On the position operator for massless particles.pdf

a r X i v : q u a n t - p h / 9 6 1 2 0 2 2 v 1 5 D e c 1 9 9 6 ON THE POSITION OPERATOR FOR MASSLESS PARTICLES ALI SHOJAI ? MEHDI GOLSHANI ?? Department of Physics, Sharif University of Technology P.O.Box 11365-9161 Tehran, IRAN and Institute for Studies in Theoretical Physics and Mathematics, P.O.Box 19395-5531, Tehran, IRAN ?Email: SHOJAI@PHYSICS.IPM.AC.IR ??Fax: 98-21-8036317 POSITION OPERATOR, A. SHOJAI M. GOLSHANI 1 ON THE POSITION OPERATOR FOR MASSLESS PARTICLES A. Shojai M. Golshani ABSTRACT It is always stated that the position operator for massless particles has non-comutting components. It is shown that the reason is that the commutation relations between co- ordinates and momenta differs for massive and massless particles. The correct one for massless particles and a position operator with commuting components are derived. §1. INTRODUCTION AND SURVEY The notion of position operator has its roots in the early days of the birth of quantum mechanics. Although in the Copenhagen interpretation of quan- tum mechanics, the concept of position, and therefore path of the particle, is meaningless, nevertheless there must exist an operator called position op- erator having the property that its expectation value in the classical limit would behave classically. In other words, any macroscopic object has posi- tion. In quantum mechanics, one deals with elementary systems which means POSITION OPERATOR, A. SHOJAI M. GOLSHANI 2 any system whose state has a definite transformation under Poincare group (or under Gallileo group in the non-relativistic case). An elementary particle, then, can be defined as an elementary system which has no constituents. In this way, electron is an elementary particle while Hydrogen atom is an ele- mentary system only. In dealing with elementary systems one works only with generators of Poincare group as physical observables rather than the position of the system. Clearly it is natural to search for a position operator as an ob- ser