46.6 Covariant gauge quantization

46.6 Covariant gauge quantization

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · unitary representation · complexification

The methods used so far to handle gauge invariance sufer from various problems that can make their use awkward, most obviously the problem that they break Lorentz invariance by imposing non-Lorentz invariant conditions $(A_0=0,\nabla$ $\mathbf{A}=0)$ . Lorentz invariance can be maintained by use of a Lorentz invariant gauge condition, for example (note that the name is not a typo):

Definition (Lorenz gauge)

$A$ vector potential $A_{\mu }$ is said to be in Lorenz gauge if

$$\chi (A)\equiv -\frac{\partial A_0}{\partial t}+\nabla \cdot \mathbf{A}=0$$

Besides Lorentz invariance (Lorentz transforms of vector potentials in Lorenz gauge remain in Lorenz gauge), this gauge has the attractive feature that

Maxwell’s equations become just the standard massless wave equation. Since

$$\begin{array}{rl}\nabla \times \mathbf{B}-\frac{\partial \mathbf{E}}{\partial t}& =\nabla \times \nabla \times \mathbf{A}+\frac{{\partial }^2\mathbf{A}}{\partial t^2}-\frac{\partial }{\partial t}\nabla A_0\\ & =\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}+\frac{{\partial }^2\mathbf{A}}{\partial t^2}-\frac{\partial }{\partial t}\nabla A_0\\ & =\nabla (\nabla \cdot \mathbf{A}-\frac{\partial }{\partial t}{A}_0)-{\nabla }^2\mathbf{A}+\frac{{\partial }^2\mathbf{A}}{\partial t^2}\\ & =-{\nabla }^2\mathbf{A}+\frac{{\partial }^2\mathbf{A}}{\partial t^2}\end{array}$$

the Maxwell equation 46.3 is the massless Klein-Gordon equation for the spatial components of $\mathbf{A}$ .

Similarly,

$$\begin{array}{rl}\nabla \cdot \mathbf{E}& =\nabla \left(-\frac{\partial \mathbf{A}}{\partial t}+\nabla A_0\right)\\ & =-\frac{\partial }{\partial t}\nabla \cdot \mathbf{A}+{\nabla }^2{A}_0^2\\ & =-\frac{{\partial }^2{A}_0}{\partial t^2}+{\nabla }^2{A}_0\end{array}$$

so Gauss’s law becomes the massless Klein-Gordon equation for $A_0$

Like the temporal gauge, the Lorenz gauge does not completely remove the gauge freedom. Under a gauge transformation

$$-\frac{\partial A_0}{\partial t}+\nabla \cdot \mathbf{A}\to -\frac{\partial A_0}{\partial t}+\nabla \cdot \mathbf{A}-\frac{{\partial }^2\phi }{\partial t^2}+{\nabla }^2\phi$$

so $\phi$ that satisfy the wave equation

$$\frac{{\partial }^2\phi }{\partial t^2}={\nabla }^2\phi \text{\hspace{1em}(46.20)}$$

will give gauge transformations that preserve the Lorenz gauge condition $\chi (A)=$ 0.

The four components of $A_{\mu }$ can be treated as four separate solutions of the massless Klein-Gordon equation, and the theory then quantized in a Lorentz covariant manner. The field operators will be

$${\hat{A}}_{\mu }(t,\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}(a_{\mu }(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+a_{\mu }^{†}(\mathbf{p})e^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}$$

using annihilation and creation operators that satisfy

$$[a_{\mu }(\mathbf{p}),a_{\nu }^{†}({\mathbf{p}}^{\prime })]=\pm {\delta }_{\mu \nu }\delta (\mathbf{p}-{\mathbf{p}}^{\prime })\text{\hspace{1em}(46.21)}$$

where $\pm \mathrm{is}+1$ for spatial coordinates, −1 for the time coordinate.

Two sorts of problems however arise:

• The Lorenz gauge condition is needed to get Maxwell’s equations, but it cannot be imposed as an operator condition

$$-\frac{\partial {\hat{A}}_0}{\partial t}+\nabla \cdot \hat{\mathbf{A}}=0$$

since this is inconsistent with the canonical commutation relation, because

$$\left[{\hat{A}}_0(\mathbf{x}),-\frac{\partial {\hat{A}}_0({\mathbf{x}}^{\prime })}{\partial t}+\nabla \cdot \hat{\mathbf{A}}({\mathbf{x}}^{\prime })\right]=\left[{\hat{A}}_0(\mathbf{x}),-\frac{\partial {\hat{A}}_0({\mathbf{x}}^{\prime })}{\partial t}\right]=i\delta (\mathbf{x}-{\mathbf{x}}^{\prime })\ne 0$$

• The commutation relations 46.21 for the operators $a_0(\mathbf{p}),a_0^{†}(\mathbf{p})$ have the wrong sign. Recall from the discussion in section 26.3 that the positive sign is required in order for Bargmann-Fock quantization to give a unitary representation on a harmonic oscillator state space, with a an annihilation operator, and $a^{†}$ a creation operator.

We saw in section 46.3 that in $A_0=0$ gauge, there was an analog of the first of these problems, with $\begin{array}{r}\nabla \cdot \hat{\mathbf{E}}=0\end{array}$ an inconsistent operator equation. There ∇ · E played the role of a moment map for the group of time-independent gauge transformations. One can show that similarly, $\chi (A)$ plays the role of a moment map for the group of gauge transformations $\phi$ satisfying the wave equation 46.20. In the $A_0=0$ gauge case, we saw that $\nabla \cdot \hat{\mathbf{E}}=0$ could be treated not as an operator equation, but as a condition on states, determining the physical state space ${\mathcal{H}}_{phys}\subset \mathcal{H}$

This will not work for the Lorenz gauge condition, since it can be shown that there will be no states such that

$$\left(-\frac{\partial {\hat{A}}_0}{\partial t}+\nabla \cdot \hat{\mathbf{A}}\right)|\psi ⟩=0$$

The problem is that, unlike in the Gauss’s law case, the complex structure $J_r$ used for quantization (+$i$ on the positive energy single-particle states, $-i$ on the negative energy ones) does not commute with the Lorenz gauge condition. The gauge condition needs to be implemented not on the dual phase space $M$ (here the space of $A_{\mu }$ satisfying the massless wave equation), but on ${\mathcal{H}}_1={\mathcal{M}}_{J_r}^{+}$ where

$$\mathcal{M}\otimes \mathbf{C}={\mathcal{M}}_{J_r}^{+}\oplus {\mathcal{M}}_{J_r}^{-}$$

is the decomposition of the complexification of ${\mathcal{H}}_1$ into negative and positive energy subspaces. The condition we want is thus

$$\chi (A{)}^{+}=0$$

where $\chi (A{)}^{+}$ is the positive energy part of the decomposition of $\chi (A)$ into positive and negative energy components.

This sort of gauge condition can be implemented either before or after quantization, as follows:

• One can take elements of ${\mathcal{H}}_1$ to be ${\mathbf{C}}^4.$ -valued functions $\alpha =({\alpha }_0(\mathbf{p}),\alpha (\mathbf{p}))$ on ${\mathbf{R}}^3$ with Lorentz invariant indefinite inner product

$$⟨\alpha ,{\alpha }^{\prime }⟩={\int }_{{\mathbf{R}}^3}(-\overline{{\alpha }_0(\mathbf{p})}{\alpha }_0^{\prime }(\mathbf{p})+\overline{\mathbf{\alpha }(\mathbf{p})}\cdot {\mathbf{\alpha }}^{\prime }(\mathbf{p}))d^3\mathbf{p}\text{\hspace{1em}(46.22)}$$

Here each ${\alpha }_{\mu }(\mathbf{p})$ is defined as in equation 43.7 for the single component field case. The subspace satisfying $\chi (A{)}^{+}=0$ will be the subspace ${\mathcal{H}}_1^{\prime }\subset$ ${\mathcal{H}}_1$ of ${\alpha }_{\mu }$ satisfying

$$-p_0{\alpha }_0(\mathbf{p})+\sum _{j=1}^3{p}_j{\alpha }_j(\mathbf{p})=0$$

This subspace will in turn have a subspace ${\mathcal{H}}_1^{\prime \prime }\subset {\mathcal{H}}_1^{\prime }$ corresponding to $A_{\mu }$ that are gauge transforms of 0, $i$.e., with Fourier coeficients satisfying

$${\alpha }_0(\mathbf{p})=p_{0}f(\mathbf{p}),{\alpha }_j(\mathbf{p})=p_{j}f(\mathbf{p})$$

for some function $f(\mathbf{p})$ . Both of these subspaces carry an action of the Lorentz group, and so does the quotient space

$${\mathcal{H}}_1^{\prime }/{\mathcal{H}}_1^{\prime \prime }$$

One can show that the indefinite inner product 46.22 is non-negative on ${\mathcal{H}}_1^{\prime }$ and null on ${\mathcal{H}}_1^{\prime \prime }$ , so positive definite on the quotient space. Note that this is an example of a symplectic reduction, although in the context of an action of an infinite dimensional complex group (the positive energy gauge transformations satisfying the massless wave equation). One can construct the quantum theory by applying the Bargmann-Fock method to this quotient space.

• One can instead implement the gauge condition after quantization, first quantizing the four components of $A_{\mu }$ as massless fields, getting a state space $H$ (which will not have a positive definite inner product), then defining ${\mathcal{H}}^{\prime }\subset \mathcal{H}$ to be the subspace of states satisfying

$${\left(-\frac{\partial {\hat{A}}_0}{\partial t}+\nabla \cdot \hat{\mathbf{A}}\right)}^{+}|\psi ⟩=0$$

where the positive energy part of the operator is taken. The state space ${\mathcal{H}}^{\prime }$ in turn has a subspace ${\mathcal{H}}^{\prime \prime }$ of states of zero norm, and one can define

$${\mathcal{H}}_{phys}={\mathcal{H}}^{\prime }/{\mathcal{H}}^{\prime \prime }$$

${\mathcal{H}}_{phys}$ will have a positive definite Hermitian inner product, and carry a unitary action of the Poincar´e group. It can be shown to be isomorphic to the physical state space of transverse photons constructed using the Coulomb gauge.

This sort of covariant quantization method is often referred to as the “Gupta-Bleuler” method, and is described in more detail in many quantum field theory textbooks.

In the Yang-Mills case, each of the methods that we have discussed for dealing with the gauge symmetry runs into problems:

• In the $A_0=0$ gauge, there again is a symmetry under the group of timeindependent gauge transformations, and a moment map µ, with $\mu =0$ the Yang-Mills version of Gauss’s law. The symplectic reduction however is now a non-linear space, so the quantization method we have developed does not apply. Gauss’s law can instead be imposed on the states, but it is dificult to explicitly characterize the physical state space that this gives.

• A Yang-Mills analog of the Coulomb gauge condition can be defined, but then the analog of equation 46.13 will be a non-linear equation without a unique solution (this problem is known as the “Gribov ambiguity”).

• The combination of fields $\chi (A)$ now no longer satisfies a linear wave equation, and one cannot consistently restrict to a positive energy subspace and use the Gupta-Bleuler covariant quantization method.

Digression. There is a much more sophisticated Lorentz covariant quantization method (called the “BRST method”) for dealing with gauge symmetry that uses quite diferent techniques. The theory is first extended by the addition of non-physical (“ghost”) fermionic fields, giving a theory of coupled bosonic and fermionic oscillators of the sort we studied in section 33.1. This includes an operator analogous to $Q_1$ , with the property that $Q_1^2=0$ . One can arrange things in the electromagnetic field case such that

$${\mathcal{H}}_{phys}=\frac{\{|\psi ⟩:Q_1|\psi ⟩=0\}}{\{|\psi ⟩:|\psi ⟩=Q_1|{\psi }^{\prime }⟩\}}$$

In the BRST method one works with unconstrained Lorentz covariant fields, but non-unitary state spaces, with unitarity only achieved on a quotient such as ${\mathcal{H}}_{phys}$ . This construction is related to the Gupta-Bleuler method in the case of electromagnetic fields, but unlike that method, generalizes to the Yang-Mills case. It can also be motivated by considerations of what happens when one imposes a gauge condition in a path integral (this is called the “Faddeev-Popov method”).

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