Intuitive meaning of Yang-Mills

Question

Is it fair to say that the "new" thing about Yang-Mills equations is that they "bend" the probability amplitude locally like mass bends space in general relativity?

Answer 1

Err, no. Yang himself said:

What Mills and I were doing in 1954 was generalising Maxwell's theory. We knew of no geometrical meaning of Maxwell's theory and were not looking in that direction. To a physicist, gauge potential is rooted in our description of the electromagnetic field. Connection is a geometrical concept I learnt about 1970.

Now Maxwells equations can be written transparently in a covariant manner on 4d spacetime, curved or flat (and in fact, in higher dimensions too) as:

$dF = 0$ and $\delta F = J$

Where $F$ is the EM field strength and $J$ is the EM current and $d$ is the exterior derivative. Here, we are using the language of differential forms, which are just anti-symmetric covariant tensor fields in the language of GR. (Or as I prefer to think of them, as cofields).

Whilst the Yang-Mills equations for a connection $\nabla$ are:

$d^{\nabla} F = 0$ and $\delta^{\nabla} F= J$

Here, $F$ is the YM field strength and $J$ is the YM current.

These equations obviously generalise the above (just look at them!) But more precisely, they reduce to EM when we look at the associated vector bundle of the trivial principal bundle with gauge structure group a circle by it's fundamental representation. In particular, the gauge algebra is just the reals and the connection will be trivial, hence the covariant exterior derivative $d^{\nabla}$ reduces to just the ordinary exterior $d$ and the same for the covariant exterior coderivative, $\delta^{\nabla}$, which reduces to the ordinary coderivative $\delta$.

Answer 2

The work of C.N. Yang and R. Mills, which took place in the early fifties, was based on the bold assumption that the symmetric properties of particles are essentially local. This hypothesis constitutes in a way the basis of a gauge theory from a modern point of view. This hypothesis thus establishes a bridge between global symmetries and local symmetries exactly as Einstein's theory of general relativity does, which connects the different reference frames. What is remarkable is that the theory of general relativity deals only with the force of gravity while the Yang-Mills theory deals only with the strong nuclear force. Thus, Yang and Mills treated a microscopic problem in a similar way to Einstein's for a problem of an extremely distant scale. This similarity between these initially seemingly distant theories allowed the concept of a Gauge symmetry to be promoted as a deep foundation for the subsequent formulation of larger, more unifying theory.