Parastatistics and the quark model

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Abstract
An extension of ordinary parastatistics is considered which makes use of all the representations of the parastatistics algebra obtained from the usual ansatz. Govorkov's demonstration that such an extension, for parastatistics of order 2, implies a U(2) symmetry, is generalized for parastatistics of order p. The parastatistics algebra, restricted to N dynamical states, is characterized by the irreducible representations of U(N), S O(2N), and S O(2N + 1) which it contains. It is shown that these representations have multiplicities equal to the dimensions of associated representations of U(p), O(p) and C(p), respectively, where C(p) is a subalgebra of the enveloping algebra of O(p), but is not a Lie algebra. The symmetric group S(p) also appears, as a subalgebra of the enveloping algebra of C(p). It is shown how a nondegenerate vacuum state may be defined for the generalized parastatistics algebra of order p, and how to construct state vectors corresponding to arbitrary numbers of quarklike particles and antiparticles. Such states belong to irreducible representations of U(N), and can be obtained by the application of one kind of creation and annihilation operators to certain basic states, here called reservoir states, which correspond to the different irreducible representations of S O(2N + 1). The specialization to parastatistics of order 3 is discussed in detail with the application to a quark model of the hadrons in view. It is shown how to define isospin and hypercharge in a significant way in this model, which, however, differs in some respects from Gell‐Mann's well‐known 3‐fermion model, and also from Greenberg's 3‐parafermion model. Some of the physical implications are examined.

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