Skip to main content

Topics in Perturbative QFT and the p-adic Holographic Correspondence

Ma 191cSpring 2018-19Caltech

Aspects of partition functions, path integrals and renormalization group flow, leading to a toy model of the p-adic holographic correspondence.

Overview

This course discusses aspects of partition functions, path integrals and renormalization group flow in statistical mechanics, quantum mechanics and scalar quantum field theories. Topics expected to be covered include: functional integrals, Feynman diagrams, correlation functions, lattice systems, renormalization group (RG) flow, p-adic (conformal) field theories, Bruhat-Tits tree as discrete anti-de Sitter space, and a toy model of the p-adic holographic correspondence.

Course prerequisites

  • calculus and classical mechanics
  • familiarity with statistical and quantum physics, as well as abstract algebra and p-adic numbers, will be useful but is not strictly imposed

Meeting time and location

  • Tuesday and Thursday, 9:00 – 10:25 am
  • 289 Linde Hall

Instructor

  • Sarthak Parikh (he/him) · 202 Kellogg/Linde Hall · Office hours by appointment

Grading

  • This class is offered Pass/Fail.
  • Student presentations: 60%
  • Class participation: 40%

Lecture outline

  1. Course overview.
  2. Overview of classical statistical mechanics, (grand) canonical ensembles, partition function, expected values, coarse-graining, thermodynamic limit.
  3. Correlation functions, generating function, one-dimensional Ising model, transfer matrix method, exact solution, spin-correlations, correlation length, high and low temperature behavior.
  4. Odd/even-point spin correlation functions, position space RG in one dimension.
  5. Fixed points of RG flow, flow diagrams; phases, phase transitions, 1st and 2nd order phase transitions, para/ferromagnetic and condensation phase diagrams.
  6. Comments on RG flow in higher-dimensional theories, higher-dimensional Ising model, connected correlation functions, magnetization, various boundary conditions for Ising model, phase transitions, critical phenomena and exponents.
  7. Critical behavior and exponents -ctd-, scaling/hyperscaling laws, Kadanoff block-spin RG flow.
  8. Continuous spin model, phi-fourth theory, Gaussian integral.
  9. Overview, multi-dimensional Gaussian integrals, Wick contractions, field strength renormalization, Wick contractions in interacting phi-fourth theory.
  10. Feynman diagrams, Feynman rules in position space, n-point Green’s functions, connected and disconnected diagrams, free energy vs partition function.
  11. Feynman diagrams & free energy in interacting lattice theory, propagation of particles, Minkowski spacetime, Minkowski metric, quadratic form and real normed division algebras.
  12. Pseudo-Riemannian manifolds, Lorentzian inner product, O(d,1) and subgroups, proper Lorentz group SO⁺(d,1), conformal group, Poincaré group, SL(2,K) as a double cover of SO⁺(d,1) for K=R,C,H,O. Fields: field bundles, field fibers, field histories as sections of field bundles, field configurations, examples of field bundles (scalar fields and vector potentials), general discussion on path integrals in QFT, connection between Euclidean QFT and statistical field theory.
  13. Tree-level Feynman diagrams and classical expansion, functional path integral, Feynman propagator, causality, first glimpses of AdS/CFT.
  14. Wilsonian low-energy effective theories, finite dimensional functional integrals in cut-off QFTs, RG flow equation in energy basis, a Wilsonian definition of perturbative scalar QFT, asymptotic locality in position space, renormalization scheme, counterterms.
  15. Renormalizability theorem for scalar QFT; AdS/CFT continued.
  16. Large N, anti-de Sitter space, hyperbolic space, isometry group, Poincaré coordinates, homogeneous spaces as quotients, conformal group, Iwasawa decomposition, irreps of conformal group, primary operators, conformal transformations, conformal covariance.
  17. Conformal invariance and low-point conformal correlators, CFT data, AdS/CFT dictionary examples, prescription for correlators, toy example of AdS/CFT, mass-dimension relation, semi-classical bulk dynamics, bulk-to-bulk and bulk-to-boundary propagators.
  18. Bulk Feynman diagrams, 1/N expansion, generalized free field theory, nonlocal/local kinetic term; p-adic numbers, Bruhat-Tits tree as quotient geometry and discrete Riemann symmetric space, CFTs on the projective line over p-adic numbers, lattice bulk theory on Bruhat-Tits tree, p-adic holography.
  19. Student presentations (Angus Gruen: Spontaneous symmetry breaking in QFT; Eitan Borgnia: Dirac operators on graphs and spectral triples; William Xu: Partition function of the 2d Ising model).

Suggested books and reference material

  • M. Kardar, Statistical Physics of Particles and Statistical Physics of Fields
  • A. Zee, Quantum Field Theory in a Nutshell
  • Gerald Folland, Quantum Field Theory: A Tourist Guide for Mathematicians
  • Kevin Costello, Renormalization and Effective Field Theory
  • AdS/CFT: D’Hoker and Freedman (hep-th/0201253), Penedones (arXiv:1608.04948)
  • p-adic AdS/CFT: Gubser, Knaute, Parikh, Samberg, and Witaszczyk (arXiv:1605.01061), Heydeman, Marcolli, Saberi, and Stoica (arXiv:1605.07639)
  • Bruhat-Tits trees: appendix of Brekke and Freund (link)