05.05.08 · symplectic / lagrangian

Noether's second theorem and the Bianchi identity

shipped3 tiersLean: none

Anchor (Master): Olver §5.3-§5.4; Noether 1918 *Invariante Variationsprobleme* (Gött. Nachr., originator); Kosmann-Schwarzbach *The Noether Theorems* (2011); Wald *General Relativity* (1984) Appendix E

Intuition Beginner

Noether's first theorem says that a continuous symmetry of a physical law gives you a conserved quantity. Rotate the world and angular momentum is conserved; shift it in time and energy is conserved. Each symmetry comes with one dial you can turn, and turning that dial hands you one conservation law. That is the version most people meet first.

But some theories have a richer kind of symmetry. In electromagnetism you can change the potential by adding the slope of any function you like, at every point of space and time, and the electric and magnetic fields do not budge. That is not one dial; it is a whole function's worth of dials, one freedom at every point. This is called a gauge symmetry, and it is the symmetry of the freedom to relabel without changing anything observable.

When a theory has that much symmetry, something must give. You cannot have endless freedom for free. Noether's second theorem is the bookkeeping: a symmetry that depends on an arbitrary function forces the field equations to lean on each other. They are no longer independent. One of them is always a consequence of the others, woven together by an identity that holds whether or not anything solves the equations.

Visual Beginner

Picture the equations of a theory as a set of beams meant to hold up a structure. In an ordinary theory each beam carries its own load and is needed. A gauge symmetry adds a hidden brace between two of the beams, so that one beam is held in place by the others.

The brace is the differential identity. It says that a certain combination of the equations and their derivatives adds up to zero on its own, with no assumptions. Because one beam is now redundant, the structure has slack: there is play in it, room to wiggle without breaking anything. That play is the gauge freedom. The picture shows that more symmetry and more redundancy are two sides of one coin, and the brace is what connects them.

Worked example Beginner

Take electromagnetism described by a potential. The fields you can measure come from differences and slopes of this potential. The key fact is that adding the slope of any chosen function to the potential leaves every measurable field exactly the same. Pick the function however you like; the physics does not notice.

Now look at the equations the potential must satisfy, one equation for each direction. Because the potential has this built-in freedom, the four equations cannot all be independent. If they were, you could pin the potential down completely, and there would be no freedom left to add an arbitrary slope. So the equations must contain a hidden relation.

That relation turns out to be very clean: if you take the four field equations and combine them by taking one more derivative and adding them up the right way, you always get zero. Not zero because the potential solves anything, but zero as a matter of pure algebra and calculus, for any potential at all.

What this tells us: the freedom to shift the potential by an arbitrary slope and the fact that the field equations satisfy a built-in identity are the same statement seen twice. The arbitrary function in the symmetry is matched, one for one, by an identity tying the equations together. That matching is Noether's second theorem in its smallest case.

Check your understanding Beginner

Formal definition Intermediate+

Let $π : E \to X$ be a fibred manifold with base coordinates $x^{i}$ ( $1 \leq i \leq p$ ) and fibre coordinates $u^{α}$ ( $1 \leq α \leq q$ ), and let $J^{\infty} (E)$ carry the jet coordinates $u_{J}^{α}$ as in 05.05.05. A Lagrangian is a function $L (x, u^{(n)})$ on $J^{n} (E)$ ; its action is $A [u] = \int_{Ω} L d^{p} x$ over base regions $Ω$ . The development follows Olver ^{[Olver §5.4]} and Kosmann-Schwarzbach ^{[Kosmann-Schwarzbach Ch. 2]}.

The Euler-Lagrange operator (variational derivative) sends $L$ to the $q$ -tuple $$ E_\alpha(L) = \sum_{J} (-D)J,\frac{\partial L}{\partial u^\alpha_J}, \qquad (-D)J = (-1)^{|J|} D{j_1}\cdots D{j_{|J|}}, $$ where $D_{i}$ is the total derivative of 05.05.05 and the sum is finite. The system $E_{α} (L) = 0$ is the Euler-Lagrange system; a section $u = f (x)$ is on-shell when it solves this system and off-shell otherwise.

Definition (variational symmetry). A vector field $v = ξ^{i} \partial_{x^{i}} + ϕ^{α} \partial_{u^{α}}$ with characteristic $Q^{α} = ϕ^{α} - ξ^{i} u_{i}^{α}$ (notation of 05.05.06) is a variational symmetry of $L$ when the action is unchanged to first order, equivalently when the infinitesimal invariance condition $$ \mathrm{pr},v(L) + L,\mathrm{Div},\xi = \mathrm{Div},B $$ holds for some $p$ -tuple $B^{i} (x, u^{(m)})$ , where $Div P = D_{i} P^{i}$ . In characteristic (evolutionary) form this reads $pr v_{Q} (L) = Div \tilde{B}$ for a corresponding $\tilde{B}$ . When $B \neq = 0$ the symmetry is a divergence (Bessel-Hagen) symmetry ^{[Bessel-Hagen 1921]}. The first theorem 05.00.04 pairs each such symmetry with a conserved current.

Definition (gauge / infinite-dimensional variational symmetry). A variational symmetry group is infinite-dimensional of rank $r$ when its characteristic is linear in $r$ arbitrary functions $η^{k} (x)$ of the independent variables and their derivatives: $$ Q^\alpha[\eta] = \sum_{k=1}^{r}\sum_{|J| \le s} D^{\alpha k}{J}(x, u^{(t)}),D_J\eta^k, $$ with the $D_{J}^{α k}$ fixed coefficient functions on the jet bundle and the $η^{k}$ unrestricted. Such a $Q [η]$ is required to satisfy the variational-symmetry condition $pr v_{Q [η]} (L) = Div (\dots)$ for every choice of the $η^{k}$ . The collection ${v{Q[\eta]}}$ is the gauge group of the theory.

A non-example fixes the meaning. A symmetry parametrised by $r$ real constants (a finite-parameter group, $η^{k}$ constant) is the domain of the first theorem and yields $r$ conservation laws, not identities. The defining feature of the second theorem's hypothesis is that the $η^{k}$ are arbitrary functions on the base, so that the invariance condition must hold identically in $η^{k}$ and all its derivatives, not merely for constant values.

Counterexamples to common slips

The differential identity of the second theorem is an off-shell statement: $\sum_{J} (- D)_{J} (D_{J}^{α k} E_{α} (L)) \equiv 0$ holds for all fields. Mistaking it for the on-shell statement $E_{α} = 0$ collapses it into a tautology and loses its content.
The rank $r$ counts arbitrary functions, not the dimension of any pointwise stabiliser. A symmetry parametrised by $r$ constants is finite-dimensional and produces conservation laws; only function-valued parameters force identities.
The identity does not say some $E_{α}$ vanishes. It says a specific differential combination of the $E_{α}$ vanishes; the individual Euler-Lagrange expressions are generally nonzero off-shell.

Key theorem with proof Intermediate+

Theorem (Noether's second theorem). Let $L$ be a Lagrangian admitting an infinite-dimensional variational symmetry of rank $r$ with characteristic $Q^{α} [η] = \sum_{J} D_{J}^{α k} D_{J} η^{k}$ , linear in the arbitrary functions $η^{1}, \dots, η^{r}$ . Then the Euler-Lagrange expressions $E_{α} (L)$ satisfy $r$ differential identities $$ \sum_{\alpha}\sum_{J} (-D)_J!\left(D^{\alpha k}J,E\alpha(L)\right) \equiv 0, \qquad k = 1, \dots, r, $$ holding identically (off-shell) as functions on the jet bundle. The statement and proof follow Olver ^{[Olver §5.4]}.

Proof. The basic variational identity, obtained by integrating $pr v_{Q} (L)$ by parts (moving all total derivatives off $Q$ onto $\partial L / \partial u_{J}^{α}$ ), reads $$ \mathrm{pr},v_Q(L) = \sum_\alpha Q^\alpha,E_\alpha(L) + \mathrm{Div},A(Q), $$ where $A (Q)$ is a $p$ -tuple bilinear in $Q$ and the derivatives of $L$ , and $E_{α} (L) = \sum_{J} (- D)_{J} (\partial L / \partial u_{J}^{α})$ is the Euler-Lagrange expression. This holds for any characteristic $Q$ and is the defining property of $E_{α}$ .

By the variational-symmetry hypothesis applied to $Q [η]$ , the left side is itself a total divergence: $pr v_{Q [η]} (L) = Div \tilde{B} [η]$ for every $η$ . Subtracting, $$ \sum_\alpha Q^\alpha[\eta],E_\alpha(L) = \mathrm{Div}\bigl(\tilde B[\eta] - A(Q[\eta])\bigr) =: \mathrm{Div},W[\eta]. $$ The current $W [η]$ is linear in $η$ and its derivatives, because $Q [η]$ , $A$ , and $\tilde{B}$ are. Now insert the explicit form $Q^{α} [η] = \sum_{J} D_{J}^{α k} D_{J} η^{k}$ and integrate the left side by parts a second time, moving the derivatives $D_{J}$ off $η^{k}$ : $$ \sum_\alpha Q^\alpha[\eta],E_\alpha(L) = \sum_{k}\sum_J \bigl(D_J\eta^k\bigr) D^{\alpha k}J E\alpha(L) = \sum_k \eta^k \sum_J (-D)J!\bigl(D^{\alpha k}J E\alpha(L)\bigr) + \mathrm{Div},V[\eta], $$ where $V [η]$ collects the boundary terms generated by this integration by parts and is again linear in $η$ . Writing $E^{k} := \sum_{J} (- D)_{J} (D_{J}^{α k} E_{α} (L))$ (summation over $α$ implied), the two expressions for $\sum_\alpha Q^\alpha[\eta] E\alpha(L)$ give $$ \sum_k \eta^k,\mathcal{E}^k = \mathrm{Div}\bigl(W[\eta] - V[\eta]\bigr). $$ The left side is an undifferentiated linear function of the $η^{k}$ ; the right side is a total divergence linear in $η^{k}$ and its derivatives. Choose any $η^{k}$ of compact support in the interior of a region $Ω$ and integrate over $Ω$ : the right side integrates to a boundary term that vanishes, so $\int_{Ω} \sum_{k} η^{k} E^{k} d^{p} x = 0$ for every compactly supported $η$ . By the fundamental lemma of the calculus of variations, $E^{k} \equiv 0$ for each $k$ . Since no equation of motion was used at any step, the identity holds off-shell. $□$

Bridge. The single integration by parts that defines $E_{α} (L)$ is run twice here: once to expose $\sum_{α} Q^{α} E_{α}$ , and once more to strip the arbitrary functions $η^{k}$ off the characteristic. The first theorem stops after the first integration, reading $Div W = 0$ on-shell as a conservation law; the second theorem continues, using the arbitrariness of $η$ to force an off-shell identity. This builds toward the gauge examples below, where $E^{k} \equiv 0$ becomes $\partial_{μ} (\partial_{ν} F^{μν}) \equiv 0$ and the contracted Bianchi identity $\nabla_{μ} G^{μν} \equiv 0$ . It appears again in 05.00.04, whose conserved Noether current degenerates to a current that is conserved identically (a so-called improper or null current) precisely when the symmetry is a gauge symmetry, so that the first theorem returns no new physical charge. The construction reuses the prolongation and characteristic of 05.05.06: the gauge characteristic $Q [η]$ is a prolongation coefficient made linear in an arbitrary function, and the adjoint of that linear differential operator is what carries $E_{α}$ into the identity. The unifying content is that a symmetry's arbitrary-function dependence and a dependency among the field equations are exchanged by formal adjointness of one differential operator.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

The electromagnetic Lagrangian is $L = - \frac{1}{4} F_{μν} F^{μν}$ with $F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}$ . Show that the Euler-Lagrange expression with respect to $A_{ν}$ is $E^{ν} = \partial_{μ} F^{μν}$ (up to sign convention), and state the gauge characteristic $Q_{ν} [η]$ for $A_{ν} \mapsto A_{ν} + \partial_{ν} η$ .

Hint

Vary $L$ in $A_{ν}$ : $\partial L / \partial (\partial_{μ} A_{ν}) = - F^{μν}$ . The gauge transformation shifts $A_{ν}$ by $\partial_{ν} η$ , so $Q_{ν} = \partial_{ν} η$ .

Answer

Since $L = - \frac{1}{4} F_{ρ σ} F^{ρ σ}$ and $F_{ρ σ}$ is linear in $\partial A$ , $\partial L / \partial (\partial_{μ} A_{ν}) = - F^{μν}$ and $\partial L / \partial A_{ν} = 0$ . Hence $E^{ν} = \partial L / \partial A_{ν} - \partial_{μ} (\partial L / \partial (\partial_{μ} A_{ν})) = \partial_{μ} F^{μν}$ . The gauge transformation $A_{ν} \mapsto A_{ν} + \partial_{ν} η$ has infinitesimal characteristic $Q_{ν} [η] = \partial_{ν} η$ , linear in the single arbitrary function $η$ , so the rank is $r = 1$ .

Rubric: full credit for $E^{ν} = \partial_{μ} F^{μν}$ and $Q_{ν} = \partial_{ν} η$ ; partial credit for either.

Exercise 4 (medium, symbolic).

Using Exercise 3, apply Noether's second theorem to electromagnetism: with $Q_{ν} [η] = \partial_{ν} η$ and $E^{ν} = \partial_{μ} F^{μν}$ , form the identity $E = \sum_{ν} (- D) (coeff)$ and show it gives $\partial_{ν} \partial_{μ} F^{μν} \equiv 0$ .

Hint

Here $Q_{ν} [η] = \partial_{ν} η$ means the coefficient of $D_{ν} η = \partial_{ν} η$ is $δ^{\cdot}$ ; the second theorem's $E = - \partial_{ν} E^{ν}$ . Use antisymmetry of $F^{μν}$ .

Answer

The characteristic is $Q_{ν} = \partial_{ν} η = D_{ν} η$ , so the coefficient $D_{(ν)}^{ν} = δ$ multiplies $D_{ν} η$ ; the second-theorem identity is $E = \sum_{ν} (- D_{ν}) (E^{ν}) = - \partial_{ν} E^{ν} \equiv 0$ , i.e. $\partial_{ν} E^{ν} = 0$ . Substituting $E^{ν} = \partial_{μ} F^{μν}$ , $$ \partial_\nu\partial_\mu F^{\mu\nu} = 0, $$ which holds identically: $\partial_{ν} \partial_{μ}$ is symmetric in $(μ, ν)$ while $F^{μν}$ is antisymmetric, so the contraction vanishes for any $A$ . This is the divergence of the field equations vanishing off-shell.

Rubric: full credit for $\partial_{ν} \partial_{μ} F^{μν} \equiv 0$ with the symmetric-times-antisymmetric reason; partial credit for reaching $\partial_{ν} E^{ν} = 0$ .

Exercise 5 (medium, short-answer).

Explain why Noether's second theorem implies the Euler-Lagrange system of a gauge theory is underdetermined, and why this means the Cauchy (initial-value) problem is not well-posed without a gauge choice.

Hint

A differential identity among the equations means one equation is a consequence of the others. Count: fewer independent equations than unknowns.

Answer

Each identity $E^{k} \equiv 0$ expresses one differential combination of the $E_{α}$ as identically zero, so the field equations are not independent: given the others, one is automatically satisfied (to within the identity). The system therefore imposes fewer independent constraints than there are field components, leaving residual freedom — precisely the gauge freedom. Concretely, the gauge transformation $u \mapsto u + Q [η]$ carries solutions to solutions for arbitrary $η$ , so the initial data do not determine a unique evolution: any two solutions differing by a gauge transformation that vanishes at the initial time but is nonzero later share the same initial data. Well-posedness is restored by imposing a gauge condition (e.g. Lorenz gauge $\partial_{μ} A^{μ} = 0$ , harmonic coordinates in gravity) that removes the redundancy and makes the reduced system determined.

Rubric: full credit for the dependence-among-equations argument and the non-uniqueness of evolution; partial credit for either.

Exercise 6 (hard, symbolic).

For a Yang-Mills theory with gauge field $A_{μ} = A_{μ}^{a} T_{a}$ and field strength $F_{μν}^{a}$ , the field equation is $E_{a}^{ν} = D_{μ} F_{a}^{μν}$ (gauge-covariant divergence). The gauge characteristic is $Q_{ν}^{a} [η] = D_{ν} η^{a} = \partial_{ν} η^{a} + f^{a}_{b c} A_{ν}^{b} η^{c}$ . Show that Noether's second theorem gives the identity $D_{ν} D_{μ} F_{a}^{μν} \equiv 0$ and explain what makes it hold.

Hint

The second theorem's identity is the formal adjoint of $Q_{ν}^{a} [η] = D_{ν} η^{a}$ applied to $E_{a}^{ν}$ : $E_{a} = - D_{ν} E_{a}^{ν}$ . For the vanishing, use $D_{[μ} D_{ν]} = \frac{1}{2} [D_{μ}, D_{ν}] = \frac{1}{2} F_{μν}$ acting in the adjoint and antisymmetry of $F$ .

Answer

The characteristic $Q_{ν}^{a} [η] = D_{ν} η^{a}$ is linear in the arbitrary functions $η^{a}$ , with the covariant derivative $D_{ν}$ as the differential operator. The second theorem produces the identity by taking the formal adjoint: $E_{a} = - D_{ν} E_{a}^{ν} \equiv 0$ , i.e. $$ D_\nu D_\mu F^{\mu\nu}a = 0. $$ Splitting $D\nu D_\mu = D_{(\nu}D_{\mu)} + D_{[\nu}D_{\mu]} $, t h esy mm e t r i c p a r t co n t r a c t s t oz er o a g ain s tt h e an t i sy mm e t r i c$ F^{\mu\nu} $, an d t h e an t i sy mm e t r i c p a r t i s$ \tfrac12[D_\nu, D_\mu] = \tfrac12 F_{\nu\mu} $a c t in g in t h e a d j o in t r e p r ese n t a t i o n, g i v in g$ \tfrac12 f_a{}^{bc}F_{\nu\mu,b}F^{\mu\nu}{}c = 0 $b y an t i sy mm e t r y o f$ f $in$ b,c $p ai r e d w i t h t h esy mm e t r i c$ F{b}F_{c} $co n t r a c t i o n . H e n ce t h eco v a r ian t d i v er g e n ceo f t h e Y an g - M i l l s f i e l d e q u a t i o n s v ani s h es i d e n t i c a l l y, t h e n o n - A b e l ianana l o g u eo f$ \partial_\nu\partial_\mu F^{\mu\nu} = 0$.

Rubric: full credit for $D_{ν} D_{μ} F_{a}^{μν} \equiv 0$ with both the symmetric-contraction and commutator arguments; partial credit for the identity statement with one argument.

Exercise 7 (hard, short-answer).

Contrast Noether's first and second theorems on the status of the resulting current. For a finite-parameter symmetry the first theorem gives a current conserved on-shell; for a gauge symmetry the associated current is conserved off-shell (identically). Explain why, and what this says about the physical charge.

Hint

Trace the proof: the first theorem reads $Div W = \sum_{α} Q^{α} E_{α}$ , which vanishes on-shell. For a gauge symmetry, the second integration by parts makes the bracket vanish identically.

Answer

In both cases the variational identity gives $Div W [\cdot] = \sum_{α} Q^{α} E_{α} (L)$ . For a finite-parameter symmetry $Q$ is fixed, so the right side vanishes only when $E_{α} = 0$ , i.e. on-shell; the current $W$ is conserved on solutions and its integral is a genuine conserved charge. For a gauge symmetry $Q = Q [η]$ is linear in an arbitrary function, and the second theorem shows $\sum_{α} Q^{α} [η] E_{α} = Div (\dots)$ holds because of the off-shell identity $E^{k} \equiv 0$ ; the corresponding Noether current is a total divergence of an antisymmetric object (a superpotential), so it is conserved identically and its charge reduces to a boundary integral. The "charge" of a gauge symmetry is therefore not a bulk dynamical quantity but a surface term — the deep reason gauge symmetries carry no independent local conservation law and instead encode redundancy.

Rubric: full credit for the on-shell vs. off-shell distinction, the superpotential/boundary-term structure, and the conclusion about the charge; partial credit for the on-shell/off-shell contrast alone.

Lean formalization Intermediate+

Mathlib has the manifold and multilinear-derivative libraries but no variational bicomplex or Noether apparatus; the block fixes intended signatures, with the gap detailed in Mathlib gap analysis.

import Mathlib.Geometry.Manifold.ContMDiff.Basic
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs

variable {p q : ℕ}

/-- The Euler-Lagrange expression E_α(L) = ∑_J (−D)_J (∂L/∂u^α_J),
    stated against the (absent) jet/total-derivative apparatus. -/
noncomputable def eulerLagrange
    (L : (Fin p → ℝ) → (ℕ → Fin q → ℝ) → ℝ) : Fin q → ℝ :=
  fun _ => sorry  -- ∑_J (−1)^{|J|} D_J (∂L/∂u^α_J)

/-- A gauge characteristic Q^α[η] = ∑_J D^{α k}_J · D_J η^k,
    linear in r arbitrary functions η^k of the base. -/
structure GaugeGenerator (p q r : ℕ) where
  coeff : (Fin p → ℝ) → (Fin q → ℝ) → Fin r → ℝ  -- the D^{α k}_J, schematically

/-- Noether's second theorem: a rank-r gauge symmetry forces
    r off-shell identities ∑_J (−D)_J (D^{α k}_J E_α(L)) ≡ 0. -/
theorem noether_second
    (L : (Fin p → ℝ) → (ℕ → Fin q → ℝ) → ℝ)
    (g : GaugeGenerator p q r) : True := by
  trivial  -- placeholder: ∑_J (−D)_J (D^{α k}_J E_α(L)) = 0 for each k

The sorry and placeholder rest on the variational-bicomplex gap: the Euler-Lagrange operator, the integration-by-parts identity $pr v_{Q} (L) = Q^{α} E_{α} + Div A$ , and the formal adjoint that carries $Q [η]$ into the identity. Each sits on the jet apparatus of 05.05.05 and is a Mathlib-contribution-sized target named in Mathlib gap analysis.

Advanced results Master

The second theorem converts a gauge freedom into an off-shell dependence among the field equations, and the two flagship instances are electromagnetism and general relativity. Two worked derivations exhibit the mechanism, and the structural results frame it through formal adjointness and the boundary-term subtlety.

Electromagnetism. For $L = - \frac{1}{4} F_{μν} F^{μν}$ the Euler-Lagrange expression is $E^{ν} = \partial_{μ} F^{μν}$ , and the gauge characteristic of $A_{ν} \mapsto A_{ν} + \partial_{ν} η$ is $Q_{ν} [η] = \partial_{ν} η = D_{ν} η$ , rank $r = 1$ . The differential operator carrying $η$ into $Q$ is $D_{ν}$ ; its formal adjoint is $- D_{ν}$ , so the second theorem reads $$ \mathcal{E} = -D_\nu E^\nu = -\partial_\nu\partial_\mu F^{\mu\nu} \equiv 0, $$ the divergence of Maxwell's equations vanishing identically because $\partial_{ν} \partial_{μ}$ is symmetric and $F^{μν}$ antisymmetric. The matched conservation law of the first theorem is the conservation of the source current $\partial_{ν} J^{ν} = 0$ , which holds as a consequence of the identity once $E^{ν} = J^{ν}$ is imposed: charge conservation is forced by gauge invariance, not assumed ^{[Bessel-Hagen 1921]}.

General relativity. Take the Einstein-Hilbert Lagrangian $L = - g R$ as a functional of the metric $g_{μν}$ . Its Euler-Lagrange expression is the Einstein tensor density: $E^{μν} = - g (R^{μν} - \frac{1}{2} g^{μν} R) = - g G^{μν}$ ^{[Wald App. E]}. The gauge group is the diffeomorphism group; an infinitesimal diffeomorphism along $η^{ρ} (x)$ acts on the metric by the Lie derivative $δ g_{μν} = L_{η} g_{μν} = \nabla_{μ} η_{ν} + \nabla_{ν} η_{μ}$ , so the characteristic is $Q_{μν} [η] = \nabla_{μ} η_{ν} + \nabla_{ν} η_{μ}$ , rank $r = p$ (one arbitrary function per coordinate). The differential operator $η_{ν} \mapsto \nabla_{μ} η_{ν} + \nabla_{ν} η_{μ}$ has formal adjoint $- 2 \nabla_{μ}$ , and the second theorem gives $$ \mathcal{E}^\nu = -2,\nabla_\mu\bigl(G^{\mu\nu}\bigr) \equiv 0, $$ the contracted Bianchi identity $\nabla_{μ} G^{μν} = 0$ . It holds for every metric, off-shell, and is the differential identity forced by the $p$ arbitrary functions of diffeomorphism invariance. On-shell it forces $\nabla_{μ} T^{μν} = 0$ : local energy-momentum conservation in general relativity is a consequence of the diffeomorphism gauge symmetry, not an independent postulate.

Yang-Mills. For $L = - \frac{1}{4} F_{μν}^{a} F_{a}^{μν}$ with non-Abelian gauge group, $E_{a}^{ν} = D_{μ} F_{a}^{μν}$ and the gauge characteristic is $Q_{ν}^{a} [η] = D_{ν} η^{a}$ , rank $r = dim g$ . The adjoint of the covariant operator $D_{ν}$ gives $E_{a} = - D_{ν} E_{a}^{ν} = - D_{ν} D_{μ} F_{a}^{μν} \equiv 0$ , which vanishes because the symmetric part of $D_{ν} D_{μ}$ kills the antisymmetric $F^{μν}$ and the antisymmetric part is the curvature $\frac{1}{2} F_{ν μ}$ acting in the adjoint, self-contracting to zero. The Abelian, gravitational, and non-Abelian identities are three instances of one formal-adjoint computation.

The boundary-term subtlety. The Noether current of a gauge symmetry is a total divergence of an antisymmetric superpotential: $W^{μ} [η] = D_{ν} (U^{[μν]} [η])$ , so $D_{μ} W^{μ} \equiv 0$ identically and the charge $\int_{Σ} W^{μ} d S_{μ} = \oint_{\partial Σ} U^{[μν]} d S_{μν}$ reduces to a surface integral. In gravity this is the route to the ADM and Komar mass: the "charge" of an asymptotic symmetry lives at infinity. The bulk current carries no local information, which is the field-theoretic content of the statement that a gauge symmetry has no independent conserved charge.

Synthesis. Noether's second theorem is the conversion of a gauge freedom into an off-shell dependence among field equations, and a single formal adjoint is the conversion factor. The argument runs along four linked stages, each carried by the integration-by-parts machinery of the Euler-Lagrange operator. First, the variational identity $pr v_{Q} (L) = \sum_{α} Q^{α} E_{α} (L) + Div A$ exposes the Euler-Lagrange expressions, holding for any characteristic. Second, the gauge hypothesis makes the left side a divergence for every arbitrary $η$ , so $\sum_{α} Q^{α} [η] E_{α}$ is a total divergence. Third, a second integration by parts strips the differential operator off $η$ and lands it, as its formal adjoint, on the $E_{α}$ , and the arbitrariness of $η$ forces the resulting combination $E^{k} = \sum_{J} (- D)_{J} (D_{J}^{α k} E_{α})$ to vanish identically. Fourth, the identity makes the system underdetermined and the Noether current a superpotential divergence, so the gauge charge is a boundary term. Electromagnetism gives $\partial_{ν} \partial_{μ} F^{μν} \equiv 0$ , gravity gives $\nabla_{μ} G^{μν} \equiv 0$ , and Yang-Mills gives $D_{ν} D_{μ} F_{a}^{μν} \equiv 0$ , the three faces of the one adjoint identity, and the same integration by parts that defines $E_{α}$ is what runs twice to produce both the first theorem's conservation law and the second theorem's dependency.

Full proof set Master

Proposition (the contracted Bianchi identity from diffeomorphism invariance). Let $L = - g R$ be the Einstein-Hilbert Lagrangian on a manifold $M$ , regarded as a functional of the metric $g_{μν}$ , with Euler-Lagrange expression $E^{μν} = - g G^{μν}$ , $G^{μν} = R^{μν} - \frac{1}{2} g^{μν} R$ . Then $\nabla_{μ} G^{μν} = 0$ identically, as a consequence of the invariance of $\int_{M} L$ under diffeomorphisms.

Proof. Under an infinitesimal diffeomorphism generated by a compactly supported vector field $η^{ρ}$ , the action $A [g] = \int_{M} - g R d^{p} x$ is invariant because it is the integral of a scalar density and $η$ has compact support: $δ A = 0$ . The first variation of any metric functional is $$ \delta\mathcal A = \int_M E^{\mu\nu},\delta g_{\mu\nu},d^p x, $$ where $E^{μν} = - g G^{μν}$ is the Einstein tensor density; this is the standard variational computation, in which the variation of the Ricci scalar contributes only a boundary term that vanishes for compactly supported $δ g$ ^{[Wald App. E]}. For the diffeomorphism variation $δ g_{μν} = L_{η} g_{μν} = \nabla_{μ} η_{ν} + \nabla_{ν} η_{μ}$ , so $$ 0 = \delta\mathcal A = \int_M \sqrt{-g},G^{\mu\nu}\bigl(\nabla_\mu\eta_\nu + \nabla_\nu\eta_\mu\bigr),d^p x = 2\int_M \sqrt{-g},G^{\mu\nu}\nabla_\mu\eta_\nu,d^p x, $$ using the symmetry $G^{μν} = G^{ν μ}$ . Integrating by parts with the metric-compatible connection (so that $\nabla_{μ} (- g V^{μ}) = - g \nabla_{μ} V^{μ}$ integrates to a vanishing boundary term for compactly supported $V$ ), $$ 0 = -2\int_M \sqrt{-g},\bigl(\nabla_\mu G^{\mu\nu}\bigr)\eta_\nu,d^p x. $$ Since $η_{ν}$ is an arbitrary compactly supported covector field, the fundamental lemma of the calculus of variations forces $\nabla_{μ} G^{μν} = 0$ at every point. No field equation was used: $δ A = 0$ holds for the metric off-shell because diffeomorphism invariance is an identity of the action, not of its critical points. Hence the contracted Bianchi identity holds for every metric. $□$

Proposition (off-shell conservation of the gauge Noether current). Let $v_{Q [η]}$ be a rank- $r$ gauge variational symmetry of $L$ with associated Noether current $W^{μ} [η]$ satisfying $D_{μ} W^{μ} [η] = \sum_{α} Q^{α} [η] E_{α} (L)$ . Then there exist functions $U^{[μν]} [η] = - U^{[ν μ]} [η]$ (a superpotential) such that $W^{μ} [η] = D_{ν} U^{[μν]} [η] + (terms proportional to E_{α})$ , so that on-shell $W^{μ} = D_{ν} U^{[μν]}$ and $D_{μ} W^{μ} \equiv 0$ identically.

Proof. By the second theorem, $\sum_{α} Q^{α} [η] E_{α} = Div V [η]$ where $V [η]$ is the boundary current from stripping $η$ off $Q [η]$ ; combined with $D_{μ} W^{μ} = \sum_{α} Q^{α} [η] E_{α}$ this gives $D_{μ} (W^{μ} - V^{μ}) = 0$ identically in $η$ . Write $S^{μ} := W^{μ} - V^{μ}$ , a current that is identically conserved and linear in the arbitrary $η$ and its derivatives. An identically conserved current that is linear and local in an arbitrary function is, by the converse to the Poincaré lemma for the variational bicomplex (the local exactness of identically conserved currents), the total divergence of an antisymmetric tensor: $S^{μ} = D_{ν} U^{[μν]} [η]$ for some $U^{[μν]} = - U^{[ν μ]}$ . Then $W^{μ} = D_{ν} U^{[μν]} + V^{μ}$ , and since $V^{μ}$ is built from the variational identity it is proportional to the $E_{α}$ and their derivatives, vanishing on-shell. Thus on-shell $W^{μ} = D_{ν} U^{[μν]}$ , and $D_{μ} W^{μ} = D_{μ} D_{ν} U^{[μν]} \equiv 0$ because $D_{μ} D_{ν}$ is symmetric while $U^{[μν]}$ is antisymmetric. The conserved charge $\int_{Σ} W^{μ} d S_{μ} = \oint_{\partial Σ} U^{[μν]} d S_{μν}$ reduces to a surface integral by Stokes' theorem. $□$

Connections Master

Noether's first theorem 05.00.04 is the companion this unit completes: the same variational identity $Div W = \sum_{α} Q^{α} E_{α} (L)$ underlies both, but where a finite-parameter symmetry stops at an on-shell conservation law, a gauge symmetry runs the integration by parts once more to force an off-shell identity, so the two theorems are the two readings of one identity according to whether the symmetry parameter is a constant or an arbitrary function.

The gauge characteristic $Q^{α} [η]$ of this unit is the prolongation characteristic of 05.05.06 made linear in arbitrary functions of the base: the same $Q^{α} = ϕ^{α} - ξ^{i} u_{i}^{α}$ that fed the determining equations there, now with $ϕ^{α}$ depending on $η^{k}$ and its derivatives, and the formal adjoint of the operator $η \mapsto Q [η]$ is what carries the Euler-Lagrange expressions into the Noether identity.

The Euler-Lagrange operator and the integration-by-parts identity are built on the total derivative of 05.05.05: $E_{α} (L) = \sum_{J} (- D)_{J} (\partial L / \partial u_{J}^{α})$ is assembled from the iterated total derivative, the variational identity is its defining adjointness property, and the differential identities $E^{k} \equiv 0$ are statements in the same jet-bundle algebra, so the entire second theorem lives in the variational subcomplex of the jet bundle introduced there.

Historical & philosophical context Master

Emmy Noether proved both of her theorems in the 1918 paper Invariante Variationsprobleme, published in the Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen ^{[Noether 1918]}. The work was prompted by the conservation-of-energy puzzle in Einstein's new general relativity: David Hilbert and Felix Klein had noticed that energy conservation in the theory behaved unlike its counterpart in other field theories, and they enlisted Noether, then in Göttingen, to clarify the situation. Her answer separated two cases. The first theorem (Satz I) treats finite-parameter symmetry groups and assigns to each generator a conservation law. The second theorem (Satz II) treats groups depending on arbitrary functions and shows that they force dependencies among the Lagrange expressions — the field equations are not independent. Klein's own 1918 note on the structure of general relativity drew directly on this analysis, and the "improper conservation laws" of gravitation that had puzzled the physicists were exactly the identically-conserved superpotential currents the second theorem predicts.

The second theorem was long the less-cited of the two; physicists absorbed the first theorem as "Noether's theorem" and the second was rediscovered piecemeal in the gauge-theory literature. Erich Bessel-Hagen's 1921 paper Über die Erhaltungssätze der Elektrodynamik ^{[Bessel-Hagen 1921]}, written on Klein's suggestion, extended the framework to divergence symmetries and worked out the conformal conservation laws of electrodynamics, fixing a gap in Noether's handling of symmetries that change the Lagrangian by a divergence. Peter Olver's Applications of Lie Groups to Differential Equations (1986; second edition 1993) ^{[Olver §5.4]} gave the modern jet-bundle formulation of both theorems in terms of the characteristic and the Euler-Lagrange operator, which is the form used in this unit, and identified the second theorem as the statement that the gauge generators and the differential dependencies are formal adjoints. Yvette Kosmann-Schwarzbach's The Noether Theorems (2011) ^{[Kosmann-Schwarzbach Ch. 3]} provides a critical translation of the 1918 paper and traces the long delayed recognition of the second theorem, which in the constrained-Hamiltonian language of Dirac and Bergmann underlies the entire treatment of gauge systems: the differential identities of Noether II are the source of the first-class constraints that generate gauge transformations in the canonical formalism.

Bibliography Master

@article{Noether1918,
  author  = {Noether, Emmy},
  title   = {Invariante Variationsprobleme},
  journal = {Nachrichten von der K{\"o}niglichen Gesellschaft der Wissenschaften zu G{\"o}ttingen, Mathematisch-Physikalische Klasse},
  year    = {1918},
  pages   = {235--257}
}

@article{Klein1918,
  author  = {Klein, Felix},
  title   = {{\"U}ber die Differentialgesetze f{\"u}r die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie},
  journal = {Nachrichten von der K{\"o}niglichen Gesellschaft der Wissenschaften zu G{\"o}ttingen, Mathematisch-Physikalische Klasse},
  year    = {1918},
  pages   = {171--189}
}

@article{BesselHagen1921,
  author  = {Bessel-Hagen, Erich},
  title   = {{\"U}ber die Erhaltungss{\"a}tze der Elektrodynamik},
  journal = {Mathematische Annalen},
  volume  = {84},
  year    = {1921},
  pages   = {258--276}
}

@book{OlverLieGroups,
  author    = {Olver, Peter J.},
  title     = {Applications of Lie Groups to Differential Equations},
  series    = {Graduate Texts in Mathematics},
  volume    = {107},
  publisher = {Springer},
  year      = {1993},
  edition   = {2nd}
}

@book{KosmannSchwarzbach2011,
  author    = {Kosmann-Schwarzbach, Yvette},
  title     = {The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century},
  series    = {Sources and Studies in the History of Mathematics and Physical Sciences},
  publisher = {Springer},
  year      = {2011},
  note      = {translated by Bertram E. Schwarzbach}
}

@book{Wald1984,
  author    = {Wald, Robert M.},
  title     = {General Relativity},
  publisher = {University of Chicago Press},
  year      = {1984}
}

@article{DiracGeneralized1950,
  author  = {Dirac, Paul A. M.},
  title   = {Generalized Hamiltonian Dynamics},
  journal = {Canadian Journal of Mathematics},
  volume  = {2},
  year    = {1950},
  pages   = {129--148}
}

Prerequisites

05.05.06
05.05.05
05.00.04

Tier anchors

beginner: Olver *Applications of Lie Groups to Differential Equations* §5.3 (a gauge freedom forces the field equations to depend on one another)
intermediate: Olver §5.3-§5.4 (variational symmetry groups, Noether's two theorems, the second theorem and differential identities); Kosmann-Schwarzbach *The Noether Theorems* (2011) Ch. 2-3
master: Olver §5.3-§5.4; Noether 1918 *Invariante Variationsprobleme* (Gött. Nachr., originator); Kosmann-Schwarzbach *The Noether Theorems* (2011); Wald *General Relativity* (1984) Appendix E

References

Olver — Applications of Lie Groups to Differential Equations (2nd ed., 1993) · §5.3 variational symmetries and Noether's first theorem; §5.4 the second theorem and differential identities among the Euler-Lagrange equations
Noether — Invariante Variationsprobleme (1918) · Nachr. König. Gesell. Wiss. Göttingen, math-phys. Kl. 235-257; Satz II (the second theorem) and the dependencies among the Lagrange expressions
Kosmann-Schwarzbach — The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (2011) · Ch. 2-3 translation and commentary on Noether 1918; the second theorem and its reception
Bessel-Hagen — Über die Erhaltungssätze der Elektrodynamik (1921) · Math. Ann. 84, 258-276; divergence (Bessel-Hagen) symmetries and the conformal conservation laws of electrodynamics
Wald — General Relativity (1984) · Appendix E variational derivation of the Einstein equation; §3.2 the contracted Bianchi identity from diffeomorphism invariance

Estimated time

beginner: 17m
intermediate: 48m
master: 86m