Review Article | Open Access
Alexei A. Deriglazov, Walberto Guzmán Ramírez, "Recent Progress on the Description of Relativistic Spin: Vector Model of Spinning Particle and Rotating Body with Gravimagnetic Moment in General Relativity", Advances in Mathematical Physics, vol. 2017, Article ID 7397159, 49 pages, 2017. https://doi.org/10.1155/2017/7397159
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 Zitterbewegung; (B) spin-induced noncommutativity 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 ultrarelativistic limit, and equations with improved behavior.
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 [1–8]. The equations are necessary for current applications of general relativity on various space-time scales: for analysis of Lense-Thirring precession , for accounting spin effects in compact binaries and rotating black holes [10, 11], and in discussion of cosmological problems; see  and references therein. Closely related problem consists in establishing classical equations that could mimic quantum mechanics of an elementary particle with spin in a semiclassical approximation [13–17]. 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 [18, 19]. In accelerator physics  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 . Basic equations of spintronics are based on heuristic and essentially semiclassical considerations [22, 23]. 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  and in the book of Corben . 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 collecting and tying 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 [24–33]. In  we constructed final Lagrangian for a spinning particle with an arbitrary magnetic moment. In  we presented the Lagrangian minimally interacting with gravitational field, while in [26, 27] it has been extended to the case of nonminimal 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 . The works [27–33] 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 [34, 35]. 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 and 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 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 ultrarelativistic limit. In particular, three-dimensional acceleration of MPTD particle increases with velocity and becomes infinite in the limit. Therefore we examine the nonminimal interaction; this gives a generalization of MPTD equations to the case of a rotating body with gravimagnetic moment . We show that a rotating body with unit gravimagnetic moment has an improved behavior in the ultrarelativistic regime and is free from the problems detected in MPTD equations.
Notation 1. Our variables are taken in arbitrary parametrization ; then . The square brackets mean antisymmetrization, . For the four-dimensional quantities we suppress the contracted indexes and use the notation , , , and . Notations for the scalar functions constructed from second-rank tensors are , . When we work in four-dimensional Minkowski space with coordinates , we use the metric , then , and so on. Levi-Civita tensors in four and three dimensions are defined by and . Suppressing the indexes of three-dimensional quantities, we use bold letters, , , , and so on.
The covariant derivative is . The tensor of Riemann curvature is .
2. Lagrangian Form of Mathisson-Papapetrou-Tulczyjew-Dixon Equations of a Rotating Body
Equations of motion of a rotating body in curved background are formulated usually in the multipole approach to description of the body; see  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, , implies certain equations for the multipoles. The first results were reported by Mathisson [1, 2] and Papapetrou . They have taken the approximation which involves only first two terms (the pole-dipole approximation). Manifestly covariant equations were formulated by Tulczyjew  and Dixon . In the current literature they usually appear in the form given by Dixon ((6.31)–(6.33) in ), we will refer to them as Mathisson-Papapetrou-Tulczyjew-Dixon equations.
We discuss MPTD equations in the form studied by Dixon (our is twice that of Dixon)In the multipole approach, is called representative point of the body; we take it in arbitrary parametrization (contrary to Dixon, we do not assume the proper-time parametrization (we will be interested in ultrarelativistic behavior of a body. The proper-time parametrization has no sense when ); that is, we do not add the equation to the system above). is associated with inner angular momentum, and 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 [5, 6, 8]. After adding (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 . The most interesting property of the resulting equation is the emergence of an effective metric instead of the original metric .
Let us start from some useful consequences of the system (2)–(4). Take derivative of the constraint, , and use (2) and (3); this gives the expressionwhich can be written in the formContract (5) with . Taking into account the fact that , this gives . Using this in (6) we obtainContracting (3) with and using (4) we obtain ; that is, square of spin is a constant of motion. Contraction of (5) with gives . Contraction of (5) with gives . Contraction of (2) with , gives ; that is, is one more constant of motion, say , const (in our vector model developed below this is fixed as ). Substituting (7) into the equations (2)–(4) we now can exclude from these equations, modulo to the constant of motion .
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 . To improve this point, we note that (7) acquires a conventional form (as the expression for conjugate momenta of in the Hamiltonian formalism), if we add to the system (2)–(4) one more equation, which fixes the remaining quantity . To see how the equation could look, we note that for nonrotating body (pole approximation) we expect equations of motion of spinless particle, , , and . Independent equations of the system (2)–(5) in this limit read , . Comparing the two systems, we see that the missing equation is the mass-shell condition . Returning to the pole-dipole approximation, an admissible equation should be , where must be a constant of motion. Since the only constant of motion in arbitrary background is , we write (we could equally start with ; assuming that this equation can be resolved with respect to , we arrive essentially at the same expression)With this value of , we can exclude from MPTD equations, obtaining closed system with second-order equation for (so we refer to the resulting equations as Lagrangian form of MPTD equations). We substitute (7) into (2)–(4); this gives the systemwhere (8) is implied. They determine evolution of and for each given function .
It is convenient to introduce the symmetric matrix composed of the “tetrad field” of (7):Since this is composed of the original metric plus (spin and field-dependent) contribution , we call the effective metric produced along the world-line by interaction of spin with gravity. Eq. (11) implies the identityso we can replace in (9)–(11) by . In particular, (9) readsAdding the consequences found above to the MPTD equations (2)–(4), we have the systemwith given in (7). In Section 13.2 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 factor , where the effective metric appeared. The equation for trajectory (14) became singular for the particle’s velocity which annihilates this factor, . We call this the critical velocity. The observer-independent scale of special relativity is called, as usual, the speed of light. The singularity determines behavior of the particle in ultrarelativistic limit. To clarify this point, consider the standard equations of a spinless particle interacting with electromagnetic field in the physical time parametrization , . Then the factor is just ; 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 ; 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, . So the singularity implies (we point out that the factor can be hidden using the singular parametrization; for instance, in the proper-time parametrization this would be encoded into the definition of , ) that during its evolution in the external field the spinless particle can not exceed the speed of light .
3. Three-Dimensional Acceleration and Speed of Light in General Relativity
The ultrarelativistic behavior of MPTD particle in an arbitrary gravitational field will be analyzed by estimation of three-acceleration as . 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 manifoldTo 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 . We first recall the most simple part of this problem, which consist in determining basic differential quantities of three-dimensional geometry: infinitesimal distances, time intervals, and velocity . 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 ultrarelativistic particles turns out to be important for analysis of near horizon geometry of extremal black holes (see ) and for accurate accounting of the corresponding corrections to geodesic motion near black hole [40–48].
3.1. Coordinate-Independence of Speed of Light
Consider an observer that labels events by some coordinates of pseudo-Riemann space (18) to describe the motion of a point particle in a gravitational field with given metric . Formal definitions of the three-dimensional quantities can be obtained representing four-interval in block-diagonal formThis prompts introducing infinitesimal time interval and distance as follows:Therefore the conversion factor between intervals of the coordinate time and the time measured by laboratory clock isFrom (20) it follows that laboratory time coincides with coordinate time in the synchronous coordinate systems where metric acquires the forms and . If metric is not of this form, we can not describe trajectory using the laboratory time as a global parameter. But we can describe it by the function and then determine various differential characteristics (such as velocity and acceleration) using the conversion factor (22). For instance, three-velocity of the particle isso it is convenient to introduce the notationThe definitions of and are consistent: . Three-dimensional geometry is determined by the metric . In particular, square of length of a vector is given by . Using these notations, the infinitesimal interval acquires the form similar to special relativityThis equality holds in any coordinate system . Hence a particle with the propagation law has the speed , and this is a coordinate-independent statement. The value of the constant , introduced by hand, is fixed from the flat limit: (20) implies when .
These rather formal tricks are based  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, . Second, for simultaneous events, the four-interval coincides with distance, . Assuming that the same holds in general relativity, let us analyze infinitesimal time interval and distance between two events with coordinates and . The world-line is associated with laboratory clock placed at the spatial point . So the time interval between the events and measured by the clock isConsider the event infinitesimally closed to the world-line . To find the event on the world-line which is simultaneous with , we first look for the events and which have null-interval with , . The equation with has two solutions and then and . Second, we compute the middle pointBy definition (in the flat limit the sequence , , of events can be associated with emission, reflection, and absorption of a photon with the propagation law ; then the middle point (27) should be considered simultaneous with ), the event with the null-coordinate (27) is simultaneous with the event ; see Figure 1. By this way we synchronized clocks at the spatial points and . According to (27), the simultaneous events have different null-coordinates, and the difference obeys the equationConsider a particle which propagated from to . Let us compute time interval and distance between these two events. According to (27), the eventat the spatial point is simultaneous with ; see Figure 2. Equation (29) determines the event (at spatial point ) simultaneous with . So the time interval between and coincides with the interval between e and is given by (30). Distance between and coincides with the distance between and ; the latter is given in (31).
According to (26) and (27), the time interval between the events and (29) isSince the events and (29) are simultaneous, this equation gives also the time interval between and . Further, the difference of coordinates between the events and (29) is . As they are simultaneous, the distance between them isSince (29) occurs at the same spatial point as , this equation gives also the distance between and . Equations (30) and (31) coincide with the formal definitions presented above, (20) and (21).
3.2. 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 . To analyze this, we first use geodesic equation to obtain the derivative of coordinate of the velocity vector.
If we take the proper time to be the parameter, geodesics obey the systemwhereDue to this definition, the system (32) obeys the identity . The system in this parametrization has no sense for the case we are interested in, . So we rewrite it in arbitrary parametrization this yields the equation of geodesic line in reparametrization-invariant formUsing the reparametrization invariance, we set ; then (20) and (22) imply , and spatial part of (35) readswhere we have denotedDirect computation of the derivative on l.h.s. of (36) leads to the desired expressionwhereand three-dimensional Christoffel symbols are constructed with help of three-dimensional metric , where is considered as a parameterWe have , so the inverse metric of turns out to be . Note that ; that is, in the limit the matrix turns into the projector on the plane orthogonal to .
If we project the derivative (38) on the direction of motion, we obtain the expressionDue to the first and second terms on r. h. s., this expression does not vanish as . Note that this remains true for stationary metric, , or even for static metric, , ! The reason is that the derivative in our three-dimensional geometry consists of three contributions: variation rate of the vector field itself, variation of basis in the passage from to , and variation of the metric during the time interval . Excluding the last two contributions, we obtain the variation rate of velocity itself, that is, an accelerationFor the special case of stationary field, , our definition reduces to that of Landau and Lifshitz; see page 251 in . Complementing in (38) up to the acceleration, we obtain three-dimensional acceleration of the particle moving along the geodesic line (35):This is the second Newton law for geodesic motion. Contracting this with , we obtain the longitudinal accelerationThis implies as .
Let us confirm that is the only special point of the function (44). Using (33), (21)–(25), and (40) and the identitieswe can present the right hand side of (44) in terms of initial metric as follows:The quantity has been defined in (37). Excluding according to this expression, we obtainFor the stationary metric, , (47) acquires a specially simple formThis shows that the longitudinal acceleration has only one special point, as . Hence the spinless particle in the stationary gravitational field can not overcome the speed of light. Then the same is true in general case (46), 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 (44) can be repeated for a more general equation. Using the factor we construct the reparametrization-invariant derivativeConsider the reparametrization-invariant equation of the formand suppose that the three-dimensional geometry is defined by according to (20)–(23). Then (50) implies the three-accelerationand the longitudinal accelerationThe spatial part of the force is , where is given by (37), and the connection is constructed with help of the three-dimensional metric according to (40). For the geodesic equation in this notation we have . With this (51) and (52) coincide with (43) and (44).
Eq. (52) shows that potentially dangerous forces are of degree four or more, .
3.3. Parallel Transport in Three-Dimensional Geometry
Now we consider an arbitrary vector/tensor field in the space with three-dimensional geometry determined by a nonstatic metric . Variation rate of the field along a curve should be defined in such a way that it coincides with (42) for the velocity. Let us show how this definition follows from natural geometric requirements . splitting preserves covariance of the formalism under the following subgroup of general-coordinate transformations: , . Under these transformations is a scalar function and is a vector while and are tensors; then the conversion factor (22) is a scalar function and the velocity (23) is a vector. So it is convenient to introduce the usual covariant derivative of a vector field in the direction of with the Christoffel symbols (40). By construction, the metric is covariantly constant, . For the field along the curve we have the covariant derivative in the direction of To define the variation rate of , 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 , we have . For the constant fields in our case it is natural to demand the same (necessary) condition: . Taking into account the fact that , this condition can be written as follows:This will be satisfied, if a constant field is defined by the equationThis 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 (42) for acceleration.
4. Behavior of Ultrarelativistic 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 equationThe singularity determines behavior of the particle in ultrarelativistic limit. The effective metric is composed of the original one plus (spin and field-dependent) contribution, . 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 to define the three-dimensional geometry (20)–(23). 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 (58) to write (61) in the formwith defined in (37). Using , we rewrite the last term as follows:As and are space-like vectors, the last term is nonnegative; this implies . Let us confirm that generally this term is nonvanishing function of velocity; then . Assuming the contrary, that this term vanishes at some velocity, thenWe analyze these equations in the following special case. Consider a space with covariantly constant curvature . Then , and using (60) we conclude that is an integral of motion. We further assume that the only nonvanishing part is the electric part of the curvature, . Then the integral of motion acquires the formLet us take the initial conditions for spin such that ; then this holds at any future instant. Contrary to this, the system (64) implies . Thus, the critical speed does not always coincide with the speed of light and, in general case, we expect that is both field and spin-dependent quantity.
The second problem is that acceleration of MPTD particle grows up in the ultrarelativistic limit. In the spinless limit (59) turns 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 field and of the effective metric in the left hand side. Due to the dependence of the tetrad field on the spin-tensor , the singularity presented in (60) causes the appearance of the term proportional to 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 of in (59):where is the inverse for . Using (60) we obtainUsing this expression, (67) reads (three-dimensional geometry is defined now by ; in this case we can not use (51) and (52) to estimate the acceleration)where we denotedIt will be sufficient to consider static metric with . Then three-dimensional metric and velocity areTaking , the spatial part of (69) with this metric readswith defined in (37), for the caseIn the result, we have presented the equation for trajectory in the form convenient for analysis of acceleration; see (36). Using the definition of three-dimensional covariant derivative (42), we present the derivative on the l.h.s. of (72) as follows:We have denotedThe matrix has the inverseCombining these equations, we obtain the three-acceleration of our spinning particleFinally, using manifest form of from (70) we haveThe longitudinal acceleration is obtained by projecting on the direction of velocity; that is,where . As the speed of the particle closes to the critical velocity, the longitudinal acceleration diverges due to the first term in (79). In resume, assuming that MPTD particle sees the original geometry , we have a theory with unsatisfactory behavior in the ultrarelativistic limit.
Let us consider the second possibility; that is, we take to construct the three-dimensional geometry (20)–(23). With these definitions we have, by construction, , 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 the spatial part of (69) implieswhere, from (70), is given byEquation (80) is of the form (36), so the acceleration is given by (43) and (44) where, for the present case, :With given in (81), the longitudinal acceleration vanishes as .
Let us resume the results of this section. Assuming that spinning particle probes the three-dimensional space-time geometry determined by the original metric , we have a theory with unsatisfactory ultrarelativistic limit. First, the critical speed, which the particle can not overcome during its evolution in gravitational field, can be more than the speed of light. The same observation has been made from analysis of MPTD particle in specific metrics [49–53]. Second, the longitudinal acceleration grows up to infinity in the ultrarelativistic limit. Assuming that the particle sees the effective metric as the space-time metric, we avoided the two problems. But the resulting theory still possesses the problem. The acceleration (82) contains the singularity due to ; that is, at 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 (some models of doubly special relativity predict rainbow geometry at Planck scale [54–57]): there is no unique space-time manifold for the Universe of spinning particles: each particle will probe its 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 ultrarelativistic regime. This will be achieved within the framework of vector model of spinning particle, which we shall describe in the subsequent sections.
5. Vector Model of Nonrelativistic 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 we introduce the two-component wave function , . The spin operators act on as -matrices and are defined bywhere stands for the Pauli matrices; they form a basis of the vector space of traceless and Hermitian -matrices:Their basic algebraical properties areNote that the commutators (87) of -matrices are the same for the angular momentum vector. The spin operators, being proportional to the Pauli matrices, have similar properties, in particularConsider Coulomb electric and a constant magnetic fields. The electromagnetic potential can be taken in the forms and . Then evolution of an electron immersed in this fields is described by the equationThe 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 is called gyromagnetic ratio of the electron (quantum electrodynamics gives due to radiative corrections). The vector 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 .
Equation (91) 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 in the third term, while Zeeman effect requires the factor 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 . 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 Schrödinger equation .
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 operators , the problem can be formulated as follows. We look for a classical-mechanics system which, besides the position variables , 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 (90) and with the classical brackets . Then canonical quantization of these variables will yield spin operators with the desired properties (89) and (90). According to this, typical spinning particle model consists 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 (there is an elegant formalism developed by Berezin and Marinov [59, 60] based on using of anticommuting (Grassmann) variables for the description of spin; we present here another formulation based on commuting variables, without appealing to a rather formal methods of the Grassmann mechanics). In fact, different spinning particles discussed in the literature differ by the choice of the inner space. An exceptional case is the rigid particle  which consists of only position variables, but with the action containing higher derivatives. The model yields the Dirac equation ; hence it also can be used for description of spin.
It should be noted that (91) is written in the laboratory system, so we do not state that our classical variable 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 . The number of variables and their algebra are different from the number of spin operators and their commutators, (89). Maybe the most natural way to arrive at the operator algebra (89) is to consider spin as a composite quantity:where are coordinates of a phase space equipped with canonical Poisson bracket. This immediately induces -algebra for ,