Physics 523

Class meets on Tuesdays and Thursdays at 5:30 in room 5.
The grader is Mr. Zhixian Yu, his email address.


You can and should vote early in the University of New Mexico's Student Union Building, Lobo A&B, from Saturday, October 20, 2018 until Saturday, November 3, 2018. SUB voting is open Monday thru Saturday from 8 a.m. to 8 p.m.

Because pdf's of textbooks often are available for free online at addresses that you probably know better than I do, I have decided to use two books this autumn. One is the first volume of Steven Weinberg's trilogy The Quantum Theory of Fields.
The other is Anthony Zee's Quantum Field Theory in a Nutshell.

I encourage students to read the first chapter of Weinberg's book before the first class or during the first week of classes. That chapter is a history of the invention of quantum field theory. It reads like a novel with equations.
I plan to cover the first eight or nine chapters in class during the fall, but I will let you read chapter 1 by yourselves and will skip or discuss lightly the starred sections and others that can be left to a second reading. The key chapters are 2, 5, and 6.

First homework assignment: Weinberg defines linear, unitary and antilinear, antiunitary operators and their adjoints on page 51. Show that the adjoint of an operator \(X\) that is either linear & unitary or antilinear & antiunitary is the inverse \(X^{-1}\).
Solutions.

Second homework assignment: (A) Suppose \( | \vec p, \vec k \rangle = a^\dagger(\vec p) a^\dagger(\vec k) | 0 \rangle \), where \( |0\rangle\) is the vacuum and \( \vec p \ne \vec k \), is a state consisting of two spin-zero bosons of momenta \( \vec p\) and \( \vec k \). Simplify the expressions \( a(\vec q) | \vec p, \vec k \rangle \) and \( \langle \vec p, \vec k | a^\dagger( \vec q) \).
The commutation relations for spin-zero bosons are \( [ a(p), a^\dagger(p') ] = \delta^{(3)}(\vec p -\vec p') \).
(B) Now work the same problem for fermions. Suppose \( | \vec p, +; \vec k, + \rangle = a^\dagger(\vec p,+) a^\dagger(\vec k,+) | 0 \rangle \), where \( |0\rangle\) is the vacuum and \( \vec p \ne \vec k \), is a state consisting of two spin-one-half fermions of momenta \( \vec p\) and \( \vec k \) with spins in the +z direction. Simplify the expressions \( a(\vec q,+) | \vec p, +; \vec k, + \rangle \) and \( \langle \vec p, +; \vec k, + | a^\dagger( \vec q, +) \).
The anticommutation relations for fermions are \( \{ a(p, s), a^\dagger(p',s') \} = [ a(p, s), a^\dagger(p',s') ]_+ = \delta_{s s'} \delta^{(3)}(\vec p -\vec p') \).


Third homework assignment: Use the commutation relations $$ [ a_i(\pmb p), a^\dagger_j(\pmb p') ] = \delta_{i j} \, \delta^3( \pmb p - \pmb p'), \qquad [ a_i(\pmb p), a_j(\pmb p') ] = 0, \qquad [ a^\dagger_i(\pmb p), a^\dagger_j(\pmb p') ] = 0 $$ for \( i,j = 1,2 \) to derive the commutation relations for the complex operators $$ a(\pmb p) = ( a_1(\pmb p) + i a_2(\pmb p) )/\sqrt{2} \quad \mbox{and} \quad b(\pmb p) = ( a_1(\pmb p) - i a_2(\pmb p) )/\sqrt{2} $$ and their adjoints. Due Thursday 20 September.

Fourth homework assignment: Gamma matrices obey the anticommutation relations \begin{equation} \{ \gamma^a , \gamma^b \} ={} 2 \, \eta^{a b} . \end{equation} Define the matrices \( \mathcal{J}^{a b} \) as \begin{equation} \mathcal{J}^{a b} ={} - \frac{i}{4} \, [ \gamma^a , \gamma^b ] . \end{equation} Show that \begin{equation} [ \mathcal{J}^{a b}, \gamma^c ] ={} -i \, \gamma^a \, \eta^{ b c} + i \, \gamma^b \, \eta^{ a c} . \end{equation} Due Tuesday 2 October.

Fifth homework assignment: The hamiltonian for a free real field \begin{equation} \phi(x) ={} \int \frac{d^3 p}{\sqrt{(2\pi)^3 2p^0}} \Big[ a(p) e^{i p \cdot x} + a^\dagger(p) e^{-i p \cdot x} \Big] \label {free real field} \end{equation} is \begin{equation} H ={} \frac{1}{2} \int \pi^2(x) + (\nabla \phi(x))^2 + m^2 \phi^2(x) \, d^3x \label {H} \end{equation} where \( \pi(x) ={} \dot \phi(x) \) is the momentum conjugate to the field. Show that \begin{equation} H ={} \int d^3p \, \sqrt{\vec p^2 + m^2} \, \Big( a^\dagger(p) a(p) + \frac{1}{2} \delta^3( \vec 0) \Big) . \label {H =} \end{equation}
Due to my misstatement of the problem, HW5 will be due on Tuesday 30 October.

Sixth homework assignment: Derive the second and third terms in the amplitude for boson-boson scattering, equation (53) of sw6.pdf. Follow the method outlined 30-50 minutes into my lecture of 16 October . Due Thursday 1 November.

Seventh homework assignment: Show that Belinfante's energy-momentum tensor (3.49) is symmteric. Due Thursday 8 November. Go vote.

Extra-credit problem: Find a group whose structure constants are imaginary or complex.

Videos of lectures of Sidney Coleman.

Notes on Lie groups.
Notes groups.
Notes on chapter 5 of QTFI.
Notes on chapter 6 of QTFI.
Notes on chapter 6 of QTFI.
Notes on chapters 5-7 of QTFI.


Lecture one: The very few, very basic principles of quantum field theory: quantum mechanics, special relativity, and the simplest implementation of the concepts of field and particle. Basic ideas of quantum mechanics and of Lie groups. Structure constants.
Video of first lecture.

Lecture two: Structure constants of the rotation group and of the Lorentz group. Demonstration of the structure constants of the rotation group. Representations of the Lorentz group.
Video of lecture 2.

Lecture three: How the quantum states of particles transform under Lorentz transformations. The little group. How the states of massive particles transform under Lorentz transformations. The little group for massive particles. The little group for massless particles is ISO(2).
Video of lecture 3.

Lecture 4: The Wigner rotation \(W\) of a Lorentz transformation \(\Lambda\) that is a rotation, i.e., \( \Lambda = R\), is the rotation \(R\), that is, \( W = R\). The little group for massless particles. A word about representations of the group of translations. Creation and annihilation operators. States of many particles. The S matrix. Normal ordering.
Video of lecture 4.

Lecture 5: How fields transform under Lorentz transformations and translations. (Much of this material will be done more clearly in lecture 6. See the notes on section 5.1 above.) Why certain quantities are Lorentz invariant.
Video of lecture 5.

Lecture 6: How fields transform under Lorentz transformations and translations. States. Creation and annihilation operators. How fields transform. Translations. Boosts. Rotations.
Video of lecture 6.

Lecture 7: How fields transform under Lorentz transformations and translations. Conditions on spinors from Poincaré covariance under translations, boosts, and rotations. Application to spin-zero fields. Why spin-zero fields describe bosons.
Video of lecture 7.

Lecture 8: Review of particles and antiparticles as arising from two fields of the same kind having the same mass. Example of the interaction of a spin-zero charged boson with the electromagnetic field. How the process \( a + b \to \gamma + \gamma\) can arise. The commutation relations of charged fields with the charge operator. Parity for charged spin-zero fields. The parity of a state of a spin-zero boson and its antiparticle is even, i.e., positive. Why the way fields transform under rotations is related to their statistics.
Video of lecture 8.

Lecture 9: How scalar fields transform under charge conjugation and time reversal. How massive vector fields transform under Lorentz transformations and translations and how that leads to explicit formulas for their spinors.
Video of lecture 9.

Lecture 10: How massive vector fields transform under Lorentz transformations and translations and how that leads to explicit formulas for their spinors. Expansion of field of a spin-one boson. The spin-statistics theorem for spin-one bosons. The battery failed ater 55 minutes.
Video of lecture 10.

Lecture 11: The Lorentz group and its (1/2,1/2) representation. Clifford algebras. Dirac matrices.
Video of lecture 11.

Lecture 12: Derivation of formulas for Dirac spinors. Spin-statistics theorem for spin-one-half particles.
Video of lecture 12.

Lecture 13: Rest of derivation of formulas for Dirac spinors. Spin-statistics theorem for spin-one-half particles. Parity of Majorana neutrinos.
Video of lecture 13.

Lecture 14: Dyson's expansion of the S matrix. First steps to Feynman diagrams.
Video of lecture 14.

Lecture 15: Time-dependent perturbation theory. A real scalar field that interacts with itself cubically. Lowest-order scattering of 2 bosons into 2 bosons. In 2d-order perturbation theory, there are 3 amplitudes which we add together. The Feynman propagator for scalar fields.
Video of lecture 15.

Lecture 16: The Feynman propagator for spin-one-half fields. More about the \(\phi^3\) theory and 2-to-2 scattering.
Video of lecture 16.

Lecture 17: More about the Feynman propagator for spin-one-half fields. Application to fermion-boson scattering. Feynman's propagator for spin-one fields.
Video of lecture 17.

Lecture 18: The Feynman rules. Application to fermion-antifermion scattering. Canonical variables.
Video of lecture 18.

Lecture 19: The principle of least action in field theory. Noether's theorem linking a symmetry of the action density to the conservation of a physical quantity.
A camera was not avaliable; there is no video of lecture 19.

Lecture 20: More about the principle of least action in field theory and Noether's theorem linking a symmetry of the action density to the conservation of a physical quantity. Internal symmetry. Energy-momentum tensor and the conservation of energy and momentum. The Belinfante energy-momentum tensor, which is symmetric. Conservation of angular momentum.
Video of lecture 20.

Lecture 21: Global \(U(1)\) symmetry. Abelian gauge invariance. Coulomb-gauge quantization.
A memory card for the camera was not avaliable; there is no video of lecture 21.

Lecture 22: Feynman rules for QED. Application to electron-positron scattering.
Video of lecture 22.

Lecture 23: Application of Feynman rules for QED to electron-positron scattering. Why there's a minus sign in the t-channel amplitude. Gamma-matrix trace identities. Application to electron-positron to muon-anti-muon scattering.
Video of lecture 23.

Kevin Cahill, cahill@unm.edu, 505-205-5448
Last modified: Fri Nov 9 18:58:56 MST 2018