Quantum Field Theory: Lecture Log

This is the lecture log for the Quantum Field Theory classes PHY 396 K and PHY 396 L taught in 2026/27 by professor Vadim Kaplunovsky.

Most lectures should be video recorded and the records available on Canvas. For the few lectures that did not get recorded because of technical glitches, I shall scan the notes I have used in class and links the scans to this page.

Navigation: Fall, Spring, Last regular lecture.

QFT 1, Fall 2026 semester

August 25 (Tuesday):

Syllabus and admin: course content, textbooks, prerequisites, homework, exams and grades, etc.
General introduction: reasons for QFT; field-particle duality.
Lagrangian mechanics: Lagrangian and action; least action principle; Euler–Lagrange equations; multiple dynamical variables; counting the degrees of freedom.
Introduction to classical fields: Definition of a classical field; Lagrangian density.

August 27 (Thursday):

Introduction to classical fields: Definition of a classical field; Lagrangian density; Euler–Lagrange equations for fields; Klein–Gordon example; multiple fields; complex fields; Landau–Ginzburg example; higher space derivatives and non-local Lagrangians for non-relativistic fields.
Relativistic fields: relativistic sign conventions; Einstein summation convention; relativistic ℒ and field equations; Klein–Gordon example; multiple scalar fields.
Relativistic electromagnetic fields: the 4–tensor F^μν=−F^νμ and the relativistic form of Maxwell equations; the 4–vector potential A^μ and the gauge transforms; the Lagrangian formulation; current conservation and gauge invariance of the action; counting the EM degrees of freedom.
Canonical quantization v. functional quantization.

September 1 (Tuesday):

Review of canonical quantization: Hamiltonian formalism in classical mechanics; quantization, operators, and commutation relations; Poisson brackets and commutator brackets.
Introduction to quantum fields: Hamiltonian formalism for the classical fields; quantum fields; equal-time commutation relations; quantum Hamiltonian, Heisenberg equtions, and the quantum Klein-Gordon equation.
Quantum fields and particles: expanding free relativistic scalar fields into modes; creation and annihilation operators for a bunch of harmonic oscillators; eigenvalues of the free quantum field's Hamiltonian.

September 3 (Thursday):

Finished Quantum fields and particles: identifying the identical bosons; the Fock space; Casimir effect (briefly).
General identical bosons: bosonic Fock space and its occupation number basis; creation and annihilation operators; wave-function language vs. Fock-space language; one-body operators; two-body operators; non-relativistic quantum fields; “second quantization”.

September 8 (Tuesday):

Lorentz groups and mass shells: rotation group SO(3); general Lorentz group O(3,1); continuous subgroup SO⁺(3,1); other Lorentz subgroups; mass shell for the particle momenta; invariant measure on the mass shell; relativistic normalization of states and operators.
Relativistic quantum fields: Expanding a free time-dependent scalar field into products of plane waves and creation/annihilation operators; massive vector field; charged scalar field and antiparticles; general free fields.
Intro to relativistic causality: no superluminal particles or signals; signals in quantum mechanics; QFT: relativistic causality requires [Ô(x),Ô'(y)]=0 for (x-y)²≤0.

September 10 (Thursday):

Relativistic causality: local operators in QFT; bosons and fermions; proof of relativistic causality for the free scalar field; going forward and backward in time; causality for interacting fields.
Feynman propagators: why and how of the time-ordering; defining the propagator; relation to D(x-y); scalar propagator is a Green's function of the Klein–Gordon equation; Green's functions in momentum space; regulating the integral over the poles: Feynman's choice and other types of Green's functions.

Extra lecture on September 11 (Friday):

Bose–Einstein condensate and superfluidity: naive Bose–Einstein condensate; the coherent state; classical and quantum fluctuation fields δφ(x); Bogolyubov transform; ground state; fluctuation spectrum; non-local forces between helium atoms and the `rotons'; fluctuation spectrum in a moving condensate and the superfluidity.

September 15 (Tuesday):

Feynman propagators as Green's functions: evaluating the ∫ dk₀ integrals for the Feynman propagator; other kinds of Green's functions; Feynman propagators for vector, spinor, etc., fields.
Tachyons: tachyons in QM; tachyon field and vacuum instability; interactions and scalar VEVs (vacuum expectation values); no tachyons in the right vacuum state.
Introduction to symmetries: symmetry groups; symmetries of classical fields: on-shell and off-shell symmetries, continuous and discrete symmetries, internal and spacetime symmetries, global and local symmetries; symmetries in quantum mechanics: representation by unitary operators, rotation symmetry and its generators, Campbell–Baker–Hausdorff formula and proof that the D(R) operators represent the SO(3) group.

September 17 (Thursday):

Finished Introduction to symmetries: scalar, vector, and tensor operators under the rotation symmetry; general Lie groups and their generators; Lie algebras of the generators; representations of general groups and algebras; multiplets.
Noether theorem: global continuous symmetries and conserved currents; generators and currents for the SO(N) example; symmetry charges in the quantum theory; representation of the SO(N) symmetry in the Fock space; the phase symmetry and the net number of particles minus antiparticles; proof of the theorem; examples of Noether currents; translation symmetry and the stress-energy tensor; symmetrizing the Noether stress-energy tensor for non-scalar fields.

Extra lecture on September 18 (Friday):

Vortices and other kinds of topological defects: domain walls as topological defects; co-dimension; rotation and vorices in the superfluid; vortex energy; vortices as topological defects of co-dimension=2; magnetic monopoles as topological defects (co-dimension=3); (briefly) Yang–Mills instantons.

September 22 (Tuesday):

Local phase symmetry: local symmetry and covariant derivatives; gauge field and gauge transforms; coupling charged scalar fields to electromagnetism;algebra of covariant derivatives; covariant field equations; [D_μ,D_ν]=iQF_μν.
Aharonov–Bohm effect: covariant Schrödinger equation for a charged quantum particle; gauge dependence of evolution kernels; Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry.
Magnetic monopoles: electric-magnetic duality; magnetic monopoles; Dirac's charge quantization; Heuristic picture; Dirac construction; gauge bundles.

September 24 (Thursday):

Finished Magnetic monopoles: electric-magnetic duality in QFT; angular momentum in presence of a monopole; spin-statistics theorem for dyons.
Non-abelian local symmetries: covariant derivatives and matrix-valued connections; non-abelian gauge transforms; Gell-Mann matrices and the component gauge fields; infinitesimal gauge transforms in components; non-abelian tension fields; gauge transforms of the tension fields; the adjoint representation; Yang–Mills theory and normalization of the gauge fields.

September 29 (Tuesday):

Local symmetries: Yang–Mills theory and normalization of the gauge fields; gauge theories with matter; simple and compact Lie groups and their Lie algebras; Lie-algebra-valued gauge fields and gauge invariant Lagrangians; multiplets, representations, and matter fields; covariant derivatives for general multiplets; the adjoint multiplet; multiple gauge groups; Standard Model example.
Started Lorentz symmetry: generators and representations; unitary but infinite particle representations; little groups and Wigner theorem; massive particles have definite spins; massless particles have definite helicities; tachyons have nothing; maybe Wigner theorem in D≠4 dimensions.

October 1 (Thursday):

Lorentz symmetry: little groups and Wigner theorem; massive particles have definite spins; massless particles have definite helicities; tachyons have nothing; Wigner theorem in D≠4 dimensions; Lorentz multiplets of fields; (j₊,j_-) multiplets; Weyl spinors and Spin(3,1)=SL(2,C); vectors and bispinors; tensors.
Dirac spinors and spinor fields: Dirac spinor representation of the Lorentz symmetry; Dirac equation and its covariance.

Extra lecture on October 2 (Friday):

Conformal symmetry: definition; complex language in Euclidean 2D; conformal symmetry group and its generators; conformal algebra in D>2 dimensions, Euclidean or Minkowski.
Conformal field theories and their application: world-sheet QFT in string theory; condenced matter at a critical point; conformal window of QCD; AdS/CFT duality.

October 6 (Tuesday):

Dirac spinor fields: Dirac conjugation and Dirac Lagrangian; Hamiltonian for the quantum Dirac field.
Grassmann numbers and classical fermionic fields.
Fermionic algebra and Fock space: Hilbert stace of one fermionic mode; multiple modes; fermionic Fock space; wave functions and operators; fermionic particles and holes; holes as quasiparticles.

October 8 (Thursday):

Finish fermionic Fock space: Fermi sea, extra particles and holes; interacting particles and holes.
Relativistic electrons and positrons: naive diagonalization of the Dirac Hamiltonian; positrons as holes in the Dirac sea; expanding the Dirac fields into creation and annihilation operators; energy and charge of the Dirac sea.
Charge conjugation symmetry: C:e⁻↔e⁺; C:Φ(x)→Φ^*(x); C:Ψ(x)→γ²Ψ^*(x); neutral particles and C–parity; Majorana fermions; counting degrees of freedom.

Plan for October 13 (Tuesday):

Dirac, Majorana, and Weyl fermions: counting degrees of freedom; Majorana–Weyl equivalence; Majorana mass term; massless and massive neutrinos.
Parity and other discrete symmetries: parity; CP; time reversal (briefly); CPT theorem; baryogenesys and Sakharov's criteria.
Chiral symmetries: vector and axial symmetries of a massless fermion; Weyl fermions and chiral symmetries; U(N)_L⊗U(N)_R symmetry of N massless fermions; maybe chiral symmetry in QCD; chiral gauge theories; electroweak example.

Tentative plan for October 15 (Thursday):

Finish Chiral symmetries: chiral symmetry in QCD; chiral gauge theories; electroweak example.
Relativistic causality for the fermions: commuting and anticommuting fields; checking anticommutativity of free Dirac fields at spacelike separation.
Spin-statistics theorem: stating the theorem; plane waves and lemmas; proving the theorem; generalize to d≠4 dimensions; proving the lemmas (skipped).
Feynman propagator for Dirac fermions.
Give out the midterm exam.

QFT 2, Spring 2027 semester

Too early to plan.