03 December 2012

Why is local symmetry called 'gauge' symmetry in quantum field theory?

What follows is the English version of a recent post of mine in Italian, Le teorie di calibro di Weyl, aimed at reaching a larger audience for the answer to the question in the title, which seems to be quite an unknown issue, albeit an unimportant one, at least from the physics point of view.
But then I stumbled upon the Thanksgiving 2012 post of Sean Carroll, where you can read:
[...] it’s called a gauge field, because Hermann Weyl introduced an (unhelpful) analogy with the “gauge” measuring the distance between rails on railroad tracks.
and the fact that Sean is certainly a non-average physicist leads me to believe that it may not only be just an unknown matter, but a misknown one: I don't know where Sean read about the gauge as a metaphor for the rails distance, but the story is quite different from that — and by far more interesting.
My original post was intended as a summary of an issue I already pointed out here and there in the comments section of different blogs (all of them in italian). I can't remember for sure where I read of it, most probabily in the Gravitation book, but it is not such a mystery since the wikipedia page on gauge theory, both in English and in Italian, says all, and even more, what I know about it.
 
A starting point can be the question of which was, historically, the very first gauge theory. Such a question mixes up words and meanings, so the answer needs to clarify.
For sure the Maxwell's classical electrodynamics is the oldest theory among the ones we nowadays say they have a gauge symmetry, but at the time of Maxwell the term "gauge symmetry" was unknown and such a symmetry was actually intended as a mere redundancy of the potentials fields with respect to the physical fields.
The theory that for the first time was expressed in term of a local symmetry is General Relativity. But, again, not when it was first presented by Einstein, where the now-we-know-is-a gauge symmetry was just stated as the Principle of general covariance, i.e. as a coordinate change invariance.
 
Entering Weyl.
 
The gauge story begun when he recast Einstein's General Relativity using a different but equivalent formalism, according to which the principle of general covariance can be seen as the invariance of the theory under an arbitray local rotation of the tetrad, the base of the tangent bundle vector space.
But even here the term gauge was not introduced yet.
Eventually it was introduced a moment later, by Weyl of course, while trying to extend his tetrads formalism. The idea was to require the theory to be invariant under an arbitrary local change not only in the tetrad orientation, but also in its scale factor. Such a new symmetry was supposed to generate the Maxwell equations for the electromagnetic four-potential, in the very same way in which the tetrad orientation symmetry generates the Einstein equations for the gravitational field. This is why Weyl introduced the name gauge for such a symmetry, the scale factor being the "size" of the base's vectors as measured by a gauge (and no railroad analogy was involved).
Unfortunately, such an attempt didn't work in his original formulation. The idea behind was of great value though. It turned out, in fact, that electrodynamics could actually be represented as a local symmetry field theory: the two key points were to keep the idea of a rotational symmetry (like the one of the tetrads) and to give up the idea of a symmetry of the tangent bundle. The Maxwell four-potential, now we know, had to be seen as the Lie generator of a U(1) rotational symmetry of an additional "abstract" fiber bundle attached to the space-time manifold. More generally, all fundamental interactions are today understood as the effect of a symmetry each with respect to its own Lie group SU(3) × SU(2) × U(1). Despite the fact that no size nor scale was involved any more, the name gauge got stuck to all such a local symmetry field theories.
 
But our story has much more than just historical and etymological lesson to be learned.
 
Usually in quantum field theory lectures the gauge symmetry idea is presentes as an upgrading procedure from a global symmetry to a local one. The prototypical example is the phase e of the wave function, which is assigned a space dependence eiφ(x) to. The power of such extension is clear, since this requirement is just enough, alone, to get the Maxwell electrodynamics equations, via minimal coupling.
But anyway such a recipe seems to come out of the blue: why should we devise such a point-to-point change of phase symmetry? Moreover, such a requirement is usually intended as a stricter constraint to the theory, since the transformation class under which the theory should remain unchanged is wider. So what?
Well, the real thing behind "gauging the global symmetry" lies precisely in the reason which moved Weyl to formulate its gauge theory of electrodynamics, which in turns boils down to the principle of general covariance.
When Einstein requires that equations of physics should be invariant with respect to any coordinates change, he is actually requiring a stronger symmetry, compared the previous requirement that equations should be invariant under just inertial reference changes. But the meaning of such a requirement is definitely the very opposite of imposing a stricter constraint: the meaning is just to relax the constraint that physical law are valid just within a small, arbitrary subset of reference frames! In the very same sense, the requirement that a field theory should have a local symmetry has to be intended as the release of the constraint that the bases of the Lie algebra of the gauge group should be rigidly oriented everywhere.
Hence, just like the gravitational field is an inertial effect due to the "connection" (in the technical sense of differential geometry) between two point at a finite distance which are not reciprocally inertial, in the same way the electromagnetic field is a U(1)-inertial effect due to the connection between two point at a finite distance which have the algebra bases not aligned each other.

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