## Dirac Singletons in a Quantum Theory over a Galois Field

**Authors:** Felix M. Lev

Dirac singletons are exceptional irreducible representations (IRs) of the so(2,3) algebra found by Dirac.
As shown in a seminal work by Flato and Fronsdal, the tensor product of singletons can be decomposed into massless IRs of
the so(2,3) algebra and therefore each massless particle (e.g. the photon) can be represented as a
composite state of singletons. This poses a fundamental problem of whether
only singletons can be treated as true elementary particles. However, in standard quantum theory (based on
complex numbers) such a possibility encounters difficulties since one has to answer the following
questions: a) why singletons have not been observed and b) why the photon is stable and its decay into singletons
has not been observed. We show by direct calculations that in a quantum theory over a Galois field (GFQT),
the decomposition of the tensor product of singletons IRs contains not only
massless IRs but also special massive IRs which have no analogs in standard theory. In the case of supersymmetry
we explicitly construct a complete set of IRs taking part in the decomposition of the tensor product of supersingletons.
Then in GFQT one can give natural explanations of a) and b).

**Comments:** 37 Pages. Well known results on singletons in standard theory are generalized to the case of a quantum theory over a Galois field.

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### Submission history

[v1] 2012-03-06 00:55:39

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