Hamilton's principle/Applications/The electromagnetic Lagrangian - Revision history

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	after integrating by parts and discarding the boundary term. For arbitrary		after integrating by parts and discarding the boundary term. For arbitrary
	<math>\alpha</math>, ~~the fundamental lemma of~~		<math>\alpha</math>, [[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] forces
	[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] forces

	:<math>\partial_\alpha j^\alpha = 0		:<math>\partial_\alpha j^\alpha = 0
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	So the map from velocities to momenta is not invertible. The Hessian		So the map from velocities to momenta is not invertible. The Hessian
	<math>\partial^2\mathcal{L}/\partial(\partial_0 A_\alpha)\,		<math>\partial^2\mathcal{L}/\partial(\partial_0 A_\alpha)\,
	\partial(\partial_0 A_\beta)</math> is singular, and ~~the Legendre transform of~~		\partial(\partial_0 A_\beta)</math> is singular, and [[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] fails exactly as that
	[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] fails exactly as that
	page warned it would.		page warned it would.

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			{{DISPLAYTITLE:The electromagnetic Lagrangian}}
	{{Plebeian\|Hamilton's principle/Explainers/Electromagnetism from an action}}		{{Plebeian\|Hamilton's principle/Explainers/Electromagnetism from an action}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/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 ==

[[File:OPL-HP-F-26.png|thumb|right|400px|Counting physical polarisations from gauge redundancy.]]

[[Hamilton's principle/Applications/Classical field theory]] ended with a complex
scalar field invariant under a constant phase rotation
<math>\phi \to e^{i\alpha}\phi</math>. Now demand invariance when
<math>\alpha</math> depends on position:

:<math>\phi(x) \to e^{i\alpha(x)}\,\phi(x)
\qquad\qquad\mathrm{(1)}</math>

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

:<math>\partial_\beta\phi \to
e^{i\alpha}\bigl( \partial_\beta\phi + i\,\partial_\beta\alpha\;\phi \bigr)
\qquad\qquad\mathrm{(2)}</math>

Repairing it requires a new field <math>A_\beta</math> transforming so as to
cancel the unwanted term:

:<math>A_\beta \to A_\beta - \partial_\beta\alpha
\qquad\qquad\mathrm{(3)}</math>

and the '''covariant derivative'''

:<math>D_\beta\phi = \bigl( \partial_\beta + i A_\beta \bigr)\phi
\qquad\qquad\mathrm{(4)}</math>

which transforms as <math>D_\beta\phi \to e^{i\alpha}D_\beta\phi</math>, exactly
as <math>\phi</math> does. Replacing every <math>\partial_\beta</math> by
<math>D_\beta</math> 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 ==

[[File:OPL-HP-F-37.png|thumb|right|400px|Identities among field equations from a local symmetry.]]

<math>A_\beta</math> must have its own dynamics, and its Lagrangian is fixed by
the same method. Build the gauge-invariant combination

:<math>F_{\alpha\beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha
\qquad\qquad\mathrm{(5)}</math>

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

:<math>\mathcal{L} = -\tfrac{1}{4}\,F_{\alpha\beta}F^{\alpha\beta}
- j^\alpha A_\alpha
\qquad\qquad\mathrm{(6)}</math>

with the second term the coupling to a current. Applying the field equation (2)
of [[Hamilton's principle/Applications/Classical field theory]] with
<math>A_\alpha</math> as the field gives

:<math>\partial_\alpha F^{\alpha\beta} = j^\beta
\qquad\qquad\mathrm{(7)}</math>

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'''

:<math>\partial_\alpha F_{\beta\gamma}
+ \partial_\beta F_{\gamma\alpha}
+ \partial_\gamma F_{\alpha\beta} = 0
\qquad\qquad\mathrm{(8)}</math>

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 ==

[[File:OPL-HP-F-46.png|thumb|right|400px|Maxwell's equations split into dynamics and identities.]]

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

:<math>\Delta S = \int j^\alpha\,\partial_\alpha\alpha\; d^4x
= -\int (\partial_\alpha j^\alpha)\,\alpha\; d^4x
\qquad\qquad\mathrm{(9)}</math>

after integrating by parts and discarding the boundary term. For arbitrary
<math>\alpha</math>, the fundamental lemma of
[[Hamilton's principle/Mathematics/The fundamental lemma]] forces

:<math>\partial_\alpha j^\alpha = 0
\qquad\qquad\mathrm{(10)}</math>

'''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
<math>A_\beta</math>:

:<math>\pi^\beta = \frac{\partial\mathcal{L}}{\partial(\partial_0 A_\beta)}
= -F^{0\beta}
\qquad\qquad\mathrm{(11)}</math>

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

So the map from velocities to momenta is not invertible. The Hessian
<math>\partial^2\mathcal{L}/\partial(\partial_0 A_\alpha)\,
\partial(\partial_0 A_\beta)</math> is singular, and the Legendre transform of
[[Hamilton's principle/Mathematics/The Legendre transform]] fails exactly as that
page warned it would.

The reason is not an accident of the Lagrangian. <math>\partial_0 A_0</math> does
not appear in <math>\mathcal{L}</math>, because it would have to enter through
<math>F_{00}</math>, which is zero by antisymmetry — which is itself forced by
gauge invariance.

== The two facts are one fact ==

By [[Hamilton's principle/Mathematics/Noether's second theorem]], invariance
under a group depending on an arbitrary function <math>\alpha(x)</math> produces
an identity among the field equations. Here that identity is

:<math>\partial_\beta\bigl( \partial_\alpha F^{\alpha\beta} - j^\beta \bigr) = 0
\qquad\qquad\mathrm{(12)}</math>

which holds identically, before any field equation is imposed, because
<math>\partial_\beta\partial_\alpha F^{\alpha\beta}</math> vanishes by
antisymmetry and <math>\partial_\beta j^\beta</math> vanishes by (10).

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

{| class="wikitable"
! quantity !! count
|-
| components of <math>A_\alpha</math> || 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
[[Hamilton's principle/Mathematics/The Legendre transform]] the failure of the
transform was named as a limitation; here it is explained.

== Gauge fixing ==

Because <math>A_\alpha</math> carries unphysical freedom, solving (7) requires
removing it. Imposing a condition such as
<math>\partial_\alpha A^\alpha = 0</math> 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
[[Hamilton's principle/Mathematics/Holonomic constraints]] — a chart on a space
whose points are gauge classes, not potentials.

== The charged particle ==

For a single particle of charge <math>e</math> in a given field, the action is

:<math>S = \int \left[ -mc^2\sqrt{1 - v^2/c^2}
- e\varphi + e\,\mathbf{v}\cdot\mathbf{A} \right] dt
\qquad\qquad\mathrm{(13)}</math>

and the Euler-Lagrange equations of
[[Hamilton's principle/Mathematics/Several dependent variables]] give the Lorentz
force. Two features are worth noting.

The canonical momentum is <math>\mathbf{p} + e\mathbf{A}</math>, not
<math>\mathbf{p}</math>. 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 <math>I</math> by an integral over a
divergence admits the same group.

<blockquote>
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 [[Hamilton's principle/Sources]].
</blockquote>

== Next ==

[[Hamilton's principle/Applications/The Einstein-Hilbert action]] repeats the
construction for a group of arbitrary functions acting on spacetime itself.

== References ==

<references />

← Older revision		Revision as of 04:58, 16 September 2026
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	[[File:OPL-HP-F-26.png\|thumb\|right\|400px\|Counting physical polarisations from gauge redundancy.]]		[[File:OPL-HP-F-26.png\|thumb\|right\|400px\|Counting physical polarisations from gauge redundancy.]]

	[[Hamilton's principle/Applications/Classical field theory]] ended with a complex		[[Hamilton's principle/Applications/Classical field theory\|Classical field theory]] ended with a complex
	scalar field invariant under a constant phase rotation		scalar field invariant under a constant phase rotation
	<math>\phi \to e^{i\alpha}\phi</math>. Now demand invariance when		<math>\phi \to e^{i\alpha}\phi</math>. Now demand invariance when
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	with the second term the coupling to a current. Applying the field equation (2)		with the second term the coupling to a current. Applying the field equation (2)
	of [[Hamilton's principle/Applications/Classical field theory]] with		of [[Hamilton's principle/Applications/Classical field theory\|Classical field theory]] with
	<math>A_\alpha</math> as the field gives		<math>A_\alpha</math> as the field gives

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	after integrating by parts and discarding the boundary term. For arbitrary		after integrating by parts and discarding the boundary term. For arbitrary
	<math>\alpha</math>, the fundamental lemma of		<math>\alpha</math>, the fundamental lemma of
	[[Hamilton's principle/Mathematics/The fundamental lemma]] forces		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] forces

	:<math>\partial_\alpha j^\alpha = 0		:<math>\partial_\alpha j^\alpha = 0
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	<math>\partial^2\mathcal{L}/\partial(\partial_0 A_\alpha)\,		<math>\partial^2\mathcal{L}/\partial(\partial_0 A_\alpha)\,
	\partial(\partial_0 A_\beta)</math> is singular, and the Legendre transform of		\partial(\partial_0 A_\beta)</math> is singular, and the Legendre transform of
	[[Hamilton's principle/Mathematics/The Legendre transform]] fails exactly as that		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] fails exactly as that
	page warned it would.		page warned it would.

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	== The two facts are one fact ==		== The two facts are one fact ==

	By [[Hamilton's principle/Mathematics/Noether's second theorem]], invariance		By [[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]], invariance
	under a group depending on an arbitrary function <math>\alpha(x)</math> produces		under a group depending on an arbitrary function <math>\alpha(x)</math> produces
	an identity among the field equations. Here that identity is		an identity among the field equations. Here that identity is
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	This is the promised connection. On		This is the promised connection. On
	[[Hamilton's principle/Mathematics/The Legendre transform]] the failure of the		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] the failure of the
	transform was named as a limitation; here it is explained.		transform was named as a limitation; here it is explained.

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	depend on it, and a dependence signals an error. It is the field-theoretic		depend on it, and a dependence signals an error. It is the field-theoretic
	analogue of choosing generalised coordinates in		analogue of choosing generalised coordinates in
	[[Hamilton's principle/Mathematics/Holonomic constraints]] — a chart on a space		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]] — a chart on a space
	whose points are gauge classes, not potentials.		whose points are gauge classes, not potentials.

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	and the Euler-Lagrange equations of		and the Euler-Lagrange equations of
	[[Hamilton's principle/Mathematics/Several dependent variables]] give the Lorentz		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] give the Lorentz
	force. Two features are worth noting.		force. Two features are worth noting.

Line 208:		Line 208:
	Provenance: every equation on this page is derived here or is standard. No		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		primary source for Maxwell is cited; the Maxwell row is not part of this volume's
	survey. See [[Hamilton's principle/Sources]].		survey. See [[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Applications/The Einstein-Hilbert action]] repeats the		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]] repeats the
	construction for a group of arbitrary functions acting on spacetime itself.		construction for a group of arbitrary functions acting on spacetime itself.