Archive for August, 2009

Electrodynamics on a Principal Bundle II

August 16, 2009

Suppose we had a principal U(1)-bundle \pi:P\to M with a connection \omega with curvature \Omega.

The Lie algebra \mathfrak{u}(1) is just the set of imaginary numbers i\mathbb{R} with trivial Lie bracket {[},{]}=0. The local potential is a real-valued 1-form A_{U} defined by \omega_{U}=iA_{U}. The local field strength F_{U} is defined by \Omega_{U}=iF_{U}.

A change of gauge is given by g_{UV}=e^{i\lambda} with \lambda:U\cap V\to\mathbb{R}. We see that local connections are related by \omega_{V}=e^{-i\lambda}\omega_{U}e^{i\lambda}+e^{-i\lambda}de^{i\lambda}=\omega_{U}+id\lambda, so that local potentials are related by A_{V}=A_{U}+d\lambda. Local curvatures are related by \Omega_{V}=e^{-i\lambda}\Omega_{U}e^{i\lambda}=\Omega_{U}, so that local field strengths are related by F_{U}=F_{V}. This means that the field strength is globally defined on M.

By the Bianchi identity we have d\Omega=0 so dF=0, so the homogeneous Maxwell equation comes along for free. We can get the inhomogeneous Maxwell equation by requiring that d*F=*J.

Now, consider the action U(1) on \mathbb{C} given by multiplication e^{i\lambda}z. Associated to our principal U(1) bundle we get a vector bundle with fiber \mathbb{C} with an induced connection \nabla locally given by \nabla=d+\omega_U=d+iA_U. We will write sections of the associated bundle as \psi. We can define the d’Alembert operator \square=*\nabla*\nabla+\nabla*\nabla*. If we require the Klein-Gordon equation, \square\psi=m^2\psi, 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 O(1,3), i.e. matrices B such that B^{T}\eta B=\eta where \eta=diag(1,-1,-1,-1), or equivalently \eta(Bv,Bw)=\eta(v,w) for any events v,w in Minkowski spacetime. This group has 4 connected components coming from det(B)=\pm1 and B_{00}>0 or B_{00}<0. The component containing the identity is called the proper, orthochronous Lorentz group L=L_{+}^{\uparrow}. Physically it contains all rotations, and boosts (Lorentz tranformations) and so dim(L)=6.

We can cover L by the simply connected group SL(2,\mathbb{C}), i.e. 2\times2 complex matrices A with det(A)=1. First we identify Minkowski spacetime \mathbb{R}^{4} with the space of 2\times2 Hermitian matrices, i.e. matrices H such that \overline{H}^{T}=H, in such a way that if H is the Hermitian matrix identified with the event x then det(H)=|x|^{2}. Then we can define a covering map \Lambda:SL(2,\mathbb{C})\to L by identifying \Lambda(A)x with AH\overline{A}^{T}. We have that \Lambda(A)\in L since
|\Lambda(A)x|^{2}=det(AH\overline{A}^{T})=det(A)det(H)det(A)=det(H)=|x|^{2}. It can be shown that \Lambda is a 2-1 homomorphism of Lie groups.

Now, there are two important irreducible representations for SL(2,\mathbb{C}) on \mathbb{C}^{2}, the “spin \frac{1}{2}” representations given by multiplication A\left(\begin{array}{c} z_{1}\\ z_{2}\end{array}\right) and multiplication by the adjoint \overline{A}^{T}\left(\begin{array}{c} z_{1}\\z_{2}\end{array}\right). The Dirac representation is the direct sum of these representations \left(\begin{array}{cc}A& 0\\ 0&\overline{A}^{T}\end{array}\right) \left(\begin{array}{c}z_{1}\\z_{2}\\z_{3}\\z_{4}\end{array}\right).

Let \pi:FM\to M be the orthonormal frame bundle for spacetime. Its fibers F_{m}M are ordered orthonormal bases of T_{m}M, or equivalently isometries p:\mathbb{R}^{4}\to T_{m}M. There is a right action of O(1,3) given by right composition pB which makes the frame bundle an O(1,3)-bundle. We say that M is space and time orientable iff FM has 4 components and a choice of component FM_{0} is a space and time orientation. Then the restriction \pi:FM_{0}\to M is an L-bundle.

The solder form is an \mathbb{R}^{4}-valued 1-form \phi on FM_{0} given by \phi_{p}(X)=p^{-1}(\pi_{*}(X)). The torsion of a connection \theta on FM_{0} is \Theta=d\phi+\theta\wedge\phi. It turns out that there is a unique connection whose torsion is \Theta=0. This is the Levi-Civita connection \theta.

A spin structure on M is a manifold SM and a smooth map \lambda:SM\to FM_{0} such that \pi\circ\lambda:SM\to M is an SL(2,\mathbb{C})-bundle with \lambda(pA)=\lambda(p)\Lambda(A). We can define a connection \tilde{\theta} on SM by \tilde{\theta}=\Lambda_{*}^{-1}\lambda^{*}\theta where \Lambda_{*} is the isomorphism of Lie algebras induced by \Lambda:SL(2,\mathbb{C})\to L.

Now consider sections \psi of the vector bundle associated to SM by the Dirac representation. Dirac’s idea was to introduce an operator \not\hspace{-4pt}D such that \not\hspace{-4pt}D^{2}=\square, 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 v^{2}=\eta(v,v). It turns out that the smallest representation \gamma of this Clifford algebra is 4-dimensional which is why we need a 4-dimensional representation of SL(2,\mathbb{C}) as well. Then we can define the Dirac operator as \not\hspace{-4pt}D=\eta(\gamma,\nabla) where \nabla is the connection associated to \tilde{\theta} and we inner product them somehow.

In more detail for the d’Alembertian on Minkowski spacetime, \square=\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial z^{2}}, define

\not\hspace{-4pt}D=\left(\begin{array}{cccc} 1& 0& 0& 0\\ 0 & 1& 0& 0\\ 0 & 0& -1& 0\\ 0 & 0& 0& -1\end{array}\right)\frac{\partial}{\partial t}+\left(\begin{array}{cccc} 0& 0& 0& 1\\ 0 & 0& 1& 0\\ 0 & -1& 0& 0\\ -1& 0& 0& 0\end{array}\right)\frac{\partial}{\partial x}

+\left(\begin{array}{cccc} 0& 0& 0& -i\\ 0 & 0& i& 0\\ 0 & i& 0& 0\\ -i& 0& 0& 0\end{array}\right)\frac{\partial}{\partial y}+\left(\begin{array}{cccc}0& 0& 1& 0\\ 0 & 0& 0& -1\\ -1& 0& 0& 0\\ 0 & 1& 0& 0\end{array}\right)\frac{\partial}{\partial z}

We can work out that \not\hspace{-4pt}D^{2}=\square.

Then we demand that the Dirac equation holds, \not\hspace{-4pt}D\psi=m\psi. This gives us a theory of a spin-\frac{1}{2} 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 Spin(1,3)=SL(2,\mathbb{C}). 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 Spin_C(1,3)=U(1)\times SL(2,\mathbb{C})/(\mathbb{Z}/2).

We can splice a G_{1}-bundle \pi_{1}:P_{1}\to M with a G_{2}-bundle \pi_{2}:P_{2}\to M. Define P=\{(p_{1},p_{2})\in P_{1}\times P_{2}:\pi_{1}(p_{1})=\pi_{2}(p_{2})\} and \pi:P\to M by \pi(p_{1},p_{2})=\pi_{1}(p_{1})=\pi_{2}(p_{2}). This is a G_{1}\times G_{2}-bundle with (p_{1},p_{2})(g_{1},g_{2})=(p_{1}g_{1},p_{2}g_{2}). Given connections \omega_{1},\omega_{2} on P_{1},P_{2}, we can define a connection \omega on P by \omega=\pi^{1*}\omega_{1}\oplus\pi^{2*}\omega_{2} with \pi^{i}:P\to P_{i} given by \pi^{i}(p_{1},p_{2})=p_{i}.

Splice together our U(1)-bundle P with SM and also splice \omega with \tilde{\theta}. Consider the representation of U(1)\times SL(2,\mathbb{C}) on \mathbb{C}^{4} given by combining the Dirac representation with multiplication by e^{i\lambda}. This structure is \mathbb{Z}/2-invariant so defines a Spin_C(1,3) -bundle. We get an associated vector bundle with an associated connection and Dirac operator \not\hspace{-4pt}D. A charged electron coupled to electromagnetism is then a section \psi for which the Dirac equation \not\hspace{-4pt}D\psi=m\psi holds.

Electrodynamics on a Principal Bundle I

August 16, 2009

I want to switch gears and talk about some mathematical physics. Actually, I’m going to cross-post some exposition I wrote for a gauge theory seminar that we held at Stony Brook.

Maxwell’s equations in relativistically covariant form are

\partial_{\mu}F^{\mu\nu}=J^{\nu}
\partial_{[\lambda}F_{\mu\nu]}=0

Since F_{\mu\nu}=-F_{\nu\mu} we can define a 2-form F=F_{\mu\nu}dx^{\mu}dx^{\nu}. We can also define a 1-form J=J_\mu dx^\mu. Then we can re-express Maxwell’s equations using exterior differentiation and the Hodge star.

d*F=*J
dF=0

The continuity equation d*J=0 then follows from the inhomogeneous Maxwell equation. We expect from the homogeneous Maxwell equation that F=dA. In fact this is only true locally. This means that for every event m in our spacetime M there is an open set U with m\in U\subset M and a 1-form A_U on U with F|_U=dA_U. This follows from Poincare’s lemma.

We cannot say the A exists globally. For instance if F=sin\phi d\phi d\theta, the area form of the unit sphere in spherical coordinates, then dF=cos\phi d\phi d\phi d\theta=0 since d\phi d\phi=0 by antisymmetry of wedge product of 1-forms. Also, taking \Sigma to be the unit sphere, we know that \int_\Sigma F=4\pi. However, by Stokes’ Theorem, if F=dA then \int_\Sigma F=\int_\Sigma dA=\int_{\partial \Sigma} A=0\neq 4\pi. So, we cannot have F=dA globally.

Physically we interpret this as a magnetic monopole with magnetic charge 4\pi and worldline, the time axis, r=0. Mathematically, what is happening is that the complement of the time axis has nontrivial topology. Specifically its second de Rham cohomology is nontrivial. Intuitively, there is a kind of 2-dimensional, spherical “hole” in the complement of the time axis.

In addition to being nonglobal, the potential A is defined only up to addition of a closed 1-form since d(A+d\lambda)=dA=F. We would like to find a global mathematical object corresponding to the potential which doesn’t depend on our “choice of gauge”. This is our motivation for understanding connections on principal bundles.

We will assume G is a group of matrices. A principal G-bundle is a smooth surjection of manifolds \pi:P\to M with a free transitive right action R of G on P such that \pi^{-1}\pi(p)=pG and for any m\in M there is an open set U with m\in U\subset M and a diffeomorphism T_U=\pi\times t_U:\pi^{-1}(U)\to U\times G called a “local trivialization” such that t_U(pg)=t_U(p)g. Local trivializations correspond to the physical notion of “choice of gauge”.

Intuitively, P is a manifold composed of copies of the group G parametrized by the base space M. A good example is the boundary of the Mobius strip which can be thought of as a \mathbb{Z}/2-bundle over S^1.

A useful notion is that of a local section \sigma_U:U\to P with U an open set with U\subset M such that \pi(\sigma_U(m))=m. It can be shown that there is a canonical 1-1 correspondence between local sections \sigma_U and local trivializations T_U.

Define transition functions g_{UV}:U\cap V\to G by g_{UV}(m)=t_U(p)t_V(p)^{-1} where \pi(p)=m. This is well defined since t_U(pg)t_V(pg)^{-1}=t_U(p)gg^{-1}t_V(p)^{-1}=t_U(p)t_V(p)^{-1}. Transition functions correspond to the physical notion of “change of gauge”. We can relate any two local sections by \sigma_V=\sigma_U g_{UV}.

Let \mathfrak{g} be the Lie algebra for G. A connection \omega is a \mathfrak{g}-valued 1-form on P such that if If X\in\mathfrak{g} and \tilde{X} is the tangent field on P given by \tilde{X}_{p}=\frac{d}{dt}pe^{tX}|_{t=0}, then \omega(\tilde{X})=X. Also we require that R(g)^{*}(\omega)=g^{-1}\omega g.

We define local connections on M by \omega_U=\sigma_U^*\omega. Local connections are related by \omega_{V}=g_{UV}^{-1}\omega_{U}g_{UV}+g_{UV}^{-1}dg_{UV}.

We define curvature \Omega=d\omega+\frac{1}{2}[\omega,\omega] meaning \Omega(X,Y)=d\omega(X,Y)+\frac{1}{2}[\omega(X),\omega(Y)]. We can define local curvature by \Omega_U=\sigma_U^*\Omega. Local curvatures are then related by \Omega_V=g_{UV}^{-1}\Omega_U g_{UV}. The Bianchi identity says d\Omega=[\omega,\Omega].

We are now in a position to define electrodynamics on a principal bundle.

Jones’ Polynomial

August 6, 2009

In the last post we investigated the linking number and writhe. These were numerical invariants of oriented links and framed knots. Now I will introduce new invariants which take their values as polynomials.

For a given crossing, we can perform an operation called resolving or smoothing the crossing. We can do this in two ways.

0-smoothing

0-smoothing

1-smoothing

1-smoothing

Let us suppose that there is a polynomial invariant of links <L> in variables A,B,C so that concentrating on a neighborhood of a crossing in a diagram for L, we have that the following relation, called the skein relation, holds.

Skein Relation

Skein Relation

Performing smoothings on all crossings reduces a link diagram to some number of circles in the plane. Let’s require that adding a circle \bigcirc to a link diagram L gives <L\bigcirc>=C<L>. Finally we require a normalization, that for the empty link <>=1. From this we can deduce that the bracket of n circles is <\bigcirc\cdots\bigcirc>=C^n.

We need to check invariance under Reidemeister moves. Let’s start with Reidemeister 2.

Reidemeister 2 Calculation

Reidemeister 2 Calculation

Thus, in order for the bracket to be invariant we must have A^2+ABC+B^2=0 and AB=1. Solving for B,C in terms of A, we get B=A^{-1},C=-A^2-A^{-2}.

The nice thing now is that Reidemeister 3 comes along for free by using invariance under Reidemeister 2.

Reidemeister 3 Calculation

Reidemeister 3 Calculation

Performing Reidemeister 1 on the other hand does not leave the bracket invariant. However, we can see that opposite Reidemeister 1 moves cancel so that the bracket is invariant under the framed Reidemeister 1 move.

Reidemeister 1 Calculation

Reidemeister 1 Calculation

Consequently, the bracket is an invariant of framed links whose values are polynomials in A and A^{-1}. To calculate it, take a blackboard diagram for the framed link and apply the skein relation, the circle relation and the normalization relation until you reach the answer.

The bracket was introduced by Kauffman as an elementary way to define Jones’ polynomial, an invariant of oriented links which was originally derived using some difficult algebra. We can define the Jones’ polynomial by V(L)=-A^{-3TotWr(L)}<L>|_{A=q^{1/4}}. Here, TotWr(L) the total writhe is the sum of signs of all crossings in the diagram and it is this factor which makes V(L) now invariant under Reidemeister 1 moves.

The Kauffman bracket and Jones’ polynomial are very closely related, in a similar way to how the writhe and linking numbers are closely related. Following the discovery of the Jones’ polynomial, there was a great deal of interest in knot theory. The Jones’ polynomial showed new connections between topology on the one hand and representation theory and quantum physics on the other.