Abstract

In this thesis we study an example of a recently proposed technique of integral quantization by looking at the Poincaré group in (1+1)-space-time dimensions, denoted P_+^{\uparrow}(1,1), which contains the affine group of the line as a subgroup. The cotangent bundle of the quotient of P_+^{\uparrow}(1,1) by the affine group has the natural structure of a physical phase space. We do an integral quantization of functions on this phase space, using coherent states coming from a certain representation of P_+^{\uparrow}(1,1). The representation in question corresponds to the "zero-mass'' or "light-cone'' situation, which when restricted to the affine subgroup gives the unique unitary irreducible representation of that group. This representation is also the one naturally associated to the above mentioned coadjoint orbit. The coherent states are labelled by points of the affine group and are obtained using the action of that group on a specially chosen vector in the Hilbert space of the representation. They satisfy a resolution of the identity, which can be computed using either the left or the right Haar measure of the affine group. The integral quantization is done using both choices and we obtain a relationship between the two quantized operators corresponding to the same phase space function.