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.
