Gauge Invariance

As we saw in the previous section, electric and magnetic fields can be written in terms of scalar and vector potentials, as follows:

$${\bf E} = - \nabla\phi - \frac{\partial {\bf A}}{\partial t},$$ (2.299)

$${\bf B} = \nabla\times{\bf A}.$$ (2.300)

However, this prescription is not unique. There are many different potentials that can generate the same fields. This phenomenon is known as gauge invariance. The most general transformation that leaves the ${\bf E}$ and ${\bf B}$ fields unchanged in Equations (2.299) and (2.300) is

$$\phi \rightarrow \phi +\frac{\partial \psi}{\partial t},$$ (2.301)

$${\bf A} \rightarrow {\bf A} - \nabla\psi,$$ (2.302)

where $\psi({\bf r},t)$ is a general scalar field known as the gauge field. A particular choice of the gauge field is termed a choice of the gauge.

We are free to choose the gauge so as to make our equations as simple as possible. As before, the most sensible gauge for the scalar potential is to set it to zero at infinity:

$$\phi({\bf r},t) \rightarrow 0 \mbox{\hspace{1cm}as~~$\vert{\bf r}\vert\rightarrow \infty$} .$$ (2.303)

For steady fields, we found that

$$\nabla\cdot {\bf A} = 0.$$ (2.304)

[See Equation (2.262).] This choice is known as the Coulomb gauge. We can still use this gauge for time-varying fields.

Equation (2.299) can be combined with the field equation [see Equation (2.54)]

$$\nabla\cdot {\bf E} = \frac{\rho}{\epsilon_0}$$ (2.305)

(which remains valid for time-varying fields) to give

$$-\nabla^2 \phi - \frac{\partial \,(\nabla\cdot{\bf A})}{\partial t} = \frac{\rho}{\epsilon_0}.$$ (2.306)

(See Section A.21.) With the Coulomb gauge, $\nabla\cdot {\bf A}=0$, the previous expression reduces to

$$\nabla^2 \phi = -\frac{\rho}{\epsilon_0},$$ (2.307)

which is just Poisson's equation. [See Equation (2.99).] Thus, we can immediately write down an expression for the scalar potential generated by time-varying fields. It is exactly analogous to our previous expression for the scalar potential generated by steady fields:

$$\phi({\bf r},t)= \frac{1}{4\pi\,\epsilon_0} \int \frac{\rho({\bf r}',t)}{\vert{\bf r} - {\bf r}'\vert} \,dV'.$$ (2.308)

[See Equation (2.18).] However, this apparently simple result is extremely deceptive. Equation (2.308) is a typical action at a distance law. If the charge density changes suddenly at ${\bf r}'$ then the potential at ${\bf r}$ responds immediately. However, special relativity forbid information from propagating faster than the speed of light in vacuum, because this would violate causality. (See Section 3.2.10.) How can these two statements be reconciled? The crucial point is that the scalar potential cannot be measured directly, it can only be inferred from the electric field. In the time dependent case, there are two parts to the electric field; that part that comes from the scalar potential, and that part that comes from the vector potential. [See Equation (2.299).] So, if the scalar potential in some region responds immediately to some distance rearrangement of charge density then it does not necessarily follow that the electric field also has an immediate response. What actually happens is that the change in the part of the electric field that comes from the scalar potential is balanced by an equal and opposite change in the part that comes from the vector potential, so that the overall electric field remains unchanged. This state of affairs persists at least until sufficient time has elapsed for a light signal to travel from the distant charges to the region in question. Thus, causality is not violated, because it is the electric field, and not the scalar potential, that carries physically accessible information.

It is clear that the apparent action at a distance nature of Equation (2.308) is highly misleading. This suggests, very strongly, that the Coulomb gauge is not the optimum gauge in the time dependent case. A more sensible choice is the so-called Lorenz gauge:

$$\nabla\cdot {\bf A} = -\epsilon_0\, \mu_0 \,\frac{\partial \phi}{\partial t}.$$ (2.309)

Substituting the Lorenz gauge into Equation (2.306), we obtain

$$\epsilon_0\,\mu_0\,\frac{\partial^2\phi}{\partial t^2} - \nabla^2\phi = \frac{\rho} {\epsilon_0}.$$ (2.310)

It turns out that this is a three-dimensional wave equation in which information propagates at the speed of light in vacuum. (See Section 2.4.4.) Thus, the Lorenz gauge makes manifest the fact that information carried by electric and magnetic fields propagates at the velocity of light in vacuum, which implies that causality is not violated.