nLab symmetry of a Lagrangian density

Definition

Throughout, let

1. $\Sigma$ be a smooth manifold of dimension $p+1$, thought of as spacetime;

2. $E \overset{fb}{\longrightarrow} \Sigma$ a fiber bundle thought of as a field bundle

3. $\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)$ a Lagrangian density.

Definition

(evolutionary vector fields)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle. Then an evolutionary vector field $v$ on $E$ is “variational vertical vector field” on $E$, hence a smooth bundle homomorphism out of the jet bundle (def. )

$$\array{ J^\infty_\Sigma E && \overset{v}{\longrightarrow} && T_\Sigma E \\ & {}_{\mathllap{jb_{\infty,0}}}\searrow && \swarrow_{\mathllap{}} \\ && E }$$

to the vertical tangent bundle $T_\Sigma E \overset{}{\to} \Sigma$ of $E \overset{fb}{\to} \Sigma$.

In the special case that the field bundle is a trivial vector bundle over Minkowski spacetime as in example , this means that an evolutionary vector field is a tangent vector field (example ) on $J^\infty_\Sigma(E)$ of the special form

$$\begin{aligned} v & = v^a \partial_{\phi^a} \\ & = v^a\left( (x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots \right) \partial_{\phi^a} \end{aligned} \,,$$

where the coefficients $v^a \in C^\infty(J^\infty_\Sigma(E))$ are general smooth functions on the jet bundle (while the cmponents are tangent vectors along the field coordinates $(\phi^a)$, but not along the spacetime coordinates $(x^\mu)$ and not along the jet coordinates $\phi^a_{,\mu_1 \cdots \mu_k}$).

We write

$$\Gamma_E^{ev}\left( T_\Sigma E \right) \;\in\; \Omega^{0,0}_\Sigma(E) Mod$$

for the space of evolutionary vector fields, regarded as a module over the $\mathbb{R}$-algebra

$$\Omega^{0,0}_\Sigma(E) \;=\; C^\infty\left( J^\infty_\Sigma(E) \right)$$

of smooth functions on the jet bundle.

An evolutionary vector field (def. ) describes an infinitesimal change of field values depending on, possibly, the point in spacetime and the values of the field and all its derivatives (locally to finite order, by prop. ).

This induces a corresponding infinitesimal change of the derivatives of the fields, called the prolongation of the evolutionary vector field:

Proposition

(prolongation of evolutionary vector field)

Let $E \overset{fb}{\to} \Sigma$ be a fiber bundle.

Given an evolutionary vector field $v$ on $E$ (def. ) there is a unique tangent vector field $\hat v$ on the jet bundle $J^\infty_\Sigma(E)$ such that

1. $\hat v$ agrees on field coordinates (as opposed to jet coordinates) with $v$:

   $$(jb_{\infty,0})_\ast(\hat v) = v \,,$$

   which means in the special case that $E \overset{fb}{\to} \Sigma$ is a trivial vector bundle over Minkowski spacetime that $\hat v$ is of the form

   (1)$$\hat v \;=\; \underset{ = v }{ \underbrace{ v^a \partial_{\phi^a} }} \,+\, \hat v^a_{\mu} \partial_{\phi^a_{,\mu}} + \hat v^a_{\mu_1 \mu_2} \partial_{\phi^a_{,\mu_1 \mu_2}} + \cdots$$

2. contraction with $\hat v$ (def. ) anti-commutes with the total spacetime derivative (def. ):

   (2)$$\iota_{\hat v} \circ d + d \circ \iota_{\hat v} = 0 \,.$$

In particular Cartan's homotopy formula (prop. ) for the Lie derivative $\mathcal{L}_{\hat v}$ holds with respect to the variational derivative $\delta$:

(3)$$\mathcal{L}_{\hat v} = \delta \circ \iota_{\hat v} + \iota_{\hat v} \circ \delta$$

Explicitly, in the special case that the field bundle is a trivial vector bundle over Minkowski spacetime (example ) $\hat v$ is given by

(4)$$\hat v = \underoverset{n = 0}{\infty}{\sum} \frac{d^n v^a}{ d x^{\mu_1} \cdots d x^{\mu_n} } \partial_{\phi^a_{\mu_1 \cdots \mu_n}} \,.$$

Proof

It is sufficient to prove the coordinate version of the statement. We prove this by induction over the maximal jet order $k$. Notice that the coefficient of $\partial_{\phi^a_{\mu_1 \cdots \mu_k}}$ in $\hat v$ is given by the contraction $\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}$ (def. ).

Similarly (at “$k = -1$”) the component of $\partial_{\mu_1}$ is given by $\iota_{\hat v} d x^{\mu}$. But by the second condition above this vanishes:

$$\begin{aligned} \iota_{\hat v} d x^\mu & = d \iota_{\hat v} x^\mu \\ & = 0 \end{aligned}$$

Moreover, the coefficient of $\partial_{\phi^a}$ in $\hat v$ is fixed by the first condition above to be

$$\iota_{\hat v} \delta \phi^a = v^a \,.$$

This shows the statement for $k = 0$. Now assume that the statement is true up to some $k \in \mathbb{N}$. Observe that the coefficients of all $\partial_{\phi^a_{\mu_1 \cdots \mu_{k+1}}}$ are fixed by the contractions with $\delta \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \wedge d x^{\mu_{k+1}}$. For this we find again from the second condition and using $\delta \circ d + d \circ \delta = 0$ as well as the induction assumption that

$$\begin{aligned} \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_{k+1}} \wedge d x^{\mu_{k+1}} & = \iota_{\hat v} \delta d \phi^a_{\mu_1 \cdots \mu_k} \\ & = d \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k} \\ & = d \frac{d^k v^a}{d x^{\mu_1} \cdots d x^{\mu_k}} \\ & = \frac{d^{k+1}v^a }{d x^{\mu_1} \cdots d x^{\mu_{k+1}}} d x^{\mu_{k+1}} \,. \end{aligned}$$

This shows that $\hat v$ satisfying the two conditions given exists uniquely.

Finally formula (3) for the Lie derivative follows from the second of the two conditions with Cartan's homotopy formula $\mathcal{L}_{\hat v} = \mathbf{d} \circ \iota_{\hat v} + \iota_{\hat v} \circ \mathbf{d}$ (prop. ) together with $\mathbf{d} = \delta + d$.

Proposition

(evolutionary vector fields form a Lie algebra)

Let $E \overset{fb}{\to} \Sigma$ be a fiber bundle. For any two evolutionary vector fields $v_1$, $v_2$ on $E$ (def. ) the Lie bracket of tangent vector fields of their prolongations $\hat v_1$, $\hat v_2$ (def. ) is itself the prolongation $\widehat{[v_1, v_2]}$ of a unique evolutionary vector field $[v_1,v_2]$.

This defines the structure of a Lie algebra on evolutionary vector fields.

Proof

It is clear that $[\hat v_1, \hat v_2]$ is still vertical, therefore, by prop. , it is sufficient to show that contraction $\iota_{[v_1, v_2]}$ with this vector field (def. ) anti-commutes with the horizontal derivative $d$, hence that $[d, \iota_{[\hat v_1, \hat v_2]}] = 0$.

Now $[d, \iota_{[\hat v_1, \hat v_2]}]$ is an operator that sends vertical 1-forms to horizontal 1-forms and vanishes on horizontal 1-forms. Therefore it is sufficient to see that this operator in fact also vanishes on all vertical 1-forms. But for this it is sufficient that it commutes with the vertical derivative. This we check by Cartan calculus, using $[d,\delta] = 0$ and $[d, \iota_{\hat v_i}]$, by assumption:

$$\begin{aligned} {[ \delta, [ d,\iota_{[\hat v_1, \hat v_2]}] ]} & = - [d, [\delta, \iota_{[\hat v_1, \hat v_2]}]] \\ & = - [d, \mathcal{L}_{[\hat v_1, \hat v_2]}] \\ & = -[d, [\mathcal{L}_{\hat v_1}, \iota_{\hat v_2}] ] \\ & = - [d, [ [\delta, \iota_{\hat v_1}], \iota_{\hat v_2} ]] \\ & = 0 \,. \end{aligned}$$

Now given an evolutionary vector field, we want to consider the flow that it induces on the space of field histories:

Definition

(flow of field histories along evolutionary vector field)

Let $E \overset{fb}{\to} \Sigma$ be a field bundle and let $v$ be an evolutionary vector field (def. ) such that the ordinary flow of its prolongation $\hat v$ (prop. )

$$\exp(t \hat v) \;\colon\; J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)$$

exists on the jet bundle (e.g. if the order of derivatives of field coordinates that it depends on is bounded).

For $\Phi_{(-)} \colon U_1 \to \Gamma_\Sigma(E)$ a collection of field histories (hence a plot of the space of field histories) the flow of $v$ through $\Phi_{(-)}$ is the smooth function

$$U_1 \times \mathbb{R}^1 \overset{\exp(v)(\Phi_{(-)})}{\longrightarrow} \Gamma_\Sigma(E)$$

whose unique factorization $\widehat{\exp(v)}(\Phi_{(-)})$ through the space of jets of field histories (i.e. the image $im(j^\infty_\Sigma)$ of jet prolongation)

$$\array{ && im(j^\infty_\Sigma) &\hookrightarrow& \Gamma_\Sigma(J^\infty_\Sigma(E)) \\ & {}^{\mathllap{\widehat{\exp(v)}(\Phi_{(-)})}} \nearrow & \Big\downarrow^{\mathrlap{\simeq}} \\ U_1 \times \mathbb{R}^1 &\underset{ \exp(v)(\Phi) }{\longrightarrow}& \Gamma_{\Sigma}(E)_{} }$$

takes a plot $t_{(-)} \;\colon\; U_2 \to \mathbb{R}^1$ of the real line (regarded as a super smooth set), to the plot

(5)$$(\exp(t(-) \hat v) \circ j^\infty_\Sigma(\Phi_{(-)}) \;\colon\: U_1 \times U_2 \longrightarrow \Gamma_\Sigma\left( J^\infty_\Sigma(E) \right)$$

of the smooth space of sections of the jet bundle.

(That $\exp(t(-) \hat v)$ indeed flows jet prolongations $j^\infty_\Sigma(\Phi(-))$ again to jet prolongations is due to its defining relation to the evolutionary vector field $v$ from prop. .)

Definition

(infinitesimal symmetries of the Lagrangian and conserved currents)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ).

Then

1. an infinitesimal symmetry of the Lagrangian is a variation $v$ (def. ) which arises as the prolongation $\hat v$ (prop. ) of an evolutionary vector field $v$ (def. ) such that the Lie derivative $\mathcal{L}_v$ of the Lagrangian density along $\hat v$ is a total spacetime derivative

   $$\mathcal{L}_v \mathbf{L} = d \tilde J_v$$

2. an on-shell conserved current is a horizontal $p$-form $J \in \Omega^{p,0}_\Sigma(E)$ whose total spacetime derivative vanishes on the prolonged shell (?)

   $$d J\vert_{\mathcal{E}^\infty} = 0 \,.$$

Proposition

(Noether's theorem I)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ).

If $v$ is an infinitesimal symmetry of the Lagrangian (def. ) with $\mathcal{L}_v \mathbf{L} = d \tilde J_v$, then

$$J_v \coloneqq \tilde J_v - \iota_v \Theta_{BFV}$$

is an on-shell conserved current (def. ), for $\Theta_{BFV}$ a presymplectic potential (?) from def. .

Proof

By Cartan's homotopy formula for the Lie derivative (prop. ) and the decomposition of the variational derivative $\delta \mathbf{L}$ (?) and the fact that contraction $\iota_{\hat v}$ with the prolongtion of an evolutionary vector field vanishes on horizontal differential forms (1) and anti-commutes with the horizontal differential (2), by def. , we may re-express the defining equation for the symmetry as follows:

$$\begin{aligned} d \tilde J_v & = \mathcal{L}_v \mathbf{L} \\ & = \iota_v \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}} + \mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}} \\ & = \iota_v \delta_{EL} \mathbf{L} + d \iota_v \Theta_{BFV} \end{aligned}$$

which is equivalent to

$$d(\underset{= J_v}{\underbrace{\tilde J_v - \iota_v \Theta_{BFV}}}) = \iota_v \delta_{EL}\mathbf{L}$$

Since, by definition of the shell $\mathcal{E}$, the form $\frac{\delta_{EL} \mathbf{L}}{\delta v}$ vanishes on $\mathcal{E}$ this yields the claim.

Proposition

(flow along infinitesimal symmetry of the Lagrangian preserves on-shell space of field histories)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ).

For $v$ an infinitesimal symmetry of the Lagrangian (def. ) the flow on the space of field histories (example ) that it induces by def. preserves the space of on-shell field histories (from prop. ):

$$\array{ \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} &\hookrightarrow& \Gamma_\Sigma(E) \\ {\mathllap{\exp(\hat v)\vert_{\delta_{EL}\mathbf{L} = 0} }} \uparrow && \uparrow {\mathrlap{\exp(\hat v)}} \\ \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} &\hookrightarrow& \Gamma_\Sigma(E) }$$

Proof

By def. a field history $\Phi \in \Gamma_\Sigma(E)$ is on-shell precisely if its jet prolongation $j^\infty_\Sigma(E)$ (def. ) factors through the shell $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ (?). Hence by def. the statement is equivalently that the ordinary flow (prop. ) of $\hat v$ (def. ) on the jet bundle $J^\infty_\Sigma(E)$ preserves the shell. This in turn means that it preserves the vanishing locus of the Euler-Lagrange form $\delta_{EL} \mathbf{L}$.

For this it is sufficient to show that the derivative of the components of the Euler-Lagrange form along $\hat v$ vanish on the prolonged shell

$$\hat v \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) \;=\; 0 \phantom{AA} on \mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E) \,.$$

This is the statement of Olver 95, theorem 5.53.

References

• Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.