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  Trace of derivatives of unitary operators

+ 4 like - 0 dislike
1334 views

I have been studying some lecture notes on the non-linear sigma model and I came up with some difficulties involving a trace. I have the following unitary operator U=exp(iτΦ(x)fπ)

How do I calculate Tr(μUμU)? The problem had arisen when it was shown (without proof) that the kinetic term of the sigma-meson field was given by 12μσμσ=fπ4Tr(μUμU). The sigma-meson field is σ=fπcos(Φ(x)fπ)=fπ+O(Φ2) and the pion field is π=fπˆΦsin(Φ(x)fπ)=Φ(x)+O(Φ3)

This post imported from StackExchange Physics at 2015-01-22 11:33 (UTC), posted by SE-user Judas503
asked Jan 21, 2015 in Theoretical Physics by Judas503 (20 points) [ no revision ]
What matrices are the τ? Isospin? How do the notes you're reading define their normalization?

This post imported from StackExchange Physics at 2015-01-22 11:33 (UTC), posted by SE-user 0celo7

1 Answer

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I'm not sure how the τ matrices are normalized, but we can get quite far without that. The first thing we should do is take the derivative μU=ifπ(τμΦ)U

and μU=ifπ(τμΦ)U
(I'm making the guess that the τ are Hermitian.) Then μUμU=f2π(τμΦ)(τμΦ)
Note that the Us cancel out because they are unitary and commute with the τ. We then write μUμU=f2πμΦiμΦjτiτj
If we know the normalization tr(τiτj), then we can complete the expression.

This post imported from StackExchange Physics at 2015-01-22 11:33 (UTC), posted by SE-user 0celo7
answered Jan 21, 2015 by 0celo7 (50 points) [ no revision ]
I'm not entirely convinced that τ commutes with the unitary. Could you provide a proof of this fact please?

This post imported from StackExchange Physics at 2015-01-22 11:33 (UTC), posted by SE-user Phoenix87
@Phoenix87 I guess the commutation relations could make that tricky. I'm having an off day today. I can fix this, but I have to make an assumption and I'm not sure I can make it. I'll assume (U)=(U). Then we have μU=(i/f)(τμΦ)U as usual but μU=(i/f)U(τμΦ) because transpose changes the order of matrices. Then, using the cyclicity of the trace, we can get the unitaries to cancel.

This post imported from StackExchange Physics at 2015-01-22 11:33 (UTC), posted by SE-user 0celo7
μ[τΦ(x)] in general will not commute with τΦ(x), in which case the first step will be incorrect.

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