# Physics 523 and 524

Class meets on Mondays and Wednesdays at 5:30 in room 184. (Last semester class met on Tuesdays and Thursdays at 5:30 in room 5.)
The grader is Mr. Zhixian Yu, his email address.

Videos of lectures of Sidney Coleman.

Notes on Lie groups.
Notes on groups.
Class notes.
Notes on general relativity.
Notes on the renormalization group.
Notes on functional derivatives.
Derivation of one of Glauber's dentities.
Note on why $$u$$ and $$v$$ spinors are so different.
Kinoshita's five-loop calculation of the magnetic moment of the electron.
Notes on spontaneous symmetry breaking
Notes on effective field theories and on the Hawking temperature of an accelerating frame
Notes on maximally symmetric spaces and conformal algebra
Notes on Monte Carlo methods
Solutions to some Monte Carlo problems.
Basic remarks about Wilson loops.
Basic remarks about Grassmann variables and fermionic path integrals.
Please tell me of any typos or errors in these notes.
Weinberg on Dirac brackets.
Weinberg on Dirac brackets and QED.
Weinberg on axial-gauge quantization of nonabelian gauge theories.

First homework assignment: Derive Feynman's trick $$\frac{1}{AB} ={} \int_0^1 \frac{dx}{[ (1-x)A + x B ]^2} .$$ Due Thursday, Feb. 14th.

Second homework assignment: Use SW's pseudounitarity condition (class notes 1.209, SW's 5.4.32) \begin{equation} \beta D^\dagger(\Lambda) \beta = D^{-1}(\Lambda) \label {class notes 1.209, SW's (5.4.32)} \end{equation} to show that \begin{equation} \overline u(\vec p,s') u(\vec p,s) ={} \frac{m}{p^0} \, \delta_{s s'} . \end{equation} Hint: The definition of the spinors (class notes 1.211, SW's 5.5.12 & 13) helps here.
Due Thursday, Feb. 21st.

Third homework assignment: Show that \begin{equation} \overline L_\ell {\gamma^a D^\ell_a} L_\ell = [\frac{1}{2} ( 1 + \gamma_5 ) L ]^\dagger i \gamma^0 {\gamma^a D^\ell_a} \frac{1}{2} ( 1 + \gamma_5 ) L = \overline L {\gamma^a D^\ell_a} \frac{1}{2} ( 1 + \gamma_5 ) L . \end{equation} Due Thursday, March 7.

Fourth homework assignment: Using the differential forms \begin{equation} \begin{split} P_i ={}& \partial_i, \quad J_{ik} ={} (x_i \, \partial_k - x_k \, \partial_i ), \\ D ={}& x^i \, \partial_i, \quad \mbox{and} \quad K^i ={} ( \eta^{ik} \, x^j \, x_j - 2 x^i \, x^k) \, \partial_k \end{split} \end{equation} of the generators of the conformal algebra in flat space, show that \begin{equation} [ K^i, K^j] ={} 0, \quad [ D, P^i ] ={} - P^i, \quad [ D, J_{ij}] = 0, \quad \end{equation} \begin{equation} [D,K^i]= K^i, \quad [J^{ij}, K^\ell] = - \eta^{i \ell} K^j + \eta^{j \ell} K^i, \quad \mbox{and} \quad [K^i,P^j] ={} 2 J^{i j} + 2 \eta^{ij} D. \end{equation}
Due Thursday 4 April 2019.

Videos of lectures of 524 for 2019.

Lecture 1: The Landau-Ginzburg theory of phase transitions. Order parameters. First-order phase transitions (discontinuous). Second-order phase transitions (continuous). Use of the Gibbs free energy. In an ideal ferromagnet at temperatures below the critical temperature, the magnetization is proportional to the square root of $$T - T_c$$. Landau and Ginzburg represented the spin density as a field and showed that the correlation function of spins is a Yukawa potential with a range that diverges as $$(T-T_c)^{-1/2}$$. The basic ideas about functional differentiation. These are in the online notes. Video of first lecture.

Lecture 2: The Landau-Ginzburg theory of phase transitions. Susceptibility diverges as $$1/|T - T_c|$$. One of Glauber's identities. Derivation posted in online notes. S-matrix for electromagnetic field radiated by a classical current. Video of lecture 2.

Lecture 3: Renormalization and counterterms. Derivation of the one-loop correction to the phton propagator. Vacuum polarization. A Feynman trick. Use of trace identities. The Wick rotation. Video of lecture 3.

Lecture 4: Renormalization and counterterms. More about the one-loop correction to the phton propagator. Vacuum polarization. The Wick rotation. Dimensional regularization. Use of counterterms. How to take the limit in which the dimension of spacetime goes to 4. Video of lecture 4.

Lecture 5: How the Dirac field and its spinors $$u$$ and $$v$$ transform. More about vacuum polarization and its effects. Video of lecture 5.

Lecture 6: Effective field theories. Integrating over fields with masses huge compared to those of the standazrd model. The linear sigma model. Goldstone's theorem. Video of lecture 6.

Lecture 7: Goldstone's theorem for $$SU(2)$$ and $$SU(3)$$. The abelian Higgs mechanism. The $$SU(2)$$ and $$SO(n)$$ Higgs mechanisms. Video of lecture 7.

Lecture 8: Review of effective field theory. Goldstone's theorem and the Higgs mechanism for $$SO(n)$$. Majorana masses. Hypercharge. The Glashow-Weinberg-Salam-Ward model and the standard model. Video of lecture 8.

Lecture 9: How the gauge bosons get their masses. Identification of the electric charge as $$Q = T_3 + Y/2$$ and the cosine of the the weak mixing angle as $$\cos \theta_w = M_W/M_Z$$. Electroweak interactions of the quarks and leptons. Masses of the quarks and leptons. Video of lecture 9.

Lecture 10: Electroweak interactions of the quarks and leptons. Beta decay. Masses of the quarks and leptons. The CKM matrix. A CP-breaking phase. How a high-energy theory naturally explains the lightness of the masses of the neutrinos. Video of lecture 10.

Lecture 11: How derivatives transform under variations. More about how a high-energy theory naturally explains the lightness of the masses of the neutrinos. The Majorana condition on Dirac spinors. How that condition leads to the Majorana condition on Majorana fields. Interactions of neutrinos going thru matter. Quantization of fields in flat and curved space. Bogoliubov transformations. How curved space makes particles. Video of lecture 11.

Lecture 12: What we mean by a field. Action and equation of motion for a scalar field in curved space. Expansion of scalar field in terms of solutions of its equation of motion. A scalar product for mode functions in flat and curved space. Properties of the scalar product: hermiticity and other properties. Expansion of mode functions in terms of solutions of the equation of motion. Expansion of the field in terms of solutions of the equation of motion in different coordinate systems. Relations between the annihilation and creation operators in different coordinate systems. Bogoliubov coefficients. Bogoliubov transformations. How curved space makes particles. Equation of motion in an accelerated frame of reference. Video of lecture 12.

Lecture 13: Lorentz transformation to an accelerated coordinate system. Rindler coordinates. Mean value of two massless scalar fields in the vacuum. Mean value of two massless scalar fields at a nonzero temperature. Mean value of two massless scalar fields in the vacuum of an inertial frame that instantaneously is the rest frame of the accelerating frame. A comparison of the two reveals that a detector in a frame uniformly accelerating with acceleration $$\alpha$$ feels a temperature $$T = \hbar \alpha/(2 \pi c k_B)$$, a result due to Hawking, Davies, and Unruh. (The missing factor of 2 in the last formulas of the lecture is due to my failure to properly latex Mathematica's formula for $$B_\beta$$.)
Notes on Hawking radiation. Video of lecture 13.

Lecture 14:
Summary of lecture on acceleration and temperature. Computation of the temperature due to the surface gravity of the Earth. Isometries. Two forms of Wilhelm Killing's characterization of an isometry. Maximally symmetric spaces. Examples of the 2-sphere and the 2-hyperboloid in 3-space and of the 3-sphere and the 3-hyperboloid in 4-space. Connection with $$\Lambda CDM$$ cosmologies. The Lie derivative. Properties of maximally symmetric spaces in 2, 3, and 4 dimensions. Killing's condition that two metrics be conformally related. Characterization of a conformal transformation.
Notes on maximally symmetric spaces with the typo in $$y_2$$ fixed. Video of lecture 14.

Lecture 15:
A conformal relation between two metrics. Condition on Killing vectors for conformal symmetry. Formula for conformal factor. Condition on Killing vectors for conformal symmetry in flat space. Formula for conformal factor in flat space. Conformal symmetries of flat space: translations, Lorentz transformations, dilations, and conformal transformations. Differential forms of the generators of conformal symmetries. Their Lie algebra.
Notes on maximally symmetric spaces and on conformal symmetry. Video of lecture 15.

Lecture 16:
Review of Killing's condition for a conformal symmetry in flat space. The commutation relations of the conformal group. Isomorphism between the conformal group of Minkowski space and the group $$SO(d,2)$$. Derivation of the conformal invariance of Maxwell's action. Remarks about the conformal invarince of pure (no matter) nonabelian gauge theory and of the theory of a free massless scalar field.
Notes on maximally symmetric spaces and on conformal symmetry. Video of lecture 16.

Lecture 17:
Review of Killing's condition for conformal symmetry in flat space. Derivation of the conditions for a Killing vector in 1, 2, and 3 or more dimensions. In 1 dimension the Killing vector is completely arbitrary. In 2-d euclidian space, the two new coordinates can be the real and imaginary parts of an arbitrary analytic or antianalytic function of the old coordinates. Light-cone coordinates. In 2-d Minkowski space, the new coordinates can be arbitrary functions of the light-cone coordinates. In 3 or more dimensions, a Killing vector must be a quadratic function of the old coordinates. The parameters of the quadratic function are those of a translation, a dilation, a Lorentz transformation, and of a special conformal transformation.
Notes on maximally symmetric spaces and on conformal symmetry.
The book Conformal Field Theory by Di Francesco, Mathieu, and Sénéchal is the best reference I know of for conformal field theory.
Video of lecture 17.

Lecture 18:
Finite conformal transformations in two-dimensional euclidian space. Analytic and antianalytic functions. Quantum statistical mechanics in euclidian space. The Monte Carlo method. The Metropolis step. Why it works. Quantum field theory on a spacetime lattice. Michael Creutz's website. Video of lecture 18.

Lecture 19:
Some remarks about Wilson lines and Wilson loops. Video of lecture 19.

Lecture 20:
More about Wilson loops, Wilson lines, and the Wilson action. The class notes basic remarks about Wilson loops fill in some gaps in the video. The notes on Creutz's examples are here: Solutions to some Monte Carlo problems. Video of lecture 20.

Lecture 21:
Announcement of Paul Steinhardt's talk on May 11th at Bookworks on Rio Grande in Albuquerue and brief summary of his theory of a cyclic universe. Sign conventions for and more details about gauge fields and Faraday tensor. Yet more about Wilson loops, Wilson lines, and the Wilson action. A scalar field on a lattice. Grassmann variables and how to integrate over them. The notes are on lattice field theory and on Grassmann variables. The notes on Creutz's examples are here: Solutions to some Monte Carlo problems. Video of lecture 21.

Lecture 22:
More about the use of Grassmann variables and how to integrate over them. Video of lecture 22.

Lecture 23:
Gauge fixing in abelian and nonabelian gauge theories. The Coulomb or radiation gauge, axial gauges, the temporal gauge. Gauss's law generates gauge transformations. Physical states are invariant under gauge transformations. Wilson's loop in the temporal gauge. Some of the history of lattice gauge theory. Speculation about confinement of quarks and gluons. Video of lecture 23.

Lecture 24:
Constraints and Dirac brackets. Video of lecture 24.

Lecture 25:
Constraints and Dirac brackets. Application to QED. Images of Weinberg's equations 8.2.1-8.2.5. Images of Weinberg's equations 8.2.6-8.3.3. Images of Weinberg's equations 8.3.4-8.3.10. Images of Weinberg's equations 8.3.11-8.4.13. Images of Weinberg's section 8.4.14-8.5.2. Video of lecture 25.

Lecture 26:
More about Dirac brackets and the quantization of abelian and nonableian gauge theories. Vector and axial-vector currents and the U(1) anomaly. Anomalies and the measure of path integration. Forms and nonabelian gauge theory. Solitons, instantons, and homotopy groups. Video of lecture 26.
Lectures by Chatterjee on anomalies. Adel Bilal's notes on anomalies.

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.

Eighth homework assignment: For the theory of spin-zero bosons with Lagrange density $$L ={} \frac{1}{2} \left[ \partial_a \phi \partial^a \phi - m^2 \phi^2 \right] - \frac{g}{4!} \phi^4 ,$$ to lowest order in the coupling constant $$g$$ find the differential and total cross-sections for the scattering of bosons with momenta $$p$$ and $$k$$ into momenta $$p'$$ and $$k'$$. Due Thursday 6 December 2018.

Ninth homework assignment: Read section ( 6.2) of the Class notes on the abelian Higgs mechanism. What are the masses of the particles of the two scalar fields in the theory with lagrangian (6.5)? What are the masses of the physical fields of the theory with lagrangian (6.11)? The equation numbers refer to those of the Class notes. Due Thursday 13 December 2018.

Extra-credit problem: Find a group whose structure constants are imaginary or complex. Videos of lectures of 523 for 2018.

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.

Lecture 24: Application of gamma-matrix trace identities to electron-positron to muon-anti-muon scattering. Interpretation of squared delta function. Box normalization of states. Density of final states. Flux of incoming particles. Evaluation of energy delta function. Calculation of differential and total cross-sections.
Video of lecture 24.

Lecture 25: Comparison of $$e^+ e^-\to \mu^+ \mu^-$$ pair production with $$e^- \mu^- \to e^- e^- \mu^-\mu^-$$ elastic scattering. Crossing symmetry. Nonabelian gauge theory. Covariant derivatives. The Yang-Mills-Faraday field strength tensor. Action for a nonabelian gauge theory. QCD.
Video of lecture 25.

Lecture 26: Action for a nonabelian gauge theory. QCD. The standard model. The Higgs mechanism. The Glashow-Salam-Weinberg model of the electroweak interactions.
Video of lecture 26.

Lecture 27: The Glashow-Salam-Weinberg model of the electroweak interactions. Path integrals for transition amplitudes. Gaussian integrals and Trotter's formula. Path integrals in quantum mechanics. Path integrals for quadratic actions.
Video of lecture 27.

Lecture 28: Bohm-Aharanov effect. Path integrals in statistical mechanics. Mean values of time-ordered products. Quantum field theory on a lattice.
Video of lecture 28.

Lecture 29: Quantum field theory on a lattice. Finite-temperature field theory. Perturbation theory. Application to quantum electrodynamics.
Video of lecture 30.

Lecture 30: Grassmann variables and fermionic path integrals. Spontaneous symmetry breaking. Goldstone bosons. Abelian Higgs mechanism. Scattering of spinless bosons in $$\lambda \phi^4$$ theory.
Video of lecture 30.

Lecture 31: Renormalization group. Renormalization and interpolation. Renormalization group in quantum field theory. Renormalization group in lattice field theory. Renormalization group in condensed-matter physics.
Video of lecture 31.

Kevin Cahill, cahill@unm.edu, 505-205-5448
Last modified: Thu Nov 21 19:04:06 MST 2019