Grassmann Variables

M. El Baz, Y. Hassouni

published in J. Phys. A37 (2004) 4361 Generalized Zk-graded Grassmann variables are used to label coherent states related to the nilpotent representation of the q-oscillator of Biedenharn and Macfarlane when the deformation parameter is a root of unity. These states are then used to construct generalized Grassmann representatives of state vectors. Recently in [1] we have constructed coherent states for k-fermions using Kerner’s [2] Z3-graded extension of the Grassmann variables [3]. These results were obtained in the case where the deformation parameter is a primitive cubic root of unity, i.e., k = 3. In order to obtain similar results in the generic case (by generic we mean here that q, the deformation parameter, is an arbitrary k th root of unity, i.e., for an arbitrary positive integer k) one should use Zk-graded generalizations of the Grassmann variables. Unfortunately, up to our knowledge, such structures have not been yet constructed in the spirit of Kerner’s variables. There exist however in the literature another point of view and other generalized Zk-graded Grassmann variables [4]. In this letter we investigate on the use of these latter variables for the description of k-fermions. Namely, we will construct k-fermionic coherent states labeled by Zk-graded Grassmann variables. The coherent states will be used to derive a space of (Zk-graded) Grassmann representatives in which state vectors are represented as ”holomorphic ” functions of the Grassmann variable. There exist many deformations of the harmonic oscillator which, for some values of the deformation parameter, give rise to k-fermions. In this letter we will illustrate the construction using the oscillator deformation of Biedenharn [5] and Macfarlane [6]. This q-oscillator is described by the operators {N, a, a +} with the following relations

Leonardo Angelini

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

A phase space theory for fermions has been developed using Grassmann phase space variables which can be used in numerical calculations for cold Fermi gases and for large fermion numbers. Numerical calculations are feasible because Grassmann stochastic variables at later times are related linearly to such variables at earlier times via c‐number stochastic quantities. A Grassmann field version has been developed making large fermion number applications possible. Applications are shown for few mode and field theory cases.

M. A. Naĭmark

Lajos Diósi

Grassmannian local hidden variables are shown to generate all possible quantum correlations in a bipartite quantum system. Grassmann representation of fermions, common in field theory, opens a related perspective. Although Grassmann hidden variables can not challange Bell's locality theorem, they can become an interesting mathematical tool to investigate entanglement.