrabarti, Pryce, and Fradkin-Good spin operators in different physical situations and concluded that interaction with electromagnetic potentials allows to distinguish between various spin operators experimentally.
In principle, Hamiltonian H spin−em can be obtained, if one impose a non covariant supplementary condition on spin, 2S i0 P 0 + S ij P j = 0, where P 0 ∼ −mc in the leading approximation. On a first glance, any covariant spinsupplementary condition [8,34,[43][44][45] would give H spin−em−cov and the discrepancy factor of 1 2 . In the next section we study this issue in the framework of vector model of a spinning particle [4]. We show that the vector model provides an answer on a pure classical ground, without appeal to the Dirac equation. In a few words, it can be described as follows. The relativistic vector model involves a second-class constraints, which should be taken into account by passing from the Poisson to Dirac bracket. The emergence of a higher non linear classical brackets that accompany the relativistic Hamiltonian (2) is a novel point, which apparently has not been taken into account in literature. If we pretend to quantize the model, it is desirable to find a set of variables with the canonical brackets. The relativistic Hamiltonian (2), when written in the canonical variables, just gives (1).

III. VECTOR MODEL OF SPINNING PARTICLE IN THE PARAMETRIZATION OF PHYSICAL TIME.
To find the classical brackets that accompany H cov we need a systematically developed model of a spinning particle. Here we consider the vector model and briefly describe the construction of the Hamiltonian and the brackets in a stationary electromagnetic field. For a detailed discussion of the model, see [4].
Configuration space of the vector model of spinning particle is parameterized by a point x µ (τ ) of a world-line and a vector ω µ (τ ) attached to that point. The configuration-space variables are taken in an arbitrary parametrization τ of the world-line. The conjugate momenta of the variables are denoted by p µ and π µ , correspondingly. Frenkel spin-tensor in the vector model is a composite quantity, S µν = 2(ω µ π ν − ω ν π µ ). The free Lagrangian can be written in a number of equivalent forms [11,19]. To describe the spin-field interaction through the gyromagnetic ratio g, we use the Lagrangian with an auxiliary variable λ(τ ) where Dω µ ≡ω µ − λ eg 2c F µν ω ν . The auxiliary variable provides a homogeneous transformation law of Dω under the reparametrizations, D τ ′ ω = dτ dτ ′ Dω. The matrix N µν is the projector on the plane orthogonal to ω ν , N µν = η µν − ωµων ω 2 . The parameter m is mass, while α determines the value of spin. The value α = 3 2 4 is fixed by quantization conditions and corresponds to an elementary spin one-half particle. In the spinless limit, α = 0 and ω µ = 0, the functional (3) reduces to the well known Lagrangian of the relativistic particle, 1 2λẋ 2 − λ 2 m 2 c 2 + e c Aẋ. Frenkel considered the case g = 2 and found approximate equations of motion neglecting quadratic and higher terms in spin, fields and field gradients. Equations of motion obtained from (3) coincide with those of Frenkel in this approximation [34].
To find relativistic Hamiltonian in the physical-time parametrization 6 , we use the Hamiltonian action associated with (3). This reads [4], dτ pẋ + πω − λ i T i , where λ i are Lagrangian multipliers associated with the primary constraints T i . The variational problem provides both equations of motion and constraints of the vector model in an arbitrary parametrization. Using the reparametrization invariance of the functional, we take physical time as the evolution parameter, τ = x 0 c = t, then the functional reads whereP 0 = p 0 − e c A 0 and P i = p i − e c A i is U (1) -invariant canonical momentum. We can treat the term associated with λ as a kinematic constraint of the variational problem. Following the known prescription of classical mechanics, we solve the constraint, and substitute the result back into Eq. (4), this gives an equivalent form of the functional where the substitution (5) is implied in the last two terms as well. The sign in front of the square root (5) was chosen according to the right spinless limit, L = −mc −ẋ µẋ µ . The expression in square brackets is the Hamiltonian. The variational problem implies the first-class constraints T 2 ≡ ωπ = 0, T 5 ≡ π 2 − α ω 2 = 0. They determine gauge symmetries and physical observables of the theory. The quantities x i (t), P i (t) and S µν (t) have vanishing Poisson brackets with the constraints and hence are candidates for observables. The set where represents a pair of second class constraints. In all expressions below the symbol P 0 represents the function (8). The constraints imply the spin-supplementary condition as well as the value-of-spin condition S µν S µν = 8α.
To represent the Hamiltonian in a more familiar form, we take into account the second-class constraints by passing from Poisson to Dirac bracket. As the constraints involve conjugate momenta of the position x, this leads to nonvanishing brackets for the position variables. In the result, the position space is endowed, in a natural way, with a noncommutative structure which originates from accounting of spin degrees of freedom. For the convenience, an exact form of Dirac brackets of our observables is presented in the Appendix. Since the Dirac bracket of any quantity with second-class constraints vanishes, we can omit them from the Hamiltonian. The first-class constraints can be omitted as well, as they do not contribute into equations of motion for physical variables. In the result we obtain the relativistic Hamiltonian Note that the Dirac brackets (21), (20) encode the most part of spin-field interaction, on this reason we have arrived at a rather simple form for the physical Hamiltonian. Equations of motion follow from this Hamiltonian with use of the Dirac bracket 7 : dz dt = {z, H cov } D .

IV. FIRST RELATIVISTIC CORRECTIONS AND FINE STRUCTURE OF HYDROGEN SPECTRUM
To quantize our relativistic theory we need to find quantum realization of highly non linear classical brackets (21). They remain non canonical even in absence of interaction. For instance, the first equation from (21) in a free theory reads {x i , x j } = 1 2mcp 0 S ij . It is worth noting that non relativistic spinning particle [3,23] implies the canonical brackets, so the deformation arises as a relativistic correction induced by spin of a particle. Technically, the deformation arises from the fact that the constraints (7), used to construct the Dirac bracket, mixes up the space-time and inner-spin coordinates. Concerning quantum realization of the brackets in a free theory and relativistic covariance of the resulting quantum mechanics, see [19]. In an interacting theory, the explicit form of the brackets is not known. Therefore we quantize the interacting theory perturbatively, considering c −1 as a small parameter and expanding all quantities in power series. Let us consider the approximation o(c −2 ) neglecting c −3 and higher order terms. For the Hamiltonian (10) we have H cov ≈ mc 2 + P 2 2m − P 4 8m 3 c 2 − eg 8mc (F S). Since the last term is of order (mc) −1 , resolving the constraint S µν P ν = 0 with respect to S i0 we can approximate P 0 = mc, then S i0 = 1 mc S ij P j . Using this expression we obtain Due to the second and fourth terms, we need to know the operatorsP i andx i up to order c −2 , whileŜ ij ∼Ŝ should be found up to order c −1 . With this approximation, the commutators [x,x], [x,P], and [P,P] can be computed up to order c −2 , while the remaining commutators can be written only up to c −1 . Therefore, we expand the right hand sides of Dirac brackets (21) in this approximation Only the first bracket acquires a non standard form in the leading approximation. An operator realization of these brackets on the space two-component Weyl spinors readŝ By construction of a Dirac bracket, the operatorŜ i0 automatically obeys the desired commutators up to order c −1 .
The operatorx i coincides with the positive-energy part of Pryce (d) operator in the Foldy-Wouthuysen representation, see [19] for details. We substitute these operators into the classical Hamiltonian (11). Expanding A 0 (x) in a power series, we obtain an additional contribution of order c −2 to the potential due to non commutativity of the position operator The contribution has the same structure as fifth term in the Hamiltonian (10). In the result, the quantum Hamiltonian up to order c −2 readsĤ The first three terms corresponds to an increase of relativistic mass. The last two terms coincides with those in Eq.
(1). We could carry out the same reasoning in the classical theory, by asking on the new variables z ′ that obey the canonical brackets as a consequence of Eq. (12). In the desired approximation they are P i = P ′i − e c A i (x ′j ), x i = x ′i − 1 4m 2 c 2 S ′ij P ′j and S ij = S ′ij . In the result, we have shown that non-commutativity of electron's position at the Compton scale is responsible for the fine structure of hydrogen atom.

V. CONCLUSIONS
Vector model of relativistic spinning particle gives an example of a noncommutative system. In the leading approximation, the non commutative geometry induced by spin affects only the brackets of position variables, see the first equation from (12). The "parameter of noncommutativity" is being proportional to spin-tensor. As a consequence, canonical quantization of the model in leading approximation gives the Pauli Hamiltonian. Our calculations show that 1) classical interaction of spin with electromagnetic field can be described by manifestly covariant term S µν F µν , accompanied by the covariant spin-supplementary condition S µν P ν = 0; 2) phase space is endowed with a non-trivial symplectic structure (Dirac brackets), in particular, position variables become non-commutative due to non-vanishing Dirac brackets; 3) the Thomas precession automatically appears in the equations of motion [4] due to non-trivial Dirac bracket, without modification of the Hamiltonian; 4) quantization of the vector model for free electron leads to one-particle relativistic quantum mechanics with positiveenergy states. The free Hamiltonian acts in the space of two-component spinors and readsĤ phys = p 2 + m 2 c 2 , position operator of a free electron is the Pryce's d-type [19,20]; 5) quantization of the model in the case of a stationary electromagnetic background formally leads to the Hamiltonian which, up to o(c −2 ) order, coincides with the positive energy part of Dirac Hamiltonian in the Foldy-Wouthuysen representation. It would be interesting to compare high-order terms; 6) non-commutativity of position operator results in the Thomas 1/2-correction of spin-orbital interaction coming from eA 0 (x) term 8 .
In the considered approximation our HamiltonianĤ phys (F ) coincides with the Pauli Hamiltonian for the case of stationary fields. Therefore, within this approximation there is no any difference between standard and noncommutative approach to the spin-orbital interaction except a conceptual one. However, in the case of non-stationary fields the classical Hamiltonian changes form. Further studies of time-dependent electromagnetic fields and next order corrections may give suggestions for the experimental searches of effects produced by non-commutativity.
Acknowledgments The work of AAD has been supported by the Brazilian foundations CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnológico -Brasil) and FAPEMIG (Fundação de Amparoà Pesquisa do Estado de Minas Gerais -Brasil). The work of AMPM is suported by FAPEMIG (Demanda Universal 2015).
Electromagnetic field: Spin-tensor: Here S i is three-dimensional spin-vector of Frenkel and D i is dipole electric moment.