nLab action functional



Variational calculus


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



In physics the dynamics of a physical system may be encoded by a nonlinear functional – called the action functional – on its configuration space:

For emphasis the description of dynamics by action functionals is called the Lagrangean approach. Another formulation of dynamics in physics that does not involve an action functional explicitly is Hamiltonian mechanics on phase space. At least in certain classes of cases the relation and equivalence of both approaches is understood. Generally the formulation of quantum field theory in terms of action functionals suffers from a lack of precise understanding of what the path integral over the action functional really means.

Action functional on a napkin
Taken from A Zee, Fearful Symmetry


Let H\mathbf{H} be the ambient (∞,1)-topos with a natural numbers object and equipped with an additive continuum line object 𝔸 1\mathbb{A}^1 (see there). Let CHC \in \mathbf{H} be the configuration space of a physical system. Then an action functional is a morphism

exp(iS()):C𝔸 1/ \exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbb{A}^1 / \mathbb{Z} \,

(here \hbar refers to Planck's constant).

If H\mathbf{H} is a cohesive (∞,1)-topos then there is an intrinsic differential of the action functional to a morphism

dexp(iS()):C dR𝔸 1/. \mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbf{\flat}_{dR}\mathbb{A}^1/\mathbb{Z} \,.

The equation

dexp(iS())=0 \mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) = 0

is the Euler-Lagrange equation of the system. It characterizes the critical locus of SS is the covariant phase space inside the configuration space: the space of classically realized trajectories/histories of the system. If H\mathbf{H} models derived geometry then this critical locus is presented by a BRST-BV complex.

Local action functionals (traditional theory)

An action functional is called local if it arises from integration of a Lagrangian.

In traditional theory this is interpreted as follows: an action functional S:C𝔸 1S : C \to \mathbb{A}^1 is called local if

  • the configuration space CC is the space C=Γ X(E)C = \Gamma_X(E) of sections of a fiber bundle EXE \to X over some parameter space (spacetime XX);

  • there is a Lagrangian density J (E)Ω dimX(X)J_\infty(E) \to \Omega^{\dim X}(X) on the jet bundle of EE;

  • on a section/field configuration ϕ:XE\phi : X \to E the action SS takes the value

    S(ϕ)= XL(j (ϕ)), S(\phi) = \int_X L(j_\infty(\phi)) \,,

    where j (ϕ)=(ϕ, iϕ,)j_\infty(\phi) = (\phi, \partial_i \phi, \cdots) is the jet-prolongation of ϕ\phi (the collection of all its higher partial derivatives).

Consider action functional for on a configuration space of smooth functions from the line to a smooth manifold XX.

We can consider

  1. S(q)= a bL(q,q˙)dt S(q) = \int_a^b L(q,\dot{q}) \,\mathrm{d}t , where qq is a path through configuration space, on the time interval [a,b][a,b], with derivative q˙=dq/dt\dot{q} = \mathrm{d}q/\mathrm{d}t. When minimising the action, we fix the values of q(a)q(a) and q(b)q(b).
  2. L(q,q˙)= S(q,q˙)dxdydz L(q,\dot{q}) = \int_{S} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z , where now qq is a configuration of fields on SS, which is a region of space. We fix boundary conditions on the boundary of SS (typically that qq and q˙\dot{q} go to zero if SS is all of space).
  3. S(q)= R(q,q˙)dxdydzdt S(q) = \int_{R} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t , where now qq is a configuration of fields on RR, which a region of spacetime, with time derivative q˙=q/t\dot{q} = \partial{q}/\partial{t}. We fix boundary conditions on the boundary of RR.

The formulation of (3) above is still not manifestly coordinate independent. However, dxdydzdt\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t is simply the volume form on spacetime and q˙\dot{q} is merely one choice of coordinate on state space and could just as easily be replaced by a derivative with respect to any timelike coordinate on spacetime (or drop coordinates altogether).

Extended local action functionals in (higher) gauge theory

For gauge theories and higher gauge theories the configuration spaces of the physical system are in general not plain manifolds or similar, but are orbifolds or more generally smooth groupoids, smooth ∞-groupoids. (An exposition of and introduction to much of the following is at geometry of physics.)

For instance for GG a Lie group and BG conn\mathbf{B}G_{conn} the smooth moduli stack of GG-principal connections (see at connection on a bundle), then the smooth groupoid of GG-gauge field configurations is the internal hom/mapping stack [Σ,BG conn][\Sigma, \mathbf{B}G_{conn}] \in Smooth∞Grpd (or some concretification thereof, see at geometric of physics – differential moduli: this is the smooth groupoid whose objects are GG-gauge field-configurations on Σ\Sigma (connections on GG-principal bundles over Σ\Sigma), and whose morphisms are gauge transformations between these. The infinitesimal approximation to this smooth ∞-groupoid, its ∞-Lie algebroid is the (off-shell) BRST complex of the theory. The tangent to the nn-fold higher gauge transformations becomes the nn-fold ghosts in the BRST complex.

More generally GG here can by any smooth ∞-group, such as the circle n-group B n1U(1)\mathbf{B}^{n-1}U(1) or the String 2-group or the Fivebrane 6-group, and so on, in which case [Σ,BG conn][\Sigma, \mathbf{B}G_{conn}] is the smooth ∞-groupoid of higher gauge field, gauge transformations between these, higher gauge transformations between those, and so on.

Notice that this means in particular that in higher geometry a gauge theory is a sigma-model quantum field theory: one whose target space is not just a plain manifold but is a moduli stack of gauge field configurations.

A gauge invariant action functional is then a morphism of smooth ∞-groupoids

exp(iS()):[Σ,BG conn]U(1). \exp( i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}] \to U(1) \,.

This is of particular interest, again, if it is local. In fact, in this context now we can also ask that it is “extended” in the sense of extended topological quantum field theory: that we have an action functional not only in top dimension, being a function, but also in codimension 1, being a prequantum bundle, and in higher codimension, being a prequantum n-bundle.

This is notably the case for all (higher) gauge theories of infinity-Chern-Simons theory type, such as ordinary Chern-Simons theory and such as ordinary Dijkgraaf-Witten theory, as well as its higher generalizations. In these cases the action functional exp(iS()):[Σ,BG conn]\exp(i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}] arises itself from transgression of an extended Lagrangian that is defined on the universal moduli stack of gauge field configurations BG conn\mathbf{B}G_{conn} itself, namely from a universal characteristic class in higher nonabelian differential cohomology of the form

L:BG connB nU(1) conn. \mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} \,.

Here B nU(1) conn\mathbf{B}^n U(1)_{conn} is the universal smooth moduli infinity-stack for circle n-bundles with connection. Such a morphism of moduli stacks locally takes a connection differential form AA to a Chern-Simons form CS(A)CS(A), but globally it sends the underlying principal bundle to a circle (n-1)-group principal ∞-bundle and accordingly acts globally on the connection. This is hence a fully local Lagrangian: an extended Lagrangian. Alternatively, one may think of this whole morphism as modulating a prequantum circle n-bundle on the universal moduli stack BG conn\mathbf{B}G_{conn} of gauge fields itself.

For instance for ordinary Chern-Simons theory here n=3n = 3 GG is a semisimple Lie group and L\mathbf{L} is a smooth and differential refinement of the first Pontryagin class/second Chern class, or of an integral multiple of that (the “level” of the theory). In this case L\mathbf{L} may also be thought of as modulating the universal Chern-Simons circle 3-bundle. If instead GG is a discrete group then L\mathbf{L} is a cocycle in the U(1)U(1)-group cohomology and this is the extended Lagrangian of Dijkgraaf-Witten theory.

This extended Lagrangian becomes an extended action functional after transgression: the operation of fiber integration in ordinary differential cohomology refines to a morphism of moduli stacks of the form

exp(2πi Σ k()):[Σ k,B nU(1) conn]B nkU(1) conn, \exp(2 \pi i \int_{\Sigma_k} (-)) \colon [\Sigma_k, \mathbf{B}^n U(1)_{\mathrm{conn}} ] \to \mathbf{B}^{n-k}U(1)_{conn} \,,

where Σ\Sigma is an oriented closed manifold of dimension kk. This morphism locally simply takes a differential n-form to its ordinary integration of differential forms over Σ k\Sigma_k, but globally it takes the correct higher holonomy of circle n-bundles with connection.

Combining this with an extended ∞-Chern-Simons theory Lagrangian L:BG connB nU(1) conn\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} as above yields for each dimension kk a prequantum circle n-bundle on the space of gauge field configurations over Σ k\Sigma_k, by forming the transgression composite

exp(iS())exp(2πi Σ k[Σ k,L]):[Σ k,BG conn][Σ k,L][Σ k,B nU(1) conn]exp(2πi Σ k())B nkU(1) conn. \exp(i S(-)) \coloneqq \exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, \mathbf{L}]) \;\; \colon \;\; [\Sigma_k, \mathbf{B}G_{conn}] \stackrel{[\Sigma_k, \mathbf{L}]}{\to} [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_k}(-))}{\to} \mathbf{B}^{n-k}U(1)_{conn} \,.

This morphism locally takes the local differential form incarnation AA of a connection on an ∞-bundle to the exponentiation of the integration of differential forms ΣCS(A)\int_\Sigma CS(A) of some higher Chern-Simons form, but globally it computes the correct higher holonomy of the higher circle n-bundle with connection over the universal moduli stack of fields, as modulated by the extended Lagrangian L\mathbf{L}.


For spacetime field theory:

For branes:

action functionalkinetic actioninteractionpath integral measure
exp(S(ϕ))μ=\exp(-S(\phi)) \cdot \mu = exp((ϕ,Qϕ))\exp(-(\phi, Q \phi)) \cdotexp(I(ϕ))\exp(I(\phi)) \cdotμ\mu
BV differentialelliptic complex +antibracket with interaction +BV-Laplacian
d q=d_q =QQ +{I,}\{I,-\} +Δ\hbar \Delta

extended prequantum field theory

0kn0 \leq k \leq n(off-shell) prequantum (n-k)-bundletraditional terminology
00differential universal characteristic maplevel
11prequantum (n-1)-bundleWZW bundle (n-2)-gerbe
kkprequantum (n-k)-bundle
n1n-1prequantum 1-bundle(off-shell) prequantum bundle
nnprequantum 0-bundleaction functional


Historical and expository accounts of the “principle of extremal action” or “action principle”, for short:

  • Agamenon R. E. Oliveira, History of Two Fundamental Principles of Physics: Least Action and Conservation of Energy, Advances in Historical Studies 3 2 (2014) [doi:10.4236/ahs.2014.32008]

  • Walter Dittrich, The Development of the Action Principle – A Didactic History from Euler-Lagrange to Schwinger, SpringerBriefs in Physics, Springer (2021) [doi:10.1007/978-3-030-69105-9]

  • Douglas Cline, Variational Principles in Classical Mechanics, University of Rochester (2021) [pdf, online version]

Textbook account in the context of gauge theories:

Lecture notes with more details are in the section Lagrangians and Action functionals of

Discussion of extended higher local action functional for (higher) gauge theories of generalized ∞-Chern-Simons theory type are discussed in

The extended local action functionals for ordinary 3d Chern-Simons theory/Dijkgraaf-Witten theory and for 7d String 2-group Chern-Simons theory are constructed in

and discussed further in

A comprehensive discussion in a general context of higher differential geometry is in

Last revised on December 23, 2023 at 09:03:35. See the history of this page for a list of all contributions to it.