Gauge system

Physically relevant variables, corresponding to observables, are those that are independent of the choice of local frames. A classical observable can be represented as a function on the constraint surface. This corresponds to a function that has a weakly vanishing Dirac bracket with the first class constraints of the system.

A gauge transformation is a transformation of the variables induced by a change in the arbitrary frame of reference. Under gauge transformations physically relevant variables are thus invariant.

It is when the constraint matrix does not have an inverse, that the system is a gauge system. According to a conjucture by Dirac, all first class constraints act as gauge generators. Second class constraints can however not be treated as gauge generators, or as generators of any transformations of physical relevance whatsoever. This is because second class constraints do not preserve all the constraints of the system, and may thus map an allowed state of the system on a nonallowed state.

Gauge symmetries are unusual symmetries in the sense that a gauge transformation does not transform from one solution to another solution, but rather from one description of a solution to another description of a solution.

In field theory, because of the infinite number of degrees of freedom, it is possible to distinguish between proper and improper gauge transformations. Proper gauge transformations correspond to constraints that are preserved in time, whereas the improper gauge transformations correspond to conservation laws. Improper gauge transformations thus cannot be ``eliminated'', while proper gauge transformations can be ``eliminated'' by gauge fixing, and are thus physically irrelevant. Proper first class constraints thus generate gauge transformations, while improper first class constraints generate ``conventional'' symmetry transformations. We now can recognize our earlier distinction between local and global symmetry transformations, in the distinction between proper and improper gauge transformations.

In a system with $N$ first class constraints $G_{A}$, the most general generator of proper gauge transformation can be expressed as

$$G = \sum_{A=1}^N \epsilon^A G_{A} \approx 0$$ (47)

When $N$ is finite, there are only proper gauge transformations, but if it is infinite, there can be improper gauge transformations as well. Go to the limit where

$$G = \int dx \epsilon(x) G(x) \approx 0,$$ (48)

In the case that there are only proper transformations, $\epsilon(x)$ then belongs to the dual space of the space of constraints $G(x)$. The $G(x)$ are functions of $q(x)$ and $p(x)$, the space of $G(x)$ is thus determined by the boundary conditions on the $q(x)$ and $p(x)$. If, for example, $G$ is periodic in $x$, then $\epsilon$ is periodic as well, and the transformations are proper.

When the transformations are improper, the $\epsilon$ must still satisfy the same boundary conditions as $G$, but the functional derivatives with respect to the canonical coordinates do not exist unless a nonvanishing surface term is added to $G$.

Note that the Hamiltonian is invariant under all gauge transformations, proper and improper.