Phys777 & Phys778

What are these courses?

For a formal answer to this question, see

and

More informally, these course series introduce several interconnected concepts in theoretical physics. Although there is a sizable overlap with standard field theory courses (such as quantum field theory), bootstrap methods are inherently non-perturbative and the focus is rather on the implications of the conceptual points than the actual computations (unlike a field theory course).

Phys777 starts with some topics of the condensed matter physics with an approach from high energy physics, and discusses critical phenomena, Wiener path integrals, and RG flow. UV and IR fixed point analysis of RG flow then leads to scale and conformal symmetries, which are discussed in an introductory level for the rest of the semester. The course concludes with some modern tools of the conformal bootstrap program.

Phys778 starts with a review of special relativity and the axiomatic properties of quantum mechanics. It then moves onto conceptual discussion of Feynman path integrals, concept of quantization, s-matrix theory, and analytic structure of correlators. This leads to the contemplations of the implications of several fundamental principles (such as locality and unitarity) and the modern studies regarding these. The course concludes with an overview of on-going bootstrap programs (such as cosmological bootstrap).

Who can/should take these courses?

These are research level selective courses designed for graduate students in the department of Physics. Although the courses are rather self-contained, undergraduate level physics knowledge is prerequisite to follow them properly; that being said, any ambitious undergraduate student with a sufficient command of undergraduate physics is welcome to take these courses as well.

When/where are these courses?

I thought Phys777 (Phys778) in the Fall (Spring) semester of 2024-2025 academic year.

What are the course policies?

General:

• You accept by taking this course that you have read and agreed my policies in soneralbayrak.com/teaching/MyTeachingPolicy.
• This course is currently not offered: relevant course policies (including grading) will be available here next time the courses are offered again.

• In the last time they were offered, the relevant assessment policies were as follows:

  1.

  In Phys777, there were an oral-midterm-examination and a discussion-final-examination: see this file for further details. As part of their final examination, Phys777 students prepared review articles in the 2024 fall semester; you can check them out by downloading this zip file.

  2.

  In Phys778, the grading was conducted through a project proposal preperation and project presentation; see here for more details.

Course Material:

• As research level courses, neither Phys777 nor Phys778 can be fully followed through a book; nevertheless, many of the essential ingredients can be found in the following books:

  1.

  Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

  2.

  Elvang H, Huang Y-tin. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge: Cambridge University Press; 2015.

  3.

  V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, vol. 63. 1977

  4.

  E. S. Fradkin and M. Y. Palchik, Conformal quantum field theory in D-dimensions. Mathematics and Its Applications Book Series. Springer Dordrecht, 1996.

• Similar courses have been taught worldwide and some of them have freely available lecture notes online, including but not limit to

  1.

  Lecture notes of Slava Rychkov at various schools (EPFL, TASI, CEA Saclay, $\dots <!--\; FUNCTION\; APPLICATION\; -->$)

  2.

  Lecture notes of David Simmons-Duffin at various schools (Caltech, TASI, $\dots <!--\; FUNCTION\; APPLICATION\; -->$)

  3.

  Lecture notes of Marc Gillioz at SISSA

  4.

  Lecture notes of Shai Chester at Weizmann Institute

  5.

  Lecture notes of Joshua Qualls at National Taiwan Uni.

  6.

  Lecture notes of Bert Schellekens at Saalburg

  7.

  Lecture notes of Paul Ginsparg at Les Houches

  8.

  Lecture notes of Lorenz Eberhardt at DESY

  9.

  Lecture notes of Yu Nakayama at Taiwan Central Uni.

  10.

  Lecture notes of S. Dotsenko at Kyoto University

What will you learn in these courses?

_____________________________________

_____________________________________

(1) Concept of critical phenomena: Review of phase diagrams, ensembles, and free energy; Phenomenological approach to critical points, Landau potential, spontaneous symmetry breaking

(2) Euclidean (Wiener) Path Integrals: Gaussian path integrals, generating functions, cumulant expansion, Wick’s contraction; Correlation functions, correlation length;Critical exponents, universality

(3) Renormalization Group Flow: Position space RG, momentum space RG; Classification of fixed points; Analysis of Wilson-Fisher Fixed Point; Epsilon-expansion; Scaling of operators, relevancy under RG flow, scale invariance

(4) Conformal symmetry: Conformal symmetry as the natural extension of scale invariance; Dilation and conformal groups; Irreducible representations of the conformal group; Classification of operators under the action of the conformal group

(5) Symmetry of Maxwell Theory: Stress energy tensor, Virial current; Conformal (scale) symmetry of Maxwell Theory in $d=4$ ($d\ne 4$)

(6) Introduction to Conformal Bootstrap: Operator Product Expansion (OPE), Associativity of OPE, Crossing Symmetry; Conformal kinematics, conformal frame; Casimirs of the conformal group, conformal blocks, hypergeometric functions; Unitarity bounds, lightcone limit, spectrum of CFTs; Numerical tools in Conformal Bootstrap _____________________________________

_____________________________________

(1) Mathematical Review of Special Relativity: Lorentz group, its subgroups, and associated algebras; Relation between Lorentz and Rotation group: imaginary time and Wick rotation; Stabilizer subgroup formalism (Wigner classification); Representations of Lorentz group, spin; Lorentz covariance of physical observables

(2) Axiomatic Approach to Quantum Mechanics: Hilbert space and its properties; Pictures, Moller operators; Euclidean & Lorentzian time evaluation; Distributional nature of Lorentzian operators and basic introduction to distributions

(3) Lorentzian (Feynman) Path Integrals: Feynman’s approach to quantum mechanics: quantization via path integrals, Feynman diagrams; Connection between Feynman & Wiener path integrals; Branch cuts of Lorentzian correlators, time ordering

(4) Scattering Matrix: Haag-Rulle construction, LSZ theorem; Analyticity of S-matrix, implications of causality; Locality vs Causality vs Cluster decomposition; Generalization to homogeneous spaces: active studies

(5) Concept of Unitarity: Optical theorem, cutting rules, spectral decomposition

(6) Bootstrap Philosophy & Modern non-perturbative techniques: Perturbative vs non-perturbative, exact vs constrained results, analytic vs numeric approaches; Global symmetries and how they can be utilized; Coleman-Mandula theorem; On-shell recursion relations, BCFW, spinor helicity, amplitudes program; Positive geometry, cosmological polytopes

What is the course calendar?