Four-vector versus four-scalar representation of the Dirac wave function. (English) Zbl 1271.53049

In the standard formulation of the Dirac equation on a curved space-time, the four components of the Dirac wave function transform as complex scalars under coordinate changes. This is called the quadruplet representation of the Dirac theory (QRD). Recently, the first author [Found. Phys. 38, No. 11, 1020–1045 (2008; Zbl 1161.81339)] suggested another version of the Dirac equation where the four components of the Dirac wave function transform as a complex four-vector. This is called the tensor representation of the Dirac theory (TRD). It is the purpose of the present paper to further explore the relation between the QRD and the TRD. To that end, the authors formulate a common geometric framework. A spinor structure is defined as a complex vector bundle \(E\) with certain properties over space-time \(V\), and it is shown that any connection on \(E\) defines a Dirac equation for sections in \(E\). The QRD is recovered by choosing for \(E\) the trivial bundle, \(E= V \times \mathbb{C}^4\), and the TRD is recovered by choosing for \(E\) the complexified tangent bundle, \(E = T_{\mathbb{C}}{}V\). As a main result, it is proven that any form of the QRD can be equivalently rewritten as a TRD. Several other results on equivalence or non-equivalence of different versions of the Dirac equation are shown.


53C27 Spin and Spin\({}^c\) geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism


Zbl 1161.81339
Full Text: DOI arXiv


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