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To answer your second question, group theory has a lot of application in physics, especially in particle physics. I guess you are unfamiliar with Special unitary group. Basically, if we consider $SU(2) \times U(1)|$ Yang-Mills system along with the Higgs field, we'd describe electromagnetism along with weak force. Also, let us consider $SU(3)|$ Yang-Mills ...


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I) First of all, note that although gauge theory and BRST formulation originally only referred to Yang-Mills theory (and hence QED), they nowadays apply to general theories with so-called local gauge symmetry, cf. e.g. this Phys.SE post. The Lagrangian and Hamiltonian BRST formalism are known as Batalin-Vilkovisky (BV) formalism and ...



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