Dilation operator in gauge theories

Abstract

The electromagnetic field is expanded in a series of O(4) eigenstates of total spin, and quantized by specifying commutators on surfaces of constant ${x}_{\ensuremath{\mu}}{x}^{\ensuremath{\mu}}={R}^{2}$ in four-dimensional Euclidean space. It is demonstrated that, under an arbitrary gauge transformation, some of the O(4) eigenstates are invariant; these gauge-invariant states are labeled by SU(2) \ensuremath{\bigotimes} SU(2) total (orbital plus internal) spin quantum numbers ($A$,$B$) and with $A\ensuremath{\ne}B$. Only these gauge-invariant states are nontrivial in the absence of sources, and are quantized. The leading-twist quantum states of the dilation field theory contain the minimum number of these dilation photons. The remaining spin degrees of freedom of the electromagnetic field are most simply written as a function of the form ${\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\varphi}(x)+{x}_{\ensuremath{\mu}}\frac{\ensuremath{\psi}(x)}{{R}^{2}}$. $\ensuremath{\varphi}(x)$ is obviously devoid of physics while $\ensuremath{\psi}(x)$ is a classical field propagating between radial projections of two electrical currents ${x}_{\ensuremath{\mu}}{J}^{\ensuremath{\mu}}(x)$ and ${y}_{\ensuremath{\mu}}{J}^{\ensuremath{\mu}}(y)$ only if ${x}_{\ensuremath{\mu}}{x}^{\ensuremath{\mu}}={y}_{\ensuremath{\mu}}{y}^{\ensuremath{\mu}}$. The quantization procedure described herein may be applied to non-Abelian theories. The procedure does not lead to a gauge-invariant decomposition of a non-Abelian field, but the identification of leading-twist quantum states is preserved in the zero-coupling limit.