notes on quantization of Gauge field I — Dirac Bracket

By xil41

Dirac bracket is an extension of Poisson bracket, used to deal with the Hamiltonian with constraints. Especially it is useful in quantizing gauge theories. In detail, Dirac bracket is defined as $[A,B]_D= [A,B]_P - [A, \Phi_i]_P (\Delta^{-1})^{ij} [\Phi_j,B]_P$ , where $[,]_P$ is the Poisson bracket, $\Phi$ is some constraints such as a gauge condition, and $\Delta$ is some matrix we will talk about later.

Suppose there exits a Hamiltonian $H$ with a set of constraints $\{ \Phi_i \approx 0 \}$ (Here “ $\approx$ ” means equations hold on shell, say after evaluate all the Poisson brackets then turn on the constraints conditions. We call this “weakly equal to”). For example, in E&M theory, $L = F^{\mu \nu}F_{\mu \nu}$ . So the canonical momenta are $\Pi^\mu=F^{0\mu}$ . We immediately get a set of primary constraints that $\Pi^0 \approx 0$ (we call this a set of constraints, because the equation holds on every point of space time). These constraints are needed to hold all the time, so $[\Pi^0,H] = 0$ is demanded; hence a set of secondary constraints $\partial_i \Pi^i \approx 0$ follows. There are no more constraints here, because $[\partial_i \Pi_i,H] = 0$ reduces to $0 = 0$ (trivial). If it’s a non-trivial equation, we keep on this process until we get a “ $0 = 0$ “.

According to Dirac, we decompose those constraints into two classes: the first-class constraints and the second-class constraints. The first-class constraint means its Poison bracket with all the other constraints vanishes $[\Phi_i,\Phi_j]_P \approx 0$ on the other hand, the second-class does not, which indicates that no linear combination $\sum c_i [\Phi_i,\Phi_j]_P$ vanishes. As the E&M example above, $\Pi^0$ and $\partial_i \Pi^i$ are first-class constraints.

In quantizing the gauge field, the primary first-class constraints(such as $\Pi^0 \approx 0)$ or maybe all the first-class constraints are harmless for they can be eliminated by a choice of gauge. So we took linear combination of constraints so that as many constraints can be put into first-class as possible then eliminate them by choosing gauge. After that, we left with a set of second-class constraints.

The existence of the second-class constraints indicates some degrees of freedom are not physically important. The naive idea to deal with them is just to “throw them away” and only keeping those degrees of freedom of physical importance. For example, suppose that we have constraints $p_1 \approx 0$ , $q_1 \approx 0$ , we can just take $p_1$ and $q_1$ as identically $0$ . This is a simplest example. For generalization, Dirac proposed the following scheme:

Since for second-class constraints no linear combination $\sum c_i[\Phi_i,\Phi_j]_P$ vanishes, then $Det(\Delta) \neq 0$ , where $\Delta_{ij} \equiv [\Phi_i,\Phi_j]_P$ . Hence $\Delta$ is reversible. By introducing Dirac bracket $[A,B]_D = [A,B]_P - [A,\Phi_i]_P (\Delta^{-1})^{ij}[\Phi_j,B]_P$ , we can get all commutation relations for canonical variables. And use them to quantize gauge field theory. We can check by using the $p_1 \approx 0$ , $q_1 \approx 0$ , that this scheme is the same as we throw away $p_1$ and $q_1$ .

Still use the E&M case as an example. First as we mentioned above , $\Pi^0 = 0$ and $\partial_i \Pi^i = 0$ are first class constraints. Now we propose a gauge(here we use coulomb gauge) $\partial^i A_i = 0$ . Together with $\partial_i \Pi^i=\partial_i F^{i0} = 0$ , we can eliminate $A^0$ (In this case $A^0= 0$ ). Then we left with two second-class constraints: $\partial_i \Pi^i = 0$ and the gauge condition $\partial^i A_i = 0$ , since $\Delta^{1 \bold{x},2 \bold{y}} = -\Delta^{2 \bold{y},1 \bold{x}} = [\partial^i A_i, \partial_i \Pi^i]_P = \nabla^2 \delta( \bold{x}-\bold{y})$ , Δ^1x,1y=Δ^2x,2y = 0, Det(Δ)≠0. Finally, we use the Dirac bracket we get, [A_i, Π^j]_D= iδⁱ_jδ(x-y)+i ∂²/∂_i∂_j(1/4π|x-y|). And extend this to quantum version, to quantize E&M.

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notes on quantization of Gauge field I — Dirac Bracket

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