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
Question
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 
is assumed (here we take a slightly more restrictive point of view) to be a connected smooth manifold. The configuration space 
is defined to be a fibred manifold 
. By 
we refer to the fibred manifold of 
-jets of , whose projection 
where for 
we use 
for the canonical projection.
A field is a (smooth) section 
. A simple example that capture most of the usual cases: if we are studying mappings between manifolds 
, then we take 
the trivial fibre bundle. The 
-jet operator naturally sends 
.
A partial differential equation of order is defined to be a fibred submanifold 
. A field is said to solve the PDE if 
.
In the usual case of systems of PDEs on Euclidean space, is taken to be 
and 
the trivial vector bundle. A system of 
PDEs of order is usually taken to be 
where
is some function. We note that the domain of 
can be identified in this case with , We can then extend to 
a fibre bundle morphism.
If we assume that 
has constant rank, then 
is a fibred submanifold of , 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 
be an invariant submanifold.
More precisely, we take
Definition
A symmetry/gauge group 
is a subgroup of 
, with the property that for any 
, there exists a 
with 
.
It is important we are looking at the diffeomorphism group for , not . In general diffeomorphisms of will not preserve holonomy for sections of the form 
, a condition that is essential for solving PDEs. The condition that the symmetry operation “commutes with projections” is to ensure that 
, which in particular guarantees that 
extends to a diffeomorphism of 
with itself that commutes with projections.
From this point of view, a (system of) partial differential equation(s) is said to be -invariant if for every , we have 
.
We give two examples showing that this description agrees with the classical notions.
Gauge theory. In classical gauged theories, the configuration space is a fibre bundle with structure group 
which acts on the fibres. A section of 
induces a diffeomorphism of 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 . And the configuration space is the open submanifold of 
given by non-degenerate symmetric bilinear forms with signature (-+++). A diffeomorphism 
induces 
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 is a vector bundle and 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.
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Comment by Alejandro Ricardo Urzúa Pineda — 02/10/2011 @ 03:57 |
Hi, Willie. I wonder why you decided to attache the term “gauge” to the symmetry transformations you’ve described. The transformations you described are indeed symmetries. Though, in the literature on classical field theory (aka variational PDEs) gauge symmetries are special kinds of symmetries that come in families parametrized by smooth functions on $X$, such that the support of the diffeomorphism $g\in\mathcal{G}$ projects to the the support of the corresponding smooth function on $X$. This dependence on arbitrary smooth functions is crucial in the proof of Nöther’s second theorem: a family of gauge symmetries implies degeneracy in the equations of motion. It is these degeneracies that are then responsible for the notorious trouble of formulating well-posed initial value problems of systems with gauge symmetries, such as electrodynamics or general relativity.
BTW, you may know this already, but infinitesimal versions of the kinds of symmetries you’ve described are extensively discussed, including the distinction between global and (local) gauge symmetries, in P. Olver’s book _Applications of Lie Groups to Differential Equations_ and in Henneaux & Teitelboim’s _Quantization of Gauge Systems_, though the latter from a much more physicist point of view.
Comment by Igor Khavkine — 06/10/2011 @ 20:49 |
Yes, indeed I was using the word “gauge” in a very loose sense of the word. I was originally also going to write a few words about the issue of gauge fixing for IVPs, but I realized half-way through that the construction I had in mind doesn’t work in all generality, and so I omitted it from the post.
And thanks for the reminder of Olver’s book: I was aware of it but had forgotten about it.
Comment by Willie Wong — 07/10/2011 @ 07:45 |