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 $({ \partial}_-)^{ -2}$ 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 ${ \theta}=\frac{ \pi}{ 4}$.
Categories: hep-th