# Physics 523

Class meets on Tuesdays and Thursdays at 5:30 in room 5.

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.

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.

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.
Kevin Cahill, cahill@unm.edu, 505-205-5448