Indispensability of Ghost Fields and Extended Hamiltonian Formalism in Axial Gauge Quantization of Gauge Fields

It is shown that ghost fields are indispensable in deriving well-defined antiderivatives in pure space-like axial gauge quantizations of gauge fields. To avoid inessential complications we confine ourselves to noninteracting abelian fields and incorporate their quantizations as a continuous deformation of those in light-cone gauge. We attain this by constructing an axial gauge formulation in auxiliary coordinates $x^{\mu }=(x^{+},x^{-},x^1,x^2)$, where $x^+=x^0{\rm sin}{\theta}+x^3{\rm cos}{\theta}, x^-=x^0{\rm cos}{\theta}-x^3{\rm sin}{\theta}$and$x^+$and$A_-=A^0{\rm cos} {\theta}+A^3{\rm sin}{\theta}=0$ are taken as the evolution parameter and the gauge fixing condition, respectively. We introduce $x^{-}$-independent residual gauge fields as ghost fields and accomodate them to the Hamiltonian formalism by applying McCartor and Robertson’s method. As a result, we obtain conserved translational generators $P_{\mu }$, which retain ghost degrees of freedom integrated over the hyperplane $x^{-}=$ constant. They enable us to determine quantization conditions for the ghost fields in such a way that commutation relations with $P_{\mu }$ give rise to the correct Heisenberg equations. We show that regularizing singularities arising from the inversion of a hyperbolic Laplace operator as principal values, enables us to cancel linear divergences resulting from $({\mathrm{\partial }}_{-}{)}^{-2}$ so that the Mandelstam- Leibbrandt form of gauge field propagator can be derived. It is also shown that the pure space-like axial gauge formulation in ordinary coordinates can be derived in the limit $\theta \to \frac{\pi }{2}-0$ and that the light-cone axial gauge formulation turns out to be the case of $\theta =\frac{\pi }{4}$.