Suppose we had a principal -bundle with a connection with curvature .

The Lie algebra is just the set of imaginary numbers with trivial Lie bracket . The local potential is a real-valued 1-form defined by . The local field strength is defined by .

A change of gauge is given by with . We see that local connections are related by , so that local potentials are related by . Local curvatures are related by , so that local field strengths are related by . This means that the field strength is globally defined on .

By the Bianchi identity we have so , so the homogeneous Maxwell equation comes along for free. We can get the inhomogeneous Maxwell equation by requiring that .

Now, consider the action on given by multiplication . Associated to our principal bundle we get a vector bundle with fiber with an induced connection locally given by . We will write sections of the associated bundle as . We can define the d’Alembert operator . If we require the Klein-Gordon equation, , then we have a theory of a charged spin-0 particle coupled to electromagnetism.

In order to couple electromagnetism to more interesting particles like Dirac’s electron, we need to incorporate spin somehow.

Consider the matrix group , i.e. matrices such that where , or equivalently for any events in Minkowski spacetime. This group has 4 connected components coming from and or . The component containing the identity is called the proper, orthochronous Lorentz group . Physically it contains all rotations, and boosts (Lorentz tranformations) and so .

We can cover by the simply connected group , i.e. complex matrices with . First we identify Minkowski spacetime with the space of Hermitian matrices, i.e. matrices such that , in such a way that if is the Hermitian matrix identified with the event then . Then we can define a covering map by identifying with . We have that since

. It can be shown that is a 2-1 homomorphism of Lie groups.

Now, there are two important irreducible representations for on , the “spin ” representations given by multiplication and multiplication by the adjoint . The Dirac representation is the direct sum of these representations .

Let be the orthonormal frame bundle for spacetime. Its fibers are ordered orthonormal bases of , or equivalently isometries . There is a right action of given by right composition which makes the frame bundle an -bundle. We say that is space and time orientable iff has 4 components and a choice of component is a space and time orientation. Then the restriction is an -bundle.

The solder form is an -valued 1-form on given by . The torsion of a connection on is . It turns out that there is a unique connection whose torsion is . This is the Levi-Civita connection .

A spin structure on is a manifold and a smooth map such that is an -bundle with . We can define a connection on by where is the isomorphism of Lie algebras induced by .

Now consider sections of the vector bundle associated to by the Dirac representation. Dirac’s idea was to introduce an operator such that , i.e. the Dirac operator is the “square root” of the d’Alembert operator. A full understanding of the Dirac operator requires Clifford algebras, i.e the algebra generated over Minkowski space modulo . It turns out that the smallest representation of this Clifford algebra is 4-dimensional which is why we need a 4-dimensional representation of as well. Then we can define the Dirac operator as where is the connection associated to and we inner product them somehow.

In more detail for the d’Alembertian on Minkowski spacetime, , define

We can work out that .

Then we demand that the Dirac equation holds, . This gives us a theory of a spin- particle, an electron or positron, but we have not yet coupled it to electromagnetism.

Right now, our notion of an electron is that it is a field which takes its values in a representation of the spin group . In order to couple to the electromagnetic field, we will rather think of the electron taking its values in a representation of the charged spin group .

We can splice a -bundle with a -bundle . Define and by . This is a -bundle with . Given connections on , we can define a connection on by with given by .

Splice together our -bundle with and also splice with . Consider the representation of on given by combining the Dirac representation with multiplication by . This structure is -invariant so defines a -bundle. We get an associated vector bundle with an associated connection and Dirac operator . A charged electron coupled to electromagnetism is then a section for which the Dirac equation holds.

## Leave a Reply