Recent progress on the description of relativistic spin: vector model of spinning particle and rotating body with gravimagnetic moment in General Relativity

We review the recent results on development of vector models of spin and apply them to study the influence of spin-field interaction on the trajectory and precession of a spinning particle in external gravitational and electromagnetic fields. The formalism is developed starting from the Lagrangian variational problem, which implies both equations of motion and constraints which should be presented in a model of spinning particle. We present a detailed analysis of the resulting theory and show that it has reasonable properties on both classical and quantum level. We describe a number of applications and show how the vector model clarifies some issues presented in theoretical description of a relativistic spin: A) One-particle relativistic quantum mechanics with positive energies and its relation with the Dirac equation and with relativistic {\it Zitterbewegung}; B) Spin-induced non commutativity and the problem of covariant formalism; C) Three-dimensional acceleration consistent with coordinate-independence of the speed of light in general relativity and rainbow geometry seen by spinning particle; D) Paradoxical behavior of the Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body in ultra relativistic limit, and equations with improved behavior.


I. INTRODUCTION
Basic notions of Special and General Relativity have been formulated before the discovery of spin, so they describe the properties of space and time as they are seen by spinless test-particle. It is natural to ask, whether these notions remain the same if the spinless particle is replaced by more realistic spinning test-particle. To analyze this issue, it is desirable to have a systematic formalism for semiclassical description of spinning degrees of freedom in relativistic (Poincaré invariant) and generally covariant theories.
Search for the relativistic equations that describe evolution of rotational degrees of freedom and their influence on the trajectory of a rotating body represents a problem with almost centenary history [1][2][3][4][5][6][7]. The equations are necessary for current applications of general relativity on various space-time scales: for analysis of Lense-Thirring precession [8], for accounting spin effects in compact binaries and rotating black holes [9,10], and in discussion of cosmological problems, see [11] and references therein. Closely related problem consists in establishing of classical equations that could mimic quantum mechanics of an elementary particle with spin in a semiclassical approximation [12][13][14][15][16]. While the description of spin effects of relativistic electron is achieved in QED on the base of Dirac equation, the relationship among classical and quantum descriptions has an important bearing, providing interpretation of results of quantum-field-theory computations in usual terms: particles and their interactions. Semiclassical understanding of spin precession of a particle with an arbitrary magnetic moment is important in the development of experimental technics for measurements of anomalous magnetic moment [17,18]. In accelerator physics [19] it is important to control resonances leading to depolarization of a beam. In the case of vertex electrons carrying arbitrary angular momentum, semiclassical description can also be useful [20]. Basic equations of spintronics are based on heuristic and essentially semiclassical considerations [21]. It would be very interesting to obtain them from first principles, that is from equations of motion of a spinning particle.
Hence the further development of classical models of relativistic spinning particles/bodies represents an actual task. A review of the achievements in this fascinating area before 1968 can be found in the works of Dixon [5] and in the book of Corben [15]. Contrary to these works, where the problem was discussed on the level of equations of motion, our emphasis has been placed on the Lagrangian and Hamiltonian variational formulations for the description of rotational degrees of freedom. Taking a variational problem as the starting point, we avoid the ambiguities and confusion, otherwise arising in the passage from Lagrangian to Hamiltonian description and vice-versa. Besides, it essentially fixes the possible form of interaction with external fields. In this review we show that so called vector model of spin represents a unified conceptual framework, allowing to collect and tie together a lot of remarkable ideas, observations and results accumulated on the subject after 1968.
The present review article is based mainly on the recent works [22][23][24][25][26][27][28][29][30][31]. In [22] we constructed final Lagrangian for a spinning particle with an arbitrary magnetic moment. In [23] we presented the Lagrangian minimally interacting with gravitational field, while in [24,25] it has been extended to the case of non minimal interaction through the gravimagnetic moment. In all cases, our variational problem leads to both dynamical equations of motion and appropriate constraints, the latter guarantee the fixed value of spin, as well as the spin supplementary condition S µν p ν = 0. The works [25][26][27][28][29][30][31] are devoted to some applications of the vector model to various classical and quantum mechanical problems.
We have not tried to establish a variational problem of the most general form [32,33]. Instead, the emphasis has been placed on the variational problem leading to the equations which are widely considered the most promising candidates for description of spinning particles in external fields. For the case of electromagnetic field, the vector model leads to a generalization of approximate equations of Frenkel and Bargmann, Michel and Telegdi (BMT) to the case of an arbitrary field. Here the strong restriction on possible form of equations is that the reasonable model should be in correspondence with the Dirac equation. In this regard, the vector model is of interest because it yields a relativistic quantum mechanics with positive-energy states, and is closely related to the Dirac equation.
Concerning the equations of a rotating body in general relativity, the widely assumed candidates are the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations. While our vector model has been constructed as a semiclassical model of an elementary spin one-half particle, it turns out to be possible to apply it to the case: the vector model with minimal spin-gravity interaction and properly chosen parameters (mass and spin, see below), yields Hamiltonian equations equivalent to the MPTD equations. In the Lagrangian counterpart of MPTD equations emerges the term, which can be thought as an effective metric generated along the world-line by the minimal coupling. This leads to certain problems if we assume that MPTD equations remain applicable in the ultra-relativistic limit. In particular, three-dimensional acceleration of MPTD particle increases with velocity and becomes infinite in the limit. Therefore we examine the non-minimal interaction, this gives a generalization of MPTD equations to the case of a rotating body with gravimagnetic moment [108]. We show that a rotating body with unit gravimagnetic moment has an improved behavior in the ultra-relativistic regime and is free from the problems detected in MPTD-equations.

II. LAGRANGIAN FORM OF MATHISSON-PAPAPETROU-TULCZYJEW-DIXON EQUATIONS OF A ROTATING BODY
Equations of motion of a rotating body in curved background formulated usually in the multipole approach to description of the body, see [34] for the review. In this approach, the energy-momentum of the body is modelled by a set of quantities called multipoles. Then the conservation law for the energy-momentum tensor, ∇ µ T µν = 0, implies certain equations for the multipoles. The first results were reported by Mathisson [1] and Papapetrou [3]. They have taken the approximation which involves only first two terms (the pole-dipole approximation). A manifestly covariant equations were formulated by Tulczyjew [4] and Dixon [5]. In the current literature they usually appear in the form given by Dixon (the equations (6.31)-(6.33) in [5]), we will refer them as Mathisson-Papapetrou-Tulczyjew-Dixon equations.
We discuss MPTD-equations in the form studied by Dixon 1 ∇S µν = 2(P µẋν − P νẋµ ), (3) S µν P ν = 0. (4) In the multipole approach, x µ (τ ) is called representative point of the body, we take it in arbitrary parametrization τ (contrary to Dixon, we do not assume the proper-time parametrization 2 , that is we do not add the equation g µνẋ µẋν = −c 2 to the system above). S µν (τ ) is associated with inner angular momentum, and P µ (τ ) is called momentum. The first-order equations (2) and (3) appear in the pole-dipole approximation, while the algebraic equation (4) has been added by hand. In the multipole approach it is called the spin supplementary condition (SSC) and corresponds to the choice of representative point as the center of mass [4,5,7]. After adding the equation (4) to the system, the number of equations coincides with the number of variables.
Since we are interested in the influence of spin on the trajectory of a particle, we eliminate the momenta from MPTD equations, thus obtaining a second-order equation for the representative point x µ (τ ). The most interesting property of the resulting equation is the emergence of an effective metric G µν instead of the original metric g µν .
Let us start from some useful consequences of the system (2)- (4). Take derivative of the constraint, ∇(S µν P ν ) = 0, and use (2) and (3), this gives the expression which can be written in the form Contract (5) with g µαẋ α . Taking into account that (Pẋ) < 0, this gives (Pẋ) = − √ −P 2 −ẋTẋ. Using this in Eq. (6) we obtain Contracting (3) with S µν and using (4) we obtain d dτ (S µν S µν ) = 0, that is, square of spin is a constant of motion. Contraction of (5) with P µ gives (P Sθẋ) = 0. Contraction of (5) with (ẋθ) µ gives (P θẋ) = 0. Contraction of (2) with P µ , gives d dτ (P 2 ) = − 1 2 (P θẋ) = 0, that is P 2 is one more constant of motion, say k, √ −P 2 = k = const (in our vector model developed below this is fixed as k = mc). Substituting (7) into the equations (2)-(4) we now can exclude P µ from these equations, modulo to the constant of motion k = √ −P 2 . Thus, square of momentum can not be excluded from the system (2)- (5), that is MPTD-equations in this form do not represent a Hamiltonian system for the pair x µ , P µ . To improve this point, we note that Eq. (7) acquires a conventional form (as the expression for conjugate momenta of x µ in the Hamiltonian formalism), if we add to the system (2)-(4) one more equation, which fixes the remaining quantity P 2 . To see, how the equation could look, we note that for non-rotating body (pole approximation) we expect equations of motion of spinless particle, ∇p µ = 0, p µ =m c √ −ẋgẋẋ µ , p 2 + (mc) 2 = 0. Independent equations of the system (2)-(5) in this limit read ∇P µ = 0, P µ = √ −P 2 √ −ẋgẋẋ µ . Comparing the two systems, we see that the missing equation is the mass-shell condition P 2 + (mc) 2 = 0. Returning to the pole-dipole approximation, an admissible equation should be P 2 + (mc) 2 + f (S, . . .) = 0, where f must be a constant of motion. Since the only constant of motion in arbitrary background is S 2 , we write 3 With this value of P 2 , we can exclude P µ from MPTD-equations, obtaining closed system with second-order equation for x µ (so we refer the resulting equations as Lagrangian form of MPTD-equations). We substitute (7) into (2)- (4), this gives the system where (8) is implied. They determine evolution of x µ and S µν for each given function f (S 2 ). It is convenient to introduce the symmetric matrix G composed from the "tetrad field"T of Eq. (7) G µν = g αβT α µT β ν = g µν + h µν (S). (12) Since this is composed from the original metric g µν plus (spin and field-dependent) contribution h µν , we call G the effective metric produced along the world-line by interaction of spin with gravity. Eq. (11) implies the identitẏ so we can replace −ẋTẋ in (9)-(11) by √ −ẋGẋ. In particular, Eq. (9) reads Adding the consequences found above to the MPTD equations (2)-(4), we have the system P 2 + (mc) 2 + f (S 2 ) = 0, (16) S 2 , m(S 2 ) are constants of motion, (17) withT given in (7). In section XIII B we will see that they essentially coincide with Hamiltonian equations of our spinning particle with vanishing gravimagnetic moment. Let us finish this section with the following comment. Our discussion in the next two sections will be around the factorẋGẋ, where appeared the effective metric G µν . The equation for trajectory (14) became singular for the particle's velocity which annihilates this factor,ẋGẋ = 0. We call this the critical velocity. The observer independent scale c of special relativity is called, as usual, the speed of light. The singularity determines behavior of the particle in ultra-relativistic limit. To clarify this point, consider the standard equations of a spinless particle interacting with electromagnetic field in the physical-time parametrization x µ (t) = (ct, x(t)), = e mc 2 F µ νẋ ν . Then the factor is just c 2 − v 2 , that is critical speed coincides with the speed of light. Rewriting the equations of motion in the form of second law of Newton, we find an acceleration. For the case, the longitudinal acceleration reads (Ev), that is the factor, elevated in some degree, appears on the right hand side of the equation, and thus determines the value of velocity at which the longitudinal acceleration vanishes, a || v→c −→ 0. So the singularity implies 4 , that during its evolution in the external field the spinless particle can not exceed the speed of light c.

III. THREE-DIMENSIONAL ACCELERATION AND SPEED OF LIGHT IN GENERAL RELATIVITY
The ultra-relativistic behavior of MPTD particle in an arbitrary gravitational field will be analyzed by estimation of three-acceleration as v → c. Let us discuss the necessary notions.
By construction of Lorentz transformations, the speed of light in special relativity is an observer-independent quantity. In the presence of gravity, we replace the Minkowski space by a four-dimensional pseudo Riemann manifold To discuss the physics behind this abstract four-dimensional construction, we should establish a correspondence between the quantities computed in an arbitrary coordinates of the Riemann space and the three-dimensional quantities used by an observer in his laboratory. In particular, in a curved space we replace the Lorentz transformations on the general-coordinate ones, so we need to ensure the coordinate-independence of the speed of light for that case. It turns out that this essentially determines the relationship between the four-dimensional and three-dimensional geometries [36]. We first recall the most simple part of this problem, which consist in determining of basic differential quantities of three-dimensional geometry: infinitesimal distances, time intervals and velocity [36]. Then we define the three-dimensional acceleration which guarantees that a particle, propagating along a four-dimensional geodesic, can not exceed the speed of light. This gives us the necessary tool for discussion of a fast moving body. The behavior of ultra-relativistic particles turns out to be important for analysis of near horizon geometry of extremal black holes, see [37], and for accurate accounting of the corresponding corrections to geodesic motion near black hole [38][39][40][41][42][43][44][45][46].
A. Coordinate independence of speed of light.
Consider an observer that label events by some coordinates of pseudo Riemann space (18) to describe the motion of a point particle in a gravitational field with given metric g µν . Formal definitions of the three-dimensional quantities can be obtained representing four-interval in 1 + 3 block-diagonal form This prompts to introduce infinitesimal time interval and distance as follows: Therefore the conversion factor between intervals of the coordinate time dx 0 c and the time dt measured by laboratory clock is From Eq. (19) it follows that laboratory time coincides with coordinate time in the synchronous coordinate systems where metric acquires the form g 00 = 1 and g 0i = 0. If metric is not of this form, we can not describe trajectory x(t) using the laboratory time t as a global parameter. But we can describe it by the function x(x 0 ), and then determine various differential characteristics (such as velocity and acceleration) using the conversion factor (21). For instance, three-velocity of the particle is so it is convenient to introduce the notation The definitions of v and v are consistent Three-dimensional geometry is determined by the metric γ ij (x 0 , x). In particular, square of length of a vector is given by vγv = v i γ ij v j . Using these notation, the infinitesimal interval acquires the form similar to special relativity This equality holds in any coordinate system x µ . Hence a particle with the propagation law ds 2 = 0 has the speed v 2 = c 2 , and this is a coordinate-independent statement. The value of the constant c, introduced by hand, is fixed from the flat limit: equation (19) implies dt = cdx 0 when g µν → η µν . These rather formal tricks are based [36] on the notion of simultaneity in general relativity and on the analysis of flat limit. The four-interval of special relativity has direct physical interpretation in two cases. First, for two events which occur at the same point, the four-interval is proportional to time interval, dt = − ds c . Second, for simultaneous events the four-interval coincides with distance, dl = ds. Assuming that the same holds in general relativity, let us analyze infinitesimal time interval and distance between two events with coordinates x µ and x µ + dx µ . The world line y µ = (y 0 , y = const) is associated with laboratory clock placed at the spatial point y. So the time-interval between the events (y 0 , y) and (y 0 + dy 0 , y) measured by the clock is Consider the event x µ infinitesimally closed to the world line (y 0 , y = const). To find the event on the world line which is simultaneous with x µ , we first look for the events y µ (1) and y µ Second, we compute the middle point By definition 5 , the event (y 0 , y) with the null-coordinate (26) is simultaneous with the event (x 0 , x), see Fig. 1 on page 6. By this way we synchronized clocks at the spatial points x and y. According to (26), the simultaneous events have different null-coordinates, and the difference dx 0 obeys the equation Consider a particle which propagated from x µ to x µ + dx µ . Let us compute time-interval and distance between these two events. According to (26), the event at the spatial point x is simultaneous with x µ + dx µ , see Fig. 2 on page 7. Equation (28) determines the event A (at spatial point x) simultaneous with x µ + dx µ . So the time interval between x µ and x µ + dx µ coincide with the interval between x µ e A. Distance between x µ and x µ + dx µ coincide with the distance between x µ + dx µ and A. According to (25) and (26), the time interval between the events x µ and (28) is 5 In the flat limit the sequence y µ (1) , x µ , y µ (2) of events can be associated with emission, reflection and absorbtion of a photon with the propagation law ds = 0. Then the middle point (26) should be considered simultaneous with x 0 .
x A x x +dx FIG. 2: Time and distance between the events x µ and x µ + dx µ . Equation (28) determines the event A (at spatial point x) simultaneous with x µ + dx µ . So the time interval between x µ and x µ + dx µ coincide with the interval between x µ e A, and is given by (29). Distance between x µ and x µ + dx µ coincide with the distance between x µ + dx µ and A, the latter is given in (30).
Since the events x µ + dx µ and (28) are simultaneous, this equation gives also the time interval between x µ and x µ + dx µ . Further, the difference of coordinates between the events x µ + dx µ and (28) is dz µ = (− g0idx i g00 , dx i ). As they are simultaneous, the distance between them is Since (28) occur at the same spatial point as x µ , this equation gives also the distance between x µ and x µ + dx µ . The equations (29) and (30) coincide with the formal definitions presented above, Eqs. (19) and (20).
B. Three-dimensional acceleration and maximum speed of a particle in geodesic motion.
We now turn to the definition of three-acceleration. Point particle in general relativity follows a geodesic line, and we expect that during its evolution in gravitational field the particle can not reach the speed of light. This implies that longitudinal acceleration should vanish when speed of the particle approximates to c. To analyze this, we first use geodesic equation to obtain the derivative dv i dt of coordinate of the velocity vector. If we take the proper time to be the parameter, geodesics obey the system where Due to this definition, the system (31) obeys the identity g µν dx µ ds ∇ s dx ν ds = 0. The system in this parametrization has no sense for the case we are interested in, ds 2 → 0. So we rewrite it in arbitrary parametrization τ dτ ds d dτ this yields the equation of geodesic line in reparametrization-invariant form Using the reparametrization invariance, we set τ = x 0 , then Eqs. (19) and (21) imply √ −ẋgẋ = dt dx 0 c 2 − vγv, and spatial part of (33) reads where we have denoted Direct computation of the derivative on l. h. s. of equation (34) leads to the desired expression and three-dimensional Christoffel symbolsΓ i jk (γ) are constructed with help of three-dimensional metric γ ij (x 0 , x k ), where x 0 is considered as a parameter We have g ij γ jk = δ i k , so the inverse metric of γ ij turns out to be γ ij = g ij . Note thatM i j v j |v|→c −→ 0, that is in the limit the matrixM turns into the projector on the plane orthogonal to v.
If we project the derivative (36) on the direction of motion, we obtain the expression Due to the first and second terms on r. h. s., this expression does not vanish as vγv → c 2 . Note that this remains true for stationary metric, g µν (x), or even for static metric, g 00 = 1, g 0i = 0! The reason is that the derivative dv i dt in our three-dimensional geometry consist of three contributions: variation rate of the vector field v itself, variation of basis in the passage from x to x + dx, and variation of the metric γ ij during the time interval dt. Excluding the last two contributions, we obtain the variation rate of velocity itself, that is an acceleration For the special case of stationary field, g µν (x), our definition reduces to that of Landau and Lifshitz, see page 251 in [36]. Complementing dv i dt in Eq. (36) up to the acceleration, we obtain three-dimensional acceleration of the particle moving along the geodesic line (33) This is the second Newton law for geodesic motion. Contracting this with (vγ) i , we obtain the longitudinal acceleration This implies vγa → 0 as vγv → c 2 . Let us confirm that c is the only special point of the function (42). Using Eqs. (32), (20)- (23) and (38) together with the identities we can present the right hand side of Eq. (42) in terms of initial metric as follows The quantity v µ has been defined in (35). Excluding v 0 according to this expression, we obtain For the stationary metric, g µν (x k ), the equation (45) acquires a specially simple form This shows that the longitudinal acceleration has only one special point, vγa → 0 as vγv → c 2 . Hence the spinless particle in the stationary gravitational field can not overcome the speed of light. Then the same is true in general case (44), at least for the metric which is sufficiently slowly varied in time.
While we have discussed the geodesic equation, the computation which leads to the formula (42) can be repeated for a more general equation. Using the factor √ −ẋgẋ we construct the reparametrization-invariant derivative Consider the reparametrization-invariant equation of the form and suppose that the three-dimensional geometry is defined by g µν according to equations (19)- (22). Then Eq. (48) implies the three-acceleration and the longitudinal acceleration The spatial part of the force is where v µ is given by (35), and the connectionΓ i kl (γ) is constructed with help of the three-dimensional metric γ ij = (g ij − g0ig0j g00 ) according to (38). For the geodesic equation in this notation we have (49) and (50) coincide with (41) and (42). Eq. (50) shows that potentially dangerous forces are of degree four or more, F j ∼ (Dx) 4 .
C. Parallel transport in three-dimensional geometry. Now we consider an arbitrary vector/tensor field in the space with three-dimensional geometry determined by a non static metric γ ij (x 0 , x). Variation rate of the field along a curve x(x 0 ) should be defined in such a way, that it coincides with (40) for the velocity. Let us show, how this definition follows from a natural geometric requirements [30]. 1 + 3 splitting preserves covariance of the formalism under the following subgroup of general-coordinate transformations: . Under these transformations g 00 is a scalar function, g 0i is a vector while g ij and γ ij are tensors, then the conversion factor (21) is a scalar function and the velocity (22) is a vector. So it is convenient to introduce the usual covariant derivative ∇ k of a vector field ξ i (x 0 , x) in the direction of x k with the Christoffel symbols (38). By construction, the metric γ is covariantly constant, ∇ k γ ij = 0. For the field ξ i (x 0 , x(x 0 )) along the curve x(x 0 )) we have the covariant derivative in the direction of x 0 To define the variation rate of ξ, we need the notion of a constant field (or, equivalently, the parallel-transport equation). In Euclidean space the scalar product of two constant fields does not depend on the point where it was computed. In particular, taking the scalar product along a line x(x 0 ), we have d dx 0 (ξ, η) = 0. For the constant fields in our case it is natural to demand the same (necessary) condition: Taking into account that ∇ k γ ij = 0, this condition can be written as follows This will be satisfied, if a constant field is defined by the equation This is the parallel-transport equation in our three-dimensional geometry. Deviation from the constant field is the variation rate. Hence, when the l. h. s. does not vanish, it gives the variation rate, which we write with respect to physical time: This result is in correspondence with the definition (40) for acceleration.

IV. BEHAVIOR OF ULTRA-RELATIVISTIC MPTD-PARTICLE AND THE RAINBOW GEOMETRY INDUCED BY SPIN
As we saw above, point particle in a gravitational field propagates along a geodesic line with the speed less then speed of light. Let us study the influence of rotational degrees of freedom on the trajectory of a fast spinning particle.
Using (8) and (13), we present MPTD-equations (9)-(11) in the following form The equations for trajectory and for precession of spin become singular at the critical velocity which obeys the equatioṅ The singularity determines behavior of the particle in ultra-relativistic limit. The effective metric is composed from the original one plus (spin and field-dependent) contribution, G = g + h(S). So we need to decide, which of them the particle probes as the space-time metric. Let us consider separately the two possibilities.
Let us use g to define the three-dimensional geometry (19)- (22). This leads to two problems. The first problem is that the critical speed turns out to be slightly more than the speed of light. To see this, we use the supplementary spin condition (55) to write (58) in the form with v µ defined in (35). Using S µν = 2ω [µ π ν] , we rewrite the last term as follows: As π and ω are space-like vectors, the last term is non-negative, this implies |v cr | ≥ c. Let us confirm that generally this term is nonvanishing function of velocity, then |v cr | > c. Assume the contrary, that this term vanishes at some velocity, then We analyze these equations in the following special case. Consider a space with covariantly-constant curvature ∇ µ R µναβ = 0. Then d dτ (θ µν S µν ) = 2θ µν ∇S µν , and using (57) we conclude that θ µν S µν is an integral of motion. We further assume that the only non vanishing part is the electric part of the curvature, R 0i0j = K ij . Then the integral of motion acquires the form Let us take the initial conditions for spin such that K ij S 0i S 0j = 0, then this holds at any future instant. Contrary to this, the system (61) implies K ij S 0i S 0j = 0. Thus, the critical speed does not always coincide with the speed of light and, in general case, we expect that v cr is both field and spin-dependent quantity.
The second problem is that acceleration of MPTD-particle grows up in the ultra-relativistic limit. In the spinless limit the equation (56) turn into the geodesic equation. Spin causes deviations from the geodesic equation due to right hand side of this equation, as well as due to the presence of the tetrad fieldT and of the effective metric G in the left hand side. Due to the dependence of the tetrad field on the spin-tensor S, the singularity presented in (57) causes the appearance of the term proportional to 1 √ẋ Gẋ in the expression for longitudinal acceleration. In the result, the acceleration grows up to infinity as the particle's speed approximates to the critical speed. To see this, we separate derivative ofT in Eq. (56) where T is the inverse forT . Using (57) we obtain Using this expression, the equation (64) reads 6 d dτ where we denoted It will be sufficient to consider static metric g µν (x) with g 0i = 0. Then three-dimensional metric and velocity are Taking τ = x 0 , the spatial part of equation (66) with this metric reads with v µ defined in (35), for the case and In the result, we have presented the equation for trajectory in the form convenient for analysis of acceleration, see (34). Using the definition of three-dimensional covariant derivative (40), we present the derivative on the l.h.s. of (69) as follows We have denoted The matrix M i k has the inverseM Combining these equations, we obtain the three-acceleration of our spinning particle Finally, using manifest form of f i from (67) we have The longitudinal acceleration is obtained by projecting a i on the direction of velocity, that is whereŜ k = (T S) kµ R µναβ v ν v α (Sθv) β . As the speed of the particle closes to the critical velocity, the longitudinal acceleration diverges due to the first term in (77). In resume, assuming that MPTD-particle sees the original geometry g µν , we have a theory with unsatisfactory behavior in the ultra-relativistic limit.
Let us consider the second possibility, that is we take G µν to construct the three-dimensional geometry (19)- (22). With these definitions we have, by construction, −ẋGẋ = ( dt dx 0 ) 2 (c 2 − (vγv)), so the critical speed coincides with the speed of light. In the present case, the expression for three-acceleration can be obtained in closed form for an arbitrary curved background. Taking τ = x 0 the spatial part of (66) implies where, from (67), f i is given by Equation (78) is of the form (34), so the acceleration is given by (41) and (42) where, for the present case, With f i given in (79), the longitudinal acceleration vanishes as v → c. Let us resume the results of this section. Assuming that spinning particle probes the three-dimensional space-time geometry determined by the original metric g, we have a theory with unsatisfactory ultra-relativistic limit. First, the critical speed, which the particle can not overcome during its evolution in gravitational field, can be more then the speed of light. The same observation has been made from analysis of MPTD-particle in specific metrics [47][48][49][50][51]. Second, the longitudinal acceleration grows up to infinity in the ultra-relativistic limit. Assuming that the the particle sees the effective metric G(S) as the space-time metric, we avoided the two problems. But the resulting theory still possess the problem. The acceleration (80) contains the singularity due to , that is at v = c the acceleration becomes orthogonal to the velocity, but remains divergent. Besides, due to dependence of effective metric on spin, we arrive at rather unusual picture of the Universe with rainbow geometry 7 : there is no unique space-time manifold for the Universe of spinning particles: each particle will probe his own three-dimensional geometry. We conclude that MPTD equations do not seem promising candidate for the description of a relativistic rotating body. It would be interesting to find their generalization with improved behavior in ultra-relativistic regime. This will be achieved within the framework of vector model of spinning particle, which we shall describe in the subsequent sections.

V. VECTOR MODEL OF NON RELATIVISTIC SPINNING PARTICLE
The data of some experiments with elementary particles and atoms (Stern-Gerlach experiment, fine structure of hydrogen atom, Zeeman effect) shows that the Schrödinger equation for a one-component wave function is not adequate to describe the behavior of these systems in the presence of an electromagnetic field. This implies a radical modification of the formalism. Besides the position and the momentum, the state of an electron is specified by some discrete numbers, which are eigenvalues of suitably defined operators, called the operators of spin. The mathematical theory of these operators is similar to the formalism of angular momentum. So, intuitively, an elementary particle carries an intrinsic angular momentum called spin.
To describe a particle with spin s = 1 2 we introduce the two-component wave function Ψ α , α = 1, 2. The spin operatorsŜ i act on Ψ α as 2 × 2-matrices, and are defined bŷ where σ i stands for the Pauli matrices, they form a basis of the vector space of traceless and Hermitian 2 × 2-matrices, Their basic algebraical properties are Note that the commutators (85) of σ-matrices are the same as for the angular-momentum vector. The spin operators, being proportional to the Pauli matrices, have similar properties, in particular Consider Coulomb electric and a constant magnetic fields. The electromagnetic potential can be taken in the form Then evolution of an electron immersed in this fields described by the equation The first and second terms in the Hamiltonian correspond to the minimal interaction of a point particle with an electromagnetic potential, whereas the last two terms represent interaction of spin with electric and magnetic fields. A numeric factor g is called gyromagnetic ratio of the electron 8 . The vector eg 2mcŜ is known as magnetic moment of the particle.
The equation is written in the Schrödinger picture, that is we ascribe time-dependence to the wave function, whereas in semiclassical models we deal with dynamical variables. We recall that the time-dependence can be ascribed to operators using the Heisenberg picture. Passing to the Heisenberg picture, we could write dynamical equations for basic operators of the theory. According to Ehrenfest theorem, expectation values of the operators approximately obey the classical Hamiltonian equations [59].
The equation (89) gives the structure and properties of the energy levels of hydrogen atom in a good agreement with experiment. The fine structure of hydrogen atom fixes the factor g − 1 in the third term, while Zeeman effect requires the factor g in the last term.
To formulate the problem that we wish to discuss, we recall that quantum mechanics of a spinless particle can be obtained applying the canonical quantization procedure to a classical-mechanics system with the Lagrangian L = 1 2 mx 2 − U (x). To achieve this, we construct a Hamiltonian formulation for the system, then associate with the phase-space variables the operators with commutators resembling the Poisson brackets, and write on this base the It is natural to ask whether this ideology can be realized for the spinning particle. Since the quantum-mechanical description of a spin implies the use of three extra operatorsŜ i , the problem can be formulated as follows. We look for a classical-mechanics system which, besides the position variables x i , contains additional degrees of freedom, suitable for the description of a spin: in the Hamiltonian formulation the spin should be described, in the end, by three variables with fixed square (88) and with the classical brackets {S i , S j } = ijk S k . Then canonical quantization of these variables will yield spin operators with the desired properties properties (87) and (88). According to this, typical spinning-particle model consist of a point on a world-line and some set of variables describing the spin degrees of freedom, which form an inner space attached to that point 9 . In fact, different spinning particles discussed in the literature differ by the choice of the inner space. An exceptional case is the rigid particle [61] which consist of only position variables, but with the action containing higher derivatives. The model yields the Dirac equation [62], hence it also can be used for description of spin.
It should be noted that equation (89) is written in the laboratory system, so we do not state that our classical variable S i is a quantity defined in the instantaneous rest frame of the particle.
We intend to construct the spinning particle starting from a suitable variational problem. This is the first task we need to solve, as the formulation of a variational problem in closed form is known only for the case of a phase space equipped with canonical Poisson bracket, say {ω i , π j } = δ ij . The number of variables and their algebra are different from the number of spin operators and their commutators, (87). May be the most natural way to arrive at the operator algebra (87) is to consider spin as a composite quantity, where ω, π are coordinates of a phase space equipped with canonical Poisson bracket. This immediately induces Unfortunately, this is not the whole story. First, we need some mechanism which explains why S, not ω and π must be taken for the description of spin degrees of freedom. Second, the basic space is six-dimensional, while the spin manifold is two-dimensional (we remind that the square of spin operator has fixed value, Eq. (88)). To improve this, we look for a variational problem which, besides dynamical equations, implies the constraints According to Dirac's terminology [63][64][65][66][67], they form the first-class set, so in the model with these constraints the spin sector contains 6 − 2 × 2 = 2 physical degrees of freedom. Geometrically, the constraints determine four-dimensional SO(3) -invariant surface of the six-dimensional phase space. The constraints imply the fixed value of spin The same square of spin follows from the constraints if we put β 2 = 3 2 4α 2 , any α. The combination π 2 − β 2 + β 2 α 2 (ω 2 − α 2 ) represents the first-class constraint of the set (93). Hence the model with these constraints also has the desired number of degrees of freedom, 6 − 2 − 1 × 2 = 2. The equalities (93) determine essentially unique SO(3) -invariant three-dimensional surface of the phase space. The set (91) turns out to be more convenient for generalization to the case of a relativistic spin.
While S in (90) looks like an angular momentum, the crucial difference is due to the presence of first-class constraints, and hence of a local symmetry which we refer as spin-plane symmetry. The latter acts on the basic variables ω, π, while leaves invariant the spin variable S. Using analogy with classical electrodynamics, ω and π are similar to four-potential A µ while S plays the role of F µν . The coordinates ω of the "inner-space particle" are not physical (observable) quantities. The only observable quantities are the gauge-invariant variables S. So our construction realizes, in a systematic form, the oldest idea about spin as the "hidden angular momentum".

A. Lagrangian and Hamiltonian for the spin-sector
As the Lagrangian which implies the constraints (91), we take the expression where N ij = δ ij − ωiωj ω 2 is the projector on the plane orthogonal to ω: N ij ω j = 0, N 2 = N . The equivalent forms of the Lagrangian are There are also two (finite) local symmetries: reparametrizations t → t = σ(t) ≡ t + (t), and the scale transformations ω → χ(t)ω. Let us construct the Hamiltonian formulation of the model. Equation for the conjugated momentum reads π =

Nω √ω
Nω . This expression immediately implies (91) as the primary constraints. We also note the equality πω = L, that is H 0 = πω − L = 0. So the complete Hamiltonian is composed from the primary constraints,

and the Hamiltonian action reads
There are no of higher-stage constraints in the problem. Let us write Hamiltonian counterparts of the Lagrangian local symmetries. 1. Reparametrizations in extended phase space are They induce the transformations of dynamical variables Their infinitesimal form read 2. Scale transformations of coordinates are Since τ is not involved, the induced transformations of dynamical variables are the same, for instance ω (τ ) = χω(τ ). Presenting χ = 1 + γ, infinitesimal transformations of dynamical variables read δω = γω, δπ = −γπ δv =γ, δv 1 = 2γv 1 .
Besides the constraints (91), the variational problem (96) implies the Hamiltonian equationṡ To make the system more symmetric, we have introduced the variable ρ = 2v1 ω 2 instead of v 1 . According to general formalism of constrained systems [63][64][65], neither the dynamical equations nor the constraints determine the variables v and ρ. They enter as arbitrary functions of time into general solution for the variables ω and π, making completely undetermined their dynamics. Indeed, for any given functions v(t) and ρ(t), the equations represent a normal system for determining ω and π. Its general solution is where the integration constants b and c are subject to the conditions This implies ω 2 = √ αe 2 t 0 vdτ and π 2 = √ αe −2 t 0 vdτ . According to these expressions, the pair of orthogonal vectors ω and π rotates in their own plane (or, equivalently, in the plane determined by b and c) with the variable angular velocity prescribed by the function ρ(t). The function v(τ ) determines the variation of their magnitudes. Choosing the functions v and ρ suitably, we can make the point with radius-vector ω move along any prescribed line! We point out that the two-parametric ambiguity is in correspondence with the invariance of the action (96) under the two local symmetries described above. Summing up, all the basic variables of our model are unobservable quantities.
The spin-vector 10 S = ω × π has unambiguous evolutioṅ Note also that it is invariant of the local symmetries. Hence the spin-vector is a candidate for an observable quantity. In interacting theory S will precess under the torque exercised by a magnetic field, see below. Due to Eqs. (91), the coordinates S i obey (92).

B. Spin fiber bundle and spin-plane local symmetry
The passage from initial variables ω and π to the observables S is not a change of variables, and acquires a natural interpretation in the geometric terms. It should be noted that basic notions of the theory of constrained systems have their analogs in differential geometry. Second-class constraints imply that all true trajectories lie on a submanifold of the initial phase-space. The Dirac bracket, constructed on the base of second-class constraints, induces canonical symplectic structure on the submanifold. If the first-class constraints (equivalently, the local symmetries) are presented in the model, a part of variables have ambiguous evolution. This also can be translated into the geometric language: due to the ambiguity, the submanifold should be endowed with a natural structure of a fiber bundle. Physical variables are (functions of) the coordinates which parameterize the base of the fiber bundle. Let us describe, how all this look like in our model.
Consider six-dimensional phase space equipped with canonical Poisson bracket and three-dimensional spin space Poisson bracket on R 6 together with the map induce SO(3) Lie-Poisson bracket on R 3 As we saw above, all the trajectories ω(t), π(t) lie on SO(3) -invariant surface of R 6 determined by the constraints that is ω and π represent a pair of orthogonal vectors with their ends attached to the hyperbole y = α x .
Then the two-dimensional manifold F 2 S contains all pairs (χω, 1 χ π), χ ∈ R + , as well as the pairs obtained by rotation of these (χω, 1 χ π) in the plane of vectors (ω, π). So elements of F 2 S are related by two-parametric transformations In the result, the manifold T 4 acquires a natural structure of fiber bundle with base S 2 , standard fiber F 2 , projection map f and structure group given by the transformations (110) and (111). The adjusted with the structure of the fiber bundle local coordinates are χ, φ, and two coordinates of the vector S. By construction, the structure-group transformations leave inert points of the base, δS i = 0. Let us discuss the relationship between the structure group and local symmetries of the Hamiltonian action (96). The structure transformation (110) can be identified with the scale transformation (100). Concerning the transformation (111), let us apply it to the action (96). Inserting ω and π into the action and disregarding the total derivative, we obtain the expression The action does not change, Hence we have found one more local symmetry of the action. Its infinitesimal form reads The three infinitesimal symmetries (99), (101) and (114) are not independent on the subspace of solutions to equations of motion. To see this, we note that the following infinitesimal transformation: being applied to any variable, turns out to be proportional to equations of motion. For instance, The on-shell symmetries are considered as trivial symmetries, see [68]. Hence on the subspace of solutions the infinitesimal reparametrization can be identified with a special transformation of the structure group In the result, the number of infinitesimal symmetries coincides with the number of primary first-class constraints. Summing up, in the passage from geometric to dynamical realization, the transformations of structure group of the spin fiber bundle acts independently at each instance of time and turn into the local symmetries of Hamiltonian action. Equivalent formulations. Let us consider a slightly different Lagrangian The conjugated momentum π = Nω implies only one primary constraint ωπ = 0, then the complete Hamiltonian reads Computing d dt (ωπ) = {ωπ, H}, we obtain π 2 − α ω 2 = 0 as the secondary constraint. Hence the Lagrangian implies an equivalent formulation.
As the Lagrangian which implies the constraints (93), we could take the expression L spin = 1 2gω 2 + 1 2 gβ 2 − 1 2 λ(ω 2 − α 2 ). We remind that in this model β 2 = 3 2 4α 2 , while α is any given number. Variation with respect to auxiliary variables g(t) and λ(t) gives the equationsω 2 = g 2 β 2 and ω 2 = α 2 , the latter impliesωω = 0. In the Hamiltonian formulation these equations turn into the desired constraints. We can integrate out the variable g, presenting the Lagrangian in a more compact form This also gives the desired constraints. The last term represents kinematic (velocity-independent) constraint. So, we might follow the known classical-mechanics prescription and exclude λ as well. But this would lead to loss of the manifest rotational invariance of the formalism. The spin fiber bundle corresponding to this formulation turns out to be the group manifold SO (3), see [69] for details.

C. Canonical quantization and Pauli equation
To test our formulation, we show that our spinning particle yields the Pauli equation in a stationary magnetic field with the vector potential A. Consider the action The configuration-space variables are x i (t), and ω i (t). Here x i represents the spatial coordinates of the particle with the mass m, charge e and gyromagnetic ratio g. In our classical model g appeared as a coupling constant of ω with the magnetic field B = ∇ × A in the last term of Eq. (121). At the end, it produces the Pauli term in the quantum-mechanical Hamiltonian. Let us construct Hamiltonian formulation for the model. Equations for the conjugated momenta p i and π i reads Eq. (123) implies the primary constraints ωπ = 0 and π 2 − α ω 2 = 0. The complete Hamiltonian, There are no of higher-stage constraints in the formulation. Besides the constraints, the Hamiltonian (124) implies the dynamical equationsẋ As a consequence of these equations, the spin-vector S i = ijk ω j π k have unambiguous evolutioṅ This is the classical equation for precession of spin in an external magnetic field. Due to Eqs. (93), the coordinates S i obey (92). Equations (125) imply the second-order equation for x i Note that in the absence of interaction, the spinning particle does not experience a self-acceleration. The last term gives non vanishing contribution into the trajectory in unhomogeneous field and can be used for semiclassical description of Stern-Gerlach experiment. Since S 2 ∼ 2 , the S-term disappears from Eq. (128) at the classical limit → 0. Then Eq. (128) reproduces the classical motion of a charged particle subject to the Lorentz force. Precession of spin. Let us denote − ge 2mc B = ω p , then Eq. (127) readṡ The vectorṠ is orthogonal to the plane of ω p and S at any instant. Besides, contracting Eq. (129) with S we see that magnitude of spin does not change, S 2 = const. In the result, the end point of S rotates around the axis ω p . Let S(0) = S 0 is the initial position of spin. We present this vector as a sum of longitudinal and transversal parts with respect to ω p , S 0 = S 0⊥ + S 0|| . Then for the constant vector ω p , the general solution to Eq. (129) is Hence the magnitude of vector ω p from Eq. (129) is just the frequency of precession. Equation of trajectory (128) in the constant magnetic field isv = 2 g [ω p × v], that is particle's velocity precesses with the frequency 2 g ω p . For a particle with classical gyromagnetic ratio g = 2, the two frequences coincide and the angle between velocity and spin preserves during the evolution. For the anomalous magnetic moment, g = 2, the frequences are different. The spin precession relative to the velocity is used in a cyclotron experiments for measurement of anomalous magnetic moment [17,18].
Canonical quantization. We quantize only the physical variables x i , p i , S i . Their classical brackets are As the last two terms in (124) does not contributes into equations of motion for the physical variables, we omit them. This gives the physical Hamiltonian The first equation from (131) implies the standard quantization of the variables x and p, we According to the second equation from (131), we look for the wave-function space which is a representation of the group SO(3). Finite-dimensional irreducible representations of the group are numbered by spin s, which is related with the values of Casimir operator as follows: S 2 ∼ s(s + 1). Then Eq. (92) fixes the spin s = 1 2 , and S i must be quantized byŜ i = 2 σ i . The operators act on the spinor space of two-component complex columns Ψ. Quantum Hamiltonian is obtained from Eq. (132) replacing classical variables by the operators. This immediately yields the Pauli equation, that is Eq. (89) with E = 0.

VI. WHY WE NEED A SEMICLASSICAL MODEL OF RELATIVISTIC SPIN?
Dirac equation. We expect that a semiclassical relativistic model of spin should be closely related to the Dirac equation normally used to describe the relativistic spin in quantum theory. The consistent description of relativistic spin is achieved in quantum electrodynamics, where the Dirac equation is considered as a quantum field theory equation. But it also admits a quantum-mechanical interpretation and thus represents an example of relativistic quantum mechanics. This is of interest on various reasons. In particular, namely being considered as a quantummechanical equation, the Dirac equation gives the correct energy levels of hydrogen atom. As we saw in Sect. V A, dynamical equations for expectation values of operators in quantum mechanics should resemble the Hamiltonian equations of the corresponding classical system. Let us discuss these equations in the Dirac theory.
Under the infinitesimal Lorentz transformation δx µ = ω µ ν x ν , the Dirac spinor Ψ = (Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 ) transforms as follows: where and the 4 × 4 γ -matrices can be composed from σ -matrices of Pauli We use the representation with hermitian γ 0 and antihermitian γ i . The matrices do not commute with each other, and the basic formula for their permutation is as follows The Dirac equation in an external four-potential A µ turns out to be covariant under the transformation (133). Applying the operator γ µ (p µ − e c A µ ) − mc to (137), we see that the Dirac equation implies the Klein-Gordon equation with non-minimal interaction where For the latter use, let us analyze commutators of the matrices involved. The commutators of γ -matrices can not be presented through themselves, but produce γ µν -matrices as they are written in (134). The set γ µ , γ µν forms a closed algebra As it was tacitly implied in Eq.
assuming γ µ ≡ J 5µ , γ µν ≡ J µν . Vector model of spinning particle with SO(2, 3) covariant spin-space has been constructed in [70]. Observer-independent probability. Ψ can be used to construct the adjoint spinorΨ = Ψ † γ 0 with the transformation law δΨ = i 4Ψ γ µν ω µν . ThenΨΨ is a scalar,Ψγ µ Ψ ia a vector 11 and so on. The vector turns out to be a conserved current, that is on solutions to the Dirac equation. The time-component of the vector is Ψ † Ψ. Assuming that symbols x i represent the position of a particle, the quantity is identified with relativistic-invariant probability to find a particle in the infinitesimal volume d 3 x at the instant t = x 0 c . To confirm this interpretation, we first note that the probability density Ψ † Ψ is a positive function. Second, due to the continuity equation (141), integral of the density over all space does not depend on time: for the solutions Ψ that vanish on spatial infinity. Third, P coincides with the manifestly Lorentz-invariant quantity when it computed over equal-time surface x 0 = const of Minkowski space. This implies an observer-independence of the probability P : all inertial observers, when they compute P using their coordinates, will compute the same number (143). However, it is well known that adopting the quantum-mechanical interpretation, we arrive at a rather strange and controversial picture. We outline here the results of analysis on the applicability of quantum-mechanical treatment to the free Dirac equation made by Schrödinger 12 in [71]. We multiply the Dirac equation on γ 0 , representing it in the Schrödinger-like form where α i = γ 0 γ i and β = γ 0 are Dirac matrices. ThenĤ may be interpreted as the Hamiltonian. Passing from the Schrödinger to Heisenberg picture, the time derivative of an operator a is i ȧ = [a, H], and for the expectation values of basic operators of the Dirac theory we obtain the equationṡ Some properties of the equations are in order. 1. The wrong balance of the number of degrees of freedom. The first equation in (145) implies that the operator cα i represents the velocity of the particle. Then physical meaning of the operator p i becomes rather obscure in the classical limit.
2. Zitterbewegung. The equations (145) can be solved, with the result for The first and second terms are expected and describe a motion along the straight line. The last term on the r.h.s. of this equation states that the free electron experiences rapid oscillations with higher frequency 2H ∼ 2mc 2 . 3. Velocity of an electron. Since the velocity operator cα i has eigenvalues ±c, we conclude that a measurement of a component of the velocity of a free electron is certain to lead to the result ±c. 4. Operator of relativistic spin. We expect that in the Dirac theory can be constructed the relativistic generalization of the spin operator (82). The question on the definition of a conventional spin operator has been raised a long time ago [73,74] and is under discussion up to date [75][76][77].
Many people noticed that in the Dirac theory it is possible to construct another operators that obey to a reasonable equations, see [15,52]. Presenting these equations, Feynman accompanied them with the following comment (see p. 48 in [72]): "The following relations may be verified as true but their meaning is not yet completely understood, if at all: ...".
In view of all this, it seem desirable to construct a semiclassical model of spin that will be as close as possible to the Dirac equation. By this we mean the model which, being quantized, yields the Dirac equation. In the following sections, we will see how the vector model clarifies the issues discussed above. In a few words, this can be resumed as follows. As we already saw above, the vector model is necessarily invariant under the spin-plane local symmetry which determines its physical sector formed by observables. We show that observables of the vector model have an

VII. SPIN-TENSOR OF FRENKEL
To construct the relativistic spinning particle, we need a Lorentz-covariant description of the spin fiber bundle (237). We remind that our construction involves basic and target spaces as well as the map f : R 6 (ω, π) → R 3 (S), see Eqs.
(106)-(108). We embed this SO(3) -covariant construction into its suitably chosen SO(1, 3) -covariant extension. Let us start from the three-vector ω. We assume that relativistic spin can be described by a vector ω µ of Minkowski space such that ω µ = (0, ω) for the particle at rest in the laboratory frame. This condition expresses the Correspondence Principle: relativistic physics should approximate to the Newton physics in the limit of small velocities. To represent this condition in a covariant form in an arbitrary frame, we assume that in our model there exists a four-vector p µ which for the particle at rest has the components (p 0 , 0). For the case of a free particle, the natural candidate is a vector proportional to the particle's four-velocity. For the particle in external field, the form of this vector is dictated by the structure of interaction, see below. With this p µ , the Lorentz-invariant statement pω = 0 is equivalent to the condition that ω µ = (0, ω) for the particle at rest. Following the same lines, we also assume the condition pπ = 0 for the conjugated momentum π µ for ω µ . Hence we replace the basic space R 6 (ω, π) by direct product of two Minkowski spaces with the following natural action of the Lorentz group on it: The relativistic generalization of the surface (108) is given by the following SO(1, 3) -invariant surface of the phase space M × M Below we denote these constraints T 2 , T 5 , T 3 and T 4 . As in non relativistic case, we have two first-class constraints ωπ = 0 and π 2 − α ω 2 = 0. The constraints pω = 0 and pπ = 0 are of second class, so we expect 8 − 2 × 2 − 2 = 2 physical degrees of freedom in the spin-sector.
Let us consider the target space. To generalize the map S i = ijk ω j π k to the case of four-dimensional quantities, we rewrite it in an equivalent form, using the known isomorphism among three-vectors and antisymmetric 3 × 3 -matrices Then The last equality has an evident generalization to the four-dimensional case: S µν = 2(ω µ π ν − ω ν π µ ). Hence the target space R 3 (S) should be extended to the six-dimensional space R 6 (D, S) of antisymmetric 4 × 4 matrices. We present them as follows: or, equivalently Lorentz group naturally acts on this space This equation determines transformation rules of the columns D and S. They transform as three-vectors under the subgroup of rotations of the Lorentz group. The embedding (151) of three-dimensional spin-vector S into the fourdimensional spin-tensor has been suggested by Frenkel [12]. So we call S µν the Frenkel spin-tensor. The vector D is called dipole electric moment of the particle [16]. Now we are ready to define the covariant version of the map (107) f : M(ω µ ) × M(π ν ) → R 6 (S µν ); (ω µ , π ν ) → S µν = 2(ω µ π ν − ω ν π µ ).
Consider the image S µν (ω, π) of a point of the surface (147). Using the identity S µν S µν = 8(ω 2 π 2 − (ωπ) 2 ) together with the equations (147), we obtain five covariant equations which determine the spin-surface S 2 in an arbitrary Lorentz frame As (S µν p ν )p µ ≡ 0, we have only four independent equations imposed on six variables, therefore the spin-surface has dimension 2, as it should be. Denote F S ∈ T 4 preimage of a point S µν of the base, F S = f −1 (S µν ), that is the standard fiber, see Fig. 3 on page 24. Its points are related by the structure-group transformations (110) and (111). Consider the rest frame of the vector p µ , that is p µ = (p 0 , 0) in this frame. The surface (147) acquires the form and can be identified with the non relativistic spin-surface (237). Being restricted to this surface, the map (153) reads Hence in the rest frame the dipole electric moment vanishes, while the spatial part of spin-tensor coincides with the non-relativistic spin. We conclude that SO(3) -construction (106)-(237) is embedded into SO(1, 3) -covariant scheme. As in non relativistic case, the basic variables ω µ and π ν do not represent observable quantities, only S µν does. This may be contrasted with [32,105,106], where the equation S µν = 0 assumed to be the first-class constraint of the Dirac formalism. In the result, S µν turns out to be unobservable quantity.

VIII. FOUR-DIMENSIONAL SPIN-VECTOR, PAULI-LUBANSKI VECTOR AND BARGMANN-MICHEL-TELEGDI VECTOR
On the pure algebraic grounds, spin-tensor of Frenkel turns out to be equivalent to a four-dimensional vector. So the latter could also be used for the description of relativistic spin. Here we discuss the relevant formalism.
Levi-Civita symbol with 0123 = 1 obeys the identities Given an antisymmetric matrix J µν = −J νµ and a vector p µ , we define the vectors When p and J represent generators of the Poincaré group, the vector (161) is called Pauli-Lubanski vector. It turns out to be useful for the classification of irreducible representations of the Poincaré group [99][100][101].
The tensor J µν and its dual, * J µν ≡ 1 2 µνab J ab , can be decomposed on these vectors as follows: To prove (163), we contract (160) with p a p i J jk . Eq. (164) follows from (163) contracted with abµν . The definitions imply the identity relating the square of s µ with a "square" of J µν Frenkel spin-tensor obeys S µν p ν = 0, that is Φ µ = 0, and can be used to construct four-vector of spin (below we also call it Pauli-Lubanski vector) In the free theory p µ is independent on S µν , so this equation is linear on S µν and can be inverted. According to Eq.
(163) we have that is the two quantities are mathematically equivalent, and we could work with s µ instead of S µν . The equation (165) implies proportionality of their magnitudes. In the interacting theory p µ contains S µν , so (166) becomes a non linear equation. Let us compare spatial components of s µ with the non-relativistic spin-vector S. In the rest system of p µ , p µ = (p 0 , 0), −p 2 = |p 0 |, we have s 0 = 0 and that is the two vectors coincide. This explains our normalization for s µ , Eq. (161). Under the Lorentz boost, S transforms as the spatial part of a tensor whereas s µ transforms as a four-vector. So the two spins are different in all Lorentz frames except the rest frame. The relation between them in an arbitrary frame follows from Eq. (167) A four-dimensional vector s µ bmt with the property u µ s µ bmt = 0, where u µ represents a four-velocity of the particle, has been successively used by Bargmann Michel and Telegdi to analyze the spin precession in uniform magnetic field, see [102] for details. In our vector model, even in the case of interaction, the condition ps = 0 implies us = 0. So we expect that our equations of motion for s µ should represent a generalization of the Bargmann-Michel-Telegdi equations to the case of an arbitrary electromagnetic field.
In summary, the relativistic spin can be described by the Frenkel spin-tensor (150) composed by the dipole electric moment D and the spin S. In our vector model the Frenkel tensor is a composite quantity, see (153). In the rest frame of the vector p µ we have D = 0, while S coincides with the vector of non-relativistic spin. Intuitively, the Frenkel tensor shows how the non relativistic spin looks like in an arbitrary Lorentz frame.

IX. LAGRANGIAN OF RELATIVISTIC SPINNING PARTICLE
A. Variational problem for the prescribed Dirac's constraints In the previous section we have discussed only the spin-sector of a spinning particle. To construct a complete theory, we add the position x µ (τ ) and its conjugated momentum p µ (τ ) taken in an arbitrary parametrization τ . This implies that we deal with the reparametrization-invariant theory. So besides the spin-sector constraints (147) we expect also the mass-shell condition T 1 = p 2 + (mc) 2 = 0. Let us look for the Hamiltonian action which could produce these constraints. According to general theory [63,64,66], it has the form dτ pẋ + πω − (H 0 + λ i T i ) where T i are the primary constraints of the theory. We expect H 0 = 0 due to the reparametrization invariance. As the suitable primary constraints, let us take p 2 + (mc) 2 + π 2 − a ω 2 , T 2 , T 3 and T 4 . Thus we consider the Hamiltonian variational problem Due to the Poisson bracket {T 2 , T 5 } = 2T 5 , in this formulation T 5 = 0 appears as the secondary constraint. To arrive at the Lagrangian action, we could follow the standard prescription. Excluding the conjugate momenta from S H according to their equations of motion, we obtain an action with the auxiliary variables λ i . Excluding them, one after another, we obtain various equivalent forms of the Lagrangian action. To simplify these computations, we proceed as follows. First, we note that the constraints ωπ = 0 and pω = 0 always appear from the Lagrangian which involves the projector N , that is we use Nẋ and Nω instead ofẋ andω. So we set λ 2 = λ 3 = 0 in Eq. (170). Second, we present the remaining terms in (170) in the matrix form where λ = λ4 λ1 . The matrix appeared in (171) is invertible, the inverse matrix is Eq. (171) is the Hamiltonian variational problem of the form pq− λ1 2 (pAp+M 2 ), the latter follows from the Lagrangian −M −qA −1q . This allows us to exclude the variable λ 1 . As it was combined above, we then replaceẋ,ω by Nẋ, Nω and obtain To exclude the remaining auxiliary variable λ, we compute variation of (174) with respect to λ, this gives the equation which determines λ λ ± = (ẋNẋ +ωNω) ± (ẋNẋ +ωNω) 2 − 4(ẋNω) 2 2(ẋNω) .
We substitute λ + into (174) and use λ + λ − = 1. Then (174) turns into the following action The matrix N µν is the projector on the plane orthogonal to ω ν In the spinless limit, α = 0 and ω µ = 0, the functional (177) reduces to the expected Lagrangian of spinless particle, −mc −ẋ µẋ µ . It is well known that the latter can be written in equivalent form using the auxiliary variable λ(τ ) as follows: 1 2λẋ 2 − λ 2 m 2 c 2 . Similarly to this, (177) can be presented in the equivalent form In summary, besides the "minimal" Lagrangian (177) we have obtained two its equivalent formulations given by Eqs. (174) and (179). The Lagrangians provide the appearance of equation pπ = 0 as the primary constraint. In turn, this seems crucial to introduce an interaction consistent with the constraints.

B. Interaction and the problem of covariant formalism
In the formulation (177) without auxiliary variables, our model admits the minimal interaction with electromagnetic field and with gravity. As we detaily shown below, this does not spoil the number and algebraic structure of constraints presented in the free theory. Interaction with an electromagnetic potential is achieved by adding the standard term The minimal interaction with gravity is achieved [23,31] by covariantization of (177). We replace η µν → g µν , and usual derivative by the covariant one,ω Velocitiesẋ µ , ∇ω µ and projector N µν transform like contravariant vectors and covariant tensor, so the action is manifestly invariant under the general-coordinate transformations.
To introduce an interaction of spin with electromagnetic field, we use [22] the formulation (179) with the auxiliary variable λ 1 . We add to the action (179) the term (180) and replacė We have denoted µ = g 2 , where g is gyromagnetic ratio, this agreement simplifies many of equations below. So we restore g only in the final answers. λ 1 in this expression provides the homogeneous transformation law of Dω under the reparametrizations, D τ ω = ∂τ ∂τ D τ ω. The interaction of spin with gravity through the gravimagnetic moment will be achieved in the formulation (174), see below.
Concerning the interaction of spin with electromagnetic field, let us briefly discuss an issue with nearly a century of history, that is not completely clarified so far. While the complete relativistic Hamiltonian of the covariant formulation will be obtained below, its linear on spin part can be predicted from a symmetry considerations. Indeed, the only Lorentz and U (1) -invariant term which involves F and S is F µν S µν . Using the covariant condition (156) we obtain This can be compared with spin part of the Hamiltonian (89) with g = 2 They differ by the famous and troublesome factor 13 of 1 2 . The same conclusion follows from comparison of equations of motion of the two formulations [9]. As we saw in Sect. V A, the expression (184) has very strong experimental support. The question, why a covariant formalism does not lead directly to the correct result, has been raised in 1926 [12], and remain under discussion to date [9,107,108].
Following the work [29], in Sect. XI we show that the vector model provides an answer to this question on a pure classical ground, without appeal to the Thomas precession, Dirac equation or Foldy-Wouthuysen transformation. In a few words it can be described as follows. The relativistic vector model involves a second-class constraints (T 3 and T 4 of Eq. (147)), which we take into account by passing from the Poisson to Dirac bracket. So in the covariant formulation we arrive at the relativistic Hamiltonian (183) accompanied by non canonical classical brackets. To construct the quantum mechanics, we could work with the relativistic Hamiltonian, but in this case we need to find quantum realization of the non canonical brackets. Equivalently, we can find the variables with canonical brackets and quantize them in the standard way. The relativistic Hamiltonian (183), when written in the canonical variables, just gives (184).

C. Particle with the fundamental length scale
Our basic model yields the fixed value of spin, as it should be for an elementary particle. Let us present the modification which leads to the theory with unfixed spin, and, similarly to Hanson-Regge approach [32], with a mass-spin trajectory constraint. Consider the following Lagrangian where l is a parameter with the dimension of length. The Dirac procedure yields the Hamiltonian which turns out to be combination of the first-class constraints p 2 + m 2 c 2 + π 2 ω 2 l 2 = 0, ωπ = 0 and the second-class constraints pω = 0, pπ = 0. The Dirac procedure stops on the first stage, that is there are no of secondary constraints. As compared with (177), the first-class constraint π 2 − α ω 2 = 0 does not appear in the present model. Due to this, square of spin is not fixed, S 2 = 8(ω 2 π 2 − ωπ) ≈ 8ω 2 π 2 . Using this equality, the mass-shell constraint acquires the form similar to the string theory It has a clear meaning: the energy of the particle grows with its spin. The model has four physical degrees of freedom in the spin-sector. As the independent gauge-invariant degrees of freedom, we can take three components S ij of the spin-tensor together with any one product of conjugate coordinates, for instance, ω 0 π 0 .

D. Classification of vector models
While we concentrate on the model specified by Eqs. (147), it is instructive to discuss other sets of constraints that could be used for construction of a spinning particle. The equation (165) relating the Poincaré and Lorentz spins turns out to be useful in what follows. 1. Our basic model (147) with two degrees of freedom implies S µν p ν = 0. Then Eq. (188) implies proportionality of the two spins, 8s 2 = S 2 , whereas their magnitudes are fixed due to the constraints T 2 and T 5 . The variables x µ , p µ and S µν have vanishing brackets with first-class constraints, so they are candidates for observables. 2. The model with the constraints ω 2 = α 2 and π 2 = β 2 instead of π 2 = α ω 2 is essentially equivalent to the basic model. The relationship between two models can be probably established using the conversion scheme [53,54]. 3. Let us replace T 3 ≡ pω = 0 by T 3 ≡ pω − √ ω 2 = 0 in the set (147). These constraints appear in the model of rigid particle. T 4 and T 5 can be taken as the second-class constraints, while T 2 and T 3 form the first-class subset. As a consequence, the model has two degrees of freedom. The Poincaré and Lorentz spins are proportional and have fixed magnitudes. The variables x µ and S µν have non vanishing brackets with first-class constraints. After canonical quantization, the constraint T 3 = 0 turns into the Dirac equation. Hence this semiclassical model can be used to study the relation among classical observables and operators of the Dirac theory. 4. Hanson and Regge developed their model of a relativistic top [32] on the base of antisymmetric tensor S µν without making of any special assumptions on its inner structure. The tensor is subject to first-class constraints S µν p ν = 0. This implies phase space with 2×6−2×3 = 6 degrees of freedom as well as proportional spins with unfixed magnitude. A similar vector model could be constructed starting from the Hamiltonian action where S µν = 2(ω µ π ν − ω ν π µ ). The variables x µ and S µν are not observables in this model. 5. To avoid the unobservable character of original variables in the model (189), we could replace S µν p ν = 0 by the pair of second-class constraints pω = pπ = 0. They provide S µν p ν = 0 and 8 − 2 = 6 degrees of freedom. 6. Adding the first-class constraint ωπ = 0 to the model of Item 5 we arrive at the Lagrangian (185) with four degrees of freedom. 7. There are models based on the light-like vector ω µ [109,110]. Consider the first-class constraints ω 2 = 0, ωπ = 0, π 2 (pω) 2 = const, then s 2 = const, S 2 = 0.
This implies two degrees of freedom. The Poincaré and Lorentz spins, while are fixed, do not correlate one with another. The variables x µ , S µν and s µ are not observable quantities. We note also that S µν p ν = 0, this complicates the analysis of non relativistic limit. 8. Let us replace π 2 (pω) 2 = const by √ π 2 (pω) = const in the set (190). Similarly to Item 3, this constraint may be classical analog of the Dirac equation. This model still has not been studied.
A common for the models 5-8 is the problem whether they admit an interaction with external fields. Concerning the Hanson-Regge model, in their work [32] they analyzed whether the spin-tensor interacts directly with an electromagnetic field, and concluded on impossibility to construct the interaction in a closed form. In our vector model an electromagnetic field interacts with the part ω µ of the spin-tensor.

X. INTERACTION WITH ELECTROMAGNETIC FIELD
In this rather technical section we demonstrate that our variational problem yields a model of spinning particle with expected properties. In particular, our equations of motion generalize an approximate equations of Frenkel and Bargmann-Michel-Telegdi to the case of an arbitrary electromagnetic field.

A. Manifestly covariant Hamiltonian formulation
As we saw in previous section, interaction with an arbitrary electromagnetic field can be described within the action where the term accounts the spin-field interaction. Let us construct Hamiltonian formulation of the model. Conjugate momenta for x µ , ω µ and λ are denoted as p µ , π µ and p λ . We use also the canonical momentum P µ ≡ p µ − e c A µ . Contrary to p µ , the canonical momentum is U (1) gauge-invariant quantity. Since p λ = ∂L ∂λ = 0, the momentum p λ represents the primary constraint, p λ = 0. Expressions for the remaining momenta, p µ = ∂L ∂ẋµ and π µ = ∂L ∂ωµ , can be written in the form where T = [ẋNẋ + DωN Dω] 2 − 4(ẋN Dω) 2 . The functions K µ and R µ obey the following remarkable identities Due to Eq. (178), contractions of the momenta with ω µ vanish, that is we have the primary constraints ωπ = 0 and Pω = 0. One more primary constraint, Pπ = 0, is implied by (195). Hence we deal with a theory with four primary constraints. Hamiltonian is obtained excluding velocities from the expression where λ i are the Lagrangian multipliers for the primary constraints T i . To obtain its manifest form, we note the equalities P 2 = 1 2λ 2 [ẋNẋ −ẋK], π 2 = 1 2λ 2 [DωN Dω − DωR], and Pẋ + πDω = 2L 1 , where L 1 is the first line in Eq. (191). Then, using (195) we obtain Further, using Eqs. (195) we have where appeared the Frenkel spin-tensor S µν . Using (198) and (197) in (196), the Hamiltonian reads The fundamental Poisson brackets {x µ , p ν } = η µν and {ω µ , π ν } = η µν imply {S µν , S αβ } = 2(η µα S νβ − η µβ S να − η να S µβ + η νβ S µα ). According to Eq. (201), the spin-tensor is generator of Lorentz algebra SO(1, 3). As ωπ, ω 2 and π 2 are Lorentzinvariants, they have vanishing Poisson brackets with S µν . To reveal the higher-stage constraints we write the equationsṪ i = {T i , H} = 0. The Dirac procedure stops on third stage with the following equations We have denoted and the function a is written in (209). The last equation from (203) turns out to be a consequence of (205) and (206) and can be omitted. Due to the secondary constraint T 5 appeared in (204) we can replace the constraint T 1 on the equivalent one This can be compared with Eq. (138). The Dirac procedure revealed two secondary constraints written in Eqs. (208) and (204), and fixed the Lagrangian multipliers λ 3 and λ 4 , the latter can be substituted into the Hamiltonian. The multipliers λ 0 , λ 2 and the auxiliary variable λ have not been determined. H vanishes on the complete constraint surface, as it should be in a reparametrization-invariant theory. We summarized the algebra of Poisson brackets between constraints in the Table I. The constraints p λ , T 1 , T 2 and T 5 form the first-class subset, while T 3 and T 4 represent a pair of second class. The presence of two primary first-class constraints p λ and T 2 is in correspondence with the fact that two lagrangian multipliers remain undetermined within the Dirac procedure.
Below we will use the following notation. In the equation which relates velocity and canonical momentum will appear the matrix T Using the identity S µα F αβ S βν = − 1 2 (S αβ F αβ )S µν we find the inverse matrix The two functions are related as follows: b = 2a[2 + (µ − 1)a(SF )] −1 . The vector Z µ is defined by This vanishes for homogeneous field, ∂F = 0. The evolution of the basic variables obtained according the standard ruleż = {z, H}. The equations reaḋ Neither constraints nor equations of motion do not determine the variables λ and λ 2 , that is the interacting theory preserves both reparametrization and spin-plane symmetries of the free theory. As a consequence, all the basic variables have ambiguous evolution. x µ and P µ have one-parametric ambiguity due to λ while ω and π have twoparametric ambiguity due to λ and λ 2 . The variables with ambiguous dynamics do not represent observable quantities, so we need to search for the variables that can be candidates for observables. We note that (213) imply an equation for S µν which does not contain λ 2Ṡ This proves that the spin-tensor is invariant under local spin-plane symmetry. The remaining ambiguity due to λ contained in Eqs. (212) and (214) is related with reparametrization invariance and disappears when we work with physical dynamical variables x i (t). So we will work with x µ , P µ and S µν . We remind that our constraints imply the algebraic restrictions on spin-tensor Equations (212) and (214), together with (215), form a closed system which determines evolution of a spinning particle.
The quantities x µ , P µ and S µν , being invariant under spin-plane symmetry, have vanishing brackets with the corresponding first-class constraints T 2 and T 5 . So, obtaining equations for these quantities, we can omit the corresponding terms in the Hamiltonian (199). Further, we can construct the Dirac bracket for the second-class pair T 3 and T 4 . Since the Dirac bracket of a second-class constraint with any quantity vanishes, we can now omit T 3 and T 4 from (199). Then the relativistic Hamiltonian acquires an expected form (compare it with the square of Dirac equation (138)) The equations (212) and (214) follow from this H with use of Dirac bracket,ż = {z, H} DB . The Dirac brackets in physical-time parametrization will be computed in Sect. X C. The brackets in arbitrary parametrization can be found in [28]. We could also use the constraint S µν P ν = 0 to represent S 0i through S ij , then

B. Comparison with approximate equations of Frenkel and Bargmann-Michel-Telegdi
Lagrangian equations. We can exclude the momenta P and the auxiliary variable λ from the equations of motion. This yields second-order equation for the particle's position. To achieve this, we solve the first equation from (212) with respect to P and use the identities (SF Z) µ = − 1 2 (SF )Z µ ,T µ ν Z ν = b a Z µ , this gives P µ = 1 λT µ νẋ ν − µcZ µ . Then the condition S µν P ν = 0 reads 1 λ (STẋ) µ = µc(SZ) µ . Using this equality, P 2 can be presented as P 2 = 1 λ 2 (ẋGẋ)+µ 2 c 2 Z 2 , where appeared the symmetric matrix The matrix G is composed from the Minkowski metric η µν plus spin and field-dependent contribution, G µν = η µν + h µν (S). So we call G the effective metric induced along the world-line by interaction of spin with electromagnetic field. We substitute P 2 into the constraint (208), this gives λ This shows that the presence of λ in Eq. (192) implies highly non-linear interaction of spinning particle with electromagnetic field. The final expression of canonical momentum through velocity is Using (219) and (220), we exclude P µ and λ from the Hamiltonian equations (212), (214) and (215). This gives closed system of Lagrangian equations for the set x, S. It is convenient to work with reparametrization-invariant derivative Then we have the dynamical equations the Lagrangian counterpart of the condition S µν P ν = 0, as well as the value-of-spin condition, S µν S µν = 8α. The equations contains the effective (spin and position-dependent) mass m r , this can lead to certain geometric effects [78].
In the absence of interaction we obtain an expected dynamics The trajectory is a straight line, while S µν is a constant tensor. Discussion. Eq. (222) and (224) show how spin modifies the classical equation of a point particle subject to Lorentz force m d dτ Let us discuss qualitatively the corresponding contributions. Canonical momentum P µ = p µ − e c A µ of a spinless particle is proportional to its velocity, P µ = mc µ . Interaction of spin with electromagnetic field modifies the relation between the two quantities, see Eq. (220). Contribution of anomalous magnetic moment µ = 1 to the difference betweenẋ µ and P µ is proportional to J c 3 ∼ c 3 , while the term with a gradient of field is proportional to The interaction also modifies the constraints. In particular, the condition S µνẋ ν = 0 of a free theory turns into S µν P ν = 0 with P ν =ẋ ν . This has an important consequence. If we adopt the standard special relativity notions of time and distance, the components S 0i vanish in the frame P µ = (P 0 , 0) instead of the rest frame. Hence our model predicts small dipole electric moment of the particle immersed in an external field (for an experimental estimations, see [79]).
Other important point is the emergence of an effective metric (218) for the particle in flat space. As we saw above, the incorporation of the constraints (215) into a variational problem, as well as the search for an interaction consistent with them turn out to be rather non trivial tasks, and the action (191) is probably the only solution of the problem. So, the appearance of effective metric (218) in equations of motion seems to be unavoidable in a systematically constructed model of spinning particle. An important consequences will be discussed in Sect. XII.
Summing up, in general case the Lorentz force is modified due to the presence of (time-dependent) radiation mass m r (219), the tetrad fieldT , the effective metric G and due to two extra-terms on right hand side of (222).
Consider the "classical" value of magnetic moment µ = 1. ThenT = η and G = η. The Lorentz force is modified due to the presence of time-dependent radiation mass m r , and two extra-terms on right hand side of (222).
Homogeneous field. The structure of our equations simplifies significantly for the homogeneous field ∂ α F µν = 0, then Z µ = 0. Contraction of (224) with F µν yields (SF )˙= 0, that is S µν F µν turns out to be the conserved quantity. This impliesṁ r =ȧ =ḃ = 0. Hence the Lorentz force is modified due to the presence of time-independent radiation mass m r , the tetrad fieldT and the effective metric G. The equations (222) and (224) read They simplify more in the parametrization which implies Since Gẋẋ =ẋ 2 + O(S 2 ), in the linear approximation on S this is just the proper-time parametrization.
The equations become even more simple when µ = g 2 = 1. Let us specify the equation of precession of spin to this case, taking physical time as the parameter, τ = t. Then (224) reduces to the Frenkel condition, S µνẋ ν = 0, while (223) readsṠ µν = e √ −ẋ 2 mrc 2 (F S) [µν] . We decompose spin-tensor on electric dipole moment D and Frenkel spin-vector S according to (151), then D = − 2 c S × v, while variation rate of S is given by Comparison with Bargmann-Michel-Telegdi equations. BMT-equations arë Obtaining their equations in homogeneous field, Bargmann, Michel and Telegdi supposed that the motion of a particle is independent from the motion of spin. Besides they looked for the equation linear on s µ and F µν . It is convenient to introduce BMT-tensor dual to s µ Due to (234) this obeys the equatioṅ Taking the proper-time parametrization and neglecting non linear on F and S terms in our equations (227) and (228), we obtain (233) and (236). Exact solution to equations of motion in a constant magnetic field. Comparing Eqs. (227) and (233) we conclude that spin-field interaction modifies the Lorentz-force equation even for the homogeneous magnetic field. To estimate the influence, it is convenient to work with four-dimensional spin-vector (166) instead of spin-tensor. The constraint S µν P ν = 0 implies s µẋ µ = 0, so s µ can be identified with BMT-vector of spin. As a consequence of Eqs. (212) and (214), it obeys the equatioṅ For the homogeneous magnetic field the equations (212) and (237) has been solved exactly [28], a qualitative picture of motion for µ = 1 can be described as follows. Besides oscillations of spin first calculated by Bargmann, Michel and Telegdi, the particle with anomalous magnetic moment experiences an effect of magnetic Zitterbewegung of the trajectory. Usual circular motion in the plane orthogonal to B is perturbed by slow oscillations along B with the amplitude of order of Compton wavelength, P P 0 λ C . The Larmor frequency and the frequency of spin oscillations are also shifted by small corrections.

C. Parametrization of physical time and physical Hamiltonian
Equations for physical variables x i (t), P i (t) and S µν (t) follow from the formula of derivative of parametric function, dz dt = cż x 0 , after the substitution of (212) and (214) on the right hand side. Our task here is to find a conventional Hamiltonian for these equations. Consider the Hamiltonian action associated with the Hamiltonian (199), dτ pẋ + πω −λ i T i . The variational problem provides both equations of motion and constraints of the vector model in 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 where it is convenient to denoteP 0 = p 0 − e c A 0 . We can treat the term associated with λ as a kinematic (that is velocity-independent) constraint of the problem. According to the standard classical-mechanics prescription [65], we solve the constraintP and substitute the result back into Eq. (238), this gives an equivalent form of the functional where the substitution (239) is implied in the last two terms as well. The expression in square brackets is the Hamiltonian. The sign in front of the square root in (239) was chosen according to the right spinless limit, H = c P 2 i + m 2 c 2 + eA 0 . The variational problem implies the first-class constraints T 2 = ωπ = 0, T 5 = π 2 − α ω 2 = 0 and the second-class constraints where In all expressions below the symbol P 0 represents the function (242).
To represent the Hamiltonian from (240) in a more familiar form, we take into account the second-class constraints by passing from Poisson to Dirac bracket To compute the Dirac brackets of our variables, we use an auxiliary Poisson brackets shown in table II. We will use the notation (209) and Using the table, we obtain {T 3 , T 4 } = eu 0 2caP 0 . Then Dirac brackets among the physical variables x i (t), P i (t) and S µν (t) are To continue, let us restrict to the case of a stationary electromagnetic field, then constraints do not depend explicitly on time. Dirac bracket of any quantity with second-class constraints vanish, so they can be omitted from the Hamiltonian. So we omit the last two terms in (240). The first-class constraints T 2 and T 5 can be omitted as well, as they do not contribute into equations of motion for physical variables. In the result we obtain the physical Hamiltonian The equations of motion that we discussed at the beginning of this section follow from this Hamiltonian according the rule dz dt = {z, H ph } D . Note that the Dirac brackets encode the most part of spin-field interaction, on this reason we have arrived at a rather simple form of physical Hamiltonian. The inclusion of an interaction into the geometry of phase-space and the resulting non commutative geometry is under intensive investigation in various models [56, 80? -98].

XI. SPIN-INDUCED NON COMMUTATIVITY OF POSITION AND FINE STRUCTURE OF HYDROGEN SPECTRUM
Here we discuss how the vector model resolves [29] the problem of covariant formalism described in Sect. IX B.
To quantize our relativistic theory we need to find quantum realization of highly non linear classical brackets (245)-(249). They remain non canonical even in absence of interaction. For instance, Eq. (245) in a free theory reads {x i , x j } = 1 2mcp 0 S ij . We emphasize that non relativistic model has canonical brackets (131), so the deformation arises as a relativistic correction induced by spin of a particle. Technically, the deformation is due to the fact that the constraints pω = pπ = 0 of relativistic theory, used to construct the Dirac bracket, mixes up space-time and inner-space coordinates.
Quantum realization of the brackets in a free theory will be obtained in Sect. XIV, while in an interacting theory its explicit form is unknown. Therefore we quantize the interacting theory perturbatively, considering c −1 as a small parameter and expanding all quantities in a power series. Let us consider the approximation O(c −2 ), that is we neglect c −3 and higher order terms. For the Hamiltonian (250) we have H ph ≈ mc 2 + P 2 2m − P 4 8m 3 c 2 − eµ 4mc (F S). Since the last term is of order c −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 together with Eq. (148) we obtain, up to order c −2 Due to the second and fourth terms, we need to know the operatorsP i andx i up to order c −2 , whileŜ ij 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 (245)-(249) in this approximation An operator realization of these brackets readŝ By construction of a Dirac bracket, the operatorŜ i0 automatically obeys the desired commutators up to order c −1 . So we do not worried on this operator in the computations above.
We substitute these operators into the classical Hamiltonian (251). 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 (251). In the result, the quantum Hamiltonian up to order c −2 reads (we remind that µ = g 2 ) The first three terms corresponds to an increase of relativistic mass. The last two terms coincides with those in Eq. (89). In the result, we have shown that non commutativity of electron's position in the vector model of spin is responsible for the fine structure of hydrogen atom. We could carry out the same reasoning in classical theory, by asking on the new variables z that obey the canonical brackets (131) as a consequence of equations (252). In the desired approximation they are P i = P i , x i = x i − 1 4m 2 c 2 S ij P j and S ij = S ij , that is the first relativistic corrections modify only the position variable.

XII. ULTRA-RELATIVISTIC SPINNING PARTICLE IN ELECTROMAGNETIC BACKGROUND
Let us compare the Lagrangian equations of spinning (222) and spinless (226) particle. For the spinning particle with µ = 1, the relativistic-contraction factor (see (221)) contains the effective metric (218) instead of the Minkowski metric η µν . In the result, equations for trajectory (222) and for precession of spin (223) became singular at the critical velocity which obeys the equationẋGẋ = 0. As we saw above, the singularity determines behavior of the particle in ultra-relativistic limit. The effective metric is composed from the Minkowski one plus (spin and field-dependent) contribution, G = η + h(S). So we need to decide, which of them should be used to construct the three-dimensional geometry discussed in Sect. III. We first test the usual special-relativity notions, v i = dx i dt , a i = dv i dt and va = v i a i , that is we suppose that the particle sees η as the space-time metric. We show that in this case acceleration vanishes at the critical speed which is different from the speed of light. Then we estimate the ultra-relativistic limit using G to define the three-dimensional geometry (19)- (22). Then v cr = c, but since G depends on spin, particles with different spins will probe slightly different three-dimensional geometries.
Ultra relativistic limit within the usual special-relativity notions. It will be sufficient to estimate the acceleration in the uniform and stationary field. We take τ = t in equations (222)-(224) and compute the time derivative derivative on l. h. s. of equations (222) with µ = 1, 2, 3. Then the equations read where v µ = (c, v). Eqs. (262) and (218) imply We compute the time-derivatives in Eq. (260) We note that all the potentially divergent terms (two last terms in (264) and in (265)), arising due to the contribution fromṠ ∼ 1 √ −vGv , disappear on the symmetry grounds. We substitute non vanishing terms into (260) obtaining the expression where the matrix has the inverseM with the propertyM Applying the inverse matrix we obtain the acceleration For the particle with non anomalous magnetic moment (µ = 1), the right hand side reduces to the Lorentz force, so the expression in braces is certainly non vanishing in the ultra-relativistic limit. Thus the acceleration vanishes only when v → v cr , where the critical velocity is determined by the equation vGv = 0. Let us estimate the critical velocity. Using the consequence (ẋSFẋ) = −b(µ − 1)(ẋF SSFẋ) of the supplementary spin condition, and the expression S µ α S αν = −4 π 2 ω µ ω ν + ω 2 π µ π ν , we write As π and ω are space-like vectors, the last term is non-negative, so v cr ≥ c. We show that generally this term is nonvanishing function of velocity, then v cr > c. Assume the contrary, that this term vanishes at some velocity, then Ultra-relativistic limit within the geometry determined by effective metric. As we saw above, if we insist to preserve the usual special-relativity definitions of time and distance, the speed of light does not represent special point of the equation for trajectory. Acceleration of the particle with anomalous magnetic moment generally vanishes at the speed slightly higher than the speed of light. Hence we arrive at a rather surprising result that speed of light does not represent maximum velocity of the manifestly relativistic equation (266). This state of affairs is unsatisfactory because the Lorentz transformations have no sense above c, so two observers with relative velocity c < v < v cr will not be able to compare results of their measurements.
To keep the condition v cr = c, we use formal similarity of the matrix G appeared in (218) with space-time metric. Then we can follow the general-relativity prescription of Sect. III to define time and distance in the presence of electromagnetic field. That is we use G of Eq. (218) to define the three-dimensional geometry (19)- (22). The effective metric depends on x i via the field strength F (x 0 , x i ), and on x 0 via the field strength as well as via the spin-tensor S(x 0 ). So the effective metric is time-dependent even in stationary electromagnetic field. With these definitions we have, by construction, −ẋGẋ = ( dt dx 0 ) 2 (c 2 − (vγv)), so the critical speed coincides with the speed of light. The intervals of time and distance are given now by Eq. (19) and (20), they slightly differ from those in empty space.
In the present case, the expression for three-acceleration can be obtained in closed form in an arbitrary electromagnetic field. We present Eq. (222) in the form (47) Then the acceleration is given by (49). The first two terms on right hand side of (273) give potentially divergent contributions arising from the pieceṠ ∼ , we obtain the acceleration , we conclude that the longitudinal acceleration (50) vanishes in this limit.
In summary, to preserve the equality v cr = c, we are forced to assume that particle in electromagnetic field probes the three-dimensional geometry determined with respect to the effective metric instead of the Minkowski metric. Similarly to Sect. IV, this implies rather unusual picture of the Universe filled with spinning matter. Since G depends on spin, each particle will probe his own three-dimensional geometry. In principle this could be an observable effect. With the effective metric (218), the equation (19) implies that the time of life of muon in electromagnetic field and in empty space should be different.

XIII. INTERACTION WITH GRAVITATIONAL FIELD
A. Lagrangian of spinning particle with gravimagnetic Moment As we saw in Sect. IX B, minimal interaction with gravitational field can be achieved by direct covariantization of the action (177). Remarkably, this leads to MPTD equations, see Sect. XIII B below. Since they become problematic in ultra-relativistic regime, we are forced to look for a non minimal interaction that could suitably modify the equations in this regime. To understand, how they might look, we use the remarkable analogy existing between the gravitational and electromagnetic fields. Hamiltonian formulations of the two minimally interacting theories become very similar if we identify electromagnetic field strength with the Riemann tensor contracted with spin, F µν ∼ R µναβ S αβ . In particular, Hamiltonian action for both theories is where P µ = p µ − e c A µ for electromagnetic field and P µ = p µ −Γ β αµ ω α π β for gravitational field. According to Eq. (199), non minimal interaction through gyromagnetic ratio 2µ implies the contribution − eµ 2c F µν S µν into the third term. So we expect that non minimal interaction with gravity could be achieved replacing this term by λ1 32 κR αβµν S αβ S µν . By analogy with the magnetic moment, the coupling constant κ is called gravimagnetic moment [9,108]. Thus we consider the variational problem [25] S κ = dτ p µẋ µ + π µω µ − λ 1 2 P 2 + κR αβµν ω α π β ω µ π ν + (mc) 2 + π 2 − α ω 2 + λ 2 (ωπ) + λ 3 (P ω) + λ 4 (P π)] .
(277) on the space of independent variables x µ , p ν , ω µ , π ν and λ a . Let us look for the Lagrangian which in the phase space implies the variational problem (277). First, we note that the constraints πω = P ω = 0 always appear from the Lagrangian which depends on Nẋ and Nω instead ofẋ andω. So we set λ 2 = λ 3 = 0 in (277). Second, we present the remaining terms in (277) in the form where we have introduced the symmetric matrix The matrix appeared in (278) is invertible, the inverse matrix is When κ = 0 we have K µν = (1−λ 2 ) −1 g µν , and (280) coincides with the matrix appeared in the free Lagrangian (173). Third, we remind that the Hamiltonian variational problem of the form pq− λ1 2 pAp follows from the reparametrizationinvariant Lagrangian qA −1q . So, we tentatively replace the matrix appeared in (173) by (280) and switch on the minimal interaction of spin with gravity,ω → ∇ω. This gives the following Lagrangian formulation of spinning particle with gravimagnetic moment [26]: Let us show that it does give the desired Hamiltonian formulation (277). The matrixes σ, K and N are symmetric and mutually commuting. Canonical momentum for λ vanishes and hence represents the primary constraint, p λ = 0. Conjugate momenta for x µ and ω µ are p µ = ∂L ∂ẋ µ and π µ = ∂L ∂ω µ respectively. Due to the presence of Christoffel symbols in ∇ω µ , the conjugated momentum p µ does not transform as a vector, so it is convenient to introduce the canonical momentum the latter transforms as a vector under general transformations of coordinates. Manifest form of the momenta is as follows: where L 0 is the second square root in (282). They immediately imply the primary constraints ωπ = 0 and P ω = 0. From the expressions we verify that their sum does not depend on velocities and hence gives one more constraint Then where the first and second terms have been identically rewritten in the general-covariant form. From (284) and (285) we obtain H 0 = Pẋ + π∇ω − L = 0, so the Hamiltonian is composed from primary constraints After the change of variables λ → λ 4 = 1 2 λ 1 λ, we arrive at the Hamiltonian appeared in the variational problem (277). Hamiltonian equations of motion. Variation of the Hamiltonian action (277) with respect to λ a gives the algebraic equations P 2 + κR αβµν ω α π β ω µ π ν + (mc) 2 + π 2 − α ω 2 = 0, while variations with respect to the remaining variables yield dynamical equations which can be written in the covariant form as follows Eq. (291) has been repeatedly used to obtain the final form of equations (292)-(294). Computing time-derivative of the algebraic equations (290) and using (291)-(294) we obtain the consequences Here and below we use the following notation. The gravitational analogy of electromagnetic field strength is denoted In the equation which relates velocity and momentum will appear the matrix Using the identity we find inverse of the matrix T .
The vector Z µ is defined by This vanishes in a space with homogeneous curvature, ∇R = 0. The time-derivatives of (289), (295) and (296) do not yield new algebraic equations. Due to (295) we can replace the constraint (289) on P 2 + κR αβµν ω α π β ω µ π ν + (mc) 2 = 0. The obtained expressions for λ 3 and λ 4 can be used to exclude these variables from the equations (291)-(294). The constraints T 1 , T 2 and T 5 form the first-class subset, while T 3 and T 4 represent a pair of second class.
Neither constraints nor equations of motion do not determine the functions λ 1 and λ 2 , that is the non-minimal interaction preserves both reparametrization and spin-plane symmetries of the theory. The presence of λ 1 and λ 2 in the equations (293) and (294) implies that evolution of the basic variables is ambiguous, so they are not observable. To find the candidates for observables, we note once again that (293) and (294) imply an equation for S µν which does not contain λ 2 . So we rewrite (291) and (292) in terms of spin-tensor and add to them the equation for S µν , this gives the systemẋ Besides, the constraints (289), (290) and (295) imply The equations (306) imply that only two components of spin-tensor are independent, as it should be for spin one-half particle. Eq. (304), contrary to the equations for ω and π, does not depend on λ 2 . This proves that the spin-tensor is invariant under local spin-plane symmetry. The remaining ambiguity due to λ 1 is related with reparametrization invariance and disappears when we work with physical dynamical variables x i (t). Equations (302)-(304), together with (305) and (306), form a closed system which determines evolution of a spinning particle with gravimagnetic moment. The Hamiltonian equations can be equally obtained computingż = {z, H}, where z = (x, p, ω, π), with the Hamiltonian given in square brackets of Eq. (277). Our original variables fulfill the usual Poisson brackets {x µ , p ν } = δ µ ν and {ω µ , π ν } = δ µ ν , then {P µ , P ν } = R σ λµν π σ ω λ , {P µ , ω ν } = Γ ν µα ω α , {P µ , π ν } = −Γ α µν π α . For the quantities x µ , P µ and S µν these brackets imply {S µν , S αβ } = 2(g µα S νβ − g µβ S να − g να S µβ + g νβ S µα ).
We can simplify the Hamiltonian introducing the Dirac bracket constructed with help of second-class constraints Since the Dirac bracket of a second-class constraint with any quantity vanishes, we can now omit T 3 and T 4 from the Hamiltonian. The quantities x µ , P µ and S µν , being invariant under spin-plane symmetry, have vanishing brackets with the first-class constraints T 2 and T 5 . So, obtaining equations for these quantities, we can omit the last two terms in (277), arriving at the relativistic Hamiltonian The equations (302)-(304) can be obtained according the ruleż = {z, H 1 } D .
We have obtained a rather simple expression for the Hamiltonian because of the most part of spin-gravity interaction is encoded now in the Dirac brackets. The expression (310) together with the Dirac brackets could be an alternative starting point for computation of post-Newton corrections due to spin [105,106].
Lagrangian equations. Let us exclude P µ and λ 1 from the equations (303) and (304). Using (300) we solve (302) with respect to P µ . Using the resulting expression in the constraint (305) we obtain λ 1 where the effective metric now is given by Then the expression for momentum in terms of velocity implied by (302) is We substitute this P µ into (303), (304) Together with (306), this gives us the Lagrangian equations for the spinning particle with gravimagnetic moment. Comparing our equations to those of spinning particle on electromagnetic background (222)-(224), we see that the two systems have the same structure after the identification κ ∼ µ and θ µν ≡ R µναβ S αβ ∼ F µν , where µ is the magnetic moment. That is a curvature influences trajectory of a spinning particle in the same way as an electromagnetic field with the strength θ µν .
B. MPTD particle as the spinning particle without gravimagnetic moment Let us compare MPTD equations (15)- (17) with equations of our spinning particle. Imposing κ = 0 in Eqs. (302)-(306), we write them in the form withT from Eq. (300) with κ = 0. Comparing the systems, we see that our spinning particle has fixed values of square of spin and canonical momentum, while for MPTD-particle these quantities represent constants of motion. We conclude that all the trajectories of a body with given m and S 2 = β are described by our spinning particle with spin α = β 8 and with the mass equal to m 2 + f 2 (β) c 2 . In this sense our spinning particle is equivalent to MPTD-particle. We point out that our final conclusion remains true even we do not add the equation (8) to MPTD-equations: to study the class of trajectories of a body with √ −P 2 = k and S 2 = β we take our spinning particle with m = k c and α = β 8 . MPTD-equations in the Lagrangian form (9)- (11) can be compared with (314)-(315). Summing up, we demonstrated that MPTD-equations correspond to the minimal interaction of spinning particle with gravity.
C. Consistency in ultra relativistic regime implies quantized gravimagnetic moment The paradoxical behavior of MPTD-particle originates from the fact that variation rate of spin (57) diverges in the ultra-relativistic limit, ∇S ∼ 1 √ẋ Gẋ , and contributes into the expression for acceleration (77) through the tetrad field T (S). Remarkably, for the non minimal interaction with κ = 1, the undesirable term in Eq. (315) vanishes. Besides this impliesT µ ν = δ µ ν , G µν = g µν , and crucially simplifies the equations of motion 14 . The Hamiltonian equations (302)-(304) read Particle with unit gravimagnetic moment and MPTD particle have a qualitatively different behavior at low velocities. Indeed, keeping only the terms which may give a contribution in the leading post-Newton approximation, ∼ 1 c 2 , we obtain from (320), (321) the approximate equations while MPTD equations in the same approximation read Lagrangian equations are composed now by the equation for trajectory and by the equation for precession of spin-tensor These equations can be compared with (56) and (57) 3. Spin ceases to affect the trajectory in ultra-relativistic limit: the trajectory of spinning particle becomes more and more close to that of spinless particle as v → c. Besides, the spin precesses with finite angular velocity in this limit.
14 Besides S µν Pν = 0, there are known others supplementary spin conditions [4,5,7]. In this respect we point out that the MPTD theory implies this condition with certain Pν written in Eq. (15). Introducing κ, we effectively changed Pν and hence changed the supplementary spin condition. For instance, when κ = 1 and in the space with ∇R = 0, we have P µ =m c √ −ẋgẋẋ µ instead of (15).
where the force is While this expression contains derivatives of spin due toṁ r -term, the resulting expression is non singular function of velocity because ∇S is a smooth function. Hence, contrary to Eq. (67), the force now is non singular function of velocity. We take τ = x 0 in the spatial part of the system (326), this gives where f i (v) is obtained from (327) replacingẋ µ by v µ of equation (35). This system is of the form (34), so the acceleration is given by (41) and (42) With the smooth f i given in Eq. (327), the acceleration (329) remains finite while the longitudinal acceleration (330) vanishes in the limit v → c. Due to the identities (37), we have (vγ) i f i v→c −→ −(vγ) i Γ i αβẋ αẋβ , that is the trajectory tends to that of spinless particle in the limit.
In resume, contrary to MPTD-equations, the modified theory is consistent with respect to the original metric g µν . Hence the modified equations could be more promising for description of the rotating objects in astrophysics.

XIV. ONE-PARTICLE RELATIVISTIC QUANTUM MECHANICS WITH POSITIVE ENERGIES, CANONICAL FORMALISM
As we have seen above, on the classical level our vector model adequately describes spinning particle in an arbitrary gravitational and electromagnetic fields. Moreover, taking into account the leading relativistic corrections in quantized theory with interaction, we have explained the famous one-half factor in non-relativistic Hamiltonian (184), see Sect. X C. Now we turn to a systematic discussion of our model on the quantum level. In this section we construct quantum mechanics of the free theory (179) in the physical-time parametrization. This yields the Schrödinger equation (358), with the Hamiltonian c p 2 + (mc) 2 acting on a space of two-component wave functions. Note that all the solutions have positive energy. The novel point is that the naive expressions, x i and σ i , do not represent operators of position and spin of our theory. This is due to the second-class constraints pω = pπ = 0 of the relativistic theory, which guarantee the supplementary spin condition S µν p ν = 0. The constraints should be taken into account with help of Dirac bracket, this implies a deformation of classical brackets which are subject to quantization. In the result, the position and spin of a spinning particle are represented by the operators (354) and (357). The remaining sections are devoted to establishing of Lorentz covariance of the obtained quantum mechanics.
In the free Lagrangian (179) it is convenient to rescale ω → √ λω, then Repeating the computations made in Sect. X A, we arrive at the Hamiltonian action dτ pẋ + πω − H with the Hamiltonian This can be compared with (199). Recall that the constraint π 2 − α ω 2 = 0 arises as a secondary constraint, from the condition of preservation in time of the primary constraint ωπ = 0. The Hamiltonian action provides both equations of motion and constraints of the vector model in an arbitrary parametrization. Using the reparametrization invariance, we take physical time as the evolution parameter, τ = x 0 c = t, then the Hamiltonian action reads We can treat the term associated with λ as a kinematic constraint of the variational problem. According to known prescription of classical mechanics, we solve the constraint, and substitute the result back into Eq. (238), this gives an equivalent form of the functional where the substitution (334) is implied in the last two terms as well. We have excluded the variables x 0 and p 0 , the remaining variables are x i (t), p i (t), ω µ (t) and π µ (t). The expression in square brackets represents the Hamiltonian. The sign in front of the square root in (334) was chosen according to the right spinless limit. We have excluded non physical variables of the position sector and work now with x i (t), p i (t), ω µ (t) and π µ (t).
The variational problem implies the first-class constraints T 2 = ωπ = 0, T 5 = π 2 − α ω 2 = 0 and the second-class constraints In all expressions below the symbol p 0 represents the function (334). The action (335) implies the Hamiltonian equations Equations (337) describe free-moving particle with the speed less then speed of light The spin-sector variables have ambiguous evolution, because a general solution to (338) depends on an arbitrary function λ 2 . So they do not represent the observable quantities. As candidates for the physical variables of spinsector, we can take either the Frenkel spin-tensor, or, equivalently, the Pauli-Lubanski vector (166) ds µ dt = 0, s µ p ν = 0, s 2 = 3 2 4 .
(341) operators has the standard form The conversion formulas (351) between canonical and original variables have no ordering ambiguities, so we immediately obtain the operatorsX j andŜ j P L corresponding to position and Pauli-Lubanski vector of classical theory where the expression forŜ 0 P L follows from (348) and (351). Using equations (167) and (148) relating the Pauli-Lubanski vector with Frenkel spin-tensor and three-vector of spin, we obtain their quantum realization as follows: The energy operator (349) determines the evolution of a state-vector according to the Schrödinger equation as well the evolution of operators by Heisenberg equations. The scalar product we define as follows then is a probability density forx i . We emphasize that an abstract vector Ψ(t, x) of Hilbert space represents an amplitude of probability density of canonical coordinatex i . The wave function for the original coordinate x i should be constructed according to known rules of quantum mechanics. Let us introduce the operatorp 0 = −i d dx 0 . Then the Schrödinger equation equation reads (p 0 + p 2 + (mc) 2 )Ψ = 0, and applying the operatorp 0 − p 2 + (mc) 2 we obtain Klein-Gordon equation Hence all solutions to the Schrödinger equation form the subspace of positive-energy solutions to the manifestlycovariant Klein-Gordon equation for a two-component wave functions.
Let us compare our operators with known in the literature. Pryce [73] studied possible candidates for observables of the Dirac equation, they are marked as P (d), P (e) and P (c) in the Tables III and IV. He wrote his operators acting on space of Dirac spinor Ψ D , see Table III. Foldy and Wouthuysen [74] found unitary transformation which maps the Dirac equation i ∂ t Ψ D = c(α i p i + mcβ)Ψ D into the pair of square-root equations i ∂ t Ψ = cβp 0 Ψ. Applying the FW transformation, the Pryce operators acquire block-diagonal form on space Ψ, see Table IV. Our operators act on space of solutions of square-root equation (358), so we compared them with positive-energy parts (upper-left blocks) of Pryce operators in the Table IV. Our operators of canonical variablesX j = x j andŜ j correspond to the Pryce (e) (∼ Foldy-Wouthuysen ∼ Newton-Wigner [111]) position and spin operators.
However, operators of position x j and spin S j of our model areX j andŜ j . They correspond to the Pryce (d)operators.

XV. RELATIVISTIC COVARIANCE OF CANONICAL FORMALISM
While we have started from the relativistic theory (331), working in the physical-time parametrization we have lost, from the beginning, the manifest relativistic covariance. Is the quantum mechanics thus obtained is a relativistic theory? Are the scalar product (359) and probability (360) the Lorentz-invariant quantities? Are the mean values Ψ,X i Φ and Ψ,Ŝ i Φ the Lorentz-covariant quantities? To answer these questions, we follow the standard ideology of quantum theory.
First, we associate with our theory the manifestly covariant Hilbert space of states ψ which carries a representation of Poincaré group and admits conserved four-vector ∂ µ I µ (ψ) = 0 with positive null-component. We define the invariant integral over space-like surface Ω can be identified with probability. Second, using the covariant formulation (332) of the classical theory, we find quantum realization of basic variables by means of covariant operators acting in this space. The resulting construction is called a covariant formalism.
Third, we establish a correspondence between the canonical and covariant pictures which respect the scalar products (359) and (362), and show that the scalar products, mean values and transition amplitudes of canonical formalism can be computed using the covariant formalism. This proves the relativistic covariance of quantum mechanics constructed in Sect. XIV.
As we saw above, state-vectors of spinning particle belong to space of solutions to the covariant two-component Klein-Gordon equation. So it is natural to construct the covariant formalism on this base. We do this in the next subsection. The covariant formalism based on the space of solutions to the Dirac equation will be discussed in Sect. XV D.
We emphasize that quantum mechanics of Sect. XIV already has a clear physical interpretation: the state vector Ψ describes a spinning particle with positive energy inx -representation, the operatorX represents a position,Ŝ represents a spin and so on. Therefore there is no need to search physical interpretation of the covariant formalisms (negative energy states of the Dirac equation and so on), and we will not do it. We consider the covariant formalisms as an auxiliary construction that has the only aim to prove the relativistic covariance of the quantum mechanics formulated in Sect. XIV.

A. Relativistic quantum mechanics of two-component Klein-Gordon equation
We denote states and operators of covariant formalism by small letters, to distinguish them from the quantities of canonical formalism.
According to Wigner [99][100][101], with an elementary particle in quantum-field theory we associate the Hilbert space of representation of Poincare group. The space can be described in a manifestly covariant form as a space of solutions to Klein-Gordon equation for properly chosen multicomponent field ψ a (x µ ). The space of one-component fields corresponds to spin-zero particle. It is well-known, that this space has no quantum-mechanical interpretation. In contrast, two-component Klein-Gordon equation does admit the probabilistic interpretation. As we show below, the four-vector (371) represents positively defined conserved current of this equation. Using the current, we can define an invariant scalar product and hence the covariant rules for computing mean values of covariant operators defined on this space.
The two-component KG equation has been considered by Feynman and Gell-Mann [112] to describe weak interaction of spin one-half particle.
Using this formula, we have checked by direct computations that covariant operatorsp,ŝ µν andŝ µ P L transform into canonical operatorsp,Ŝ µν andŜ µ P L (recall that the spatial part ofŜ µν ,Ŝ i = 1 4 ijkŜ jk represents the classical spin S i ). This result together with Eq. (379) implies that mean values of these operators of canonical formulation are relativistic-covariant quantities.
Concerning the position operator, we first apply the inverse to Eq. (378) to our canonical coordinateX i = i ∂ ∂p i in the momentum representatioñ Our position operator (351) then can be mapped as follows: We note that pseudo-Hermitian part of operatorx i rp coincides with the imagex i V , Sincex µ rp has explicitly covariant form, this also proves covariant character of position operatorX i . Indeed, (378) means that matrix elements ofX i are expressed through the real part of manifestly covariant matrix elements Ψ,X i Φ = (ψ,x i V φ) = Re(ψ,x i rp φ).
In summary, we have proved the proposition formulated above. It could be formulated also as follows. The operatorŝ s µν andx µ rp , which act on the space of two-component KG equation, represent manifestly-covariant form of the Pryce (d)-operators.
That is dynamics of ψ is determined by (405), whileξ accompanies ψ:ξ is determined from the known ψ taking its derivative,ξ = 1 mc (σp)ψ. Evidently, the systems (370)  In non relativistic theory, spin can be described on the base of semiclassical Lagrangian (94) which is invariant under spin-plane local symmetry. The symmetry yields two first-class constraints (91) on spin-sector variables in Hamiltonian formalism. The resulting theory has an expected number of physical degrees of freedom, in particular, the only observable quantities of spin-sector turn out to be the components of spin-vector (90).
To treat spin in a manifestly Lorentz-covariant way, we extended phase space with two auxiliary degrees of freedom, adding null-components to the basic variables ω and π. To supply their auxiliary character, we used two second-class constraints, see (147). This implies drastic modification of the formalism: the constraints induce noncommutative geometry in the phase space even in a free relativistic theory. In particular, spin induces the noncommutative classical brackets (342) of position variables. This must be taken into account in construction of quantum mechanics of a spinning particle. For a spinning electron in Coulomb electric and constant magnetic field, our model yields the non relativistic Hamiltonian (259) with correct factors in front of spin-orbit and Pauli terms. Hence the spin-induced noncommutativity explains the famous one-half factor in the Pauli equation on the classical level, without appeal to the Thomas precession, Dirac equation or Foldy-Wouthuysen transformation. Besides, for a spinning body in gravitational field, the spin-induced noncommutativity clarifies the discrepancy in expressions for three-acceleration obtained by different methods, see [31].
Lagrangian of the vector model admits interaction with an arbitrary gravitational and electromagnetic fields and has reasonable properties both on classical and on quantum level. Dealing with the variational problem, we were able to determine both Hamiltonian and classical brackets of the theory in unambiguous way.
Regarding the interaction with electromagnetic field, equations (222)-(224) of our particle with a magnetic moment generalize the approximate equations of Frenkel and BMT to the case of an arbitrary field. Straightforward canonical quantization of the model yields quantum mechanics in Hilbert space of two-component Weyl spinors. All solutions to the Schrödinger equation (358) have positive energy. Since our basic variables obey non-canonical brackets, the operators which represent them have a non standard form, see (354) and (357). To establish the relativistic covariance of obtained quantum mechanics, we first developed manifestly relativistic quantum mechanics of two-component Klein-Gordon equation. Then we related states and operators of two formalisms and demonstrated on this base the relativistic invariance of the scalar product (359), and relativistic covariance of mean values of operators of canonical formalism, see Eqs. (377) and (393). Using the relationship (411) between Klein-Gordon and Dirac formalisms, we also formulated the rules (418) for computation the mean values in the framework of Dirac formalism. Here we emphasize once again, that we have not tried to find an interpretation of negative-energy states presented in the covariant KG and Dirac formalisms. The formalisms were considered as an auxiliary constructions that allow us to prove relativistic covariance of the quantum mechanics formulated in Sect. XIV.
Regarding the interaction with gravity, the minimal coupling gives the equations (316)-(318) equivalent to MPTD equations of a rotating body (15)- (17). Spin-gravity interaction induces an effective metric (312) in the Lagrangian equation (314) for a trajectory. To study it in ultra-relativistic limit, we used the Landau-Lifshitz approach to define the three-dimensional geometry, and defined on this base the three-dimensional acceleration (54) which guarantees the impossibility for a particle in geodesic motion to overcome the speed of light. The effective metric causes unsatisfactory behavior of MPTD particle in the ultra-relativistic regime. In particular, its acceleration grows with velocity and becomes infinite in the limit. To improve this, we constructed a non minimal spin-gravity interaction (282) through a gravimagnetic moment, and showed that a fast-moving body with unit gravimagnetic moment has a satisfactory behavior.