Prime statistics in particle algebras

Abstract

The importance of the fermion algebra extends to all branches of physics. It is characterized by the important property that at most one particle can be present in a quantum state with otherwise same quantum numbers. In this thesis we will deal with algebras Ad where at most d - 1 particles can be present in a quantum state with otherwise same quantum numbers. A2 is thus the fermion algebra. In the limit where d goes to infinity the algebra becomes the boson algebra. Thus, the particles obeying Ad can be considered as a generalization of bosons and fermions. Algebras Ad have some important properties. They are constructed in terms of a single annihilation operator a and a single creation operator a* satisfying certain relations. Ad has a unique d dimensional representation. In this thesis we will prove another important property of these algebras that the tensor product of two algebras Ad1 and Ad2 is isomorphic to Ad where d = d1d2. This property brings in the idea that the particle algebras of prime dimensions are fundamental. We use this valuable property to constitute the idea of prime statistics.