Dynamical Equations of Spinning Particles: Feynman’s Proof

In this letter, we discuss the extension of Feynman’s derivation of the equation of motion to the case of spinning particles. We show that a spinning particle interacts only with the electromagnetic and gravitational fields. In the absence of the electromagnetic interactions, we rederive Papapetrou’s equations for spinning particles in the background of the conformal gravity. We also find that the effect of spin coupled to non-constant electromagnetic fields leads to further corrections to the Lorentz force equations. Some discussions of these results are given at the end. RU93-10-B November 1993 This work was supported in part by the US Department of Energy. † [email protected] The derivation of the dynamical equations of the spinning particles in external fields has attracted the interests of physicists for over fifty years.1−6 The goal in attacking this problem is to study the dynamical effects of the spin precession, the spin-spin interactions (the Stern-Gerlach effects) and the spin-orbits couplings for particles in external fields. The most notable results of this search are the Bargmann-Michel-Telegdi (BMT) equations2 and the Papapetrou equations.3 Based on what Frenkel suggested,1 Bargmann, Michel and Telegdi discussed the precession of spinning particles in external electromagnetic fields,2 while Papapetrou conjectured the dynamical equations of spinning particles in general relativity by considering a rotational mass-energy distribution in the limit of vanishing volume but with the angular momentum remaining finite.3 Still various discussions in looking for the dynamics of spinning particles remain active from different points of view5,6,12,4 and all of these are less direct or attractive when compared to Feynman’s rederivation of Lorentz force equations.7 Since Dyson7 presented the Feynman’s proof of the homogeneous Maxwell equations and Lorentz force equation for a Newtonian particle, the generalizations to the case of the spinless particle (with and without the internal structure) in both special and general relativity have been studied.8−13 The conventional theories have not been completely rederived. In particular, Tanimura9 showed that the particle worldlines are not parametrized by the proper time of motion and the particle does not follow a geodesic. In this paper, we will take a further generalization to the case of spinning particles and reexamine closely the above mentioned features. We rederive both the Lorentz force equations and the Papapetrou equations in a simple and direct way. Specially, our approach offers a systematic study of both spin-charge and spin-gravitational coupling to all orders. By postulating the Poisson brackets of the Newtonian variables of particle motion, Feynman, according to Dyson, obtained the homogeneous Maxwell equations and the Lorentz force equations.7 The key points in his proof are to use the associative conditions, namely the Jacobi identity of the brackets, and the so-called second Leibniz rule:9 d dτ [A,B] = [ dA dτ , B] + [A, dB dτ ], (1) where the τ is a parameter of the particle trajectory. Note that using the Jacobi identity one may derive Eq. (1) if one assumes the existence of Hamiltonian evolution. Therefore, this procedure may be re-formulated in terms of the symplectic language, in which the symplectic two-form, by definition yields the particle’s Poisson brackets and their associated Jacobi identity. Moreover, the homogeneous Maxwell equations follow trivially from the closure of the symplectic two-form and the Lorentz force equations may be derived simply from one or two line computations. We shall use this new formulation of the Feynman’s approach to study spinning particles. Since the spin degree of freedom arises from symmetry transformations of space-time, it is generally not possible to formulate the spin variables purely in terms of Newtonian coordinates of the particle motion. This difficulty prevents one from directly employing Feynman’s proof. However, in two spatial dimensions, the canonical structure of a spinning particle is explicitly known14,15,16 and the spin degree of freedom indeed can be purely represented by the Newtonian variables of the particle motion.14,16 We shall begin with a discussion of a spinning charged particle in 2+1 dimensional flat space-time. In this paper, we assume the particle without internal structure (extension to the case of internal degrees of freedom is straightforward). For a spinning particle or an anyon with spin −s in a 2+1 dimensional flat spacetime, with the metric ηab = diag(+−−), the symplectic structure is given by16 ω = dx ∧ dpa + 1 2 sfabdp a ∧ dp + 1 2 eFabdx a ∧ dx. (2) Where pa = mẋa, p2 = ηabp apb (ǫ012 = 1), fab = ǫabcp c/(p2)3/2, and xa(a = 0, 1, 2) are the position variables of a particle(the overdot denotes the τ -derivative and m is the particle’s mass). The spin vector Sa for the particles, as shown in Ref. [14, 16], is given by S = −s p a √ p2 . (3) The closure condition dω = 0 tells us that the antisymmetric tensor Fab is a function of xa only and satisfies the homogeneous Maxwell equations: ∂cFab + ∂aFbc + ∂bFca = 0. (4) By definition, ω gives the Poisson brackets or the commutation rules as [x, x] = is(M̃f), [p, x] = i(M), [p, p] = ie(MF ), (5) where Mab = ηab + es(Ff)ab ≡ ηab + esFacf b. M̃ denotes the transpose of M . Using the second Leibniz rule (1) and Eqs. (5), we have, d dτ (M) = i ( 1 m [p, p] + [ṗ, x] ) , = 1 m (MF ) + i[ṗ, x]. (6) In principle, one may deduce from Eq. (6) the equation of motion of the spinning particle for arbitrary value of spins and external field F . It is, of course, very difficult to simplify these equations in general to obtain the desired equations. We shall instead derive the equations of motion in terms of a series in powers of spin. Since the τ derivative of pa is also hidden in the left hand side of Eq. (6), it is easy to deduce the equations of motion from the first equation of (5) by applying the Leibniz rule (1). Thus ms d dτ (M̃f) = i ( [p, x] + [x, p] ) , = (M) − (M). (7) We expand Eq. (7) into a power series of s and keep only the lowest nontrivial order: dfab dτ = e m [ (Ff) − (Ff) ] + es d dτ (fFf) + e2s m [ (FfFf) − (FfFf) ] +O(s), (8) which may be further simplified into dY a dτ = − e m F Yb − es m Y Y · Ḟ + O(s), (9) where Y a = 1 2ǫ fbc and Fa = 1 2ǫabcF bc. Thus, we have the equations of motion for both the position and spin variables (3): dpa dτ + e m F pb = 3 2 p d dτ ln(p) + e m pa p2 S(∂Fb)pc +O(s) = − e 2m pa p2 S(∂Fb)pc +O(s), (10) dSa dτ = − e m F Sb +O(s). (11) Clearly, Eq. (11) is the Bargmann-Michel-Telegdi equations in 2+1 dimensions.2,16 Since ∂F is always coupled to the spin, ∂F term will not give any contributions to order s2 to the equations of motion for the spin. We will see this again below. However, the equations (10) are physically wrong although it follows from our starting point. First of all, they are not the Hamiltonian equations, because the τ -evolution is not generated by the τ -Hamiltonian, which is proportional to the first integral of motion of Eqs. (10), i.e. A = p2 + eS · F . Secondly, Eqs. (10) are not the typical Hamiltonian equations of motion because of explicit τ -derivative appearing in the right hand side of the equations. Alternatively speaking, using the rules in Eq. (5) we cannot derive Eqs. (10) and their counterpart mẋa = pa as the Heisenberg equations. For example, ẋa 6= i−1[xa, ζA] for any constant ζ . This apparent inconsistency arises as the consequence of the spin coupled to the gradient of the external electromagnetic field, although it is perfectly consistent for a slowly varying field F , as we will demonstrated below. For a slowly varying field F , Eqs. (11) remain unchanged and Eqs. (10) yield: d dτ p + O(∂F, s) = 0, dpa dτ = − e m F pb + O(∂F, s), (12) i.e. we have the (correct) worldline-equation and the Lorentz force equations (upto possible terms which depend on gradients of F ). As shown in Ref. [16], the τ Hamiltonian G which generates τ -evolution for this system can be obtained from (2) and (12): G = − 1 2m ( p − 2eS · F −m ) +O(s). (13) Notice that Eq. (13) is not the first integral of motion of (12), but rather is that of a similar equation:6,17 dpa dτ + e m F pb = e m S∂Fb +O(s). (14) These equations in Eqs. (12) or (14) are indeed realized quantum mechanically as the Heisenberg equations:16 dηa dτ = 1 i [η, G], (15) where ηa = (xa, pa). Notice that one needs only the rules in Eq. (5) to order s to verify the equivalence between Eq. (15) and (14). However, when we used the first equation in Eq. (5) to derive Eq. (10), we actually use the rules in Eq. (5) to order s2. This suggests that the postulated symplectic structure or the Poisson brackets in Eq. (5) are not compatible with the definition of mẋa ≡ pa to order s2. In other word, it is necessary to either find the proper higher order corrections to ω or modify the relation mẋa ≡ pa to resolve the physically unacceptable situation mentioned above. In fact, one may find that if one assumes mẋa = pa+emska ≡ p+emsǫ∂b(S ·F )Yc, in stead of having the equation (7) or (10), one has a new equation by applying the rule (1) ms d dτ (M̃f) = i ( [p, x] + [x, p] + ems{[k, x] + [x, k]} ) , = (M) − (M) + ems ( ∂ka ∂pb − ∂k b ∂pa ) +O(s). (16)