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On the gravitational moments of a Dirac particle

We consider the classical theory of the Dirac massive particle in the Riemann-Cartan spacetime. We demonstrate that the translational and the Lorentz gravitational moments, obtained by means of the Gordon type decompositions of the canonical energy-momentum and spin currents, are consistently coupled to torsion and curvature, as expected.

General dynamical scheme of the Poincaré gauge theory is well established.
The Noether currents of matter fields, Σ α (the canonical energy-momentum three-form) and τ αβ (the canonical spin three-form), are coupled to the translational ϑ α and the Lorentz Γ αβ gauge potentials, respectively. These currents are thus representing the two types of gravitational charges of a matter source.
Recently, the Dirac electron theory has been analyzed in flat Minkowski spacetime [3], and the structure of the canonical energy-momentum and spin currents was studied in detail. It was shown, developing analogy with electrodynamics, that a Dirac particle is naturally characterized by two gravitational moments. In simple physical terms, one can describe them as the Ampére type ring currents induced by the two gravitational charges via spin of a particle.
Here we consider the Dirac theory with the gravitational field "switched on".
We demonstrate the consistency of the coupling of the gravitational moments to the torsion and curvature.
It is worthwhile to mention that gravitational moments of a Dirac particle were discussed also by Kobzarev and Okun [4] and Khriplovich [5] in the framework of Einstein's general relativity theory, whereas in [6,7] gravitational moments of higher spins and in lower dimensions were studied.

II. FERMIONS IN THE RIEMANN-CARTAN SPACETIME
For the description of classical Dirac particle of mass m, we will use the formalism of Clifford algebra-valued exterior forms [1]. The following matrixvalued one-or three-forms are basic objects in this approach: Exterior product yields a two-form: The coefficients σ αβ := iγ [α γ β] generate infinitesimal Lorentz transformations of spinor fields. Defining another constant matrix, γ 5 := −i γ0γ1γ2γ3, it is straightforward to prove the fundamental identities for the Clifford algebravalued objects: The Lagrangian four-form of a Dirac field Ψ is given by The Dirac field equation, which arises from the Lagrangian (2.5), reads where T := e α ⌋T α is the torsion trace one-form.
The standard Lagrange-Noether machinery in the gauge gravity, see e.g.
[2], provides a general definition of the energy-momentum and spin currents in non-flat spacetime (accounting also for the possibility of non-minimal coupling and "Pauli-type" terms): (2.10) Here Ψ A is a set of arbitrary matter fields, with ℓ αβ denoting the corresponding Lorentz group generators.
The first and second Noether theorems yield two covariant conservation laws which are fullfilled on the classical matter field equations: (2.14)

III. GRAVITATIONAL MOMENTS OF A DIRAC PARTICLE
In a theory invariant with respect to some internal gauge group G, the generalized moment is a G-valued two-form such that its exterior differential produces the polarizational part of the related Noether current, with G denoting the Lie algebra of G. On the Lagrangian level, the moment couples directly to the gauge field strength. To be specific, in the Dirac-Yang-Mills theory the Noether "isospin" current J K = −ieΨτ K Ψ couples to the gauge potential one-form A K , with e denoting the coupling constant, and τ K being generators of the gauge group. The Gordon decomposition [3] reveals a nontrivial substructure of this coupling by relating the polarization moment two-form P K = −i eh 2mc Ψτ K * σΨ directly to the gauge field strength F K .
One may expect similar results to hold when passing from internal symmetry groups to spacetime symmetries. Geometrically this means a departure from flat Minkowski spacetime by "switching on" gravity.
Technically, we can proceed along the same lines as for the Dirac-Yang-Mills theory [3]. We use (2.7)-(2.8) in order to express Ψ and Ψ in terms of the differentials, and then substitute them back into the Dirac equation and into the Noether currents (2.13)-(2.14). After some algebra, we find the squared Dirac equation in the form where the three-form S and the four-form X read, respectively, Here we have used the Ricci identity for the spinor covariant derivative (2.6): One can immediately verify that equation (3.1) can be derived from the Lagrange four-form where the convective Lagrangian and the polarizational Lagrangian read, respectively: Here M αβ is the Lorentz gravitational moment given by Ψ( * σ σ αβ + σ αβ * σ)Ψ, (3.8) and We are now in a position to find the decosmposition of the energymomentum and spin currents into the convective and polarization parts corresponding to the decomposition of the Lagrangian (3.5), αβ .
For the polarizational part, we need the derivatives with respect to curvature and torsion: 14) Here we recover the correct translational grativational moment [3] Substituting (3.14),(3.15) into (2.9),(2.10), we finally obtain the polarizational energy-momentum and spin currents of a Dirac particle in the Riemann-Cartan spacetime: The terms Σ give their explicit form: As was noticed in [3], the three-form plays a role of a "superpotential" from which all the moments of a Dirac particle are generated. For comparison, it is instructive to collect the properties of the moments in a Table I.  TABLES   TABLE I. Gauge couplings and moments of the Dirac particle Gauge model Generator Field Moment 2-form Dim Yang-Mills SU (N )

IV. DISCUSSION AND CONCLUSION
In this paper, we have generalized our previous discussion [3] of the gravitational moments to the case of the Riemann-Cartan spacetime.
Although the structure of the torsion-and curvature-dependent terms The results obtained are helpful for deepening our understanding of the classical limit of the Dirac theory. A relevant general discussion of the lowenergy limit can be found in [8] (for non-inertial frames in flat spacetime), and in [9] (for the Riemann-Cartan spacetime). This problem will be analyzed elsewhere.