Lagrangians & local gauge invariance

Quantum field theory begins by rewriting the whole of dynamics in a single object: the Lagrangian density $\mathcal L$ . Where ordinary mechanics starts from $L=T-V$ and grinds out Newton's laws through the Euler–Lagrange equation, field theory promotes the discrete coordinates $q_i(t)$ to fields $\phi_i(x)$ spread over spacetime, and the same variational machine turns a chosen $\mathcal L$ into the field's equation of motion. Feed it the scalar Lagrangian and out comes Klein–Gordon; feed it the spinor Lagrangian and out comes Dirac's; feed it the antisymmetric field strength and out come Maxwell's equations, or Proca's if the vector is massive. One template, three spins, every free relativistic wave equation. The Lagrangian is not just compact bookkeeping — its symmetries carry the physics. Noether's theorem, the deepest statement in the subject, says that every continuous symmetry of $\mathcal L$ is matched, exactly and automatically, by a conserved current: time-translation invariance gives energy, spatial invariance momentum, rotational invariance angular momentum, and a rigid phase rotation of a charged field gives conserved electric charge. Symmetry and conservation are two names for one fact. The final step builds the Standard Model. A free charged field is invariant under a global phase change — the same rotation everywhere — but not under a local one that varies from point to point, because the derivative in $\mathcal L$ trips over the position-dependent phase. Demanding local invariance anyway forces a new vector field into the theory with a rigidly fixed coupling: that field is the photon, and its masslessness is a theorem, not an input.

Key takeaways

The Lagrangian density $\mathcal L(\phi_i,\partial_\mu\phi_i)$ replaces $L=T-V$ for continuous fields; its dynamics follow from one master equation, the Euler–Lagrange field equation.
Specific choices of $\mathcal L$ reproduce the Klein–Gordon (spin 0), Dirac (spin $\tfrac12$ ), and Maxwell/Proca (spin 1) field equations — the machine is universal across spin.
Noether's theorem: every continuous symmetry of $\mathcal L$ yields a conserved current $\partial_\mu j^\mu=0$ and a conserved charge $Q=\int j^0\,d^3x$ .
A global phase rotation $\psi\to e^{iq\alpha}\psi$ is a symmetry of the free Dirac Lagrangian; its Noether charge is electric charge.
Making that phase local, $\alpha\to\theta(x)$ , breaks the symmetry — restoring it via the covariant derivative $D_\mu=\partial_\mu+iqA_\mu$ forces the electromagnetic field into existence.
The gauge boson must be massless: a term $\tfrac12 m^2A_\mu A^\mu$ breaks gauge invariance, which is exactly why the $W$ and $Z$ need the Higgs mechanism.
Key idea: a field theory is fixed by the symmetries of its Lagrangian — Noether turns each continuous symmetry into a conservation law, and demanding one symmetry hold locally forces the gauge fields that mediate the fundamental forces, the photon emerging directly from local U(1) phase invariance of the Dirac field.

The core insight

A field theory is fixed by the symmetries of its Lagrangian: Noether's theorem turns every continuous symmetry into a conservation law, and demanding that a global phase symmetry hold locally — independently at every point of spacetime — forces into existence the gauge fields that mediate the fundamental forces, with the photon emerging directly from local U(1) phase invariance of the Dirac field.

Where this lives today

The Lagrangian is not a pedagogical relic — it is the working definition of the Standard Model. The theory is stated as a single Lagrangian density, and every LHC prediction is extracted from it: the Feynman rules for each vertex are read off the interaction terms, and cross-sections follow by squaring the amplitudes. Lattice QCD discretises the same Lagrangian onto a spacetime grid and evaluates its path integral to compute the hadron spectrum from first principles. Effective field theory organises low-energy physics as the most general Lagrangian consistent with the assumed symmetries, ordered by dimension — the framework behind everything from chiral perturbation theory to beyond-Standard-Model searches, where a non-renormalisable operator would signal new interactions. Noether's theorem is tested continuously: energy–momentum and charge conservation are enforced event-by-event in every detector, and precision searches for symmetry violation (CP violation, proton decay, lepton-flavour violation) are hunts for the breakdown of specific Noether currents. And the gauge principle glimpsed here — one global symmetry made local — is the template from which the photon, the gluons, and the electroweak bosons are built, the subject of the next article.

The physical picture

Everything in a relativistic field theory descends from one line — the Euler–Lagrange field equation — applied to whatever Lagrangian density you write down. A scalar Lagrangian returns the Klein–Gordon equation; the Dirac spinor Lagrangian returns Dirac's; the antisymmetric field strength of a vector returns Maxwell's (or, with a mass term, Proca's) equations. The first figure is that machine — one input slot, one crank, three spins coming out.

Figure

The Euler-Lagrange field equation ∂_μ(∂ℒ/∂(∂_μφ)) − ∂ℒ/∂φ = 0 turns each Lagrangian into a wave equation: ℒ = ½(∂φ)² − ½m²φ² gives Klein-Gordon (□+m²)φ = 0 for spin 0, ℒ = ψ̄(iγ^μ∂_μ − m)ψ gives Dirac (iγ^μ∂_μ − m)ψ = 0 for spin ½, and ℒ = −¼F_μνF^μν + ½m²A² gives Proca ∂_μF^μν + m²A^ν = 0 for spin 1

The second idea gives the Lagrangian its authority: its symmetries are conservation laws. Slide the system forward in time and, if $\mathcal L$ does not notice, energy is conserved; slide it in space, momentum; rotate it, angular momentum. Rotate the phase of a charged field by the same angle everywhere and the conserved quantity is electric charge, carried by the current $j^\mu=\bar\psi\gamma^\mu\psi$ . Noether's theorem makes this exact: one conserved current per continuous symmetry, no exceptions, no free constants. The second figure is that dictionary — from the geometry of a symmetry to a number that never changes.

Figure

Noether's theorem pairs every continuous symmetry of ℒ with a conserved current ∂_μ j^μ = 0: time translation gives energy E, space translation gives momentum p, rotation gives angular momentum J = mr²φ̇, and a global phase ψ → e^(iα)ψ gives the conserved electric current j^μ = ψ̄γ^μψ whose charge Q = ∫j⁰d³x is constant

The core equations

A field theory is specified by its action $S$ , the spacetime integral of $\mathcal L$ . Requiring the action to be stationary under variations of the field — the principle of least action — yields the Euler–Lagrange field equation, the master relation from which every equation of motion below is read off.

S=\int \mathcal L(\phi_i,\partial_\mu\phi_i)\,d^4x,\qquad \partial_\mu\!\left(\frac{\partial\mathcal L}{\partial(\partial_\mu\phi_i)}\right)-\frac{\partial\mathcal L}{\partial\phi_i}=0 \qquad (1)

where: $\mathcal L$ — Lagrangian density ( $[\text{mass}]^4$ in natural units); $\phi_i(x)$ — the field(s), indexed by $i$ ; $\partial_\mu\phi_i$ — their derivatives; $S$ — the action. Reading it: one relation per field component; whatever you write for $\mathcal L$ , this crank returns its dynamics. The scalar case is the simplest non-trivial Lagrangian — a kinetic term quadratic in derivatives minus a mass term quadratic in the field. Applying equation (1) returns the Klein–Gordon equation, and the coefficient of the field term is fixed to be the square of the quantum's mass.

\mathcal L_{\rm KG}=\tfrac12(\partial_\mu\phi)(\partial^\mu\phi)-\tfrac12 m^2\phi^2\ \ \Longrightarrow\ \ (\partial_\mu\partial^\mu+m^2)\phi=(\Box+m^2)\phi=0 \qquad (2)

where: $\phi$ — real spin-0 scalar field; $m$ — its mass; $\Box\equiv\partial_\mu\partial^\mu$ — the d'Alembertian. Reading it: the $-\tfrac12 m^2\phi^2$ term is what puts the mass into the wave equation — drop it and you recover the massless $\Box\phi=0$ . For spin $\tfrac12$ the Lagrangian is first order in derivatives, built from the Dirac spinor $\psi$ and its adjoint $\bar\psi=\psi^\dagger\gamma^0$ . Treating $\bar\psi$ as independent, equation (1) hands back the Dirac equation in one step.

\mathcal L_{\rm D}=\bar\psi\,(i\gamma^\mu\partial_\mu-m)\,\psi\ \ \Longrightarrow\ \ (i\gamma^\mu\partial_\mu-m)\psi=0 \qquad (3)

where: $\psi$ — four-component Dirac spinor; $\gamma^\mu$ — gamma matrices, $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ ; $m$ — fermion mass. Reading it: since $\mathcal L_{\rm D}$ is independent of $\partial_\mu\bar\psi$ , varying $\bar\psi$ gives the Dirac equation directly and varying $\psi$ its adjoint. For spin 1 the field is the vector potential $A_\mu$ , and the Lagrangian is built from the antisymmetric field strength $F_{\mu\nu}$ plus an optional mass term. Equation (1) returns Proca; deleting the mass term gives source-free Maxwell.

\mathcal L_{\rm P}=-\tfrac14 F_{\mu\nu}F^{\mu\nu}+\tfrac12 m^2 A_\nu A^\nu,\quad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu\ \ \Longrightarrow\ \ \partial_\mu F^{\mu\nu}+m^2 A^\nu=0 \qquad (4)

where: $A_\mu$ — vector field; $F_{\mu\nu}$ — field strength holding $\mathbf E$ and $\mathbf B$ ; $m$ — vector-boson mass. Reading it: for $m=0$ this is $\partial_\mu F^{\mu\nu}=0$ , source-free Maxwell; the mass term is what a photon is forbidden to have. Noether's theorem converts the invariance of $\mathcal L$ under a continuous transformation $\phi\to\phi+\delta\phi$ into a conserved current, and hence a charge — the same theorem that ties spacetime symmetries to energy and momentum ties an internal phase symmetry to electric charge.

j^\mu=\frac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\,\delta\phi,\qquad \partial_\mu j^\mu=0\ \ \Longrightarrow\ \ Q=\int j^0\,d^3x=\text{const} \qquad (5)

where: $\delta\phi$ — the infinitesimal symmetry variation; $j^\mu$ — Noether current; $Q$ — conserved charge. Reading it: conservation is local ( $\partial_\mu j^\mu=0$ , a continuity equation) and global (fixed total $Q$ ) at once. The final relation is the pivot of the whole subject. A global phase rotation leaves $\mathcal L_{\rm D}$ invariant, but a local one $\theta(x)$ leaves a residue proportional to the current; cancelling it forces the covariant derivative and, with it, the gauge field.

\psi\to e^{iq\theta(x)}\psi:\ \ \delta\mathcal L=-q(\partial_\mu\theta)\,\bar\psi\gamma^\mu\psi;\qquad \partial_\mu\to D_\mu=\partial_\mu+iqA_\mu,\ \ A_\mu\to A_\mu-\partial_\mu\theta \qquad (6)

where: $\theta(x)$ — position-dependent phase; $q$ — charge; $A_\mu$ — the forced gauge field (photon); $D_\mu$ — covariant derivative. Reading it: the residue is exactly $-q(\partial_\mu\theta)j^\mu$ , so a field coupling to $j^\mu$ with a compensating transformation cancels it — that field is electromagnetism, derived not assumed.

Derivation AThe Euler-Lagrange field equation from stationary action

The goal of this derivation is to obtain the master equation of the whole subject — the field Euler–Lagrange equation (1) — from a single physical postulate, the principle of stationary action, and then to watch it manufacture a concrete wave equation. We vary the field by an arbitrary amount that vanishes on the boundary, demand the action be stationary, integrate by parts to strip the derivative off the variation, and read off the condition that must hold at every point — then apply it to the scalar Lagrangian to recover Klein–Gordon.

Step 1. Write $S=\int\mathcal L(\phi,\partial_\mu\phi)\,d^4x$ and vary the field, $\phi\to\phi+\delta\phi$ , with $\delta\phi\to 0$ on the boundary. To first order, $\delta S=\int d^4x\,\big[\tfrac{\partial\mathcal L}{\partial\phi}\delta\phi+\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\delta(\partial_\mu\phi)\big]$ . Why: nature extremises the action, so the physics lives in $\delta S=0$ , and expanding to first order turns that condition into a differential equation.

Step 2. Use $\delta(\partial_\mu\phi)=\partial_\mu(\delta\phi)$ and integrate the second term by parts: $\int d^4x\,\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\partial_\mu(\delta\phi)=-\int d^4x\,\big[\partial_\mu\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\big]\delta\phi$ , the surface term vanishing since $\delta\phi=0$ on the boundary. Why: this moves the derivative off the unknown $\delta\phi$ and onto the known coefficient, letting us factor $\delta\phi$ out of the integral.

Step 3. Collect both terms: $\delta S=\int d^4x\,\big[\tfrac{\partial\mathcal L}{\partial\phi}-\partial_\mu\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\big]\delta\phi$ . Stationarity for arbitrary $\delta\phi$ forces the bracket to vanish pointwise: $\partial_\mu\big(\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\big)-\tfrac{\partial\mathcal L}{\partial\phi}=0$ . Why: an integral of (something) $\times\delta\phi$ that vanishes for every $\delta\phi$ forces that something to be zero everywhere — the fundamental lemma of variations, converting a global condition into a local field equation.

Step 4. Test on $\mathcal L_{\rm KG}=\tfrac12(\partial_\mu\phi)(\partial^\mu\phi)-\tfrac12 m^2\phi^2$ : here $\tfrac{\partial\mathcal L}{\partial\phi}=-m^2\phi$ and $\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}=\partial^\mu\phi$ , giving $\partial_\mu\partial^\mu\phi+m^2\phi=0$ — the Klein–Gordon equation. Why: the formalism made concrete — the mass term in $\mathcal L$ becomes the mass term in the wave equation, and every free field works the same way.

Go deeper · Derivation B · 1 minNoether's theorem — a conserved current for every symmetry

Derivation BNoether's theorem — a conserved current for every symmetry

The purpose here is to prove Noether's theorem in the form used throughout particle physics — that any continuous transformation leaving the Lagrangian invariant produces a current with zero four-divergence — and to identify that current for the phase symmetry of the Dirac field, where it turns out to be the electromagnetic current. We compute the change in $\mathcal L$ under an infinitesimal symmetry, use the equation of motion to fold it into a total derivative, read off the conserved current, then specialise to $\psi\to e^{iq\alpha}\psi$ and verify conservation with the Dirac equation.

Step 1. Vary by the symmetry, $\phi\to\phi+\delta\phi$ , and expand: $\delta\mathcal L=\tfrac{\partial\mathcal L}{\partial\phi}\delta\phi+\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\partial_\mu(\delta\phi)$ . Why: a symmetry is a variation under which $\mathcal L$ is unchanged, so tracking $\delta\mathcal L$ and setting it to zero extracts the conservation law hidden in the invariance.

Step 2. On a physical field Euler–Lagrange holds, $\tfrac{\partial\mathcal L}{\partial\phi}=\partial_\mu\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}$ ; substituting folds both terms into one total derivative, $\delta\mathcal L=\partial_\mu\!\big(\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\,\delta\phi\big)$ . Why: using the equation of motion is the crucial move — it collapses two separate pieces into a single divergence, the mathematical shape of a conservation law.

Step 3. For a genuine symmetry $\delta\mathcal L=0$ , so $\partial_\mu j^\mu=0$ with $j^\mu=\tfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\,\delta\phi$ ; integrating the time component gives a constant charge $Q=\int j^0\,d^3x$ . Why: $\partial_\mu j^\mu=0$ is a continuity equation — whatever flows out of a region is exactly what its stored charge loses.

Step 4. Specialise to the Dirac field, $\psi\to e^{iq\alpha}\psi$ , i.e. $\delta\psi=iq\alpha\psi$ . Since $\tfrac{\partial\mathcal L_{\rm D}}{\partial(\partial_\mu\psi)}=i\bar\psi\gamma^\mu$ , the current is $j^\mu\propto\bar\psi\gamma^\mu\psi$ ; check with the Dirac equation, $\partial_\mu(\bar\psi\gamma^\mu\psi)=(im\bar\psi)\psi+\bar\psi(-im\psi)=0$ . Why: the Noether current of the phase symmetry is exactly the charge current the photon couples to — electric charge is conserved because the electron field has a phase symmetry.

Go deeper · Derivation C · 1 minFrom global phase invariance to the photon

Derivation CFrom global phase invariance to the photon

This derivation carries the free Dirac field from a global phase symmetry to a local one and shows that restoring the broken invariance forces the electromagnetic field into existence with no freedom left — the covariant derivative, the gauge field's transformation law, the minimal-coupling interaction, and the photon's masslessness are all consequences, not inputs. We confirm the global symmetry, localise the phase and watch $\mathcal L$ break, introduce the covariant derivative that repairs it, and test a mass term and find it forbidden. This is the germ of the local gauge principle, generalised to non-Abelian groups in the article that follows.

Step 1. The free Lagrangian $\mathcal L_{\rm D}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi$ is invariant under a global phase $\psi\to e^{iq\alpha}\psi$ ( $\alpha$ constant), since $\bar\psi\to e^{-iq\alpha}\bar\psi$ and the phases cancel in every term — the symmetry whose Noether charge is electric charge. Why: the construction demands that this already-present symmetry hold under the far stricter local condition, so it must be established first.

Step 2. Localise the phase, $\alpha\to\theta(x)$ . Now $\partial_\mu(e^{iq\theta}\psi)=e^{iq\theta}(\partial_\mu\psi+iq(\partial_\mu\theta)\psi)$ , so $\delta\mathcal L=-q(\partial_\mu\theta)\bar\psi\gamma^\mu\psi\neq 0$ . Why: the derivative cannot slide past a position-dependent phase, and the leftover is proportional to the current $j^\mu=\bar\psi\gamma^\mu\psi$ — which dictates what must cancel it.

Step 3. Replace $\partial_\mu$ by $D_\mu=\partial_\mu+iqA_\mu$ and give the new field the law $A_\mu\to A_\mu-\partial_\mu\theta$ . Then $D_\mu\psi\to e^{iq\theta}D_\mu\psi$ : the $+iq(\partial_\mu\theta)$ from the derivative and the $-iq(\partial_\mu\theta)$ from the shifted $A_\mu$ cancel, so $\bar\psi\,i\gamma^\mu D_\mu\psi$ is locally invariant, expanding to $\mathcal L_{\rm D}-q\bar\psi\gamma^\mu\psi A_\mu$ — a free electron plus the electromagnetic interaction. Why: the interaction $-qj^\mu A_\mu$ was never inserted; it fell out of insisting a free electron respect a local phase — the gauge principle in one line.

Step 4. Give $A_\mu$ its own gauge-invariant dynamics, $-\tfrac14 F_{\mu\nu}F^{\mu\nu}$ (invariant because $F_{\mu\nu}$ is antisymmetric), and test a mass term: under $A_\mu\to A_\mu-\partial_\mu\theta$ , $\tfrac12 m^2 A_\mu A^\mu$ generates an uncancellable cross-term $-m^2 A_\mu\partial^\mu\theta$ , so gauge invariance demands $m=0$ . Why: the photon's masslessness is a theorem of the symmetry; the $W$ and $Z$ masses cannot be written by hand — they require the Higgs mechanism, which breaks the symmetry spontaneously while preserving it in the Lagrangian.

Limiting cases

Case	Result	Meaning
No mass term in $\mathcal L_{\rm KG}$	$\Box\phi=0$	massless scalar; the field quantum has zero rest mass, infinite Compton range
No mass term in $\mathcal L_{\rm P}$	$\partial_\mu F^{\mu\nu}=0$	source-free Maxwell equations — light as a massless spin-1 field
Global phase only ( $\partial_\mu\theta=0$ )	no gauge field needed	rigid phase is a symmetry of the free theory; Noether current $j^\mu$ conserved, no interaction
Local phase ( $\theta=\theta(x)$ )	$D_\mu=\partial_\mu+iqA_\mu$ forced	the derivative catches $\partial_\mu\theta$ ; a gauge field must appear — this is electromagnetism
Gauge-boson mass term $\tfrac12 m^2A_\mu A^\mu$	not gauge invariant	forbidden by the principle; photon exactly massless, $m_\gamma<10^{-18}\ \mathrm{eV}$
Non-relativistic point particle	$L=T-V$ , $\tfrac{d}{dt}\tfrac{\partial L}{\partial\dot q}-\tfrac{\partial L}{\partial q}=0$	the field formalism collapses to ordinary Lagrangian mechanics — the sanity check

Common trapGlobal invariance already needs the photon, and a gauge boson can just be given a mass

Both halves are wrong, and they are the two commonest errors. First, the free Dirac Lagrangian is already invariant under a global phase $\psi\to e^{iq\alpha}\psi$ — no photon is required, and its only consequence is Noether conservation of charge. The photon is forced only by the stricter demand of local invariance, $\alpha\to\theta(x)$ , where the derivative trips over the position-dependent phase; confusing the two makes the whole gauge argument evaporate. Second, you cannot simply hand the gauge boson a mass: $\tfrac12 m^2A_\mu A^\mu$ generates an uncancellable cross-term $-m^2A_\mu\partial^\mu\theta$ under $A_\mu\to A_\mu-\partial_\mu\theta$ , explicitly breaking the invariance that produced electromagnetism. That is why the massive $W$ and $Z$ require the Higgs mechanism — spontaneous breaking supplies their mass while leaving the Lagrangian gauge invariant.

Worked examples

1 · Klein–Gordon from its Lagrangian, and reading off the mass (fully worked)

Start from $\mathcal L_{\rm KG}=\tfrac12(\partial_\mu\phi)(\partial^\mu\phi)-\tfrac12 m^2\phi^2$ . Derive the field equation, and interpret the mass by computing the range of the associated Yukawa-type force for a neutral pion, $m_{\pi^0}=135\ \mathrm{MeV}$ .

Step 1. Compute the two ingredients of the Euler–Lagrange equation. The field derivative is $\dfrac{\partial\mathcal L}{\partial\phi}=-m^2\phi$ . For the derivative slot, note $(\partial_\mu\phi)(\partial^\mu\phi)=g^{\alpha\beta}(\partial_\alpha\phi)(\partial_\beta\phi)$ , so $\dfrac{\partial\mathcal L}{\partial(\partial_\mu\phi)}=\partial^\mu\phi$ .

Step 2. Assemble Euler–Lagrange: $\partial_\mu\big(\partial^\mu\phi\big)-(-m^2\phi)=0$ , i.e. $\boxed{(\Box+m^2)\phi=0}$ , the Klein–Gordon equation. The coefficient of the field term is $m^2$ , so a plane-wave solution $\phi\propto e^{-ip\cdot x}$ obeys $-p_\mu p^\mu+m^2=0$ , the relativistic relation $E^2=\mathbf p^2+m^2$ . The mass in the Lagrangian is the mass of the particle.

Step 3. Read off the physics. A massive scalar mediates a short-range force of reach $R=\hbar/(mc)=\hbar c/(mc^2)$ . With $\hbar c=197.3\ \mathrm{MeV\,fm}$ and $m_{\pi^0}c^2=135\ \mathrm{MeV}$ ,

R=\frac{197.3\ \mathrm{MeV\,fm}}{135\ \mathrm{MeV}}=\mathbf{1.46\ fm}.

A 135 MeV scalar quantum gives a force of range ≈ 1.5 fm — the nuclear scale, exactly Yukawa's estimate for the pion. Set $m=0$ and the range diverges: a massless scalar (or the photon) mediates an infinite-range $1/r$ force.

2 · The Dirac equation and its conserved current from one Lagrangian

From $\mathcal L_{\rm D}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi$ , obtain the Dirac equation and show the phase-symmetry current $j^\mu=\bar\psi\gamma^\mu\psi$ is conserved.

Step 1. Treat $\psi$ and $\bar\psi$ as independent fields. Varying $\bar\psi$ : since $\mathcal L_{\rm D}$ contains no $\partial_\mu\bar\psi$ , the Euler–Lagrange equation is just $\dfrac{\partial\mathcal L}{\partial\bar\psi}=0$ , giving $(i\gamma^\mu\partial_\mu-m)\psi=0$ — the Dirac equation, in one line.

Step 2. Varying $\psi$ gives the adjoint equation $i\partial_\mu\bar\psi\,\gamma^\mu+m\bar\psi=0$ , i.e. $\partial_\mu\bar\psi\,\gamma^\mu=im\bar\psi$ . The two equations are Hermitian conjugates, as they must be.

Step 3. The global phase $\psi\to e^{iq\alpha}\psi$ is a symmetry ( $\delta\psi=iq\alpha\psi$ ); Noether's current is $j^\mu=\bar\psi\gamma^\mu\psi$ . Test conservation: $\partial_\mu j^\mu=(\partial_\mu\bar\psi\,\gamma^\mu)\psi+\bar\psi(\gamma^\mu\partial_\mu\psi)$ . From the two equations of motion $\gamma^\mu\partial_\mu\psi=-im\psi$ and $\partial_\mu\bar\psi\,\gamma^\mu=im\bar\psi$ , so $\partial_\mu j^\mu=(im\bar\psi)\psi+\bar\psi(-im\psi)=\mathbf{0}$ .

Step 4. The conserved charge is $Q=\int j^0\,d^3x=\int\psi^\dagger\psi\,d^3x$ (since $j^0=\psi^\dagger\psi$ ), a positive-definite, time-independent number. The Lagrangian delivers both the equation of motion and, through its phase symmetry, conservation of electric charge — one object, both results.

3 · Why a photon mass is forbidden — the range of a massive vs massless vector

Compare the range of the force mediated by a massive spin-1 field (Proca) with that of the massless photon, and connect the result to gauge invariance.

Step 1. The Proca equation $\partial_\mu F^{\mu\nu}+m^2 A^\nu=0$ with a static point source gives a Yukawa potential $V(r)\propto e^{-r/R}/r$ of range $R=\hbar/(mc)$ ; the massless limit $m\to0$ recovers the Coulomb potential $V\propto1/r$ of infinite range.

Step 2. Put in a heavy vector boson, the $W$ at $m_Wc^2=80.4\ \mathrm{GeV}=80\,400\ \mathrm{MeV}$ :

R_W=\frac{\hbar c}{m_Wc^2}=\frac{197.3\ \mathrm{MeV\,fm}}{80\,400\ \mathrm{MeV}}=\mathbf{2.45\times10^{-3}\ fm}.

This is the sub-femtometre reach of the weak force — the reason $\beta$ decay looks point-like at nuclear energies.

Step 3. The photon's mass is bounded at $m_\gamma<10^{-18}\ \mathrm{eV}$ — an effectively infinite range, consistent with the exact $1/r^2$ Coulomb law. The contrast is no accident of parameters: gauge invariance forbids $\tfrac12 m^2A_\mu A^\mu$ , so $m_\gamma=0$ is a theorem and electromagnetism is exactly long-range. The $W$ evades the theorem only because its mass comes from the Higgs mechanism, not a bare term.

Physics of the portrait

The bridge from a Lagrangian's symmetries to its conservation laws — the idea that makes the whole formalism physically potent — has one author, a mathematician invited to Göttingen to solve a puzzle in Einstein's new theory of gravity.

Origins · 1 minEmmy Noether and how this was discovered

Emmy Noether

1882–1935 · the theorem linking symmetry to conservation laws

Emmy Noether was the greatest algebraist of her generation and, by Einstein's own estimate, the most important woman in the history of mathematics — yet she lectured at Göttingen under David Hilbert's name because the faculty would not appoint a woman, and for years was unpaid. Hilbert and Felix Klein had brought her there in 1915 to untangle a paradox in general relativity: energy conservation seemed to fail in Einstein's theory. Her resolution, published in 1918 as "Invariante Variationsprobleme", was far more general than the problem that prompted it. Her first theorem proved that every continuous (global) symmetry of an action corresponds to a conserved current — time to energy, space to momentum, rotation to angular momentum, internal phase to charge. Her second theorem handled local symmetries, showing a gauge invariance produces identities among the field equations rather than independent conservation laws. Together they are the backbone of modern theoretical physics: every conservation law in the Standard Model, and the whole logic of gauge theory, is Noether's. Alongside this she founded much of abstract algebra — Noetherian rings, ideal theory, the axiomatic method. Dismissed from Göttingen by the Nazis in 1933 for being Jewish, she emigrated to Bryn Mawr College and died in 1935, aged fifty-three, days after surgery; Einstein's tribute ran in the New York Times.

History

1788

— Lagrange recasts mechanics around L = T − V in Mécanique analytique

1834

— Hamilton states the principle of stationary action

1915

— Noether proves that symmetry implies conservation, at Göttingen

1918

— Noether publishes "Invariante Variationsprobleme" with her two theorems

1929

— Weyl derives electromagnetism by demanding local U(1) phase invariance

Going further

Texts. Thomson, Modern Particle Physics, §17.2 (Lagrangians in QFT), §17.2.3 (Noether's theorem), §17.3 (local gauge invariance) · Griffiths, Introduction to Elementary Particles, 2nd ed., §10.1–10.3 (Lagrangian formulation, relativistic field theory, local gauge invariance) · Halzen & Martin, Quarks and Leptons, Ch. 14 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §2.2 (Euler–Lagrange), §2.2 and §22.3 (Noether) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 3 and Ch. 8.

Provenance. The Dirac spinor, its adjoint $\bar\psi$ , and the current $\bar\psi\gamma^\mu\psi$ are recalled from aq-pp-dirac-equation; the Klein–Gordon equation from aq-pp-klein-gordon-equation; symmetry generators and Noether's mechanical examples from aq-pp-symmetries-in-qm; index conventions ( $g^{\mu\nu}$ , $\Box=\partial_\mu\partial^\mu$ ) from aq-pp-four-vector-notation. This article owns the Lagrangian formalism, the Euler–Lagrange equation, Noether's theorem, and global gauge invariance; the local gauge principle, its non-Abelian generalisation, and the boson counting are developed in aq-pp-local-gauge-principle, the QED interaction in aq-pp-qed-introduction. A verifier should confirm: (1) Euler–Lagrange from $\delta S=0$ ; (2) $\mathcal L_{\rm KG}\Rightarrow(\Box+m^2)\phi=0$ ; (3) $\mathcal L_{\rm D}\Rightarrow(i\gamma^\mu\partial_\mu-m)\psi=0$ and $\partial_\mu(\bar\psi\gamma^\mu\psi)=0$ ; (4) $\mathcal L_{\rm P}\Rightarrow\partial_\mu F^{\mu\nu}+m^2A^\nu=0$ ; (5) the local-phase residue $\delta\mathcal L=-q(\partial_\mu\theta)\bar\psi\gamma^\mu\psi$ ; (6) the ranges $R_{\pi^0}=1.46$ fm, $R_W=2.45\times10^{-3}$ fm.

Prerequisites. The Dirac Equation · The Klein–Gordon Equation · Symmetries in Quantum Mechanics · Four-Vector Notation. Leads to. The Local Gauge Principle · Spontaneous Symmetry Breaking · The Higgs Mechanism · Introduction to QED.

Ask the tutor: Ask me to run the machine on any field you like — "derive the equation of motion for a complex scalar and find its conserved current" — and we will pick a Lagrangian, turn the Euler–Lagrange crank, and read off both the wave equation and the Noether charge together.

What is the difference between the Lagrangian L and the Lagrangian density 𝓛?

In point-particle mechanics $L=T-V$ is a single number depending on time. In field theory the variable is a field spread over space, so the natural object is a density $\mathcal L$ defined at every point, and the ordinary Lagrangian is its spatial integral $L=\int\mathcal L\,d^3x$ . The action is then $S=\int L\,dt=\int\mathcal L\,d^4x$ , a manifestly Lorentz-invariant spacetime integral. Working with the density keeps relativistic invariance visible — $\mathcal L$ is a Lorentz scalar, whereas $L$ singles out a time coordinate. In particle physics "the Lagrangian" almost always means the density.

Why are ψ and ψ̄ treated as independent fields in the Euler–Lagrange equation?

It is a bookkeeping convenience that gives the right answer. A complex field carries two real degrees of freedom, packaged either as (Re, Im) or equivalently as the pair $(\psi,\bar\psi)$ . Because the change of variables is invertible, varying $\psi$ and $\bar\psi$ separately produces the same equations as varying the real and imaginary parts, but far more compactly: varying $\bar\psi$ (which appears without derivatives in $\mathcal L_{\rm D}$ ) instantly gives the Dirac equation, varying $\psi$ gives its adjoint. The two are Hermitian conjugates, so nothing is lost or double-counted.

Does Noether's theorem apply to discrete symmetries like parity?

No — Noether's first theorem requires a continuous symmetry, because the proof differentiates the Lagrangian with respect to the symmetry parameter. Discrete symmetries such as parity ( $P$ ), charge conjugation ( $C$ ), and time reversal ( $T$ ) have no infinitesimal version and so produce no conserved Noether current. They can still yield conserved quantum numbers (parity is multiplicatively conserved in strong and electromagnetic interactions), but through the commutation of the symmetry operator with the Hamiltonian, not a continuity equation. The distinction matters: the weak interaction violates $P$ and $C$ maximally, yet the continuous symmetries and their Noether charges stay exact.

Why does the free Dirac Lagrangian have exactly the form ψ̄(iγ^μ∂_μ − m)ψ?

It is the simplest Lorentz-invariant, Hermitian, first-order expression that reproduces the Dirac equation, and each piece is forced: $\bar\psi(\cdots)\psi$ makes a Lorentz scalar; $i\gamma^\mu\partial_\mu$ is the unique scalar linear in one derivative; the factor $i$ makes the action real; and $-m\bar\psi\psi$ is the unique scalar mass term. Euler–Lagrange then returns $(i\gamma^\mu\partial_\mu-m)\psi=0$ . There is essentially no freedom once you demand a first-order, relativistically invariant, real Lagrangian for a spinor.

What exactly is "gauge invariance", and why the word "gauge"?

Gauge invariance is the statement that a redundancy in how we describe a field carries no physical consequence — descriptions related by a gauge transformation give identical predictions. In electromagnetism the potential $A_\mu$ can be shifted by $-\partial_\mu\theta$ without changing $\mathbf E$ or $\mathbf B$ ; that freedom is the U(1) gauge symmetry. The word comes from Hermann Weyl's 1918 attempt to make the local scale (German Eich, "gauge") of physical quantities arbitrary at each point. His scale idea failed, but when quantum mechanics reinterpreted the factor as a phase rather than a length, the name stuck — the modern content is local phase freedom, not scale.

If a global phase symmetry already conserves charge, what does making it local buy you?

An entire force. The global symmetry gives a conservation law and nothing more — charge is conserved, but there is no interaction. Promoting it to a local one is far stronger: it requires invariance even when the phase is chosen independently at every point, which the free theory cannot meet. The only repair is to introduce a new field $A_\mu$ with a precisely dictated coupling to the matter current — that field is the photon, and the coupling is electromagnetism. So global symmetry yields a conserved quantity; local symmetry yields a mediator. This is why the gauge principle, not Noether's theorem alone, is called the engine of the Standard Model.