We construct the algebra of the creation and destruction operators for spin 1/2 particles obeying a new exclusion principle which is “more exclusive” than Pauli’s exclusion principle: an orbital state shall not contain more than one particle, whether spin up or spin down. The consequences of this algebra are studied and applications to the Hubbard model in condensed matter physics are indicated.

We propose a new two-parameter deformation of the algebra of creation and destruction operators, which allows the construction of a new family of Hillbert spaces with positive definite inner product. This provides a continuous interpolation between two new forms of statistics named orthofermi and orthobose statistics. Positivity of the inner product over the two-parameter region is discussed.

A general analysis of bilinear algebras of creation and destruction operators is performed. Generalizing the earlier work on the single-parameterq-deformation of the Heisenberg algebra, we study two-parameter and four-parameter algebras. Two new forms of quantum statistics called orthofermi and orthobose statistics and aq-deformation interpolating between them have been found. In the Fock representation, quadratic relations among destruction operators, wherever they are allowed, are shown to follow from the bilinear algebra of creation and destruction operators. Postitivity of the Hilbert space for the four-parameter algebra has been studied in the two-particle sector, but for the two-parameter algebra, results are presented up to the four-particle sector.

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ‘infinite’, Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are: new algebras for infinite statistics,q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ‘doubly-infinite’ statistics, many representations of orthostatistics, Hubbard statistics and its variations.