Phys 517, Fall 2016

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Sep. 26	Minkowski metric, Minkowski space and its isometries, Lorentz and Poincare groups, relativistic particles and representations of Poincare group, scalar fields and Klein-Gordon equation, Klein-Gordon inner product.	PPages 1-6 of Srednicki
2	Sep. 28	Remarks on the definition of the position operator, localized states, position wave function, and probability density of localization of a particle in nonrelativistic QM, the implications for the definition probability density of localization of a scalar relativistic particle, the historical developments leading to the Dirac equation, QFT formulation of the QM of a nonrelativistic n-particle system with n fixed: The Fock space, the Number operator, the fermonic analog.	Pages 7-14 of Srednicki
3	Oct. 03	Some structural properties of the Lorentz group, O(1,3): Proper and orthocronous Lorentz transformations, the connected components of O(1,3), party, time-reversal, transformations. Rotations and boosts as basic elements of SO(1,3)^+. Infinitesimally small Lorentz transformations, generators of O(1,3), and their commutation relation, i.e., the Lie algebra so(1,3).	Pages 15-17 of Srednicki
4	Oct. 05	Generators of the Poincare group in its unitary representations and their commutation (Lie algebra) relations. Classical field theory of a real scalar field, Lagrangian density, action functional, Euler-Lagrange equation, Lagrangian density for a massive Klein-Gordon (KG) field and the derivation of its field equation, solution of the KG equation using the method of Fourier transform, the Lorentz-invariant measure and Fourier coefficients a(k).	Pages 17-18 & 22-26 of Srednicki
Homework 1	Oct. 10
5	Oct. 10	Hamiltonian formulation of classical scalar field theories, conjugate momentum and Hamiltonian density, the classical Hamiltonian for a real KG field, the classical observables and Poisson bracket for scalar field theories, the basic observables associated with the scalar field and its conjugate momentum, the complex-valued function associated with the Fourier coefficients a(k) and their complex conjugate, the basic Poisson bracket relations for a real scalar field theory, canonical quantization and the equal-time commutation relations	Pages 26-28 of Srednicki
6	Oct. 12	Quantum Hamiltonian operator for a real KG field, constructing the ground  (vacuum) state vector |0> and other basis state vectors |k1,k2,...,kn> of the Fock space using the annihilation and creation operators, a(k)  and a(k)^†, the Lorentz-invariance of the inner product of the Fock space, derivation of the transformation rule for a(k) and  |k1,k2,...,kn> under general orthocronous Poincare transformations, construction of a unitary representation of the orthocronous Poincare group in the Fock space, derivation of the transformation rule for the field operator	Pages 28-29 of Srednicki
7	Oct. 17	Identification of the Hamiltonian operator with the generator of time-translation in the Focl space, the generators of the space-translations, interpretation of |k1,k2,...,kn> in terms of the state vector for an n-particle system with each of particles having a positive energy, the field operator as a Heisenberg-picture operator, the connection between single particle state vector |k> and the first quantized positive-energy complex classical scalar field. Path-Integral Formulation of QM in 1D: The localized states in the Heisenberg picture, the propagator, the phase-space and configuration-space path-integral formulas for the propagator	Pages 43-45 of Srednicki
8	Oct. 19	Path-integral formula for the expectation values of the time-ordered product of the position and momentum operators in an arbitrary state and in the ground state of a quantum system with a discrete spectrum, Path-integral formulation of perturbation theory	Pages 45-48 of Srednicki
9	Oct. 24	Application of the path-integral formula for the calculation of the expectation values of the time-ordered product of the position operators of a simple harmonic oscillator (SHO) in one dimension.	Pages 50-52 of Srednicki
10	Oct. 26	Review of the treatment of (SHO) and a condensed notion that simplifies the related calculations; Application of the path-integral formula for the calculation of the expectation values of the time-ordered product of the field operators for a free real KG field, Feynman propagator	Pages 52-56 of Srednicki
Homework 2
11	Oct. 31	Derivation of an explicit formula the Feynman propagator, scattering setup for a real scalar field, the Lehman-Symanzik-Zimmermann (LSZ) reduction formula	Pages 35-37 of Srednicki
12	Nov. 02	Comment on the particle interpretation for interacting fields and the conditions on the validity of the LSZ reduction formula. The phi^3-field theory: Perturbation series for the path-integral Z[J] and the Feynman diagrams	Pages 37-42 & 58-63 of Srednicki
13	Nov. 07	Various examples of Feynman diagrams and their Symmetry factor, the disconnected diagrams, the expression of  the path-integral Z[J] in terms of the connected diagrams.	Pages 60-65 of Srednicki
14	Nov. 09	Calculation of <0 | phi(x) | 0> for the phi^3-field theory to leading order in the coupling constant g, the emergence of an ultraviolet divergence, the inclusion of the linear counter term and the corresponding one-point vertices, the tadpole diagrams	Pages 65-67 of Srednicki
Homework 3
15	Nov. 14	Effect of the remaining counter terms and the corresponding two-point vertex, the proof of Zphi=1+O(g^2) and Zm=1+O(g^2), Calculation of <0 | T { phi(x1) phi(x2) } | 0> & <0 | T { phi(x1) phi(x2) phi(x3) phi(x4} | 0> for the phi^3-field theory to leading order in the coupling constant g, the exact propagator and the use of LSZ formula to compute the scattering amplitude for the two-particle scattering events	Pages 67-74 of Srednicki
16	Nov. 16	Derivation of the scattering amplitude for the two-particle scattering events to leading order term in powers of the coupling constant g, i.e., O(g^2), the scattering (transfer) matrix and the Feynman rules for its computation.	Pages 74-78 of Srednicki
17	Nov. 21	Scattering Cross Section: Madelstam variables, the probability for scattering two particles into n' particles, derivation of the formula for the total cross section for this process	Pages 79-83 of Srednicki
Midterm Exam	Nov. 23
Homework 4
18	Nov. 25	Differential cross section for the scattering of 2 particles into 2 particles in the center of mass frame, the case of identical particles, Lorentz-invariant expression for the differential cross section, application to phi^3-field theory; decay rates	Pages 83-87 of Srednicki
19	Nov. 28	Dimensionless and dimensionful coupling constants in phi^n-field theories, the Lehmann-Kallen formul for the exact propagator	Pages 90-95 of Srednicki
20	Nov. 30	Calculation of the exact propagator for phi^3-field theory in d-dimensions: One-particle irreducible diagrams and their calculation to the order g^2	Pages 96-100 of Srednicki
21	Dec. 05	Calculation of the exact propagator for phi^3-field theory to the order g^2 in 6-dimensions using dimensional regularization	Pages 100-105 of Srednicki
22	Dec. 09	Calculation of the exact three-point vertex function for phi^3-field theory to the order g^2 in 6-dimensions using dimensional regularization	Pages 111-114 of Srednicki
Homework 5
23	Dec. 12	Calculation of the one-loop correction to the exact four-point vertex function for phi^3-field theory 6-dimensions, the generalization to n-point vertex function, renormalizability, superficial degree of divergence of a digram, a sufficient condition for the nonrenormalizability of a generic diagram	Pages 115-120 of Srednicki
24	Dec. 14	Perturbation theory to arbitrary order, Feynman rules for computing loop corrections to scattering amplitude, the calculation one-loop correction to tree-level scattering amplitude for two-particle elastic scattering	Pages 121-126 of Srednicki
25	Dec. 19	Effective action, computation of the diagrams of a scalar field theory using the tree diagrams of the effective theory,  the quantum equation of motion, the vacuum expectation value in the presence of a source, the notion of a quantum potential	Pages 127-130 of Srednicki
26	Dec. 21	Symmetry transformations and Nöther's theorem in classical scalar field theories, a quartic field theory involving two real scalar fields and its SO(2) and O(2) symmetries, the complex description of this theory and its symmetries, the isomorphism of SO(2) and U(1), complex realization of the reflection symmetry, the calculation of the conserved change of the theory and its physical consequences.	Pages 132-135 of Srednicki
Homework 6
27	Dec. 26	Quantum analog of the Ehrenfest and Nöther theorems: The Schwinger-Dyson equations and the Ward-Takahashi Identities; Symmetry transformations that change the Lagrangian density by a divergence and the associate conserved currents and charges, application to the spacetime translations and Lorentz transformations, the energy-momentum tensor, generators of the Poincare' group in its spin-zero representations as conserved charges	Pages 135-139 of Srednicki
28	Dec. 28	Discrete Symmetries: Symmetry transformations and Wigner symmetry representation theorem, Parity and Time-reversal operators, pseudo-scalar field, Charge-conjugation and other examples of Z_2-symmetries	Pages 140-145 of Srednicki

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.