Friday, July 4, 2014

Exact, Coexact and Harmonic (Hodge Theory)

Happy 4th of July

Today, in celebration of Independence day we'll have some mathematical fireworks :)

It is customary to learned in school about the dot product and the cross product. The dot product comes from projecting one vector onto the other, while the cross product creates a new (pseudo) vector out of two other vectors. The cross product is basically a historical accident which got accepted on due to its practical convenience but a better concept is the exterior product. Even better we can understand all of this in the framework of Clifford algebras.

Here is how it goes. We’ll work out the usual 3D space for convenience. Start with the 3 x,y,z unit vectors and call them: \( e_1, e_2, e_3\). Then introduce 2 practical rules:
  • \( e_1 e_1 = e_2 e_2 = e_3 e_3 = 1 \)
  • \( e_i e_j = - e_j e_i \) when \( i \ne j \)
Think of the unit vectors as matrices which collapse to the identity when multiplied by themselves, and anti-commutes.

Then you can have the following basis in general:
  • scalar: \( 1 \)
  • vectors: \( e_1, e_2, e_3 \)
  • bivectors: \( e_1 e_2, e_2 e_3, e_3 e_1 \)
  • trivector (pseudo scalar): \( e_1 e_2 e_3 = I \)
For two vectors \( A, B\), with \( A = a_1 e_1 + a_2 e_2 + a_3 e_3\) and \( B = b_1 e_1 + b_2 e_2 +  b_3 e_3\) the dot product is:

\(A\cdot B = \frac{1}{2}(AB + BA)\)

and the exterior product is:

\( A \wedge B = \frac{1}{2}(AB - BA) \)

and in general for two vectors:

\( A B = A \cdot B + A \wedge B \)

Here is what we can always do: given a scalar, vector, bivector, or trivector, we can multiply with \( I = e_1 e_2 e_3\) and this defines the Hodge dual \(A \rightarrow \star A \) 

So for example Hodge duality maps bivectors (which are oriented areas to preuso-vectors (the cross product vector orthogonal to the area):

\( A \wedge B = I (A \times B) \)

The Hodge dual exists not only for vectors and bivectors but for differential forms as well:

\( \star dx = dy \wedge dz, \star dy = dz \wedge dx, \star dz = dx \wedge dy \)

The unit volume is: \( vol = I = \star 1 = dx \wedge dy \wedge dz \)

and Hodge defined an inner product of any two p-forms \( \alpha , \beta \) as follows:

\( (\alpha , \beta) = \int <\alpha , \beta > \star (1) = \int \alpha \wedge \star \beta \)

last, Hodge introduces a codifferential \( \delta = {(-1)}^{n(p+1) + 1}\star d \star\)

and proved the Hodge decomposition theorem for any form \( \omega \) :

\( \omega (any form) = d \alpha (exact) + \delta \beta (coexact) + \gamma (harmonic) \)

where \( \Delta \gamma = 0\) Here \( \Delta = d \delta + \delta d\) is the Hodge Laplacian. FIREWORKS PLEASE!!!

Now here is some physics: Maxwell's equations:

Let \( A = A_\mu d x^\mu \) be the electromagnetic four potential. The electromagnetic field 2-form \(F \) is: \( F = dA\)

\( F = \frac{1}{2} F_{\mu \nu}dx^\mu dx^\nu \) with \( F_{\mu \nu}= \partial_\nu A_\mu - \partial_\mu A_\nu \)

Then Maxwell's equations are:

\( dF = 0, d \star F = \star J \)

and the electromagnetic Lagrangian is: \( L = \frac{1}{2} (F, F)\)

So why are we looking at this compact formalism for Maxwell's equations? Because electromagnetism is only one of the 4 fundamental forces in the universe: gravity, weak force, electromagnetism, strong force, and the weak and strong forces are described by Yang-Mills gauge theory which is a generalization of Maxwell's theory. Without a compact notation and a clear geometrical meaning, we have no hope of understanding Yang-Mills theory and we will be stuck forever in the La La land of using cross products, gradients, divergences, and equations in components.


    1. Are you going to discuss the heat kernel? The Laplace-Beltrami operator Δ = dδ + δd for δ = *d* defines the harmonic condition and the corresponding heat kernel. This is crucial for the Atiyah-Singer theorem and leads to results involving gauge theory on manifolds. I wrote a while back a bit on the Milnor’s theorem on spheres and bundles in 7 dimensions. This is connected to this as well, and leads to some interesting results with exotic structures in four dimensions.

      Cheers LC

      1. I was thinking to present next short exact sequences as a prerequisite for fiber bundles, but if you would like, you can have a guest post on Laplace-Beltrami operator.

        In my mind the plan after short exact sequences is to discuss curvature, first in general relativity and then in gauge theory by Yang's matrix trick. By the way, I am learning Morse theory now to get into advanced gauge theory topics.

      2. I read again my copy of Milnor’s lectures on Morse theory earlier this year. There is I think a connection with quantum physics. In particular an experiment such as the double slit experiment is a sort of topological obstruction, where paths in the path integral that enter on slit are in equivalent to those which enter the other slit. This is a topological way of thinking about quantum bits. In the case of the Aharonov-Bohm effect there is some general set of maps between the homotopy π_i(S^2) --- > G_i, say some homomorphism on the space, for i usually one and the S^2 the stereographic projection of the C^2 plane for U(1). This is maybe some way of looking at gauge theory from a topological perspective.

        Of course the more general form of this is Floer cohomology, which is a complex form of Morse theory. It seems that one could look at gauge theory according to general cohomologies of groups or gauge spaces over (R, C, H or O), so that we can in general look at SL(2, Q) for Q = (R, C, H or O) bundles as cohomologies of algebraic varieties.

        I am in some ways a bit out of my league with some of this. I have for the most part a rather basic understanding of these advanced area of mathematics. The current research in these areas is largely a bit off my horizon.

        Cheers LC

    2. Lawrence, Witten has uncovered the role of Morse theory in gauge theory, and I am after understanding this. Also Morse theory is related to 2-categories and the generalization of 3-point Feynman diagram to the string theory pants diagram.

      1. Witten has a paper related to this subject “Analytic Continuation of the Chern-Simons Theory” that is related to this. I have not read this, but it appears to relate aspects of CS theory with cycles in Morse theory. My interest is in looking at Bott periodicity in Morse theory with regards to this.

        Cheers LC

    3. I was not aware of this paper, but Witten had fundamental contributions in gauge theory:

    4. Look up the Hitchen fibration. It is similar to this; sometimes called the Higgs fibration.

      The paper by Witten I referenced is with the Chern-Simon's Lagrangian. This is the knot invariant in the S^1 ---> S^3 --- S^2, and is also a "knot topology" for the fibrations on S^4 in the 7-sphere. These different fibrations are in a sense "different knots."

      Cheers LC