Greetings--I have a few questions from(adsbygoogle = window.adsbygoogle || []).push({}); An Introduction to Quantum Field Theoryby Peskin and Schroeder.

Note: I'm not sure how to construct the contraction symbol using [tex]\LaTeX[/tex], so instead I will use the following cumbersome convention: [tex]\overbrace{\psi(x)\overline{\psi(y)}}=S_F(x-y)[/tex], they feynman propagator for a spin 1/2 particle, as in equation (4.111) of P&S.

First of all, I'm a little skeptical about the product [tex]\overline{\psi(y)}\psi(x)[/tex] where [tex]\psi(y) \equiv \psi(y)^\dag \gamma^0 [/tex] because the order seems backward. The product [tex]\overline{\psi}\psi[/tex] is ok because it is the "conventional" matrix multiplication of a row vector with a column vector to yield a scalar (with a gamma matrix inside). However, [tex]\psi \overline{\psi}[/tex] is the multriplication of a column vector by a [row vector times a gamma matrix]. This doesn't seem to make sense, and hence the commutators (say, the bottom of p. 54) with [tex]\overline{\psi}[/tex] and [tex]\psi[/tex] don't seem well defined to me. Thus, similarly, I'm unhappy with the definition of the contraction [tex]\overbrace{\psi(x)\overline{\psi(y)}}[/tex] in (4.108) on p. 116.

Moving on with the fermion Feynman rules, on p.119 P&S say "note that in our examples the Dirac indices contract together along fermion lines." I imagine this has to do with the order in which we translate Feynman diagrams into amplitudes (i.e. against the fermion arrows). However I'm not sure how this rule is "derived." Is this because one needs to move around fermion and boson operators (using anticommutation and commutation relations) such that the contractions are "nested": [tex]\overbrace{\psi\overbrace{\psi\overline{\psi}}\overline{\psi}}[/tex]?

Related to my uncomfortability with these contractions is the closed fermion loop that P&S describe at the bottom of p.120. I'm unclear about the derivation of equation (4.120) in which a closed loop of fermions yields a trace of fermion propagators. ((I'm happy with where the (-1) came from, the anticommutation relations of the [tex]\psi[/tex]s)). How does one go from the series of contractions [tex]\overbrace{\overline{\psi}\overbrace{\psi\overline{\psi}}\overbrace{\psi\overline{\psi}}\overbrace{\psi\overline{\psi}}\psi}[/tex] to atraceof these contractions? It seems like i'm not really understanding these fermion contractions.

Any assistance would be much appreciated!

Thanks,

Flip

flipt *at* stanford *dot* edu

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