Wigner distributions for finite dimensional

quantum systems: An algebraic approach

S CHATURVEDI^1,*, E ERCOLESSI², G MARMO³, G MORANDI⁴,

N MUKUNDA⁵ and R SIMON⁶

¹School of Physics, University of Hyderabad, Hyderabad 500 046, India

²Physics Department, University of Bologna, INFM and INFN, Via

Irnerio 46, I-40126, Bologna, Italy

³Dipartimento di Scienze Fisiche, University of Napoli and INFN,

Via Cinzia, I-80126, Napoli, Italy

⁴Physics Department, University of Bologna, INFM and INFN, V.le

B.Pichat 6/2, I-40127, Bologna, Italy

⁵Centre for High Energy Physics, Indian Institute of Science,

Bangalore 560 012, India

⁶The Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India

^*Corresponding author. E-mail: scsp@uohyd.ernet.in

Abstract. We discuss questions pertaining to the definition of

`momentum', `momentum space', `phase space' and `Wigner

distributions'; for finite dimensional quantum systems. For such

systems, where traditional concepts of `momenta' established for

continuum situations offer little help, we propose a physically

reasonable and mathematically tangible definition and use it for the

purpose of setting up Wigner distributions in a purely algebraic

manner. It is found that the point of view adopted here is limited

to odd dimensional systems only. The mathematical reasons which

force this situation are examined in detail.

Keywords. Wigner distribution; phase space; finite groups;

representation theory; phase point operators.

PACS Nos 03.65.-w; 03.65.Wj; 03.65.-a; 03.65.Fd