Covariant Derivative question

From: Flip Tomato (flipt_at_stanford.edu)
Date: 08/14/04 

Date: 14 Aug 2004 07:58:04 -0400

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).

Thanks,
Flip Tanedo
flipt (at) stanford (dot) edu

PS--on a totally unrelated note, I'm not very good with literature searches
yet... how do I find *review* articles in a subject that I'm interested in
studying?

Relevant Pages

  • John Baezs Strange Roles in Invariances and Intersections
    ... I have been looking through the intersection and invariance research ... this generalized literature from 1998 on, including Carlo Rovelli ... Loop quantum gravity," Living Rev. Relativity 1,, URL ... and Algebraic Field Theory and Local Quantum Field Theory and ...
    (sci.physics)
  • Re: Effectuationism and Value
    ... >> So how a principle strikes you is important rather than its experimental ... Then you should have made your context explicit by saying the invariance of ... > The Principle of Relativity says that the laws of physics are the ...
    (sci.physics.relativity)
  • Re: Well, Hobba? Re: A. Newtonian Invariance vs SR-cult fraud and corruption (Rev B)
    ... The reason is you are so confused about what invariance is ... The POR applies only to inertial frames and only to laws ... of nature; where one must ensure one understands what meant by law of ... GR is based on the principle of general invariance - not on the ...
    (sci.physics)
  • Re: Well, Hobba? Re: A. Newtonian Invariance vs SR-cult fraud and corruption (Rev B)
    ... The reason is you are so confused about what invariance is ... The POR applies only to inertial frames and only to laws ... of nature; where one must ensure one understands what meant by law of ... GR is based on the principle of general invariance - not on the ...
    (sci.physics.relativity)
  • Re: Well, Hobba? Re: A. Newtonian Invariance vs SR-cult fraud and corruption (Rev B)
    ... The reason is you are so confused about what invariance is ... The POR applies only to inertial frames and only to laws ... of nature; where one must ensure one understands what meant by law of ... GR is based on the principle of general invariance - not on the ...
    (sci.math)
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