Physics 523 & 524: Quantum Field Theory
M & W 5:30 to 6:45 in room 5
Instructor: Kevin Cahill
Office: room 176, Phone:
505-205-5448.
I am in my office most afternoons and some
evenings.
Come by any time, but call first to be sure I'm
in.
Official syllabus.
I plan to use Matthew Schwartz's book
Quantum Field Theory and the Standard Model, which Cambridge University Press
published in 2014.
Ask for the second printing (June 2014) which has
fewer typos.
The typo list for the first printing (December 2013) is
here.
The typo list for the second printing (June 2014) is
there.
I have asked the UNM bookstore to order copies of this book, but
you can order it from Amazon
for $81
or directly from Cambridge
University Press for $90 but with a 20% discount if you first go
here and sign up for
their book ads.
I will post pdf's of its first few chapters on this website
to allow students to read it before they get a paper copy.
Here are
chapters one and two of Schwartz's
book.
Here is a pdf of some pages on
delta-functions from my book.
Here is a pdf of some pages on
the Casimir effect from my book.
And here is a pdf of some pages on
tensors, and there
is a pdf of some pages on the Lorentz group,
there
is a pdf of some pages on probability and statistics,
there
is a pdf of some pages on path integrals,
there
is a pdf of some pages on functional derivatives,
and there
is a pdf of some pages on the use of Monte Carlo tricks, all from my book.
Here is a pdf of some pages on spinors.
Here is a pdf of some pages on Feynman's
fermion propagator.
Here is a pdf of appendix A of Schwartz's
book.
Here is a pdf of my notes on masses,
spontaneous symmetry breaking, the Higgs mechanism, and tree-level
electroweak interactions.
Here is a pdf of my notes on supersymmetry.
Homework due Wednesday 3 September: Problems 2.3, 2.6, and 2.7.
Homework due Wednesday 17 September: Problem 3.5.
Homework due Wednesday 1 October: Problem 5.3.
Homework due Wednesday 15 October: Problem 7.8.
Homework due Wednesday 12 November: Problem 10.5(b) and 11.1.
The h.c. in problem 10.5 refers only to the part of L that is not
already real or hermitian.
Homework due Wednesday 3 December: Problem 13.1.
Homework due Tuesday 14 April: Use Noether's theorem to derive
from the action density (25.74) the formula (25.80) for
the current that is conserved but not gauge invariant.
Solution.
Homework due Tuesday 21 April: Do the problem stated
in equations (205 & 206) of my notes
Masses and the Higgs Mechanism.
Solution.
Lectures on YouTube.
Video of lecture 1 on 18 August 2014.
Video of lecture 2 on 20 August 2014.
Video of lecture 3 on 25 August 2014.
Video of lecture 4 on 27 August 2014.
Video of lecture 5 on 3 September 2014.
Video of lecture 6 on 8 September 2014.
Video of lecture 7 on 10 September 2014.
Video of lecture 8 on 15 September 2014.
Video of lecture 9 on 17 September 2014.
Video of lecture 10 on 22 September 2014.
Video of lecture 11 on 24 September 2014.
Video of lecture 12 on 29 September 2014.
Video of lecture 13 on 1 October 2014.
Video of lecture 14 on 6 October 2014.
Video of lecture 15 on 8 October 2014.
Video of lecture 16 on 13 October 2014.
Video of lecture 17 on 15 October 2014.
Video of lecture 18 on 20 October 2014.
Video of lecture 19 on 22 October 2014.
Video of lecture 20 on 27 October 2014.
Video of part 1 and
part 2 of lecture 21 on 29 October 2014.
Video of lecture 22 on 3 November 2014.
Video of lecture 23 on 5 November 2014.
Video of lecture 24 on 10 November 2014.
Video of lecture 25 on 12 November 2014.
Video of lecture 26 on 17 November 2014.
Video of lecture 27 on 19 November 2014.
Video of lecture 28 on 24 November 2014.
Video
of lecture 29 on 26 November 2014.
Video
of lecture 30 on 1 December 2014.
Videos of
part 1 and
part 2
of lecture 31 on 3 December 2014.
Video
of lecture 32 on 8 December 2014.
2015 lectures of Quantum Field Theory II, physics 524:
Video
of lecture 1 on 13 January 2015:
path integrals and functional derivatives.
Video
of lecture 2 on 15 January 2015:
path integrals for perturbation theory and
quantum electrodynamics.
Video
of lecture 3 on 20 January 2015:
Grassmann path integrals for fermions.
Video
of lecture 4 on 22 January 2015:
More about Grassmann path integrals for fermions.
Video
of lecture 5 on 27 January 2015:
Derivation of Feynman's propagator for spin-one-half
fields from Grassmann path integrals.
Path integrals for nonabelian gauge theories.
The Faddeev-Popov trick.
Video
of lecture 6 on 29 January 2015:
The Faddeev-Popov trick and ghosts.
Path-integral derivation of Schwinger-Dyson
equations and of Ward-Takahashi identities.
Video
Lecture 7 on 3 February 2015:
The Abel-Plana formula and the Casimir effect.
Introduction to the renormalization of
simple theories of scalar fields.
A Feynman trick.
Video
Lecture 8 on 5 February 2015:
Renormalization in scalar field theory
with a cubic interaction and in scalar QED.
Wick's rotation. Area of sphere in any number
of dimensions. The basic idea of dimensional
regularization.
Video
Lecture 9 on 10 February 2015:
Renormalization by differentiation.
More about Feynman tricks, Schwinger tricks,
uses of the Gamma function, dimensions of various fields
in d dimensions, and dimensional regularization.
Video
Lecture 10 on 12 February 2015:
Renormalization by differentiation.
Correction to the photon propagator.
Uehling potential and its contribution
to the Lamb shift E(2s1/2) - E(2p1/2),
the Landau pole, and the "running" coupling "constant."
Video
Lecture 11 on 17 February 2015:
The Gordon identity and the
anomalous magnetic moment of the electron.
Video
Lecture 12 on 19 February 2015:
Mass renormalization by differentiation,
by Pauli-Villars, and by dimensional regularization.
The electron propagator.
Video
Lecture 13 on 24 February 2015:
Renormalized perturbation theory.
The systematic use of counterterms.
Two-point and three-point functions.
Z_1 = Z_2.
Video
Lecture 14 on 26 February 2015:
The covariant derivative
and Z_1 = Z_2.
Infrared divergences arise in the
scattering of charged particles.
They cancel in x-sections
that include states
with soft photons.
Video
Lecture 15 on 3 March 2015:
The running coupling constant in QED.
Unitarity of the S-matrix.
The optical theorem and its
application to decay rates
and total cross-sections.
The partial-wave
optical theorem.
Video
Lecture 16 on 5 March 2015:
Global and local internal symmetry.
Group theory and the groups U(n) and SU(n).
Representations and structure constants.
How Yang and Mills made kinetic
actions invariant under unitary transformations
that are arbitrary functions of the spacetime x.
The covariant derivative.
The covariant field strength F.
The connection as a 1-form A.
The exterior derivative d.
The field strength as
F = (d + A) A.
Basis vectors and gauge theory.
Video
Lecture 17 on 17 March 2015:
Review of the basic ideas of (Yang-Mills)
theories with local internal symmetry.
Lie groups, Lie algebras, and their representations.
The generators of a Lie algebra and their structure
constants. Why the structure constants are the same
in all representations of a Lie group.
Why they are real and totally antisymmetric when the group
is compact. The Jacobi identity and the definition
of the generators in the adjoint representation.
Schur's lemma and Casimir operators.
Some details about SU(N).
Video
Lecture 18 on 19 March 2015:
The Wilson action of a plaquette.
Why it tends to the continuum action
as the lattice spacing shrinks to zero.
Why it is gauge invariant.
Force fields and the Wilson line.
How the Wilson line responds to gauge transformations.
Gauge invariance of the Wilson loop.
Why a Wilson loop that goes to zero
with the area it encloses means a
linearly confining quark-antiquark potential.
Video
Lecture 19 on 24 March 2015:
The key ideas of nonabelian gauge theory.
Nother's theorem: a global symmetry of the action density
gives us a current that is conserved when the equations
of motion are obeyed.
In unbroken nonabelian gauge theory, the local symmetry
implies the existence of a conserved current,
but the current isn't gauge invariant.
There is a gauge-covariant current, but it isn't conserved
or gauge invariant.
In a nonabelian gauge theory,
gauge fixing leads to a determinant which
is equivalent to interactions with
scalar fermions that only
occur as internal lines in Feynman diagrams,
not as external lines. They are ghosts.
In QCD, scalar states attract, octet states repel.
Electron-positron --> hadrons.
Video
Lecture 20 on 26 March 2015:
The x-sections of electron-positron --> muon-antimuon
and electron-positron --> hadrons.
At energies well below the mass of the Z boson,
the ratio of the second to the first is about
3.67, but above 90 GeV the ratio rises to 20.09
unless we count gluons in the final hadronic states.
When we include the gluons, the ratio rises to the
experimental value of 20.79 if we set g^2/4pi = 0.12.
This quantity, called alpha-strong, falls from about 0.5
to about 0.1 as the energy rises from 1 to 100 GeV.
Overview of vacuum polarization and renormalization in QCD.
Computation of the running coupling constant,
alpha-strong, in QCD.
Video
Lecture 21 on 31 March 2015:
Discrete and continuous symmetries
of the action and of the ground state
or states. Explicit and spontaneous
breaking of symmetries.
Lie-algebra analysis of spontaneously broken,
continuous symmetries of the action.
Goldstone bosons. The boson mass spectrum
of four theories: unbroken U(1),
spontaneously broken U(1),
unbroken U(2), and
spontaneously broken U(2).
Video
Lecture 22 on 2 April 2015:
Physical observables are independent of
the renormalization scale mu.
Definition of the beta function.
Calculation of the beta function
of QCD from the already computed dependence of
the renormalization constants Z_i upon mu.
Integration of the differential equation
that describes how the running coupling constant g(mu)
and alpha_s = g^2/(4 pi) must depend upon the energy scale mu
if physical observables are to be independent of it.
A more abstract approach to Goldstone bosons.
Some consequences of the lightness of the masses
of the up and down quarks as compared to the energy scale
of QCD, which is 200 to 300 MeV.
Video
Lecture 23 on 7 April 2015:
Review of what mass terms look like
in theories of fields of spin zero,
one-half, and one.
Review of Goldstone's theorem.
The Higgs mechanism in U(1) abelian gauge theory.
The Higgs mechanism in O(n) nonabelian gauge theory.
Video
Lecture 24 on 9 April 2015:
The Higgs mechanism in SU(2) nonabelian gauge theory.
The Glashow-Salam-Weinberg model of the electroweak interactions.
The Higgs mechanism in SU_L(2)xU_Y(1) nonabelian gauge theory.
The masses of the Higgs boson, the charged W's, the neutral Z,
and the photon.
The covariant derivative that acts on left-handed doublets.
The electric charge Q = T_3 + Y.
The weak mixing angle and its relation to the coupling constants
g and g' of the gauge theory.
Video
Lecture 25 on 14 April 2015:
How the SU_L(2)xU_Y(1) covariant derivatives act on the
left- and right-handed quark and lepton fields.
Gamma-5 as a fourth spatial gamma matrix.
Why Dirac's mass term is Lorentz invariant
but not invariant under SU_L(2)xU_Y(1).
How quarks and leptons get their masses
from a gauge-invariant Yukawa-type interaction
with the Higgs field.
The CKM matrix and its singular-value decomposition.
Video
Lecture 26 on 16 April 2015:
The 3x3 mass matrix of the three generations
of down quarks and its singular-value decomposition.
The 3x3 mass matrix of the three generations
of up quarks and its singular-value decomposition.
The 3x3 mass matrix of the three generations
of charged leptons and its singular-value decomposition.
The covariant derivatives for the right- and left-handed
fermion fields and the CKM matrix.
One can remove six phases from the quark CKM matrix by redefining
the phases of the six quark fields.
The remaining phase breaks \( CP \).
In 1964, Cronin and Fitch discovered that
\( CP \) is not conserved.
The fermion masses are proportional to their
couplings to the Higgs field.
Majorana mass terms and why they are Lorentz invariant.
Neutrino oscillations.
Video
Lecture 27 on 21 April 2015:
How the CKM matrix fits into the
electroweak covariant derivative,
and how it occurs in the sextet of quark fields.
Current values of the magnitudes of the nine elements
of the CKM matrix. (Ignore what I said about
the phase delta; see the next lecture.)
Gauge invariance under SU_L(2)xU_Y(1) tells us
that there are at least six neutrino/antineutrino states for each momentum.
The minimum is three massive Majorana neutrinos.
If there are right-handed neutrinos,
then there are 12 neutrino/antineutrino states,
and there may be three massive Dirac neutrinos
or six massive Majorana neutrinos or some
intermediate combination. We don't know if
the neutrinos are Majorana or Dirac particles.
If they are Majorana particles, then neutrinoless
double beta decay can occur. It has not yet been seen,
so this important question is open.
Neutrino oscillations have told us about
the differences of the squares of the masses
of the three massive neutrinos.
Careful study of the endpoint of tritium beta decay
may tell us more. The decays of the charged pions
provide simple examples of the complete breakdown
of parity. The scattering of Ws and Zs would violate
unitarity if the Higgs boson were not less than 1 TeV;
it is 126 GeV. The quantum numbers
of the standard model are lopsided.
Video
Lecture 28 on 23 April 2015:
More about the quark CKM matrix.
The angles alpha, beta, and gamma
and the current limits on their values
from B-meson decays.
The neutrino CKM matrix and
current limits on its matrix elements.
The angles theta23 and theta12 are big.
It's easier to understand neutrino masses
if there are right-handed neutrinos.
In 1979, Kazuo Fujikawa showed that
the path-integral measure of Fermi fields
is not invariant under gamma5 rotations,
and that therefore certain currents
are not conserved. These anomalies cancel
when the hypercharges of the fields
satisfy certain conditions as in
the standard model.
The charges of anomalous currents
are not conserved.
In the standard model,
B - L is conserved,
but not B or L.
Video
Lecture 29 on 28 April 2015:
Elements of grand unification:
In the standard model,
SU_c(3) and U_Y(1) act on both left- and right-handed
fields, but SU_L(2) acts only on left-handed fields.
An internal symmetry can't rotate left-handed fields
or right-handed fields into linear combinations
of left- and right-handed fields.
Georgi and Glashow solved this problem by
writing right-handed fields as sigma_2 times
their conjugates, which are left handed.
That is, they put all the fields into
multiplets of left-handed fields.
In each family or generation,
there are 15 left-handed fields
if one omits the right-handed neutrino.
In SU(5), these 15 fields fit into a 5*
and a 10. In SO(10), the 16 left-handed fields,
including the right-handed neutrino,
fit into one 16.
The radical step of placing fields
and their conjugates into the same multiplets
implies that nucleons are no longer stable.
They have lifetimes that are proportional
to the fourth power of the masses of the
gauge bosons that mediate their decays.
Experiments have shown that the partial lifetime
of the proton due to the decay p --> pi-zero + positron
in at least 8.3 x 10^33 years.
So the gauge bosons must have masses of 10^15 GeV/c^2 or more.
Grand unification thus changes the focus of physics
from the TeV scale to 10^12 TeV and up.
This dramatic shift in energy is true
for any theory that puts the gauge groups
of the standard model into a single simple gauge group,
that is, into a group without an invariant subgroup.
The logic is that the trace of the square of any
generator must be the same, and these traces
are simple sums of the coupling constants
of SU_c(3), SU_L(2), and U_Y(1).
The resulting relations are not compatible
with the values of these coupling constants
at TeV-scale energies; one must
run these coupling constants up to about
10^15 GeV to get agreement with experiment.
Glauber showed that classical currents radiate
coherent states. So do classical currents
in QCD.
Video
Lecture 30 on 30 April 2015:
An introduction to supersymmetry:
Supersymmetric quantum mechanics.
In the simplest model, there are
two supersymmetric charges, and
the hamiltonian is twice the square of each.
The supersymmetry is exact if these charges
annihilate the vacuum, in which case the
ground-state energy is zero, and
one can solve for the wave-function
of the ground state as long as
the superpotential W(x) --> ± ∞
as x --> ± ∞ or W(x) --> ∓ ∞
as x --> ± ∞.
In this case supersymmetry is exact, and the excited states
occur in pairs of equal energy.
But when the superpotential W(x) --> ∞
as x --> ± ∞ or when W(x) --> − ∞
as x --> ± ∞, then the zero-energy wave function
is not normalizable and no state of zero energy exists.
In this case supersymmetry is dynamically broken,
and the excited states do not
occur in pairs of equal energy.
Supersymmetric quantum-mechanical systems
are easier to analyze than similar systems
without supersymmetry.
All these features have analogs in quantum field theory.
A quantum field theory with exact supersymmetry
has a ground state of zero energy.
The density of dark energy is not zero,
but it is small compared to the energy density
( ∞ )^4 of a theory of free bosons
and to that − ( ∞ )^4 of a theory of free fermions.
And the energy densities of theories with
broken supersymmetry diverge no worse than
± ( ∞ )^2. My
notes on supersymmetry
use dotted and undotted indicies, which may be
more trouble than they're worth.
The change in the action density
of a susy theory is a total divergence
of a current K.
The equations of motion imply that the
change in the action density also is
the divergence of the susy Noether current J.
So the difference of these two total divergences
vanishes, and the difference of these two currents
is the conserved susy current S = J - K.
The space integral of the time component of S
gives us the susy charges.
Their anticommutators are linear combinations
of the Pauli matrices and the 4-momentum
operators of the quantum field theory.
A superfield is a function both of a point
in spacetime and of some anticommuting variables.
Equivalently, a superfield is a function of a point
in superspace. Superfields in superspace provide
a simple way of making actions that are supersymmetric.
The word super is used too much in physics.
Video
Lecture 31 on 5 May 2015:
Introduction to general relativity:
Points locate events and are physical;
coordinates are arbitrary and metaphysical.
So theories should not be tied to any particular
system of coordinates. Contravariant vectors
transform like the differentials dx^i.
Covariant vectors transform like
derivatives with respect to the coordinates x^i.
Tensors transform like products of
contravariant and covariant vectors.
Basic ideas about contractions,
p-forms, exterior derivatives,
Stokes's theorem, the metric tensor,
Levi-Civita symbols, Maxwell's equations
in a gravitational field, the action
and field equations of general relativity,
weak gravitational fields, and time dilation.