Invariance quantum group of the fermionic oscillator

The fermionic oscillator defined by the algebraic relations cc∗ + c∗c = 1 and c = 0 admits the homogeneous group O(2) as its invariance group. We show that, the structure of the inhomogeneous invariance group of this oscillator is a quantum group. Quantum field theory which describes the ultimate behavior of elementary particles and fields in physics fundamentally depends on the concepts of the bosonic oscillator described by the algebraic relation aa∗ − a∗a = 1 (1) and the fermionic oscillator described by the algebraic relations cc∗ + c∗c = 1 (2) c = 0. (3) The algebra of the bosonic oscillator (1) is invariant under the inhomogeneous symplectic group ISp(2, R) which transforms a, a∗ and 1 into each other.The homogenous part of this group which just transforms a, a∗ into each other is Sp(2, R) ≃ SU(1, 1). For the fermionic oscillator (2) we should like to remark the importance of the relation c = 0. Fermions satisfy the Pauli Exclusion Principle, the two identical fermions can not occupy the same state. Thus c = 0. The algebra (2) describes the 1 simplest nontrivial quantum mechanical system. In this sense it is the most fundamental. Although the bosonic oscillator (1) has a classical limit in which it reduces to the harmonic oscillator, the fermionic oscillator (2) has no classical analogue. Thus a thorough understanding of all its properties is important. One important property of the algebra (2) is that it does not admit a q-deformation [1 − 4]. Another is that although it is invariant under the orthogonal group O(2) which transforms c and c∗ into each other there is no inhomogeneous classical Lie group which transforms c, c∗ and 1 into each other. In this paper we construct a quantum group [5 − 8] which achieves this purpose. We show that the structure of the inhomogeneous invariance group of the fermionic oscillator is a quantum group, that is, the matrix elements of the transformation matrix which transforms c, c∗ and 1 into each other belong to a noncommutative Hopf algebra [5−8] where the coproduct is given by the matrix product. We will develop the R-matrix formulation of this quantum group and show that the operators generating this quantum group have a two dimensional representation which we explicitly construct. The representation matrices depend on five parameters. We finally present a discussion of our results. To show that the structure of the inhomogeneous invariance ‘group’ of the fermionic oscillator is a quantum group, we consider a 3× 3 matrix A whose elements belong to an algebra A. We form the column matrix c = 