Fermion Contractioni in Feynman Diagrams

From: Flip (fliptomato_at_gmail.com)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 17:44:09 +0000 (UTC)

Greetings, I have a few questions about fermion contraction in Feynman
diagrams as I read through chapter 4 of An Introduction to Quantum
Field Theory by Peskin and Schroeder.

(1) First of all, the contraction operator is defined on p. 116
equation (4.108) as the anticommutatior of a positive frequency spinor
with a negative frequency spinor. This leads to terms like psi*psi-bar,
where psi-bar is defined as the hermetian conjugate of psi times
gamma_0. I'm concerned about these terms because it seems like the
multiplication is "backwards" -- i.e. psi-bar*psi is reasonable because
it's the matrix multiplication of a row vector times a column vector
(with some 4x4 matrix in between); however, the term psi*psi-bar is the
multiplication of a column vector times a row vector times a 4x4
matrix. Is this still a reasonable scalar product?

(2) That being said, I realized that I really did not understand these
fermion contractions when I got to p. 120 equation (4.120), where P&S
describe the amplitude for a closed fermion loop. For such a loop, one
ends up evaluating a series of contracted spinor which ends up yielding
a trace of propagators. However, I'm confused how the series of
contracted spinors yields a trace at all. (The minus sign is
understandable from the anticommutation relations.) How is the series
of contracted spinors even a "matrix" ?

(3) A page before (p. 119), P&S say "Note that in our examples the
Dirac indices ((what Dirac indices?)) contract together along the
fermion lines." I assume this is a statement about the order in which
fermions are "translated" into their mathematical amplitude... i.e.
backwards along fermion lines. I'm just not sure how this is *derived*
in the context of the book. ((In fact, I wasn't satisfied with it's
"derivation" in Griffiths, either.)) Is this rule a consequence of the
ordering of contraction operators? I.e. Contraction operators have to
be nested within one another, and this is what gives the ordering? (I
still don't see *how* though...)

Any assistance would be much appreciated!
Thanks much,
Flip
flipt *at* stanford *dot* edu

post script: I posted a similar (though not exactly the same) message
at http://www.physicsforums.com/showthread.php?t=58051 which has some
embedded LaTeX, if that helps readability, I'm not sure if this is a
violation of decorum by dual-posting to a usenet and non-usenet board
(in which case I apologize).