WZW term in quantum mechanics: single spin

7.1 Quantum spin Let us consider a simple example of how Wess-Zumino effective Lagrangian appears from the “current algebra”. To simplify the story we take an example of quantum spin S. This is a quantum mechanical system with an SU(2) spin algebra playing the role of “current algebra” of quantum field theory. We have standard spin commutation relations [S, S] = i!S, (7.1) where a, b, c take values x, y, z. We require that " S = S(S + 1), (7.2) where 2S is an integer number defining the representation (the value of spin). Let us consider the simplest possible Hamiltonian of a quantum spin in a constant magnetic field H = −"h · " S (7.3) and derive an operator equation of motion ∂t" S = i[H, " S] = −i["h · " S, " S] = " S × "h. (7.4) In the classical limit S →∞ (or !→ 0) it is convenient to write " S → S"n so that "n is a classical unit vector "n2 = 1 and equation of motion(7.4) becomes