[Physics] Classical electrodynamics as an $\mathrm{U}(1)$ gauge theory

differential-geometryelectromagnetismgauge-invariancegauge-theory

Preface: I haven't studied QED or any other QFT formally, only by occasionally flipping through books, and having a working knowledge of the mathematics of gauge theories (principal bundles, etc.).

As far as I am aware, the status of electrodynamics as an $\mathrm{U}(1)$ gauge theory comes from quantum-mechanical considerations, namely the ability to rotate the complex phase of a wave function describing a charged particle, however the "classical" gauge freedom $A\mapsto A+\mathrm{d}\chi$ follows naturally from this, and so does the $F$ electromagnetic field strength tensor as the $\mathrm{U}(1)$ connection's curvature, and this $F$ is the same $F$ as it is in classical ED.

My question is regarding if it is possible to formulate purely classical electrodynamics as an $\mathrm{U}(1)$ gauge theory, including motivation to do so (eg. not just postulating out of thin air that $A$ should be a $\mathrm{U}(1)$ connection's connection form, but giving a reason for it too)?

Best Answer

Yes, you can formulate Maxwell's equations as a classical $U(1)$ gauge theory without using the word "quantum" at all. In non-relativistic notation, we introduce a scalar potential $\phi$ and a vector potential $\vec{A}$ which are related to the electric $\vec{E}$ and magnetic $\vec{B}$ fields via \begin{eqnarray} -\nabla \phi - \frac{\partial \vec{A}}{\partial t} &=& \vec{E} \\ \nabla \times \vec{A} &=& \vec{B} \end{eqnarray} From these definitions, it is easy to check that $\vec{E}$ and $\vec{B}$ are unchanged if we transform $\phi$ and $\vec{A}$ by \begin{eqnarray} \phi &\rightarrow& \phi - \frac{\partial \lambda}{\partial t}\\ \vec{A} &\rightarrow& \vec{A} + \nabla \lambda \end{eqnarray} for an arbitrary (smooth) function $\lambda$. This is gauge invariance. The above set of statements can be reformulated in much more sophisticated ways, including saying that $(\phi,\vec{A})$ is a connection on a $U(1)$ fiber bundle.

The potential formalism was known and used well before quantum electrodynamics was discovered. The main reason for introducing the potentials classically, is calculational convenience. Instead of needing to solve for the 6 components of $\vec{E}$ and $\vec{B}$, we only need to solve for 4 components (1 in $\phi$ and 3 in $\vec{A}$).

In fact it is even better than this because we can use gauge invariance to massage the form of the equations into standard forms. For instance, we can always choose the function $\lambda$ such that the sourced Maxwell's equations take the form of four decoupled standard wave equations (the Lorenz gauge) \begin{eqnarray} -\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} + \nabla^2 \phi &=& 4\pi \rho \\ -\frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} + \nabla^2 \vec{A} &=& 4\pi \vec{J} \end{eqnarray} where $\rho$ is the charge density and $\vec{J}$ is the current density. These can be solved formally using Green's function methods, and the expressions for the electric and magnetic fields can be obtained by differentiating the potentials. This method of solution is easier than trying to directly solve for $\vec{E}$ and $\vec{B}$, because the sourced Maxwell's equations do not take the form of decoupled copies of the wave equation in standard form in those variables (the components are mixed together and there are a mix of different kinds of second spatial derivatives).

Also note that we need to solve fewer independent equations using the potential formalism. Since $\nabla \cdot \nabla \times \vec{A}=0$ identically, Maxwell's equation $\nabla \cdot B=0$ is automatically satisfied if we solve for the vector potential. Similarly, the equation $\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$ is automatically satisfied by the potential, which you can verify using the identity $\nabla \times \nabla \phi=0$ and the definition $\vec{B}=\nabla \times \vec{A}$.

Finally, as others have noted, using the potentials lets you express Maxwell's theory in terms of a local Lagrangian. This is useful at a formal level; for example it lets you derive a stress-energy tensor in a systematic way, and to explain conservation of charge as a consequence of Noether's theorem applied to the global $U(1)$ symmetry ($\lambda={\rm const}$, which leaves the potentials invariant but under which charged particles and fields will transform).


Incidentally, to make a comment that isn't directly in response to your question, but which I think is relevant: it's worth understanding why the formulation of a $U(1)$ gauge theory is useful in quantum theory. It actually is not necessary to formulate Maxwell's equations as a gauge theory quantum mechanically, but it is convenient, because the local Lagrangian formulation makes the locality of physical results manifest. However, this comes at the cost of needing to establish that the unphysical parts of the gauge field do not contribute to the final answer, established using an elaborate set of relations known as Ward-Takahashi identities, and makes unitarity difficult to establish. In other formulations (helicity-spinor variables), locality is not obvious but other properties of the theory such as unitarity are.

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