46.2 The Hamiltonian formalism for electromagnetic fields

46.2 The Hamiltonian formalism for electromagnetic fields

In order to quantize the electromagnetic field, we first need to express Maxwell’s equations in Hamiltonian form. These equations are second-order diferential equations in $t$, so we expect to parametrize solutions in terms of initial data

$$(A_0(0,\mathbf{x}),\mathbf{A}(0,\mathbf{x}))\text{ and }\left(\frac{\partial A_0}{\partial t}(0,\mathbf{x}),\frac{\partial \mathbf{A}}{\partial t}(0,\mathbf{x})\right)\text{\hspace{1em}(46.5)}$$

The problem with this is that gauge invariance implies that if $A_{\mu }$ is a solution with this initial data, so is its gauge-transform $A_{\mu }^{\phi }$ (see equation 45.1) for any function $\phi (t,\mathbf{x})$ such that

$$\phi (0,\mathbf{x})=0,\frac{\partial \phi }{\partial t}(t,\mathbf{x})=0$$

This implies that solutions are not uniquely determined by the initial data of the vector potential and its time derivative at $t=0$ , and thus this initial data will not provide coordinates on the space of solutions.

One way to deal with this problem is to try and find conditions on the vector potential which will remove this freedom to perform such gauge transformations, then take as phase space the subspace of initial data satisfying the conditions. This is called making a “choice of $\mathrm{gaug}{\mathrm{e}}^{\cdot }$ . We will begin with:

Definition (Temporal gauge)

$A$ vector potential $A_{\mu }$ is said to be in temporal gauge if $A_0=0$

Note that given any vector potential $A_{\mu }$ , we can find a gauge transformation $\phi$ such that the gauge transformed vector potential will have $A_0=0$ by solving the equation

$$\frac{\partial \phi }{\partial t}(t,\mathbf{x})=A_0(t,\mathbf{x})$$

which has solution

$$\phi (t,\mathbf{x})={\int }_0^t{A}_0(\tau ,\mathbf{x})d\tau +{\phi }_0(\mathbf{x})\text{\hspace{1em}(46.6)}$$

where ${\phi }_0(\mathbf{x})=\phi (0,\mathbf{x})$ is any function of the spatial variables $x$.

In temporal gauge, initial data for a solution to Maxwell’s equations is given by a pair of functions

$$\left(\mathbf{A}(\mathbf{x}),\frac{\partial \mathbf{A}}{\partial t}(\mathbf{x})\right)$$

and we can take these as our coordinates on the phase space of solutions. The electric field is now

$$\mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}$$

so we can also write our coordinates on phase space as

$$(\mathbf{A}(\mathbf{x}),-\mathbf{E}(\mathbf{x}))$$

Requiring that these coordinates behave just like position and momentum coordinates in the finite dimensional case, we can specify the Poisson bracket and thus the symplectic form by

$$\begin{array}{rl}\{A_j(\mathbf{x}),A_k({\mathbf{x}}^{\prime })\}& =\{E_j(\mathbf{x}),E_k({\mathbf{x}}^{\prime })\}=0\\ \{A_j(\mathbf{x}),E_k({\mathbf{x}}^{\prime })\}& =-{\delta }_{jk}{\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime })\end{array}\text{\hspace{1em}(46.7)}$$

If we then take as Hamiltonian function

$$h=\frac{1}{2}{\int }_{{\mathbf{R}}^3}(|\mathbf{E}{|}^2+|\mathbf{B}{|}^2)d^3\mathbf{x}\text{\hspace{1em}(46.8)}$$

Hamilton’s equations become

$$\frac{\partial \mathbf{A}}{\partial t}(t,\mathbf{x})=\{\mathbf{A}(t,\mathbf{x}),h\}=-\mathbf{E}(\mathbf{x})\text{\hspace{1em}(46.9)}$$

and

$$\frac{\partial \mathbf{E}}{\partial t}(t,\mathbf{x})=\{\mathbf{E}(t,\mathbf{x}),h\}$$

which one can show is just the Maxwell equation 46.3

$$\frac{\partial \mathbf{E}}{\partial t}=\nabla \times \mathbf{B}$$

The final Maxwell’s equation, Gauss’s law (46.4), does not appear in the Hamiltonian formalism as an equation of motion. In later sections we will see several diferent ways of dealing with this problem.

For the Yang-Mills case, in temporal gauge we can again take as initial data $(\mathbf{A}(\mathbf{x}),-\mathbf{E}(\mathbf{x}))$ , where these are now matrix-valued. For the Hamiltonian, we can use the trace function on matrices and take

$$h=\frac{1}{2}{\int }_{{\mathbf{R}}^3}tr(|\mathbf{E}{|}^2+|\mathbf{B}{|}^2)d^3\mathbf{x}\text{\hspace{1em}(46.10)}$$

since

$$⟨X_1,X_2⟩=tr(X_1{X}_2)$$

is a non-degenerate, positive, $SU(2)$ invariant inner product on $\mathfrak{su}(2)$ . One of Hamilton’s equations is then equation 46.9, which is also just the definition of the Yang-Mills electric field when $A_0=0$ (see equation 45.3).

The other Hamilton’s equation can be shown to be

$$\begin{array}{rl}\frac{\partial E_j}{\partial t}(t,\mathbf{x})& =\{E_j(t,\mathbf{x}),h\}\\ & =(\nabla \times \mathbf{B}{)}_j-ie{ϵ}_{jkl}[A_k,B_l]\end{array}\text{\hspace{1em}(46.11)}$$

where $B$ is the Yang-Mills magnetic field (45.4). If a covariant derivative acting on fields valued in su(2) is defined by

$${\nabla }^A(\cdot )=\nabla (\cdot )-ie[\mathbf{A},\cdot ]$$

then equation 46.11 can be written

$$\frac{\partial \mathbf{E}}{\partial t}(t,\mathbf{x})={\nabla }^A\times \mathbf{B}$$

The problem with these equations is that they are non-linear equations in $A$, so the phase space of solutions is no longer a linear space, and diferent methods are needed for quantization of the theory.

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