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Should There Be a Spin-Rotation Coupling for a Dirac Particle?

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Abstract

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E in a given reference frame were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We show that a tetrad field contains two informations about local rotation, which usually do not coincide. We compute the energy operator in the inertial and the rotating frame, using three different tetrad fields. We find that Mashhoon’s term is there if the spatial triad rotates as does the reference frame—but then it is also there in the energy operator for the inertial frame. In fact, if one uses the same given tetrad field, the Dirac Hamiltonian operators in two reference frames in relative rotation differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.

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Notes

  1. In that case, the r.h.s. of the “free” Dirac equation (6) is augmented with the term −iqγ μ V μ Ψ, with q the electric charge and V μ the four-potential. Thus the “free” Hamiltonian H is replaced by \(\mathrm {H}_{\mathrm{em}}=\mathrm{H}+q(V_{0}{\bf1}_{4}+V_{j}\alpha^{j})\), where α j≡γ 0 γ j/g 00, as is the case [22] for Dirac’s original equation. It follows that, after a local similarity transformation S, after which H becomes \(\widetilde {\mathrm{H}}\), the complete Hamiltonian Hem becomes \(\widetilde{\mathrm{H}_{\mathrm{em}}}\), with \(\widetilde{\mathrm {H}_{\mathrm{em}}}-S^{-1}\mathrm{H}_{\mathrm{em}}S=\widetilde {\mathrm{H}}-S^{-1}\mathrm{H}S\). We get similarly for the energy operator: \(\widetilde{\mathrm{E}_{\mathrm{em}}}-S^{-1}\mathrm{E}_{\mathrm {em}}S=\widetilde{\mathrm{E}}-S^{-1}\mathrm{E}S\), whence for any state Ψ and the corresponding state after application of S, \(\widetilde{\varPsi}\equiv S^{-1}\varPsi\) [noting ( ∣ ) and \((\,\,\widetilde{\mid}\,\,)\) the scalar products before and after application of S]:

    $$ (\widetilde{\varPsi}\,\widetilde{\mid}\,\widetilde{\mathrm {E}_{\mathrm{em}}}\widetilde{\varPsi})-(\varPsi\mid\mathrm {E}_{\mathrm{em}} \varPsi)=(\widetilde{\varPsi}\,\widetilde{\mid} \, \widetilde{\mathrm{E}} \widetilde{\varPsi})-(\varPsi\mid\mathrm {E}\varPsi),\quad \mbox{or}\quad \langle \widetilde{\mathrm{E}_{\mathrm {em}}}\rangle-\langle\mathrm{E}_ {\mathrm{em}} \rangle= \langle \widetilde{\mathrm{E}}\rangle-\langle\mathrm{E} \rangle. $$
    (3)

    {We use the fact that \((\widetilde{\varPsi}\,\widetilde{\mid} \, S^{-1}\mathrm{E}S \widetilde{\varPsi})=(\varPsi\mid\mathrm {E}\varPsi)\) [18].} Hence, the non-uniqueness of the operators Hem and Eem and that of the spectrum of Eem appear in strictly the same way as in the case of the “free” Dirac equation, whether the spacetime is curved or not.

  2. Nevertheless, the covariant Dirac equation being in particular covariant on a coordinate change, the evolutions of Ψ calculated from i  ħ∂ t Ψ=HΨ in one coordinate system or in another one are equivalent. Specifically, for the DFW equation, Ψ behaves as a scalar on any coordinate change [9, 10, 14], thus we have simply Ψ′((x′ν))=Ψ((x μ))—with the restriction mentioned after Eq. (26) below.

  3. There are alternative versions of the covariant Dirac equation in which the wave function is a complex vector field, for which case one may optionally decompose the wave function on the coordinate basis (the natural basis of the coordinate system) [14]. Taking this option means that the frame field on the spinor bundle coincides with the coordinate basis. Then Ψ transforms as a vector and (γ μ) as a (2 1) tensor [14].

  4. Let D be a connection on some vector bundle \({\sf E}\) with base V, let (u α ) be a frame field on TV, and let (e a ) be a frame field on \({\sf E}\). The connection matrices Γ α of D in the frame fields (u α ) and (e a ) are defined by their scalar components \((\varGamma_{\alpha})^{b}_{\ \,a} \), such that

    $$ De_a(u_\alpha) =(\varGamma_\alpha)^b_{\ \,a} e_b. $$
    (13)

    This leads immediately to (12)1. If the frame field on TV is a local coordinate basis: \(u_{\alpha}=\delta^{\mu}_{\alpha}\,\partial_{\mu}\), one may then compute the covariant derivatives D μ Ψ b of any section of \({\sf E}\), ψ=Ψ b e b , in a matrix form: D μ Ψ=∂ μ Ψ+Γ μ Ψ. Thus, this notion of a connection matrix [14] extends conveniently the usual notion of the matrices of the “spin connection” entering the covariant Dirac equation, to any connection on a general vector bundle. It has a simple relation to the definition of a connection “matrix” as a matrix of one-forms [26], \(\omega=(\omega^{b}_{\ \,a})\): if (θ β) is the dual frame of a frame field (u α ) on TV, one has \(\omega^{b}_{\ \,a}= (\varGamma_{\alpha})^{b}_{\ \,a} \theta^{\alpha}\). The covector transformation of the matrices Γ α on changing (u α ) applies for a given frame field on \({\sf E}\) in (13), thus it does not apply if \({\sf E}=\mathrm{TV}\) and \(e_{a}=\delta^{\alpha}_{a} u_{\alpha}\).

  5. When A is the constant A=γ ♯0, the hermiticity condition has been derived in the form (∀Ψ,Φ) \(\int\varPsi^{\dagger}\gamma^{\sharp0} \,\partial_{0} (\sqrt {-g} \gamma^{0} ) \varPhi\, \mathrm{d}^{3}{\bf x}=0\) by Parker [19] and by Huang & Parker [20]. A particular case of the latter integral condition has been derived by Leclerc [15].

  6. The spatial tensor Ω depends on the choice of the time coordinate t in a complex manner, whereas, on changing from t to t′, Ξ gets simply multiplied by dt/dt′. Hence, the equality Ξ=Ω is not covariant under a change of the time coordinate, so that the prescriptions Ξ=Ω corresponding to reference frames differing merely in the choice of the time coordinate are not physically equivalent. And indeed, there is a rewriting of the geodesic equation of motion in the form of Newton’s second law, in which the tensor Ω plays exactly the role played by the angular velocity tensor of a rotating frame in Newtonian theory [23]—but in this rewriting Ω has to be calculated with a time coordinate \(\hat {x}^{0}\) such that, along a world line of the congruence, we have \(d\hat {x}^{0}=c\,d\tau\), where dτ is the proper time increment. Thus, if one applies the prescription Ξ=Ω, one should impose that the time coordinate be a such one, \(\hat{x}^{0}\) with \(d\hat{x}^{0}=c\,d\tau\) [21]. For the uniformly rotating frame, the tensors Ω calculated with either t or τ differ only by O(V 2/c 2) [21] (V is defined in Sect. 5), and the same is easy to check for Ξ.

  7. Moreover, this form is generic for an alternative theory of gravitation [30], in the preferred reference frame assumed by that theory. That theory is based only on a scalar field which determines, among other things, the physical metric g, from an a priori assumed flat metric, say γ. Although it thus has two metrics, this is not a metric theory in the standard sense.

  8. Hence, by (23): for that tetrad, in the inertial frame F′ to which it is adapted, we have Ξ=0. Moreover, we have also Ω=0 from (24), for the frame F′: the Cartesian tetrad solves Variants (a) and (b) of Framework I for the inertial frame.

  9. The choice of the set (γ ♯α) does not matter, because corresponding (γ μ) fields exchange by constant similarity transformations, hence give rise to equivalent energy operators. With the standard set (Dirac’s), we have s jk=−2iϵ jkl Σ l and s 0j=2Σ′j.

  10. In particular, the Hamiltonian in the inertial frame F′ and with the adapted tetrad (43)–(44) is: \(\mathrm {H}'_{2}=\mathrm{H}_{2}+{{\boldsymbol{\omega.}}}{\bf L}\), with H2 given by Eq. (62). This is also the energy operator.

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Acknowledgement

A referee asked me to state clearly whether or not I consider that the rotating/non-rotating character of a frame of reference is decided uniquely by the tetrad. To answer this key question I rewrote Sect. 3 with more detail and with new remarks.

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  1. Laboratory “Soils, Solids, Structures, Risks”, 3SR, CNRS and Universités de Grenoble: UJF, Grenoble-INP, BP 53, 38041, Grenoble cedex 9, France

    Mayeul Arminjon

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Arminjon, M. Should There Be a Spin-Rotation Coupling for a Dirac Particle?. Int J Theor Phys 53, 1993–2013 (2014). https://doi.org/10.1007/s10773-014-2006-z

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