Hi! Former Notre Dame math student here, posting at Liviu's suggestion to expand on a conversation we were having on Facebook.
Suppose you have a fiber bundle E \rightarrow M. Interpret M as "spacetime;" then sections of E are "fields." (Particle dynamics can be recovered by taking M = \mathbb{R}.) To set up a classical dynamics on these fields, one writes down a Lagrangian L and associated action functional S = \int_M L, then obtains field equations by requiring \delta S = 0. When I first read the derivation of these Euler-Lagrange equations in a physics book, I felt like a trick had been played. It wasn't clear to me what the Lagrangian really "was," in a formal mathematical sense, and the formula \frac{d}{dt}(\delta q) = \delta \dot{q} seemed a bit magic.
As usual, the nlab came to my rescue and told me about the "variational bicomplex." (http://ncatlab.org/nlab/show/variational+bicomplex). This is a doubly-graded complex of differential forms on the infinite jet bundle j_{\infty}(E). In particular, any differential form on a finite jet bundle j_k(E) gives you an element of the variational bicomplex via pullback. And both fields and Lagrangians look like forms on finite jet bundles of E. A field is a 0-form on the 0-jet bundle. A Lagrangian is a bit more complicated- since the action functional is the integral of L over M, L must be an n-form on M... but since its values depend on the 1-jet of the field you're at, it's more like an n-form on the 1-jet bundle.
Write D for the exterior derivative on forms on j_{\infty}(E) (or, for that matter, on any finite jet bundle). We'd like to be able to split D into a sum d + \delta, where d is the derivative "along M" and \delta is the derivative along the fiber. What this requires is a splitting of the tangent space at any \infty-jet \varphi into horizontal directions and vertical directions. The vertical directions are already there, since we have a fiber bundle, so we just need the horizontal ones. Local coordinates on j_{\infty}(E) are \{x_1, \ldots, x_n, q_1, \ldots, q_k, \partial_i q_j, \partial_i \partial_j q_k, \ldots \}.
Trickily, the vectors \frac{\partial}{\partial x_i} are NOT an appropriate choice of horizontal vectors, even if the bundle E happens to be trivial! (As is always the case when M = \mathbb{R}.) This is precisely because we want an equation like \frac{d}{dt}(\delta q) = \delta \dot{q}. In other words, if we're at the jet \varphi, then when we go out from \varphi in a horizontal direction (say the x_1 direction), the coordinates q_1, \ldots, q_k of \varphi should change in a manner specified by the coordinates \partial_1 q_1, \ldots, \partial_1 q_k of \varphi.
But how should the coordinates \partial_i q_j of \varphi change themselves? Now we need to look at the 2-jet component of \varphi, and it's the same all the way up. Now we see why the full \infty-jet bundle was needed- we'd be stuck if we cut ourselves off at a finite jet bundle.
Given this horizontal/vertical splitting of tangent spaces to the \infty-jet bundle, we're within our rights to talk about a bicomplex of differential forms on j_{\infty}(E). Sections of E (i.e. fields) yield (0,0)-forms, and Lagrangians yield (n,0) forms. We may now happily take a variational derivative of L: it's just \delta L, the derivative in the vertical direction. This is an (n,1)-form, and the usual Euler-Lagrange argument beefs up to show that any (n,1)-form splits uniquely as E + d\Theta, where \Theta is an (n-1,1)-form and E is a "source form," i.e. an (n,1)-form such that, when contracted with a vertical vector (represented by a path of germs \varphi_s), the result only depends on the values of \varphi_s at the spacetime point in question and not on the higher jet components. So \delta(L) = E(L) + d\Theta; the field equations are E(L) = 0.
Lots of other nice stuff falls out of this framework: i.e. an infinitesimal symmetry of the system is a vertical vector field v such that \iota_v \delta L = d \sigma for some (n-1,0)-form \sigma. (Such terms d \sigma affect the action only by a boundary contribution, which can be assumed to be zero in your favorite way). Then you can immediately consider the (n-1,0)-form \sigma - \iota_v \Theta; being an (n-1)-form on spacetime, it represents a Noether current. To see that it's conserved, compute d(\sigma - \iota_v \Theta) = \iota_v \delta L - \iota_v d\Theta = \iota_v E(L). But, at a solution to the field equations, E(L) = 0, so Noether's theorem is just a bit of playing around with differential forms.
....................
So, what if you want to study gauge theories? Suppose you have a Lie group G and a G-principal bundle P \rightarrow M. Then "fields" should be G-connections on P. These aren't naturally sections of a bundle-- rather, they're an affine space for sections of the bundle \Omega^1(M; \mathfrak{g}) where \mathfrak{g} denotes the bundle associated to P via the adjoint representation of G. So "variations" in a gauge field look just like the variations above, where E = \Omega^1(M; \mathfrak{g}). But somehow this seems unnatural, and I'd like a more convincing way of saying this stuff in a gauge-theory setting.
Furthermore, the principal bundle P shouldn't need to be fixed. Fields should be "bundle plus connection" rather than just "connection on a fixed bundle." Apparently this is where differential cocycles come in. A differential cocycle is supposed to (roughly?) capture the notion of a differential form AND an integer cocycle representing the same real cohomology class. The form gives us the connection, and the integer cohomology class gives us the bundle. (?) Unfortunately, I don't know much about these beasts. What I'd like to know is:
(1) is there a way to set up a variational bicomplex for gauge theory, where the "sections of E" are replaced by differential cocycles?
(2) when G is trivial, you don't recover the non-gauge theory. Is there a more general framework which subsumes both gauge fields and non-gauge fields?
(0) what do I need to know about differential cohomology, cocycles, etc., to understand these things? The paper with the right definition, I think, is "Quadratic functions in geometry, topology, and M-theory" (Hopkins, Singer), but it's a bit formidable.
Comments, questions, answers, oracular enlightenment all appreciated!
-Andy Manion
1 comment:
Did you see this book on the subject
http://math.uni.lu/~michel/data/VARIATIONNAL%20BICOMPLEX.pdf
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