46.5 Space-time symmetries

46.5 Space-time symmetries

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · unitary representation · group action

The choice of Coulomb gauge nicely isolates the two physical degrees of freedom that describe photons and allows a straightforward quantization in terms of two copies of the previously studied relativistic scalar field. It does however do this in a way which makes some aspects of the Poincar´e group action on the theory hard to understand, in particular the action of boost transformations. Our choice of continuous basis elements for the space of solutions of the Maxwell equations was not invariant under boost transformations (since it uses initial data at a fixed time, see equation 46.5), but making the gauge choice $A_0=0,\nabla \cdot \mathbf{A}=0$ creates another fundamental problem. Acting by a boost on a solution in this gauge will typically take it to a solution no longer satisfying the gauge condition.

For some indication of the dificulties introduced by the non-Lorentz invariant Coulomb gauge choice, the field commutators can be computed, with the result

$$[{\hat{A}}_j(\mathbf{x}),{\hat{A}}_k({\mathbf{x}}^{\prime })]=[{\hat{E}}_j(\mathbf{x}),{\hat{E}}_k({\mathbf{x}}^{\prime })]=0$$

$$[{\hat{A}}_j(\mathbf{x}),{\hat{E}}_k({\mathbf{x}}^{\prime })]=-\frac{i}{(2\pi {)}^3}{\int }_{{\mathbf{R}}^3}\left({\delta }_{jk}-\frac{p_j{p}_k}{|\mathbf{p}{|}^2}\right)e^{i\mathbf{p}\cdot (\mathbf{x}-{\mathbf{x}}^{\prime })}{d}^3\mathbf{p}\text{\hspace{1em}(46.18)}$$

The right-hand side in the last case is not just the expected delta-function, but includes a term that is non-local in position space.

In section 46.6 we will discuss what happens with a Lorentz invariant gauge choice, but for now will just consider the Poincar´e subgroup of space-time translations and spatial rotations, which do preserve the Coulomb gauge choice. Such group elements can be labeled by $(a,R)$ , where $\mathbf{a}=\left(a_0,\mathbf{a}\right)$ is a translation in space-time, and $R\in SO(3)$ is a spatial rotation. Generalizing the scalar field case (equation 44.6), we want to construct a unitary representation of the group of such elements by operators $U(\mathbf{a},R)$ on the state space, with the $U(\mathbf{a},R)$ also acting as intertwining operators on the field operators, by:

$$\hat{\mathbf{A}}(t,\mathbf{x})\to U(a,R)\hat{\mathbf{A}}(t,\mathbf{x})U^{-1}(a,R)=R^{-1}\hat{\mathbf{A}}(t+a_0,R\mathbf{x}+\mathbf{a})\text{\hspace{1em}(46.19)}$$

To construct $U(a,R)$ we will proceed as for the scalar field case (see section 44.2) to identify the Lie algebra representation operators that satisfy the needed commutation relations, skipping some details (these can be found in most quantum field theory textbooks).

46.5.1 Time translations

For time translations, as usual one just needs to find the Hamiltonian operator $\hat{H}.$ , and then

$$U(a_0,1)=e^{i{a}_0\hat{H}}$$

Taking

$$\begin{array}{rl}\hat{H}& =\frac{1}{2}{\int }_{{\mathbf{R}}^3}:(|\hat{\mathbf{E}}{|}^2+|\hat{\mathbf{B}}{|}^2):d^3\mathbf{x}=\frac{1}{2}{\int }_{{\mathbf{R}}^3}:\left(|\frac{\partial \hat{\mathbf{A}}}{\partial t}{|}^2+|\nabla \times \hat{\mathbf{A}}{|}^2\right):d^3\mathbf{x}\\ & ={\int }_{{\mathbf{R}}^3}{\omega }_{\mathbf{p}}(a_1^{†}(\mathbf{p})a_1(\mathbf{p})+a_2^{†}(\mathbf{p})a_2(\mathbf{p}))d^3\mathbf{p}\end{array}$$

one can show, using equations 46.17, 46.18 and properties of the polarization vectors ${ϵ}_j(\mathbf{p})$ , that one has as required

$$\frac{\partial \hat{\mathbf{A}}}{\partial t}=[\hat{\mathbf{A}},-i\hat{H}]$$ $$\frac{\partial }{\partial t}(a_{\sigma }(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{)}_{|t=0}=-i{\omega }_{\mathbf{p}}{a}_{\sigma }(\mathbf{p})=[a_{\sigma }(\mathbf{p}),-i\hat{H}]$$

The first of these uses the position space expression in terms of fields, the second the momentum space expression in terms of annihilation and creation operators.

46.5.2 Spatial translations

For spatial translations, we have

$$U(\mathbf{a},1)=e^{-i\mathbf{a}\cdot \hat{\mathbf{P}}}$$

where $\hat{\mathbf{P}}$ is the momentum operator. It has the momentum space expression

$$\hat{\mathbf{P}}={\int }_{{\mathbf{R}}^3}\mathbf{p}(a_1^{†}(\mathbf{p})a_1(\mathbf{p})+a_2^{†}(\mathbf{p})a_2(\mathbf{p}))d^3\mathbf{p}$$

which satisfies

$$\nabla (a_{\sigma }(\mathbf{p})e^{i\mathbf{p}\cdot \mathbf{x}})=i\mathbf{p}{a}_{\sigma }(\mathbf{p})e^{i\mathbf{p}\cdot \mathbf{x}}=[-i\hat{\mathbf{P}},a_{\sigma }(\mathbf{p})e^{i\mathbf{p}\cdot \mathbf{x}}]$$

In terms of position space fields, one has

$$\hat{\mathbf{P}}={\int }_{{\mathbf{R}}^3}:\hat{\mathbf{E}}\times \hat{\mathbf{B}}:d^3\mathbf{x}$$

satisfying

$$\nabla {\hat{A}}_j=[-i\hat{\mathbf{P}},{\hat{A}}_j]$$

One way to derive this is to use the fact that, for the classical theory,

$${\mathbf{P}}_{EM}={\int }_{{\mathbf{R}}^3}\mathbf{E}\times \mathbf{B}{d}^3\mathbf{x}$$

is the momentum of the electromagnetic field, since one can use the Poisson bracket relations 46.7 to show that

$$\{{\mathbf{P}}_{EM},A_j(\mathbf{x})\}=\nabla A_j(\mathbf{x})$$

46.5.3 Rotations

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · group representation · complexification

We will not go through the exercise of constructing the angular momentum operator Jb for the electromagnetic field that gives the action of rotations on the theory. Details of how to do this can for instance be found in chapters 6 and 7 of [35]. The situation is similar to that of the spin $\frac{1}{2}$ Pauli-equation case in section 34.2. There we found that $\mathbf{J}=\mathbf{L}+\mathbf{S}$ where the first term is the “orbital” angular momentum, due to the action of rotations on space, while S was the infinitesimal counterpart of the $SU(2)$ action on two-component spinors.

Much the same thing happens in this case, with $L$ due to the action on spatial coordinates $x$, and S due to the action of $SO(3)$ rotations on the 3 components of the vector $A$ (see equation 46.19). We have seen that in the Coulomb gauge, the field $\hat{\mathbf{A}}(\mathbf{x})$ decomposes into two copies of fields behaving much like the scalar Klein-Gordon theory, corresponding to the two basis vectors ${ϵ}_1(\mathbf{p}),{ϵ}_2(\mathbf{p})$ of the plane perpendicular to the vector $p$. The subgroup $SO(2)\subset SO(3)$ of rotations about the axis $p$ acts on this plane and its basis vectors in exactly the same way as the internal $SO(2)$ symmetry acted on pairs of real Klein-Gordon fields (see section 44.1.1). In that case the same $SO(2)$ acts in the same way at each point in space-time, whereas here this $SO(2)\subset SO(3)$ varies depending on the momentum vector.

In the internal symmetry case, we found an operator $\hat{Q}$ with integer eigenvalues (the charge). The analogous operator in this case is the helicity operator. The massless Poincar´e group representations described in section 42.3.5 are the ones that occur here, for the case of helicity ±1. Just as in the internal symmetry case, where complexification allowed diagonalization of $\hat{Q}$ on the single-particle space, getting charges ±1, here complexification of the ${ϵ}_1(\mathbf{p}),{ϵ}_2(\mathbf{p})$ diagonalizes the helicity, getting so-called “left circularly polarized” and “right circularly polarized” photon states.

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