Abstract
In the present chapter we shall apply the physical ideas and mathematical techniques of the preceding chapter to quantum bundles associated with a Dirac quantum frame bundle D M. These are principal bundles isomorphic to the principal bundle whose typical fibre equals the covering group [BR] ISL(2,C) of the Poincaré group ISO0(3,1). We shall concentrate on the case of quantum bundles whose standard fibres carry irreducible representations of ISL(2,C) for rest mass m > 0 and spin-1/2, but the same procedures can be applied with equal ease to the case of arbitrary spin.
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Notes to Chapter 6
Rigorous proofs of this and of the other statements concerning the Wigner-type momentum representation can be found in Chapter 6 of [BL].
The proofs of this and of the remaining statements in this section can be found in Chapter 7 of [BL].
All the proofs can be found in the original paper on this subject (Prugovečki, 1980). The Wigner-type representation based on (2.1)–(2.3) has been used (Ali and Prugovečki, 1981) in setting up models of stochastically extended spin-1/2 particles in external electromagnetic fields, which are consistent as single-particle external-field models in the sense that they do not give rise to spontaneous transition of particle states into antiparticle states — as is generically the case with models based on the Dirac equation in (1.13) or in (2.5).
These formulae can be easily derived from the results reported in (Prugovečki, 1980) and in (Ali and Prugovečki, 1986c).
Naturally, the definition of the Q-operators in (2.7b) requires the off-shell extrapolation carried out in (5.2.23), and discussed in Note 13 to Chapter 5.
The Wigner-type GS spin-1/2 framework also possesses a probability current. That current is analogous to the one in (3.5.13), and is also covariant and conserved (Prugovečki, 1980).
The proofs of the results stated in this section can be found in (Prugovečki, 1980), where these results are derived from the Wigner-type representation based on (2.1)–(2.3), by using the transformation (2.4). However, all the results in this section can be also verified by direct computation, using the well-known algebraic properties [IQ,SI] of the Dirac γ-matrices in a manner totally analogous to that of deriving the corresponding results in the momentum representation based on (1.10)–(1.11).
Alternatively, one can associate, as in (Prugovečki and Warlow, 1989b), the negative energy solutions with values of υ on the backward 4-velocity hyperboloid by setting for them ψ(q,-υ) = ψ(-q,υ), so that the form (2.5) of the Dirac equation is preserved.
However, the factor 1/2m was introduced, which is not present in the conventional approach. This factor is designed to compensate the 2m factor in (2.9) (which also occurs in (1.16)). In addition, there is also the renormalization constant in (3.8), which is absolutely essential for securing the strict reproducibility of the integral kernel in (5.1.18) — from which (3.6) follows on account of the well-known algebraic relations for the Dirac γ-matrices [BL,IQ,SI].
Spin structures and their most basic properties are described in Chapter 13 of [W]. Detailed accounts of spinors and spin structures in curved (classical) spacetimes, and their use in CGR are provided by Carmeli (1982) at a level best suited to readers with a primary interest in physics and in the formulation of classical field theories, as well as by Penrose and Rindler (1986) in a form that might be more attractive to readers with a strong interest in the mathematics of the subject.
The concept of “null flag” (cf. [W]; [Penrose and Rindler, 1986]) associated with a spinor is introduced to deal with this problem, but it still leaves an irremovable ambiguity in sign in the two-to-one map between spinors and such null flags.
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© 1992 Springer Science+Business Media Dordrecht
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Prugovečki, E. (1992). Relativistic Dirac Quantum Geometries. In: Quantum Geometry. Fundamental Theories of Physics, vol 48. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7971-1_6
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DOI: https://doi.org/10.1007/978-94-015-7971-1_6
Publisher Name: Springer, Dordrecht
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