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In physics the dynamics of a physical system may be encoded by a nonlinear 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.
Let $\mathbf{H}$ be the ambient (∞,1)-topos with a natural numbers object and equipped with an additive continuum line object $\mathbb{A}^1$ (see there). Let $C \in \mathbf{H}$ be the configuration space of a physical system. Then an action functional is a morphism
(here $\hbar$ refers to Planck's constant).
If $\mathbf{H}$ is a cohesive (∞,1)-topos then there is an intrinsic differential of the action functional to a morphism
The equation
is the Euler-Lagrange equation of the system. It characterizes 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 $\mathbf{H}$ models derived geometry then this critical locus is presented by a BRST-BV complex.
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 $S : C \to \mathbb{A}^1$ 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 $J_\infty(E) \to \Omega^{\dim X}(X)$ on the jet bundle of $E$;
on a section/field configuration $\phi : X \to E$ the action $S$ takes the value
where $j_\infty(\phi) = (\phi, \partial_i \phi, \cdots)$ is the jet-prolongation of $\phi$ (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 $X$.
We can consider
The formulation of (3) above is still not manifestly coordinate independent. 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).
For gauge theories and higher gauge theories the configuration spaceds 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 $G$ a Lie group and $\mathbf{B}G_{conn}$ the smooth moduli stack of $G$-principal connections (see at connection on a bundle), then the smooth groupoid of $G$-gauge field configurations is the internal hom/mapping stack $[\Sigma, \mathbf{B}G_{conn}] \in$Smooth∞Grpd (or some concretification thereof, see at geometric of physics – differential moduli: this is the smooth groupoid whose objects are $G$-gauge field-configurations on $\Sigma$ (connections on $G$-principal bundles over $\Sigma$), 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 $n$-fold higher gauge transformations becomes the $n$-fold ghosts in the BRST complex.
More generall $G$ here can by any smooth ∞-group, such as the circle n-group $\mathbf{B}^{n-1}U(1)$ or the String 2-group or the Fivebrane 6-group, and so on, in which case $[\Sigma, \mathbf{B}G_{conn}]$ 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
This is of particular interst, 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 theoris 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 $\exp(i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}]$ arises itself from transgression of an extended Lagrangian that is defined on the universal moduli stack of gauge field configurations $\mathbf{B}G_{conn}$ itself, namely from a universal characteristic class in higher nonabelian differential cohomology of the form
Here $\mathbf{B}^n U(1)_{conn}$ 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 $A$ to a Chern-Simons form $CS(A)$, 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 $\mathbf{B}G_{conn}$ of gauge fields itself.
For instance for ordinary Chern-Simons theory here $n = 3$ $G$ is a semisimple Lie group and $\mathbf{L}$ 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 $\mathbf{L}$ may also be thought of as modulating the universal Chern-Simons circle 3-bundle. If instead $G$ is a discrete group then $\mathbf{L}$ is a cocycle in the $U(1)$-group cohomology and this is the extended Lagrangian of Dijkgraaf-Witten theory.
This extended Lagrangian becomes an extended action functional after transgression: the operaton of fiber integration in ordinary differential cohomology refines to a morphism of moduli stacks of the form
where $\Sigma$ is an oriented closed manifold of dimension $k$. This morphism locally simply takes a differential n-form to its ordinary integration of differential forms over $\Sigma_k$, but globally it takes the correct higher holonomy of circle n-bundles with connection.
Combining this with an extended ∞-Chern-Simons theory Lagrangian $\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$ as above yields for each dimension $k$ a prequantum circle n-bundle on the space of gauge field configurations over $\Sigma_k$, by forming the transgression composite
This morphism locally takes the local differential form incarnation $A$ of a connection on an ∞-bundle to the exponentiation of the integration of differential forms $\int_\Sigma CS(A)$ 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 $\mathbf{L}$.
A large class of examples of action functionals arises in ∞-Chern-Simons theory. See there for details.
action functional | kinetic action | interaction | path integral measure |
---|---|---|---|
$\exp(-S(\phi)) \cdot \mu =$ | $\exp(-(\phi, Q \phi)) \cdot$ | $\exp(I(\phi)) \cdot$ | $\mu$ |
BV differential | elliptic complex + | antibracket with interaction + | BV-Laplacian |
$d_q =$ | $Q$ + | $\{I,-\}$ + | $\hbar \Delta$ |
extended prequantum field theory
$0 \leq k \leq n$ | (off-shell) prequantum (n-k)-bundle | traditional terminology |
---|---|---|
$0$ | differential universal characteristic map | level |
$1$ | prequantum (n-1)-bundle | WZW bundle (n-2)-gerbe |
$k$ | prequantum (n-k)-bundle | |
$n-1$ | prequantum 1-bundle | (off-shell) prequantum bundle |
$n$ | prequantum 0-bundle | action functional |
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