Flip Tomato wrote:

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? I understand the gradient (1)
mathematically as a derivative operator and (2) physically as a
momentum operator--but the gauge covariant derivative doesn't have any
intuitive appeal to me other than sketchily looking like it would
involve the gauge freedom of choosing A.

In Chris Quigg's "Gauge Theories of the..." book, he writes: "Local
phase invariance may be achieved if the equations of motion and the
observables involving derivatives are modified by the introduction of
the electromagnetic field A_\mu(x). If the gradient is everywhere
replaced by the gauge covariant derivative, [this is satisfied]."

In terms of the big picture, I understand (but please correct me if
i'm wrong) that local phase invariance is a general principle that we
would like to have in quantum mechanics, so we impose it by
introducing this gauge covariant derivative. The term in the gauge
covariatn derivative, A_\mu(x), then *turns out* to be the EM
potential and lo and behold, E&M pops out of this principle of local
phase invariance. The fact that E&M naturally pops out of this
principle--this is "evidence" to believe that local phase invariance
is a reasonable "first principle"?

I know my questions are a little hazy right now as I'm still trying to
get my head around these topics--but any insight would be much
appreciated (and probably followed by more precise questions).

One of the most important principles in physics are symmetries.
Especially in quantum theory it is needed to make the link between
observables and the mathematical formalism. Unfortunately, in many
textbooks this point is not made clear from the very beginning, but
they use some handwaving arguments like "canonical quantisation". It's
a pity that this procedure became the name "canonical". By chance, it
works for some cases, for most not. For instance, it goes wrong even in
relatively simple cases, when you try to quantise the free
non-relativistic spin-0 particle in spherical coordinates.

The real good quantisation starts from symmetry principles. First of all
you like a quantum theory which implements the space-time structure
under consideration. In the case of relativistic quantum theory that is
the Einstein-Minkowski space time.

Then you have the principles of quantum theory which are independent
from the actual realisation and thus valid for all quantum systems
(relativistic, non-relativistic, two-level models etc. etc.):

(1) A system is described by states. States are represented by rays in
Hilbert space, or equivalently by a member of the projective Hilbert
space, i.e. A ray in Hilbert space is the set [psi], which is defined
for a given |psi \neq 0 \in H

[psi]=3D{lambda |psi|lambda \in C}

If the system is prepared to be in state [phi], the probability to find
it in state [psi] is given by

P_{phi}(psi)=3D|phi|psi|^2/(phi|phipsi|psi)

It is easy to see that this probability and thus all predictions about
measurements fromquantum theory is independent of the choice of
representatives of the states [phi] and [psi].

It is important to keep in mind that not the kets |psi are representing
the state of a system but the rays [psi].

(2) Observables are represented by selfadjoint operators. The possible
outcomes of an exact measurement of an observable is given by the
spectrum (generalised eigenvalues) of the corresponding operator.

I do not want to go into the very complicated question of how to prepare
systems in a certain state through measurements and filtering
processes. You might read the introductory chapter in Sakurai's book
"Modern Quantum Mechanics".

Now comes the more interesting step to fill this concepts with real
physics, and as stressed before, for this we need the notion of
symmetries. In QT, a symmetry is given by a map of the states (rays in
Hilbert space) and observables (selfadjoint operators in Hilbert space)
such that all outcomes of predictions about experiments are unchanged.

To implement the space-time structure consistently in quantum theory,
one has to make sure that the symmetry operations on space time are
symmetries of the quantum theory, you like to construct.

Now it turns out that there is a little complication from the fact that
states are rays and not vectors in Hilbert space, but at the same time
it also turns out that exactly this is crucial to get the right answer
about real systems, consistent with all experiments done so far.

The important thing is that all symmetry operations together build a
group, and thus one has to look for all ray representations of the
group. Such representations are rather complicated to deal with and
thus it is important that there is the Wigner-Bargmann theorem:

Each ray representation can be "lifted" to a unitary or antiunitary
transformation of the central extension of the universal covering group
of the symmetry group of the system.

In our case of the Poincaregroup (inhomogeneous Lorentz group) this
means that we need to find only the unitary irreducible representations
of the Poincaregroup where the homogeneous Lorentz transformations,
i.e., the group SO(1,3) is substituted by its covering group SL(2,C). A
system, described by such an unitary irred. representation is called a
free elementary particle.

Then it comes out that there are two large classes of such
representations which lead to a physically interpretable quantum
theory.=20

(a) Elementary particles with a finite mass m (m^20)
(b) Massless elementary particles: m^2=3D0

The former are further determined by the spin of the particles, which
determines how the state kets of particles at rest change under
rotations. This case is not much more involved than the
non-relativistic particles (although the Galilei group, underlying
non-relativistic physics is a little bit more complicated, since there
are non-trivial different central extensions which do not lead to
physically meaningful theories as was shown by Wigner and Ion=FC).=20

In the standard model there are only very view of these representations
necessary: Particles with spin 0 (the Higgs boson, which perhaps is a
mathematical artifact and not a real particle, but let's wait what LHC
tells us about it) and particles with spin 1/2 (Quarks and leptons) in
the Dirac representation (perhaps the neutrinos are in fact Majorana
fermions, but this is also not completely clarified todate).

The massless case is more involved. There is no spin, because these
particles cannot be at rest. Instead you have the helicity, which is a
spinlike quantity. It can be interpreted as the projection of the spin
in direction of the momentum of the particle (in an arbitrarily chosen
standard direction, mostly chosen as the z-direction). The helicities
can be 0, \pm 1/2, \pm 1, etc.

For helicity 0 and \pm 1/2 there is no more trouble than in the case of
massive particles. In a properly chosen convention, these cases can be
treated as the limits of the massive cases with the mass taken to 0.

This is not true for particles with helicities \pm 1. Here, precisely
this limit becomes ambiguous. From group theory, it is clear why: One
cannot represent a massless helicity-1-particle by a set of wave
functions. There is no function space, which realises this
representations, but only a quotient space, i.e., massless
helicity-1-fields are represented by vector fields modulo pure gauge
fields, i.e., if the fields A_{\mu}(x) is a representation, then for
any scalar fields \chi also the field=20

A_{\mu}'(x)=3DA_{\mu}(x)+\partial_{\mu} \chi(x)=20

represents the same physics. Thus, from space-time symmetry, it follows
that there must be an additional symmetry for massless
helicity-1-particles: The gauge symmetry!

Now you like to describe interacting particles. Then this gauge symmetry
must stay a symmetry under any circumstances, because otherwise you get
inconsistent with the strucure of space and time!

Taken together these principles with the demand of renormalisability,
you end up with the minimal coupling principle, i.e., you introduce the
interactions of massless helicity-one-particles by the substitution of
derivatives with covariant derivatives.

From this point of view, you can realise the gauge principle as well
with abelian as with nonabelian gauge groups. As the success of the
standard model shows, both realisations are important to describe the
elementary particles.

As a textbook about all this, I can recommend only one source:

S. Weinberg, The Quantum Theory of Fields, Vol. I+II

Vol. III is about the extension of symmetry principles to supersymmetry.

Of course the more theoretical study of Weinberg's books should be
supplemented by other texts which show in more detail how actual
calculations are to be done.

With a little bad feeling, for this purpose, I suggest

Peskin/Schroeder, An Introduction to Quantum Field theory

to be read with care. The erratum list on the textbook's homepage is a
mandatory source of clarification, although some conceptual mistakes
(especially in the chapter about the spontaneously broken linear sigma
model) are not solved there.

--=20
Hendrik van Hees Cyclotron Institute=20
Phone: +1 979/845-1411 Texas A&M University=20
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366