nLab
action functional

Idea

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

• in classical mechanics and classical field theory – by the action principle or principle of least action – the extrema of the action functional – obtained by variational calculus and given by Euler-Lagrange equations – encode the physically observable configurations ;

• in quantum mechanics and quantum field theory the evolution of the quantum states is encoded by the integral – the path integral – of the exponentiated action functional over the space of field configurations.

For emphasis the description of dynamics by action functionals is called the Lagrangean approach. Another forumlation 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.

Definition

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

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

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

$$d\exp (iS(-)):C\to {♭}_{\mathrm{dR}}{𝔸}^1/\mathbb{Z}.$$d \exp(i S(-)) : C \to \mathbf{\flat}_{dR}\mathbb{A}^1/\mathbb{Z} \,.

This is the Euler-Lagrange equation of the system. The critical locus of $S$ is the covariant phase space inside the configuration space: the space of classically realized trajectories/histories of the system. If $H$ models derived geometry then this critical locus is presented by a BRST-BV complex.

Local action functionals

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

More precisely, an action functional $S:C\to {𝔸}^1$ is called local if

• the configuration space $C$ is the space $C={\Gamma }_X(E)$ of sections of a fiber bundle $E\to X$ over some parameter space (spacetime $X$);

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

• on a section/field configuration $\varphi :X\to E$ the action $S$ takes the value

  $$S(\varphi )={\int }_{X}L(j_{\mathrm{\infty }}(\varphi )),$$S(\phi) = \int_X L(j_\infty(\phi)) \,,

  where $j_{\mathrm{\infty }}(\varphi )=(\varphi ,{\partial }_i\varphi ,\cdots )$ is the jet-prolongation of $\varphi$ (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 $X$.

We can consider

1. $S(q)={\int }_a^{b}L(q,\dot{q})\mathrm{d}t$, where $q$ is a path through configuration space, on the time interval $[a,b]$, with derivative $\dot{q}=\mathrm{d}q/\mathrm{d}t$. When minimising the action, we fix the values of $q(a)$ and $q(b)$.
2. $L(q,\dot{q})={\int }_Sℒ(q,\dot{q})\mathrm{d}x\mathrm{d}y\mathrm{d}z$, where now $q$ is a configuration of fields on $S$, which is a region of space. We fix boundary conditions on the boundary of $S$ (typically that $q$ and $\dot{q}$ go to zero if $S$ is all of space).
3. $S(q)={\int }_Rℒ(q,\dot{q})\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}t$, where now $q$ is a configuration of fields on $R$, which a region of spacetime, with time derivative $\dot{q}=\partial q/\partial t$. We fix boundary conditions on the boundary of $R$.

The formulation of (3) above is still not manifestly coordinate indepdendent. However, $\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}t$ is simply the volume form on spacetime and $\dot{q}$ 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).

Examples

A large class of examples of action functionals arises in ∞-Chern-Simons theory. See there for details.

Related concepts

• effective action functional, background field formalism