Constrained quantisation and θ-angles II Original Research Article

Certain aspects of the quantisation of 2D Yang-Mills theory are analysed with a new constrained quantisation technique, which quantises classical symplectic reduction. To apply this new method, it is necessary to take the gauge group as a Hilbert Lie group, and use the Wiener measure on a certain completion of the gauge group. The physical Hilbert space, the Hamiltonian and the Wilson loop observables for 2D Yang-Mills with a semisimple compact structure group are explicitly constructed using this technique. A new theory of θ-angle, not particular to 2D, emerges naturally. An interesting relation between Fock space coherent states and heat kernels on compact Lie groups is also found. All computations are explicit and simple in the sense that all results originate from the fundamental properties of Wiener measure. An example of SO(3) Yang-Mills is given. The possibility and difficulties of generalising our computations to higher dimensions by utilising the Levy measure (which generalises the Wiener measure) is briefly discussed.