Quantum Field Theory (524-001, 27855)
Spring 2007 from 5:30 to 6:45
on Mondays and Wednesdays in room 5 (but we may switch to 190 if the
students want to switch).
The course is a
continuation of 523, but students may
jump in without having taken 523. Students are expected to ask
questions in and out of class.
The course is designed for
graduate students in physics, especially those in optics,
condensed-matter, astro, or particle physics.
It will discuss the quantum theory of fields with an emphasis on
quantum electrodynamics and the standard model.
I plan to do in class many
computations of QED cross-sections.
The principal topics will be: The Canonical Formalism, Quantum Electrodynamics, Path-Integral Methods,
Renormalization, The Standard Model.
The textbook is The Quantum Theory of Fields,
Vol. I: Foundations, by Steven Weinberg (Cambridge University Press
1995,
reprinted with corrections 1996, 1998, 1999, 2000, 2002, 2003, &
2004
& 2005, ISBN 0-521-55001-7).
The bookstore will carry it
but probably at a high price
and in an old printing,
so I suggest ordering it elsewhere, such as
Amazon
which charges $36.90 for the latest paperback printing, which surely
has fewer typos than the hardcover editions. For those who have
an earlier printing of this book
or of volumes II and III, here are my files of typos
for Vol. I,
Vol. II,
and Vol. III.
Principle of Stationary Action
The action I is an integral over space-time
I = ∫d4x L(x)
of a Lagrange density L(x), which
usually is a function of the fields fi(x)
and their first partial-derivatives
fi,a(x) = dif(x)/dxa
with respect to xa for a = 0 (time), 1, 2, 3
L(x) = L(
fi(x), fi,a(x) )
(I am avoiding Greek letters, italic fonts, and partial-derivative
symbols to save time.)
Principle of Stationary Action
The field equations of both classical and quantum field
theories follow from the principle of stationary
action. Here is the idea: Suppose that we change the fields
fi(x) by a tiny quantity dfi(x)
fi(x) ' = fi(x)
+ dfi(x)
the action
changes only in second order (dfi(x))2
when the fields obey the field equations. Note that the dithered
partial-derivative fi,a(x)
' is
fi,a(x) '
= fi,a(x)
+ (dfi(x)),a
so that the tiny change dfi,a(x)
in the partial-derivative
fi,a(x) ' is
dfi,a(x)
= fi,a(x) '
−
fi,a(x)
= (dfi(x)),a
.
That is the change in the derivative is the derivative
of the change.
Let's require that the change dI in the action I be of second order in
the changes dfi(x)
in the fields fi(x). In
general, the change dI is
dI = d∫d4x
L(x) = ∫d4x
dL( fi(x),
fi,a(x) ) = ∫d4x
(dL/dfi
) dfi(x)
+ ( dL/dfi,a)
(dfi(x)),a
in which we sum over all the fields fi.
Now we integrate the last term by parts, throwing away the surface term
because our fields all vanish at large distances:
dI = ∫d4x
[ (dL/dfi
) − (
dL/dfi,a),a
] dfi(x)
.
We require that dI vanish for all
tiny variations dfi(x).
It follows then that
0 = (dL/dfi
) − (
dL/dfi,a),a
which is Lagrange's equation of
motion.
Joseph-Louis
Lagrange 
was born on 25 Jan 1736 in Turin, Sardinia-Piedmont (now Italy) and
died on 10
April 1813 in Paris, France.
At the age of 19, he became professor of mathematics
at the Royal Artillery School in Turin. He survived the French Revolution in part
because he believed that "one of the first principles of every
wise man is to conform strictly
to the laws of the country in which he is living, even when they are
unreasonable." When the
chemist Lavoisier was guillotined, Lagrange said, "It
took only a moment to cause this head to fall, and a hundred years will
not suffice to produce its like."
Symmetries of the Lagrange Density
Often the Lagrange density L(x) is invariant
under a scrambling of the fields among themselves
fi(x) ' = fi(x)
+ dfi(x)
at the same point x. Such a symmetry is called "internal." Each internal symmetry leads to a
conserved current ja,
that is, a combination of the fields and their derivatives
that satisfies
0 = ja,a .
The invariance of
the Lagrange density L(x) implies
0 = (dL/dfi
) dfi(x)
+ ( dL/dfi,a)
(dfi(x)),a
.
Lagrange's equation of
motion allow us to write this as
0 = (
dL/dfi,a),a
dfi(x)
+ ( dL/dfi,a)
(dfi(x)),a
= [ (
dL/dfi,a)
dfi(x)
],a .
So the conserved current ja is
ja
= (
dL/dfi,a)
dfi(x)
.
Here are some notes on fermion-fermion,
fermion-boson, and boson-boson scattering
and on fermion-antifermion
scattering and on the solution to Weinberg's problem 6.1.
Here are some notes on sections 7.3 &
7.4 of
Weinberg's book.
Here are 66 pages of my notes on Weinberg's
chapter 7.
Here are 10 more pages of my notes on chapter
7.
Here are 64 pages of my notes on chapter 8.
To view a video file of a lecture, students should download the file to
their computers.
Video
(wmv) file of lecture of Monday 29 January 2007
on Feynman diagrams for fermion-boson scattering.
Video of
lecture of Wednesday, 31 January 2007 on
current conservation, the energy-momentum
tensor, and the six currents that are conserved when this tensor is
symmetric.
Video of
lecture of Monday, 5 February 2007 on Belinfante's symmetric energy-momentum
tensor and the formula it gives for the angular-momentum
operators. This formula is the basis for understanding the spin
of the proton.
Video of
lecture of Wednesday, 7 February 2007 on boson-boson scattering.
The lecture of 9 February was snowed-out.
Video of
lecture of Monday, 12 February 2007 on fermion-anti-fermion
scattering and on boson-boson scattering in the theory of problem 6.1;
remarks about cross-sections.
Video of
lecture of Monday, 19 February 2007 on the quantization of the
theory of a massive vector boson.
Video of
the lecture of Wednesday, 21 February 2007
on the quantization of the
theory of a Dirac fermion and on constraints--regrettably without
audio.
Video of
lecture of Monday, 26 Feb. on Dirac brackets.
Video of
lecture of Wednesday, 28 Feb. on
quantum electrodynamics--regrettably
without audio.
The lectures of March 5th and 7th were postponed because of a
biophysics meeting; they will be rescheduled. The lectures of
March 12th and 14th did not occur because of Spring Break.
First
video of lecture of Monday, 19 March, second
video, and third
video on
the use of gauge fixing and Dirac brackets to quantize electrodynamics.
Video of
lecture of Wed., 21 March, on the photon propagator and the Feynman
rules for QED.
Video of
lecture of Mon., 26 March, on Compton scattering.
Video
of lecture of Wed., 28 March, on Compton scattering. (tail end
of that video)
Notes on Compton scattering.
Video
of lecture of Monday, April 2d, on path integrals.
Video
of lecture of Wednesday, April 4th, on path integrals.
Video
of lecture of Monday, April 9th, on path integrals.
Notes on how to do multiple gaussian integrals.
Video of
lecture of Wednesday, April 11th, on path integrals.
Unfortunately, the file is very noisy and jerky, probably unusable.
Video of
lecture of Monday, April 16th, on fermionic path integrals.
Video of
lecture of Wednesday, April 18th, on fermionic path integrals.
Notes on fermionic path integrals.
Video of
lecture of Monday, April 23d, on the path-integral formulation of QED.
Video of
lecture of Wednesday, April 25th, on poles and renormalization.
Video of
lecture of Monday, April 30th, on vacuum polarization in QED.
Notes on shift in the energy of
an atomic electron due to vacuum polarization.
Video of
lecture of Wednesday, May 2d, on the anomalous magnetic moment of the
electron.
Notes on magnetic moments.
Video of
lecture of Monday, May 7th, on magnetic moments and on the self-energy
of the electron.
Video of
lecture of Wednesday, May 9th, on the self-energy of the electron, on
the general theory of renormalization, and on infra-red singularities.
Notes on the self-energy of the
electron.
Notes on
non-abelian gauge theory.
More of same.
Video of
lecture of Monday, May 14th, on non-abelian gauge theory.
Notes on Goldstone
bosons and the Higgs mechanism.
Video of
lecture of Wednesday, May 16th, on the quantization of non-abelian
gauge theories, the Faddeev-Popov trick, ghosts, the Feynman rules,
Goldstone bosons, and the Higgs mechanism.