nLab
conserved current

Context

Variational calculus

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

In variational calculus

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

The context

Let XX be a spacetime of dimension nn, EXE \to X a bundle, j:EXj:\infty E \to X its jet bundle and

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

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

Proposition

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

δL=E(L)+dΘ \delta L = E(L) + d \Theta

for EE the Euler-Lagrange operator.

The covariant phase space of the Lagrangian is the locus

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

For ΣX\Sigma \subset X any (n1)(n-1)-dimensional submanifold,

δθ:=δ ΣΘ \delta \theta := \delta \int_\Sigma \Theta

is the presymplectic structure on covariant phase space

Definition

Definition

A conserved current is an element

jΩ n1,0(j E) j \in \Omega^{n-1, 0}(j_\infty E)

which is horizontally closed on covariant phase space

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

For ΣX\Sigma \hookrightarrow X a submanifold of dimension n1n-1, the charge of the conserved current jj with respect to Σ\Sigma is the integral

Q Σ:= Σj. Q_\Sigma := \int_\Sigma j \,.

Properties

Proposition

If Σ,ΣX\Sigma, \Sigma' \subset X are homolous, the associated charge is the same

Q Σ=Q Σ. Q_{\Sigma} = Q_{\Sigma'} \,.
Proof

By Stokes' theorem.

Theorem

Every symmetry induces a conserved current.

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

In higher prequantum geometry

The following discusses conserved currently in the context of higher prequantum geometry. This follows (Schreiber 13). Similar observations have been made by Igor Khavkine.

Context

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

:FieldsB nU(1) conn \mathcal{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

FieldsϕFields. \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 H\mathbf{H} of the form

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

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

For ϕ\phi an infinitesimal operation an LL locally the Lagrangian nn-form, this means

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

hence that the Lagrangian changes under the Lie derivative by an exact term. By Cartan's magic formula this means

d(αι δϕ)=ι δϕω. \mathbf{d}\left( \alpha - \iota_{\delta\phi} \mathcal{L} \right) = \iota_{\delta \phi} \omega \,.

Hence (…)

References

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

In higher prequantum theory

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

The same structure is considered in

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

Revised on September 16, 2014 10:04:48 by Urs Schreiber (185.26.182.33)