The Electromagnetic Field as Bundle Curvature

Let us explore nature's elegance and see how a field can be thought of as the curvature of a connection over a G-Bundle. Let's take the simplest Lie group we can think of, U(1), which is ofcourse, abelian. This note can be thought of as a "continuation" of the note titled "Rewriting the Maxwell's equations". First off, I'd want to give the readers an idea of what a G-Bundle is. First off, I shall give you all the idea of a vector bundle, its basically a non trivial fiber bundle constructed from glueing together many trivial bundles. now a Fiber bundle is a generalisation of a Manifold sought in order to define a "connection". Let us first assume that we have a differentiable manifold M for which at each point 'p' there exists a Tangent space T_pM, of all the vectors generated by an arbitrary vector field acting on the manifold. Now lets say we need to compare the vector at point p and another at point q, now these are two different vector spaces, and it isnt nessecary for them to overlap, so it maynot be possible. Ofcourse we can paralelly transport the vectors like we do in GR to get the Levi Civita connection, but one must remember that it too is a connectiondeefined on the tangent bundle of a Riemannian manifold. Now in order to get from p to q, we can allow the vector to "take a lift", so we lift our point of view to another manifold, one where p and q would be in the same neighbourhood. We'll call this manifold as the fibre space, and M as the base space. Now over every point of M, there exists a tangent space, now each tangent space defines what is called a 'fiber" of the fiber space, and we define a map called a "projection" from he fiber down to the point on the base manifold. Now coming back to our paralell transport problem, we now take the union of all the tangent spaces of all the points on the manifold to form the tangent bundle. Lets consider a simpler case, the trivial bundle, where the fiber space E=M×F where F is the fiber space,and 'x' is the cartesian product. Here a fiber over a point p will be given by: E_p={p}×F and the projection being: π(p,f)=p, p∈M; f∈F. Now getting back to the Tangent bundle, which we'll call TM=⋃T_pM, we define a 'section' as the embedding of the fiber space into the base manifld, or a little more simply, as a function from the base manifold to the fiber space. Let  me now define a vector bundle construction, first off, the set of all sections on E shall be denoted as:  , and  an endomorphism as :  , and the set of all endomorphisms on the bundle is deonoted as:  . Now Let us consider a Manifold M having open cover of sets {U_α}  and let ρ be a representation of a Lie group G on a vector space V. Now we glue together trivial bundles  to get a vector bundle . We define the entire bundle as the disjoint union:  . And we define transition functions between any two points of the bundle (p,v) and (p,v') i.e functions such as:  we identify the two point as:

we also see an essential "cocylce condition" i.e

Now the group structure arises from the orbit of the group on the fiber space. Now transformations along fibres occur via gauge transforms belonging to the group. now let us take a look at all the properties of a connection:-

now let us calculate the explicit expression of it:

Or we could alternatively see it as:

let us now see what "curvature" on a bundle means:

when we do calculations on local co ordinates, the secodn term vanshes because the vector fields are just basis vector fields, i.e partial derivatives, and they obviously commute. We call this curvature as the curvature two form, which looks like:

Now, Fibre bundles look like "Manifolds over Manifolds" so let us define  exterior calculus on them. Beggining with the exterior covariant derivative, defined on an E-valued p-form, or on vector valued forms i.e sections of the bundle: 

so the exterior covariant derivative looks like:

and now lets look at another analogous relation:

by virtue of equivalence of mixed partials, the second exterior derivative woud vanish i.e d²=0, but here, covariant derivatives dont commute, and the failure of their commutation is nothing but the curvature!!, ie:

Let me wrap it up, now we will assume that we are working with a G-Bundle and that D is a G-connection, then we have D=D⁰+A

where D⁰ is the standard flat connection, and the components of the vector potential live in the Lie Algebra of the Group, since:

Now if we choose U(1) as the group, then the right side of this commutes since the lie algebra of U(1) is abelian , so what we are left with is:

And that just the Electromagnetic Field!!! Pretty elegant huh?

By Vasudev Shyam (Admin at PDEP)