0 Spin , gravity , and inertia

The gravitational effects in the relativistic quantum mechanics are investigated. The exact Foldy-Wouthuysen transformation is constructed for the Dirac particle coupled to the static spacetime metric. As a direct application, we analyze the non-relativistic limit of the theory. The new term describing the specific spin (gravitational moment) interaction effect is recovered in the Hamiltonian. The comparison of the true gravitational coupling with the purely inertial case demonstrates that the spin relativistic effects do not violate the equivalence principle for the Dirac fermions. Typeset using REVTEX On leave from: Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia All high-energy physics experiments usually take place either in a curved spacetime or in a non-inertial reference frame (e.g., on Earth’s surface or in the nearby space). Hence the study of the gravitational effects on quantum-mechanical systems represents an important issue. The weakness of the gravitational interaction has justified, to a certain extent, the long practice of neglecting the gravitational and/or inertial effects in particle physics. However the technological progress, and especially the notable development of interferometric technique, has significantly changed the situation. In particular, in the famous Colella-Overhauser-Werner (COW) [1] and Bonse-Wroblewski [2] experiments, the quantummechanical phase shift due to the gravitational and inertial forces was measured, thereby verifying the validity of the equivalence principle for the non-relativistic neutron waves. The corresponding theoretical analysis was based on the Newtonian gravity and the Schrödinger equation. It is generally believed that the further improvement of the experimental technology (using the atomic interferometers, and the polarized neutrons, e.g.) will soon provide a more precise picture of the interaction of quantum particles with the gravitational field. Under these circumstances, it seems natural to study the higher order effects in the relativistic quantum mechanics, including the specific manifestations of the spin-gravity coupling. In this connection, it is worthwhile to recall that certain theoretical models predicted the violation of the equivalence principle for spinning particles, see e.g. [3–5]. A good example is provided by the model of Peres [4], in which the non-relativistic Hamiltonian of a massive Dirac particle included the additional term kh̄c~σ ·~g. This describes the gravitational dipole type interaction of spin ~σ and the gravitational acceleration vector ~g with the dimensionless coupling constant k. Similar interactions were considered very early by Kobzarev and Okun, and by Leitner and Okubo [6]. The comparison with the precision experimental data places though very weak restrictions on the value of the coupling constant k, see the review in [5]. In contrast to the ad hoc Peres’ type approach, here we will consider the standard theory of Dirac fermions in curved spacetime [7,8]. Correspondingly, the gravitationally coupled 4-spinor field ψ satisfies the covariant Dirac equation: (ih̄γDα −mc)ψ = 0. (1) Here, Dα is the spinor covariant derivative with [9] Dα = h i αDi, Di := ∂i + i 4 σ̂αβ Γi αβ . (2) We use the conventions of Bjorken and Drell [10] for the Dirac matrices γ, β, ~ α; as usual, σ̂ := iγγ. Gravitational and inertial effects are encoded in the (co)frame and the Lorentz connection coefficients hi , Γi αβ = −Γi. (3) As it is well known, at any point P , it is always possible to choose the local spacetime coordinates and the (non-holonomic, in general) frame so that hi (P ) = δ i , Γi (P ) = 0. This mathematical fact underlies the equivalence principle in accordance with which the gravitationally coupled Dirac equation (1) locally (at every point) assumes its flat space form in a suitably chosen reference frame [8]. The dynamics of the Dirac fermions in the different gravitational fields and non-inertial reference frames was studied previously [11] using approximation schemes. Here we for the first time will present some exact results. More specifically, let us confine our attention to the wide class of spacetimes described by the static metric ds = V 2 (dx) −W 2 (d~x · d~x), (4) where x = ct, and V = V (~x),W = W (~x) are the arbitrary functions of the spatial coordinates ~x. Two important particular cases belong to this family: (i) the flat Minkowski spacetime in accelerated frame: V = 1 + (~a · ~x) c2 , W = 1, (5) and (ii) Schwarzschild spacetime in isotropic coordinates: V = ( 1− GM 2c2r )( 1 + GM 2c2r )−1 , W = ( 1 + GM 2c2r )2 , (6) with r := √ ~x · ~x. Choosing the orthonormal frame, hi 0̂ = V δ i , hi â = W δ i , a, b = 1, 2, 3, (7) we find the local Lorentz connection: Γi â0̂ = ∂V WV hi , Γi â̂b = ∂W W 2 hi b̂ − ∂ W W 2 hi . (8) Hereafter, the hats distinguish the local frame indices from the spacetime coordinate ones. As a result, we have the explicit spinor derivative components [12]: D0̂ = 1 V ( ∂ ∂x0 + 1 2W ( ~ α · ~ ∇V )) , (9) Dâ = 1 W ( ∂ ∂xa + i 2W ǫabc ∂ W Σ ) . (10) Consequently, the Dirac equation (1) is recasted into the familiar Schrödinger form