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Spinor and vector representations in four-dimensional Grassmann space. (English) Zbl 0819.22015
The use of ‘superspace’ techniques for handling spacetime supersymmetries is an established tool for the construction of covariant field representations of physical interest. ‘Superfields’ are considered in terms of formal ‘Taylor expansions’ with respect to ‘anticommuting coordinates’ which satisfy a Grassmann algebra. In view of the spin- changing nature of Fermi-Bose supersymmetry, it is usual to take these coordinates also to carry a spinor representation of the Lorentz group. The alternative view, proposed in the present paper, is that the Grassmann coordinates \(\theta^ \mu\), \(\mu = 0, \dots, d\) be considered to belong to the vector representation of a (\(d\)-dimensional) Lorentz algebra. In terms of the elementary operators of multiplication by and differentiation with respect to \(\theta^ \mu\), two natural \(\text{so}(3,1)\) subalgebras can be defined, associated with commuting Clifford algebras, and under whose action wavefunctions transform as spinors. However, a third Lorentz subalgebra, the diagonal sum of the former two, transforms wavefunctions in tensor representations. Explicit expressions are given for the generators of these Lorentz algebras in the usual \(\text{sl}(2,\mathbb{C})_ R\) basis, and the corresponding coordinate bases in terms of Grassmann wavefunctions are tabulated. It is argued that, in order to fulfil the requirement that the Dirac matrices for the representation of physical spinors be Grassmann even operators, the spacetime dimension \(d\) must be greater than 4.

MSC:
22E70 Applications of Lie groups to the sciences; explicit representations
81T60 Supersymmetric field theories in quantum mechanics
15A90 Applications of matrix theory to physics (MSC2000)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
15A75 Exterior algebra, Grassmann algebras
16W55 “Super” (or “skew”) structure
15A66 Clifford algebras, spinors
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[1] DOI: 10.1103/PhysRevD.21.2823
[2] DOI: 10.1103/PhysRevD.21.2823
[3] DOI: 10.1103/PhysRevD.21.2823
[4] DOI: 10.1103/PhysRevD.21.2823
[5] DOI: 10.1103/PhysRevD.21.2823
[6] DOI: 10.1007/BF02731978
[7] DOI: 10.1007/BF02731978
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