Fermi quantization of non-Abelian gauge theory

The Yang-Mills field coupled to spinors is quantized, using a gauge-invariant Fermi addition to the Lagrangian, by introducing classical fictitious fields. Problems arising in the Fermi method are investigated. Use is made of the Schrödinger representation, which may be seen as the differential equivalent of the path-integral method. In this representation the conditions selecting good states are very simple: Good states are gauge-invariant functionals of the spatial Yang-Mills potentials ${b_{\alpha }}^i\left(\overrightarrow{\mathrm{x}}\right)$. The divergence in the naive scalar product of states is removed by the Faddeev-Popov factorization, and also by the cutoff method. Both methods produce gauge- and Poincaré-invariant results. The Faddeev-Popov weight is expressed in a power series, which converges in the whole coupling-constant plane, by seeing this weight as a Fredholm determinant. The consistency problem of the Fermi method is resolved by allowing certain operators to be self-adjoint; it is shown that gauge transformations cannot be implemented unitarily outside the good state space. We investigate the question which operators raise charge multiplets of good states from the vacuum. It is shown that this is done by the Mandelstam-displaced operators $\psi$, $\overline{\psi }$, $B$, and $\Pi$, and also by the potentials $b_{\mathrm{g}}$ which represent gauge-equivalence classes of Yang-Mills potentials, if the $b_{\mathrm{g}}$ form a linear subspace of the configuration space. In a gauge-invariant theory involving gauge-variant operators, the energy-momentum operators, defined as displacement operators, cannot be gauge-invariant. However, the effect of these operators in $G$ is the same as that of certain gauge-invariant operators, and they can be used as the physical energy-momentum. Double commutators of these operators with the potentials are found to satisfy an identity which has a formal consequence for the mass of states raised from the vacuum by the potentials.