Geometric quantisation and quantum mechanics in Dirac's front form

Abstract

We give a brief review of geometric quantisation up to and including the Blattner-Kostant-Sternberg kernal. In general this leads to symmetric operators that are not essentially self-adjoint so motivating a study of Hermitian operators as observables in a generalised quantum mechanics. We show that a generalised squaring axiom can reproduce the results of Blattner-Kostant-Sternberg quantisation. We also show that quantisation with respect to polarisations with compact leaves gives results that conflict with the nonlocal nature of quantum mechanics. We develop a front form quantum mechanics of a free scalar particle using geometric quantisation. The front and instant forms are related via unitary maps derived from the pairing which intertwines quantisations with respect to the forms. The front form position operator has a maximally symmetric component so we are compelled to work within the framework of a generalised quantum mechanics; the result in there being no Hegerfeldt type instantaneous spreading of initially localised wavefunctions in the front form. Finally we show that this model can be lifted to a many particle free field theory.