Dirac supermultiplet

Abstract

The graded extension of the de Sitter space-time algebra so(3,2) is identical to the structure defined by polynomials of order 1 and 2 in the natural coordinates ${\xi }_1, \dots , {\xi }_4$ of four-dimensional phase space. Ordinary Weyl quantization gives a representation that is unique among all the representations of the graded algebra in that the Poisson bracket relations $\{{\xi }_i, {\xi }_j\}=-C_{\mathrm{ij}}$ (which are not part of the structure of the graded algebra) are preserved. The restriction of this representation to the Lie subalgebra so(3,2) is the direct sum Di ⊕ Rac of the two singleton representations. There exists a unique, supersymmetric, interacting field theory of a single Dirac multiplet. The interaction Lagrangian has the form $\frac{1}{2}\int dy(3g{\phi }^2\overline{\psi }\chi +g^2{\phi }^6)$, where $\phi$ is the scalar Rac field, $\psi$ and $\chi$ are the spinor Di "field strength" and associated "potential," and $g$ is a real coupling constant. Applications to confinement and to composite massless particles is discussed.