Gauge fixing

To bring in gauge conditions is a way to restrict theory in order to avoid multiple counting of states, and this is done by imposing constraints on the canonical variables.

After a complete gauge fixing, there should be no first class constraints left, since upon gauge fixation the first class constraints, together with the gauge conditions, turn second class. And every set of second class constraints can be conceived as resulting from gauge-fixation of an equivalent gauge system.

The constraint manifold can be divided into equivalence classes of points, such that the points within an equivalence class are related by gauge transformations. A physical observable $A$ is an entity that takes a definite value as the physical system is in some definite state. An observable takes the same value at all the points within an equivalence class, or gauge flat, of the constraint manifold. In this way a physical state corresponds to one gauge flat on the constraint surface.

Canonical gauge fixing can be described as defining a physical phase space consisting of one representative point from each equivalence class of points on the constraint surface. This leaves a Hamiltonian system without any remaining constraints, and no proper gauge transformations can be performed. Improper gauge transformations are however still allowed, if preserving the gauge conditions. They act as ordinary symmetry transformations of the physical system.

Suppose we want to fix the gauge pertaining to one first class constraint $\chi$. If we find a condition $\phi(p,q)$ $\approx$ 0 such that the matrix

$$\left(\begin{array}{rcl} 0{\hspace{3mm}} & \{\phi,\chi\}\\ \{\chi,\phi\} & 0\nonumber \end{array} \right)$$

is invertible, $\phi$ and $\chi$ can be treated as a pair of second class constraints which can be solved, whereby the Dirac brackets for the remaining degrees of freedom can be computed. The resulting system has fewer degrees of freedom than the initial system.

As an example, consider the Coulomb gauge in electrodynamics, expressed as
$\partial \cdot A = 0$. This corresponds to $\{\phi(x),G(y)\}$ = - $\Delta \delta(x,y)$, where the constraint $G=\partial \pi$, and $\pi$ is the momentum. To see that this is a good gauge choice, one has to check that the Laplacian is invertible, i.e. that the inverse of the constraint matrix can be written as

$$C^{-1}(x,y)=\left(\begin{array}{rcl} 0 & -1\\ 1 & 0\nonumber \end{array} \right)\frac{1}{\Delta} \delta(x,y)$$

Every gauge flat must furthermore contain one and only one point, where $\phi =0$ is satisfied. Now suppose that the vector potential $A_{a}$ obeys the Coulomb gauge condition, then all other vector potentials on the same flat obey $\delta{\dot{\bf {A}}}'_{a}(x)=\delta_{a}({\bf {A}}'_{a}(x)+ \delta_{a}\Lambda(x)) = \Delta \Lambda(x)$.
The question is whether the Laplace equation $\Delta \Lambda(x)$ = 0 has a unique solution for $\Lambda(x)$.

Given suitable boundary conditions, with one vector potential on every gauge flat that obeys the Coulomb condition, the appearance of the Laplacian in a denominator can be justified. The constraint and the gauge condition can then be regarded as a pair of second class constraints which can be solved for the independent physical degrees of freedom, which are the transverse parts of the vector potential,

$$A^T_{\alpha} \equiv (\partial_{\alpha \beta}- \frac{\partial_{\alpha} \partial_{\beta}}{\Delta})A_{\beta},$$ (49)

and similarly for the electric field. The Dirac brackets can be calculated straightforwardly, but they involve the non-local inverse of the Laplacian. Thus manifest locality is lost when the Coulomb gauge is fixed.

In the above constraint formalism gauge symmetry is treated in a rather formal way. In the Standard Model for strong and electroweak interactions (section 10), it is represented in a somewhat different way, emphasizing the particle aspects.
Before we describe the Standard Model, we however first discuss the concept of chirality.