nLab conserved current

Via the variational bicomplex

The following discusses the formulation of conserved currents in terms of variational calculus and the variational bicomplex.

The context

Let $X$ be a spacetime of dimension $n$, $E \to X$ a bundle, $j_\infty E \to X$ its jet bundle and

$$\Omega^{\bullet,\bullet}(j_\infty E), (D = \delta + d)$$

the corresponding variational bicomplex with $\delta$ being the vertical and $d = d_{dR}$ the horizontal differential.

Proposition

For $L \in \Omega^{n,0}(j_\infty E)$ a Lagrangian we have that

$$\delta L = E(L) + d \Theta$$

for $E$ the Euler-Lagrange operator.

The covariant phase space of the Lagrangian is the locus

$$\{\phi \in \Gamma(E) | E(L)(j_\infty \phi) = 0\} \,.$$

For $\Sigma \subset X$ any $(n-1)$-dimensional submanifold,

$$\delta \theta := \delta \int_\Sigma \Theta$$

is the presymplectic structure on covariant phase space

Definition

Definition

A conserved current is an element

$$j \in \Omega^{n-1, 0}(j_\infty E)$$

which is horizontally closed on covariant phase space

$$d j|_{E(L) = 0} = 0 \,.$$

Definition

For $\Sigma \hookrightarrow X$ a submanifold of dimension $n-1$, the charge of the conserved current $j$ with respect to $\Sigma$ is the integral

$$Q_\Sigma := \int_\Sigma j \,.$$

Properties

Proposition

If $\Sigma, \Sigma' \subset X$ are homologous, the associated charge is the same

$$Q_{\Sigma} = Q_{\Sigma'} \,.$$

Proof

By Stokes' theorem.

Theorem

Every symmetry of the Lagrangian induces a conserved current.

This is Noether's theorem. See there for more details.

In higher prequantum geometry

The following discusses conserved currents in the context of higher prequantum geometry, closely related to Azcarraga-Izquierdo 95, section 8.1. This follows (classicalinhigher, section 3.3., going back to Schreiber 13). Similar observations have been made by Igor Khavkine.

this section needs much polishing. For the moment better see classicalinhigher, section 3.3

Context

Let $\mathbf{H}$ be the ambient (∞,1)-topos. For $\mathbf{Fields} \in \mathbf{H}$ a moduli ∞-stack of fields a local Lagrangian for an $n$-dimensional prequantum field theory is equivalently a prequantum n-bundle given by a map

$$\mathbf{L} \;\colon\; \mathbf{Fields} \longrightarrow \mathbf{B}^n U(1)_{conn}$$

to the moduli ∞-stack of smooth circle n-bundles with connection. The local connection differential n-form is the local Lagrangian itself as in traditional literature, the rest of the data in $\mathcal{L}$ is the higher gauge symmetry equivariant structure.

The following is effectively the direct higher geometric analog of the Hamiltonian version of Noether’s theorem.

Symmetries

A transformation of the fields is an equivalence

$$\mathbf{Fields} \underoverset{\simeq}{\phi}{\longrightarrow} \mathbf{Fields} \,.$$

That the local Lagrangian $\mathcal{L}$ be preserved by this, up to (gauge) equivalence, means that there is a diagram in $\mathbf{H}$ of the form

$$\array{ \mathbf{Fields} &&\underoverset{\simeq}{\phi}{\longrightarrow}&& \mathbf{Fields} \\ & {}_{\mathllap{\mathbf{L}}}\searrow &\swArrow^\simeq_\alpha& \swarrow_{\mathrlap{\mathbf{L}}} \\ && \mathbf{B}^n U(1)_{conn} } \,.$$

(With $\mathbf{L}$ equivalently regarded as prequantum n-bundle this is equivalently a higher quantomorphism. These are the transformations studied in (Fiorenza-Rogers-Schreiber 13))

For $\phi$ an infinitesimal operation an $L$ locally the Lagrangian $n$-form, this means that the Lie derivative $\mathcal{L}_{\delta \phi}$ of $L$ has a potential,

$$\mathcal{L}_{\delta \phi} L = \mathbf{d} \alpha$$

hence that the Lagrangian changes under the Lie derivative by an exact term, hence by a divergence on the worldvolume (since the degree of the Lagrangian form is the dimension of the worldvolume). This defines an infinitesimal symmetry of the Lagrangian. See also (Azcarraga-Izquierdo 95 (8.1.13)).

This is the situation of the Noether theorem for the general case of “weak” symmetries (see at Noether theorem – schematic idea – weak symmetries).

By Cartan's magic formula the above means

$$\mathbf{d}\left( \alpha - \iota_{\delta\phi} \mathbf{L} \right) = \iota_{\delta \phi} \omega \,.$$

and hence the combination $j \coloneqq \alpha - \iota_{\delta\phi} \mathbf{L}$ (a Hamiltonian form for $\delta \phi$ with respect to $\omega$) is conserved on trajectories in the kernel of the n-plectic form $\omega$ (which are indeed the classical trajectories of $\mathbf{L}$, see (Azcarraga-Izquierdo 95 (8.1.14))).

This is the first stage in the Poisson bracket Lie n-algebra, the current algebra (see there at As a homotopy Lie algebra).

Examples

Energy-momentum tensor

• energy-momentum tensor– see at geometry of physics – A first idea of quantum field theory this example

Dirac current

• Dirac current– see at geometry of physics – A first idea of quantum field theory this example

Of Green-Schwarz super $p$-brane sigma models

The WZW term of the Green-Schwarz super p-brane sigma models is invariant under supersymmetry only up to a divergence, hence here the general Noether theorem for “weak” symmetries applies and yields a current algebra which is an polyvector extension of the supersymmetry algebra. See at Green-Schwarz action functional – Conserved currents for more.

Related concepts

• conservation law

• current algebra

• hadron current

• chiral anomaly

• classical observable

• Noether theorem

• charge

References

Dickey-Lie bracket on currents

The Dickey Lie bracket on conserved currents is due to

• Leonid Dickey, Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, Vol. 12 (World Scientific 1991).

and is reviewed in

• Glenn Barnich, Marc Henneaux, section 3 of Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket, Journal of Mathematical Physics 37, 5273-5296 (1996) (arXiv:hep-th/9601124, DOI 10.1063/1.531726)

The statement that the Dickey bracket Lie algebra of currents is a central Lie algebra extension of the algebra of symmetries by de Rham cohomology of the jet bundle appears as theorem 11.2 in (Part II of)

• Alexandre Vinogradov, The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. the linear theory, Journal of Mathematical Analysis and Applications 100, 1-40 (1984) (doi90071-4))

• Alexandre Vinogradov, The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, Journal of Mathematical Analysis and Applications 100, Issue 1, 30 April 1984, Pages 41-129 (publisher)

and is stated as exercise 2.28 on p. 203 of

• Alexandre Vinogradov, Joseph Krasil'shchik (eds.), Symmetries and conservation laws for differential equations of mathematical physics, vol. 182 of Translations of Mathematical Monographs, AMS (1999)

A lift of the Dickey Lie bracket on cohomologically trivial spaces to an equivalent L-infinity equivalent L-infinity bracket is constructed, under some assumptions, in

• Glenn Barnich, Ronald Fulp, Tom Lada, Jim Stasheff, The sh Lie structure of Poisson brackets in field theory, Communications in Mathematical Physics 191, 585-601 (1998) (arXiv:hep-th/9702176)

• Martin Markl, Steve Shnider, Differential Operator Endomorphisms of an Euler-Lagrange Complex, Contemporary Mathematics, Volume 231, 1999 (arXiv:9808105)

The cohomologically non-trivial lift is discussed in

• Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges, J. High Energ. Phys. (2017) 2017: 87 (arXiv:1507.08692)

In variational calculus

A general discussion as above is around definition 9 of

• G. J. Zuckerman, Action Principles and Global Geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259–284. (pdf)

The relation of conserved currents to moment maps in symplectic geometry is highlighted for instance in

• Huijun Fan, Lecture 8, Moment map and symplectic reduction (pdf)

Higher conserved currents

Higher conserved currents are discussed for instance in

• Glenn Barnich, Friedemann Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl.Phys.B633:3-82, 2002 (arXiv:hep-th/0111246)

In higher prequantum theory

Conserved currents for Lagrangians written as WZW terms are discussed in

• José de Azcárraga, José M. Izquierdo, section 8.1 of Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics , Cambridge monographs of mathematical physics, (1995)

Building on that, in the context of higher prequantum geometry conserved currents of the WZW model and in ∞-Wess-Zumino-Witten theory are briefly indicated on the last page of

• Urs Schreiber, Higher geometric prequantum theory and The Brane Bouquet, notes for a talk at Bayrischzell 2013 (pdf notes)

• Urs Schreiber, Classical field theory via Cohesive homotopy types

The same structure is considered in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory, 2013

as higher quantomorphisms and Poisson bracket Lie n-algebras of local currents.

category: physics