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On the existence of the second Dirac operator in Riemannian space

Vladimir V Klishevich
- 21 Jan 2000 - 
- Vol. 17, Iss: 2, pp 305-318
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TLDR
In this article, the second Dirac operator is shown to be equivalent to a standard Dirac one in Riemannian spaces with a five-dimensional motion group, and it is shown that it exists in all metrics for which it exists.
Abstract
We describe a Riemannian space class where the second Dirac operator arises and prove that the operator is always equivalent to a standard Dirac one. The particle state in this gravitational field is degenerate to some extent and we introduce an additional value in order to describe a particle state completely. Some supersymmetry constructions are also discussed. As an example we study all Riemannian spaces with a five-dimensional motion group and find all metrics for which the second Dirac operator exists. On the basis of our discussed examples we hypothesize about the number of second Dirac operators in Riemannian space.

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References
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Journal ArticleDOI

Supersymmetry and Morse theory

TL;DR: In this paper, it was shown that the Morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics Hamiltonian, and some of the implications of modern ideas in mathematics for super-ymmetric theories are discussed.
Journal ArticleDOI

Generalized total angular momentum operator for the Dirac equation in curved space-time

TL;DR: In this paper, it was shown that an antisymmetric tensor satisfying the Penrose-Floyd equation f/sub munu(p/) = 0 is a Dirac tensor that can be interpreted as the square root of the ordinary total squared angular momentum Casimir operator of the rotation group.
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Space-Times Admitting Killing-Yano Tensors. II

TL;DR: In this paper, the authors derived canonical line elements admitting a simple KillingYano tensor f* ab and showed several close analogies between the vector field la = f * arp r along a geodesic with tangent field p r and the angular momentum l = rxpin the case of a spacelike, timelike and null tensor.
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Quantum numbers for Dirac spinor fields on a curved space-time

TL;DR: The most general first-order differential operator that commutes with the Dirac operator and permits the construction of quantum numbers is given in this paper, where necessary and sufficient conditions for its existence are expressed in terms of the generalized Killing tensors of Yano.