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.