Gauge invariance, geometrically

by Willie Wong

A somewhat convoluted chain of events led me to think about the geometric description of partial differential equations. And a question I asked myself this morning was

What is the meaning of gauge invariance in the jet-bundle treatment of partial differential equations?

The answer, actually, is quite simple.

Review of geometric formulation PDE
We consider here abstract PDEs formulated geometrically. All objects considered will be smooth. For more about the formal framework presented here, a good reference is H. Goldschmidt, “Integrability criteria for systems of nonlinear partial differential equations”, JDG (1967) 1:269–307.

A quick review: the background manifold X is assumed (here we take a slightly more restrictive point of view) to be a connected smooth manifold. The configuration space \mathcal{C} is defined to be a fibred manifold p:\mathcal{C}\to X. By J^r\mathcal{C} we refer to the fibred manifold of r-jets of \mathcal{C}, whose projection p^r = \pi^r_0 \circ p where for r > s we use \pi^r_s: J^r\mathcal{C}\to J^s\mathcal{C} for the canonical projection.

A field is a (smooth) section \phi \subset \Gamma \mathcal{C}. A simple example that capture most of the usual cases: if we are studying mappings between manifolds \phi: X\to N, then we take \mathcal{C} = N\times X the trivial fibre bundle. The s-jet operator naturally sends j^s: \Gamma\mathcal{C} \ni \phi \mapsto j^s\phi \in \Gamma J^r\mathcal{C}.

A partial differential equation of order r is defined to be a fibred submanifold J^r\mathcal{C} \supset R^r \to X. A field is said to solve the PDE if j^r\phi \subset R^r.

In the usual case of systems of PDEs on Euclidean space, X is taken to be \mathbb{R}^d and \mathcal{C} = \mathbb{R}^n\times X the trivial vector bundle. A system of m PDEs of order r is usually taken to be F(x,\phi, \partial\phi, \partial^2\phi, \ldots, \partial^r\phi) = 0 where

\displaystyle F: X\times \mathbb{R}^n \times \mathbb{R}^{dn} \times \mathbb{R}^{\frac{1}{2}d(d+1)n} \times \cdots \times \mathbb{R}^{{d+r-1 \choose r} n} \to \mathbb{R}^m

is some function. We note that the domain of F can be identified in this case with J^r\mathcal{C}, We can then extend F to \tilde{F}: J^r\mathcal{C} \ni c \mapsto (F(c),p^r(c)) \in \mathbb{R}^m\times X a fibre bundle morphism.

If we assume that \tilde{F} has constant rank, then \tilde{F}^{-1}(0) is a fibred submanifold of J^r\mathcal{C}, and this is our differential equation.

Gauge invariance
In this frame work, the gauge invariance of a partial differential equation relative to certain symmetry groups can be captured by requiring R^r be an invariant submanifold.

More precisely, we take

A symmetry/gauge group \mathcal{G} is a subgroup of \mathrm{Diff}(\mathcal{C}), with the property that for any g\in\mathcal{G}, there exists a g'\in \mathrm{Diff}(X) with p\circ g = g' \circ p.

It is important we are looking at the diffeomorphism group for \mathcal{C}, not J^r\mathcal{C}. In general diffeomorphisms of J^r\mathcal{C} will not preserve holonomy for sections of the form j^r\phi, a condition that is essential for solving PDEs. The condition that the symmetry operation “commutes with projections” is to ensure that g:\Gamma\mathcal{C}\to\Gamma\mathcal{C}, which in particular guarantees that g extends to a diffeomorphism of J^rC with itself that commutes with projections.

From this point of view, a (system of) partial differential equation(s) R^r is said to be \mathcal{G}-invariant if for every g\in\mathcal{G}, we have g(R^r) \subset R^r.

We give two examples showing that this description agrees with the classical notions.

Gauge theory. In classical gauged theories, the configuration space \mathcal{C} is a fibre bundle with structure group G which acts on the fibres. A section of G\times X \to X induces a diffeomorphism of \mathcal{C} by fibre-wise action. In fact, the gauge symmetry is a fibre bundle morphism (fixes the base points).

General relativity. In general relativity, the configuration space is the space of Lorentzian metrics. So the background manifold is the space-time X. And the configuration space is the open submanifold of S^2T^*X given by non-degenerate symmetric bilinear forms with signature (-+++). A diffeomorphism \Psi:X\to X induces T^*\Psi = (\Psi^{-1})^*: T^*X \to T^*X and hence a configuration space diffeomorphism that commutes with projection. It is in this sense that Einstein’s equations are diffeomorphism invariant.

Notice of course, this formulation does not contain the “physical” distinction between global and local gauge transformations. For example, for a linear PDE (so \mathcal{C} is a vector bundle and R^r is closed under linear operations), the trivial “global scaling” of a solution is considered in this frame work a gauge symmetry, though it is generally ignored in physics.