Indispensability of Ghost Fields and Extended Hamiltonian Formalism in Axial Gauge Quantization of Gauge Fields
Abstract
Description
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 $x^μ= (x^+,x^-,x^1,x^2)$, where $x^+=x^0{\rm sin}θ+x^3{\rm cos}θ, x^-=x^0{\rm cos}θ-x^3{\rm sin}θ$ and $x^+$ and $A_-=A^0{\rm cos} θ+A^3{\rm sin}θ=0$ 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 $P_μ$, 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 $P_μ$ 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 $θ\to\fracπ{2}-0$ and that the light-cone axial gauge formulation turns out to be the case of $θ=\fracπ{4}$.
17 pages and uses seceq, psfig and ptptex
17 pages and uses seceq, psfig and ptptex