A Schwinger term in q - deformed su ( 2 ) algebra ∗

An extra term generally appears in the q-deformed su(2) algebra for the deformation parameter q = exp 2πiθ, if one combines the Biedenharn-Macfarlane construction of q-deformed su(2), which is a generalization of Schwinger’s construction of conventional su(2), with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra. Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed. The notion of q-deformed algebra[1], which was originally introduced in connection with the inverse scattering problem and the Yang-Baxter equation[2], is going to be a standard machinery of theoretical physics. For example, the q-deformed su(2) for q = exp iπP/Q with mutually prime integers P and Q found a very interesting physical application to the Bloch electrons in two-dimensional lattice model[3-5]. Also, the qdeformed oscillator algebra, which was introduced by Biedenharn[6] and Macfarlane[7] to construct the q-deformed su(2) in the manner of Schwinger’s construction of conventional ∗To be published in Modern Physics Letters A