"Flip Tomato" wrote in message ...
Greetings--I'm trying to do some reading into relativistic quantum mechanics
(I've just taken quantum at the undergrad level) and I'm curious about the
covariant derivative that is used when discussing gauge invariance.

What motivates the definition of the gauge covariant derivative, other than
that it gives a nice result?

Weyl developed the idea while thinking about gravity. The latter has a
global symmetry, length scale. Weyl developed a geometry in which both
length and direction were strictly local - he called this "pure
infinitesimal geometry". The localization of the formerly global
length symmetry shows up as a new field alongside the metric, which
Weyl interpreted as the electromagnetic potential. Alongside the
general coordinate transformations at each point in spacetime one also
now has literal gauge transformations, that is, recalibration of the
local length scale. Under such a transformation, the Weyl gauge field
Am and the metric gmn change as

gmn - exp(L) gmn

Am - Am - d/dxm L

Under coordinate transformations the Am are just a covariant vector.
Thus exactly the right number of new fields are introduced, along with
the right transformation properties, to make a joint theory of light
and gravity. The conservation of energy-momentum and electric charge
are on the same logical footing in this theory.

In Weyl's geometry the covariant derivative of a tensor takes the form

Dm Tab.. = dm Tab.. + N Am Tab..

where N is the "conformal weight" of the tensor Tab.. and dm is the
usual Riemannian covariant derivative.

The theory failed for both physical and mathematical reasons - there
is no sensible Lagrangian theory leading to second order equations for
the gmn that are irreducibly coupled to the Am. Spacetime is just the
wrong dimension to make the idea workable (the spacetime dimension is
important because the volume element transforms differently depending
on the dimension). However, when the Dirac electron theory was
proposed, Weyl immediately saw that the arbitrary phase of the
electron field is of exactly the same nature as the arbitrary length
scale in his geometry, and is also associated with the conservation of
charge in nearly the same way as before - only now the covariant
derivative is applied to spinors with the form

Dm = dm + ie Am

so in a sense, in Dirac's theory the spinor field has "imaginary
weight" ie.

To sum up, the term "gauge" literally meant localized length, and
"covariant derivative" was a concept taken directly from Weyl's
modification of Ricci's "absolute differential calculus".

Nowadays of course the mathematicians have created the theory of fiber
bundles to systematize these connections and their covariant
derivatives.

-drl