What is the differential-geometric formulation of field theories?

Question

I understand that in classical mechanics, a system with $N$ degrees of freedom is modeled by an $N$ dimensional Manifold called the configuration space. We then look at the tangent and cotangent bundles which is the stage for the dynamics of the system. We then define the Lagrangian and the Hamiltonian which are smooth maps from tangent and cotangent bundle respectively and then we can derive the equations of motion of this system.

I'm trying to draw an analogy for field theory but I don't know how because it seems to me the approach changes somewhat in field theories (unless of course I'm horribly confused).

Fields are typically defined as smooth sections over spacetime and fields of course have infinite degrees of freedom so I suppose that the configuration space for fields ought to be thought of as an infinite dimensional Manifold. Now I'm assuming you can still construct tangent and cotangent bundles to infinite dimensional manifolds so are the Lagrangian and the Hamiltonian still defined similarly?

Now my question is whether I'm thinking along the right lines here? If so I'm wondering what exactly is the relationship between the spacetime manifold (over which the fields are defined) and the configuration space for fields. For instance if we consider the electromagnetic field which is a $(2,0)$ antisymmetric tensor field, is the configuration space the space of all $(2,0)$ antisymmetric tensor fields over spacetime?

I would really appreciate an elaborate explanation because I feel like I'm horribly confused here.

Answer 1

The modern way to deal with this -- often referred to via the variational bicomplex -- is to consider primarily the jet bundle of the field bundle, instead of the infinite-dimensional space of sections of the field bundle. The infinite-jet bundle is itself mildly infinite-dimensional, but if your Lagrangian density depends only on a finite order $n$ of derivatives of fields (as most do) then you may consider just the order $n$ jet bundle, which is a finite dimensional manifold.

All of the variational calculus which traditional textbooks do on the infinite-dimensional space of sections of the field bundle may be done equivalently on this finite-dimensional jet bundle. And in fact better so, since the jet bundle formulation allows to work strictly locally. For example the usual rule for determining the Euler-Lagrange variantional derivative by varying a functional on an infinite-dimensions space, then integrating by parts and then discarding boundary terms, all this is neatly captured on the jet bundle simply by applying the plain old de Rham differential on the jet bundle and decomposing the result into the piece proportional to the vertical differential of a 0-jet and a horizontally exact piece.

A neat textbook explaining this is

Ian Anderson, The variational bicomplex, (pdf)

There is more exposition on how this works at the my PhysicsForums-Insights article: Higher Prequantum Geometry II: The Principle of Extremal Action

Answer 2

The typical geometric formulation of field theories is not quite analogous to the symplectic approach taken for classical mechanics. As you point out, in classical mechanics, when considering some finite collection of particles, one models each degree of freedom by one dimension in the corresponding phase space (which is typically the cotangent bundle of the manifold which parametrizes the particle positions, e.g. $\Bbb R^{3n}$ for $n$ particles moving in $\Bbb R^3$).

You also correctly observe that an analogous approach for field theories would have to work with infinitely many dimensions: In fact, there would have to be uncountably many. I'm not aware of any mathematical framework that attempts to actually make this analogy work: Certainly one has to deal with some technical problems when trying to work with such manifolds, but I don't know if they can/have been overcome. However, there are completely different approaches to a mathematical formulation of field theory.

A simple scalar field living on a (spacetime) manifold $M$ can be modeled as a (smooth) map $M\to \Bbb R^n$ or $\Bbb C^n$ and the action is simply a functional on the space of such maps. However, these simple situations are typically not of such interest from a geometric point of view.

Things get much more interesting (and difficult!) when you consider gauge theories. A theory with gauge group $G$ can be formalized by considering a $G$-principal bundle over the manifold $M$ and the gauge fields are realized as principal connections on such bundles. The physicists' point of view, where gauge fields are regarded as living on $M$, can be recovered by using local sections of the bundle: Pulling a connection back by a section gives you back the usual physical way of thinking about gauge fields. The "field strength" corresponds to the curvature of a connection. Besides gauge fields, one can also add some scalar or spinor fields to make things even more interesting. The mathematics behind (classical) gauge theories is extremely rich and underlies several recent breakthroughs in differential geometry; I will not try to give a more detailed account but would be glad to point out some references if you are interested.