Yang's Matrix Trick

It is time to come back to the math series. Today I want to talk about a remarkable similarity spotted by Chen-Ning Yang, the same Yang from Yang-Mills' theory.

Yang-Mills gauge theory is a generalization of electromagnetism when the gauge group is non-abelian. 

Maxwell's equations can be written as:

\( F_{\alpha \beta}= \partial_\alpha A_\beta - \partial_\beta A_\alpha \)

where \( F\) is the electromagnetic tensor and \( A \) is the electromagnetic four-potential.

From Maxwell to Yang-Mills, the generalization is simply by adding the commutator of the potentials:

\( F_{\alpha \beta}= \partial_\alpha A_\beta - \partial_\beta A_\alpha + A_\alpha A_\beta - A_\beta A_\alpha \)

Now here is the magic: if you recall from a prior post the Riemann curvature tensor is:

\( R^{\delta}_{\alpha \beta \gamma} = \partial_\alpha \Gamma^{\delta}_{\beta \gamma} - \partial_\beta \Gamma^{\delta}_{\alpha \gamma} + \Gamma^{\delta}_{\alpha \mu} \Gamma^{\mu}_{\beta \gamma} - \Gamma^{\delta}_{\beta \mu} \Gamma^{\mu}_{\alpha \gamma} \)

we have the following identification which makes the Yang-Mills equation identical with the Riemann curvature:

\( A_\alpha = \Gamma^{\delta}_{\alpha \gamma}\)

\( F_{\alpha \beta} = R^{\delta}_{\alpha \beta \gamma } \)

This hints at a deeply geometrical interpretation of the gauge theory because both the Riemann curvature and Yang-Mills equations are nothing but Cartan's structural equations in disguise: 

\( F = d A + A \wedge A\)

There are 4 fundamental forces in our universe: gravity (SL(2,C)), electromagnetism (U(1) gauge theory), weak force (SU(2) gauge theory), and strong force (SU(3) gauge theory) and all four can be expressed in the form above proving that in nature curvature = force. This is easiest to understand in general relativity, but even there there is a very surprising fact requiring a big conceptual leap: even empty space can curve.

Next time we'll slowly start exploring gauge theory in depth starting with Maxwell's equations. Then all those abstract equations will become much more intuitive.