On non-trivial spectra of trivial gauge theories

It is usually said that two-dimensional (1+1) gauge theories are trivial. In the continuum there are no transverse degrees of freedom to sustain any dynamics. Likewise, in the lattice formulation, the system factorizes (after gauge fixing) and a partition function reduces to a simple one-plaquette integral [1–3]. This picture neglects boundary conditions. It is well known [4] that, e.g., QuantumMaxwell DynamicsQMD2 on a circle with a circumference L is not entirely trivial. There remains one degree of freedom which cannot be gauged away. In the Coulomb gauge this is the famous, constant in space, mode Ax(t) ≡ A(t). Its dynamics is given by the simple, one degree of freedom, hamiltonian H = − e 2 2L d dA2 . (1)