the generator of such transforms needs to vanish on physical states and this gives rise to a set of local and global Gauss-law constraints that the quantum fields and their conjugate momenta need to obey. Of these constraints, the local ones can be solved (and will be solved in the next section) and consequently a large subset of the gauge fields’ conjugate momenta is written in terms of the fermion color charges. The fact that these momenta become non-dynamical implies that their conjugate gauge fields are not physical and can be removed from the Hamiltonian. These are the fields that, in the action formalism, can be ‘gauged away’.

Indeed, from the path integral formalism we know that in a compact system of one spatial dimension almost all gauge fields can be gauged away. The only gluonic modes that remain are those corresponding to a constant spatial gauge field. Furthermore, one can gauge away all but the $N-1$ independent eigenvalues of this zero mode. Anticipating a similar scenario in the Hamiltonian approach, one expects that the following fermionic Hamiltonian will be the remnant of the Hamiltonian after the axial gauge fixing : $$H'_F = -i\sum_{x}\psi^{\dag a}_x \sigma_3 \, e^{i\varphi_a } \, 