Physics 7270: Intro To Quantum Mechanics III - Fall 2016
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First, a quick correction of a comment from last lecture: light-by-light scattering has in fact never been directly observed,
although there are experimental eorts underway. What has been seen

Physics 7270: Intro To Quantum
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CPT and Fermion Bilinears
Parity and spinors
Last time, we were finishing our discussion of parity, which acted on scalars as
P : (t, x) (t, x).
where the sign gives the intrinsic parity of the particle described by .
L

Physics 7270: Intro To Quantum
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Fermion signs, continued
Last time, we were looking at e e e e in QED, and concluded that because of fermion
anticommutation, the two diagrams contributing at leading order have a relative minus sign
between them:
tot =

Physics 7270: Intro To Quantum
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The Feynman propagator for Dirac fermions,
continued
Last time, we finished by calculating the object
0| (0) (x)|0 =
3 p
1
0|(asp usp + bsp vsp )(aqs u qs eiqx + bqs v qs eiqx)|0.
2
k
p
3 q
s,s
and noting that the onl

Physics 7270: Intro To Quantum
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A few last words on CPT, before we turn to spin-statistics. We ended with the observation that
the CPT theorem tells us that any Lagrangian which is Lorentz invariant and gives a unitary
theory must be invariant under t

Physics 7270: Intro To Quantum
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Cluster decomposition, continued
Last time, we ended our discussion by comparing connected and disconnected Feynman
diagrams. We noted that these processes are distinguishable by their singularity structure: a
disconnec

Physics 7270: Intro To Quantum
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The massive vector field, continued
Picking up from last time, we were trying to write down kinetic terms for a vector-valued field
A , and arrived at the Proca Lagrangian
1 2
1
= F
+ m2 A2 ,
4
2
where
F = A A .
This i

Physics 7270: Intro To Quantum
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Gauge symmetry and the comparator
Recall that last time, we were trying to fix up the interaction term
A
by proposing that both A and the complex scalar transform under the gauge symmetry as
A (x) A (x) + (x)
(x) ei(x)

Physics 7270: Intro To Quantum
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Gauge invariance in scalar QED
Lets pick up on the calculation we started last time for . We wrote down three diagrams,
and the total amplitude from summing them can be written in the form
= e2 X
(p3 ) (p4 ),
where X

Physics 7270: Intro To Quantum
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Before we pick up where we left o, a quick note on some abuse of notation. Weve been
writing the total cross section as the integral of the dierential cross section,
=
d
d
.
d
This statement is just the fundamental theo

Physics 7270: Intro To Quantum Mechanics
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Symmetry factors and the Dyson series
Last time, we studied the Schwinger-Dyson equations and the perturbative formulas they give for n -point
correlation functions. We found that the algebra can be replaced w

Physics 7270: Intro To Quantum Mechanics III
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Contractions and Wicks theorem
Picking up from last time, we showed that the n -field time-ordered matrix elements we need for the LSZ
formula can be written in terms of time-ordered matrix elements of fre

Physics 7270: Intro To Quantum
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The photon propagator
Recall that were calculating the propagator for vector fields,
0| cfw_A (x)A (y)|0 = i
4 peip(xy) (p).
where comes from solving the classical equations of motion. Last time we studied the
massive c

Physics 7270: Intro To Quantum
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Cluster decomposition, continued
Last time, we ended our discussion by comparing connected and disconnected Feynman
diagrams. We noted that these processes are distinguishable by their singularity structure: a
disconnec

Physics 7270: Intro To Quantum
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LSZ Reduction, continued
Back to the LSZ reduction. Last time, we ended considering the expression
f |S|i = f ; t = +|i; t =
= 2n/2
i
i
|ap3 (+). . . apn (+)ap1 ()ap2 ()|.
This is halfway to what we want - it relates S

Physics 7270: Intro To Quantum
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Scattering theory
Continuing from last time, we had just defined the cross-section and luminosity L for
quantum scattering; now a few more words on luminosity. Luminosity is generally timedependent; without getting into

Physics 7270: Intro To Quantum
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Feynman rules
We now only need one final ingredient to be able to take a quantum field theory Lagrangian
and make experimental predictions with it: the technique of Feynman diagrams, which give
an elegant construction f

Physics 7270: Intro To Quantum
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Old-fashioned perturbation theory
Before we proceed further into quantum field theory, it will be instructive to consider what
happens if we move too quickly and try to apply old-fashioned perturbation theory to
quantum

Physics 7270: Intro To Quantum
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Example: muon decay
Well continue into the example we set up last time, which was looking at the muon decay
process e e :
Right now, well mainly be concerned with dealing with the initial and final states properly,
whic

Physics 7270: Intro To Quantum
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Introduction to Quantum Fields
Coupled quantum harmonic oscillator, revisited
Last time, we considered a quantum system consisting of N particles in a one-dimensional
harmonic oscillator potential. We found that we coul

Physics 7270: Intro To Quantum
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Introduction
Welcome to Physics 7270, Introduction to Quantum Mechanics III - more commonly known
as Quantum Field Theory.
Syllabus
We started class by going over the syllabus
(http:/www.colorado.edu/physics/phys7270/ph

Physics 7270: Intro To Quantum
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Symmetry Currents and Propagators
Last time, we ended studying the following Lagrangian:
1
1 2 2
= a a m a
2
2
where a = 1, 2, 3 . This Lagrangian is symmetric under SO(3) rotations in the field space,
from which we de

Physics 7270: Intro To Quantum
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Conventions and Harmonic Oscillators
Fourier transformations
Finishing up from last time, we have a few more conventions to get through. Our convention
for Fourier transforms will be to put all of the (2) factors into t

Physics 7270: Intro To Quantum
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Causality and classical field theory
Causality revisited
We now have all the tools we need to see what happens to causality in our free scalar field
theory from last time, which we claim is fully consistent with special