Hamilton's principle/Applications/The electromagnetic Lagrangian

Plain-language version: Electromagnetism from an action

Maxwell's equations follow from an action, and the action follows almost entirely from a symmetry requirement. This page derives the electromagnetic Lagrangian and uses it as the volume's worked case of a degenerate variational problem — one where the Legendre transform fails, Noether's second theorem applies, and the two facts turn out to be the same fact.

Making the phase local

Counting physical polarisations from gauge redundancy.

Classical field theory ended with a complex scalar field invariant under a constant phase rotation $\varphi \to e^{i\alpha }\varphi$. Now demand invariance when $\alpha$ depends on position:

$\varphi (x)\to e^{i\alpha (x)}\varphi (x)\mathrm{(}\mathrm{1}\mathrm{)}$

The kinetic term breaks, because the derivative picks up an extra piece:

${\partial }_{\beta }\varphi \to e^{i\alpha }({\partial }_{\beta }\varphi +i{\partial }_{\beta }\alpha \varphi )\mathrm{(}\mathrm{2}\mathrm{)}$

Repairing it requires a new field $A_{\beta }$ transforming so as to cancel the unwanted term:

$A_{\beta }\to A_{\beta }-{\partial }_{\beta }\alpha \mathrm{(}\mathrm{3}\mathrm{)}$

and the covariant derivative

$D_{\beta }\varphi =({\partial }_{\beta }+i{A}_{\beta })\varphi \mathrm{(}\mathrm{4}\mathrm{)}$

which transforms as $D_{\beta }\varphi \to e^{i\alpha }{D}_{\beta }\varphi$, exactly as $\varphi$ does. Replacing every ${\partial }_{\beta }$ by $D_{\beta }$ restores invariance.

The demand for local phase invariance forces a new field into existence. That field is the electromagnetic potential, and (3) is the gauge transformation. Nothing about electromagnetism was assumed.

The field Lagrangian

Identities among field equations from a local symmetry.

$A_{\beta }$ must have its own dynamics, and its Lagrangian is fixed by the same method. Build the gauge-invariant combination

$F_{\alpha \beta }={\partial }_{\alpha }{A}_{\beta }-{\partial }_{\beta }{A}_{\alpha }\mathrm{(}\mathrm{5}\mathrm{)}$

which is unchanged under (3), since the two extra terms are $-{\partial }_{\alpha }{\partial }_{\beta }\alpha$ and its transpose, equal by symmetry of mixed partials. The only Lorentz and gauge invariant scalar quadratic in $F$ is $F_{\alpha \beta }{F}^{\alpha \beta }$, so

$ℒ=-\frac{1}{4}{F}_{\alpha \beta }{F}^{\alpha \beta }-j^{\alpha }{A}_{\alpha }\mathrm{(}\mathrm{6}\mathrm{)}$

with the second term the coupling to a current. Applying the field equation (2) of Classical field theory with $A_{\alpha }$ as the field gives

${\partial }_{\alpha }{F}^{\alpha \beta }=j^{\beta }\mathrm{(}\mathrm{7}\mathrm{)}$

which is the inhomogeneous pair of Maxwell's equations — Gauss's law and Ampère's law with the displacement current.

The homogeneous pair is not a field equation at all. It is the identity

${\partial }_{\alpha }{F}_{\beta \gamma }+{\partial }_{\beta }{F}_{\gamma \alpha }+{\partial }_{\gamma }{F}_{\alpha \beta }=0\mathrm{(}\mathrm{8}\mathrm{)}$

which follows from (5) by inspection. So no magnetic monopoles and Faraday's law hold automatically once the field is expressed through a potential, and they require no dynamics. The asymmetry between the two Maxwell pairs, which looks arbitrary in the standard presentation, is the difference between an equation of motion and an identity.

The action must be gauge invariant, and that constrains the source

Maxwell's equations split into dynamics and identities.

The coupling term in (6) is not obviously invariant. Under (3) it changes by

$\Delta S=\int j^{\alpha }{\partial }_{\alpha }\alpha d^{4}x=-\int ({\partial }_{\alpha }{j}^{\alpha })\alpha d^{4}x\mathrm{(}\mathrm{9}\mathrm{)}$

after integrating by parts and discarding the boundary term. For arbitrary $\alpha$, the fundamental lemma forces

${\partial }_{\alpha }{j}^{\alpha }=0\mathrm{(}\mathrm{10}\mathrm{)}$

Gauge invariance requires charge conservation. It is not an additional assumption and not a consequence of the equations of motion: an action coupling the potential to a non-conserved current is not gauge invariant and does not define a theory.

Degeneracy

Now the point of the page. Compute the momentum conjugate to $A_{\beta }$:

${\pi }^{\beta }=\frac{\partial ℒ}{\partial ({\partial }_0{A}_{\beta })}=-F^{0\beta }\mathrm{(}\mathrm{11}\mathrm{)}$

For $\beta =0$ this gives ${\pi }^0=-F^{00}=0$ identically, since $F$ is antisymmetric. The momentum conjugate to $A_0$ vanishes.

So the map from velocities to momenta is not invertible. The Hessian ${\partial }^2ℒ/\partial ({\partial }_0{A}_{\alpha })\partial ({\partial }_0{A}_{\beta })$ is singular, and the Legendre transform fails exactly as that page warned it would.

The reason is not an accident of the Lagrangian. ${\partial }_0{A}_0$ does not appear in $ℒ$, because it would have to enter through $F_{00}$, which is zero by antisymmetry — which is itself forced by gauge invariance.

The two facts are one fact

By Noether's second theorem, invariance under a group depending on an arbitrary function $\alpha (x)$ produces an identity among the field equations. Here that identity is

${\partial }_{\beta }({\partial }_{\alpha }{F}^{\alpha \beta }-j^{\beta })=0\mathrm{(}\mathrm{12}\mathrm{)}$

which holds identically, before any field equation is imposed, because ${\partial }_{\beta }{\partial }_{\alpha }{F}^{\alpha \beta }$ vanishes by antisymmetry and ${\partial }_{\beta }{j}^{\beta }$ vanishes by (10).

So one of the four equations in (7) is not independent. Counting:

quantity	count
components of $A_{\alpha }$	4
minus the identity (12)	−1
minus the gauge freedom (3)	−1
physical polarisations	2

Two, which is the observed number of photon polarisations. The vanishing momentum (11), the identity (12), and the gauge freedom (3) are three descriptions of one redundancy — and Noether's second theorem is the statement that they must appear together.

This is the promised connection. On the Legendre transform the failure of the transform was named as a limitation; here it is explained.

Gauge fixing

Because $A_{\alpha }$ carries unphysical freedom, solving (7) requires removing it. Imposing a condition such as ${\partial }_{\alpha }{A}^{\alpha }=0$ selects one representative from each gauge class and makes (7) into four wave equations with sources.

This is not part of the physics. The choice is a convenience, results must not depend on it, and a dependence signals an error. It is the field-theoretic analogue of choosing generalised coordinates in Holonomic constraints — a chart on a space whose points are gauge classes, not potentials.

The charged particle

For a single particle of charge $e$ in a given field, the action is

$S=\int [-m{c}^2\sqrt{1-v^2/c^2}-e\phi +e𝐯\cdot 𝐀]dt\mathrm{(}\mathrm{13}\mathrm{)}$

and the Euler-Lagrange equations of Several dependent variables give the Lorentz force. Two features are worth noting.

The canonical momentum is $𝐩+e𝐀$, not $𝐩$. Conjugate momentum and mechanical momentum differ whenever the Lagrangian is linear in the velocities, and the distinction matters in every quantum treatment.

The Lagrangian (13) is not gauge invariant: under (3) it changes by a total time derivative. That is harmless — a total derivative changes the action by endpoint terms only and leaves the equations of motion alone. Noether covers this too, noting that an integral differing from $I$ by an integral over a divergence admits the same group.

Provenance: every equation on this page is derived here or is standard. No primary source for Maxwell is cited; the Maxwell row is not part of this volume's survey. See Sources.

Next

the Einstein-Hilbert action repeats the construction for a group of arbitrary functions acting on spacetime itself.