TY - JOUR

T1 - Spinor Matter in a Gravitational Field

T2 - Covariant Equations à la Heisenberg

AU - Crawford, James P.

N1 - Funding Information:
This is the fundamental defining relation for the generators of the KDP algebra. (It is instructive to compare this defining relation for the generators of the KDP algebra with the defining relation for the Clifford algebra generators, Eq. ( 3.2).) Another representation of this algebra allows us to write the field equations for spin-one (vector) fields in `Drac firo,’m’ asin Eq. ( 5.3). So this same basic idea can be applied to vector field equations, includingtheYang± Millsgaugefields,atholughitisprobablethatinthis case there will be some subtleties due to both the masslessness and nonlinearity of the system. In addition, it seems likely, but remains to be shown, that the vierbein idea will generalize to include the KDP equations, in which case the derivation outlined above would proceed in exactly the same manner, yielding the Papapetrou equations in quantum operator form, but where now the proper velocity operator is given by a generator of the KDP algebra instead of by a generator of the Clifford algebra. Another potentially interesting problem to consider would be to seek exact solutions to the operator equations for various gravitational field configurations, as was done for the electromagnetic field case.(16) Active pursuit of these ideas is presently under way. The author gratefully acknowledges stimulating and fruitful conversations with Bruce Dean and especially William Thacker. In addition, an anonymous referee made valuable suggestions and supplied important references. This work was partially supported by a Penn State Research Development Grant.

PY - 1998/3

Y1 - 1998/3

N2 - A fundamental tenet of general relativity is geodesic motion of point particles. For extended objects, however, tidal forces make the trajectories deviate from geodesic form. In fact Mathisson, Papapetrou, and others have found that even in the limit of very small size there exists a residual curvature-spin force. Another important physical case is that of field theory. Here the ray (WKB) approximation may be used to obtain the equation of motion. In this article I consider an alternative procedure, the proper time translation operator formalism, to obtain the covariant Heisenberg equations for the quantum velocity, momentum, and angular momentum operators for the case of spinor fields. I review the flat spacetime results for Dirac particles in Yang-Mills fields, where we recover the Lorentz force. For curved spacetime I find that the geodesic equation is modified by an additional term involving the spin tensor, and the parallel transport equation for the momentum is modified by an additional term involving the curvature tensor. This curvature term is the "Lorentz force" of the gravitational field. The main result of this article is that these equations are exactly the (symmetrized) Mathisson-Papapetrou equations for the quantum operators. Extension of these results to the case of spin-one fields may be possible by use of the KDP formalism.

AB - A fundamental tenet of general relativity is geodesic motion of point particles. For extended objects, however, tidal forces make the trajectories deviate from geodesic form. In fact Mathisson, Papapetrou, and others have found that even in the limit of very small size there exists a residual curvature-spin force. Another important physical case is that of field theory. Here the ray (WKB) approximation may be used to obtain the equation of motion. In this article I consider an alternative procedure, the proper time translation operator formalism, to obtain the covariant Heisenberg equations for the quantum velocity, momentum, and angular momentum operators for the case of spinor fields. I review the flat spacetime results for Dirac particles in Yang-Mills fields, where we recover the Lorentz force. For curved spacetime I find that the geodesic equation is modified by an additional term involving the spin tensor, and the parallel transport equation for the momentum is modified by an additional term involving the curvature tensor. This curvature term is the "Lorentz force" of the gravitational field. The main result of this article is that these equations are exactly the (symmetrized) Mathisson-Papapetrou equations for the quantum operators. Extension of these results to the case of spin-one fields may be possible by use of the KDP formalism.

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U2 - 10.1023/A:1018768128810

DO - 10.1023/A:1018768128810

M3 - Article

AN - SCOPUS:0032365417

SN - 0015-9018

VL - 28

SP - 457

EP - 470

JO - Foundations of Physics

JF - Foundations of Physics

IS - 3

ER -