Choice of a gauge in the light of Dirac quantization

Abstract

We discuss the choice of a gauge from the point of view of Dirac quantization of constrained systems. In order to illustrate the discussion, we build up a mechanical example of a gauge theory. It consists of the motion on a straight line $y=0$ imbedded in a plane ($x$, $y$). Gauging the model introduces a third Cartesian coordinate $z$. We show there exist three classes of gauge conditions: (I) gauge conditions involving only $x$, $y$ and the momenta ${p}_{x}$ and ${p}_{y}$; (II) gauge conditions involving the $z$ coordinate; (III) gauge conditions involving the time derivative of the $z$ coordinate. Class I allows, in general, the decoupling between physical and unphysical variables and the existence of an effective Hamiltonian depending only on $x$ and ${p}_{x}$. It corresponds to the choice of a unique representative for each gauge-group orbit. Class II is a generalization of the ungauged model of a particle moving on a straight line imbedded in a plane. Separation between physical ($x$) and unphysical ($y$) degrees of freedom is not directly possible. External constraints must be imposed on physical states as well as on the measure defining the physical Hilbert space. Class III keeps all the unphysical degrees of freedom $y$ and $z$. An indefinite-metric formalism is needed to introduce a cancellation between these unphysical degrees of freedom. Application to Abelian and non-Abelian Yang-Mills theory is easily done by the correspondence $x$, $y\ensuremath{\leftrightarrow}{A}_{k}^{a}$, $z\ensuremath{\leftrightarrow}{A}_{0}^{a}$. Usual gauges are discussed according to this classification. In particular, we try to derive the class-II gauge conditions ${n}^{\ensuremath{\mu}}{A}_{\ensuremath{\mu}}^{a}$, ${n}_{0}\ensuremath{\ne}0$ from a suitable class-I condition. This is not possible in the non-Abelian case. In the Abelian case, the lightlike gauge ${n}^{2}=0$ does not lead to an effective Hamiltonian depending only on physical degrees of freedom.