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Tuesday, 31 January 2012

They said I was daft

They said I was daft to start a blog two days before a quantum field theory exam. Well, bah! I say to them. In fact if I post a few equations this might pass as some sort of study. And I want to try out MathJAX.
To illustrate the fact that symmetries lead to conservation laws consider the following simple O(N) invariant theory:
L=12μϕaμϕa
where ϕa are a set of real scalar fields with the a index running from 1 to N. To see the effect of the symmetry we vary the action under a symmetry transformation δϕa=iωA(LA)abϕb where (LA)ab is an orthogonal matrix, ωA is a set of infinitesimal parameters and A runs over the adjoint rep 1,...,12N(N1).
As a trick we let the ω coefficients depend on position, although only ωA=constant is actually a symmetry. This lets us pull out the conserved current because, since the action is invariant if ωA=constant, ω can only enter in to the first order change of the action through its derivatives. Thus δS=d4x jAμμωA
Integrating by parts and using the fact that ω is arbitrary gives μjAμ, the law of current conservation.
Carrying this out we find that the terms not proprtional to μωA vanish by the antisymmetry of LA and we are left with
δS=d4x μ(iμϕa(LA)abϕb)ωA
up to an irrelevant normal ordering constant. So the current is jAμ=iμϕa(LA)abϕb
and you can check by using the equation of motion ϕa=0 and antisymmetry of LA that it is indeed conserved.
Had we added an arbitrary O(N) invariant potential V(ϕaϕa) none of the work would have been affected, but now it would be an interacting theory instead of a noninteracting one. The symmetry of the theory guarantees that no matter how the interactions affect the dynamics of the theory the total charges QA=d3x N{jA0} are always conserved in any reaction.
Well, thanks for bearing with that. MathJAX works like a charm. I promise you it won't be this mathy every day. I don't think.