Differential Geometry and Mathematical Physics pp 693-757 | Cite as

# Elements of Quantum Gauge Theory

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## Abstract

We discuss some elements of quantum gauge theory with the main emphasis on those aspects which are related to the structure of the classical gauge orbit space. We start with the classical Faddeev-Popov path integral quantization procedure, address the famous Gribov problem and present Singer’s analysis of this problem. Next, we discuss anomalies for models of gauge fields coupled to fermionic matter in the spirit of Fujikawa, Atiyah, Singer and Witten. In the second part, we present some of our results on non-perturbative quantum gauge theory for (finite) lattice models in the Hamiltonian framework, including a discussion of the field algebra and the observable algebra, the Gauß law and the classification of representations of the observable algebra in terms of global colour charge. Finally, we include the nongeneric gauge orbit strata on quantum level via Huebschmann’s concept of a Hilbert space costratification using the generalized Segal-Bargmann transform of Hall.