Sunday, August 6, 2017

The origins of gauge theory


After a bit of absence I am back resuming my usual blog activity. However I am extremely busy and I will create new posts every two weeks from now on. I am starting now a series explaining gauge theory and today I will start at the beginning with Hermann Weyl's proposal.


In 1918 Hermann Weyl attempted to unify gravity with electromagnetism (the only two forces known at the time) and in the process he introduce the idea of gauge theory. He espouse his ideas in his book "Space Time Matter" and this is a book which I personally find hard to read. Usually the leading physics people have crystal clear original papers: von Neumann, Born, Schrodinger, but Weyl's book combines mathematical musings with metaphysical ideas in an unclear direction. The impression I got was of a mathematical, physical and philosophical random walk testing in all possible ways and directions and see where he could make progress. He got lucky and his lack of cohesion saved the day because he could not spot simple counter arguments against his proposal which could have stopped him cold in his tracks. But what was his motivation and what was his approach?

Weyl like the local character of general relativity and proposed (from pure philosophical reasons) the idea that all physical measurements are relative. I particular, the norm of a vector should not be thought as an absolute value, but as a value that can change at various point of spacetime. To compare at different points, you need a "gauge", like a device used in train tracks to make sure the train tracks remained at a fixed distance from each other. Another word he used was "calibration", but the name "gauge" stuck.

So now suppose we have a norm \(N(x)\) of a vector and we do a shift to \(x + dx\). Then:

\(N(x+dx) = N(x) + \partial_{\mu}N dx^{\mu}\)

Also suppose that there is a scaling factor \(S(x)\):

\(S(x+dx) = S(x) + \partial_{\mu}S dx^{\mu}\)

and so to first order we get that N changes by:

\(( \partial_{\mu} + \partial_{\mu} S) N dx^{\mu} \)
Since for a second gauge \(\Lambda\), \(S\) transforms like:

\(\partial_{\mu} S \rightarrow \partial_{\mu} S  +\partial_{\mu} \Lambda \)

and since in electromagnetism the potential changes like:

\(A_{\mu}  \rightarrow A_{\mu} S  +\partial_{\mu} \Lambda \)

Weyl conjectured that \(\partial_{\mu} S = A_{\mu}\).

However this is disastrous because (as pointed by Einstein to Weyl on a postcard) it implies that the clocks would change their frequencies based on the paths they travel (and since you can make atomic clocks it implies that the atomic spectra is not stable).

Later on with the advent of quantum mechanics Weyl changed his idea of scale change into that of a phase change for the wavefunction and the original objections became mute. Still more needed to be done for gauge theory to become useful.

Next time I will talk about Bohm-Aharonov and the importance of potentials in physics as a segway into the proper math for gauge theory. 

Please stay tuned.