Seminar on Mathematical Aspects of QFT

This is a backup of the website for a PhD seminar at Utrecht U from Spring 2014–winter 2015.

 

The (rather ambitious) goal was to understand quantum field theory (QFT) from a mathematician’s point of view, following the two volumes of “Quantum Fields and Strings: A Course for Mathematicians”, see references below. The goal of the seminar is to get familiar with QFT in the sense that we hope to understand the physicist’s reasoning about QFT from a more mathematical point of view.

The rough schedule consists of several parts:

  • Part I: Classical field theory, based on [DF1] and [Fr] §1–3.
  • Part II: Scalar quantum field theory, based on [Fr] §5, [De2], [Ka], [Wi1] (and [Gr], [Et]?)
  • Part III: Fermionic quantum field theory, based on [DM], [Fr] §4–5, [Wi2].
  • Part IV: Quantum Yang-Mills theory, based on ….
  • Part V: Supersymmetry and Seiberg-Witten theory, based on parts of [DF2] and [Wi3].

In case of any questions you can contact Jules.

Note: Students are welcome, but should be aware that this is not an actual course. In particular, participating does not yield any credits (ECTS).

Schedule (updated regularly)

Teal text refers to reminders of material that was treated before in the seminar.

Part I: Classical field theory

Date Venue Speaker Description References
May 26 MI 610 Shan Lagrangean mechanics. Space of trajectories. Example: nonrelativistic point particle. [Fr1] §1.4
[DF1] §1.1; [Bae]; [nC1]
Jun 5 BBL 007 Ralph Recap of some differential geometry: principal G-bundles; associated bundle construction; connections and parallel transport. [Fr1] §1.1, 3.4
Jun 12 BBL 007 Ori Euler-Lagrange equations revisited. Jet bundles. Lagrangian field theories. Example: nonrelativistic point particle revisited. [Fr1] §1.4, 2.1–2.2
[DF1] §2.2–2.3
Jun 16 MI 610 Ralph Electromagnetism. Symmetries and Noether’s theorem. [Fr1] §1.5
[DF1] §1.1, 1.3
Sep 18 MI 610 Jules Reminder: nonrelativistic point particle. Spacetime. Relativistic point particle. [Fr1] §1.1–1.2, 1.4, 1.6
[DF1] §1.1–1.2, 2.1
Sep 25 MI 610 Jules Variational bicomplex. Lagrangean field theories. Example: free scalar field. [Fr1] §2.1–2.2, 3.1; [nL2]
[DF1] §2.2–2.4, 3.1–3.3
Oct 2 MI 610 Joey Remark about locality. Noether’s theorem revisited. Hamiltonian structures. [Fr1] §1.3, 2.3–2.4, 3.2
[DF1] §1.3, 2.5–2.8
Oct 9 MI 610 Joey Energy-momentum tensor. Examples: massive scalar field; general relativity. [Fr1] Exercise 2.6
[DF1] §2.9
Oct 16 MG 025 Shan Electromagnetism revisited. Pure Yang-Mills theory. [Fr1] §1.5, 3.3–3.5
[DF1] §3.5, 4; [nL1]?
Oct 23 MI 610 Davide Classical vacua. Nonlinear σ-models. Gauged σ-models. [Fr1] §3.1–3.2, 3.6
[DF1] §2.10, 5
Oct 30 MI 610 Michael Overview of spinors. Matter: fermions. [De1] §1
[DF1] §3.4
Nov 6 No seminar; GQT school and colloquium.
Nov 10 MI 610 Ralph Dimensional reduction. Topological terms. [DF1] §2.11, 6

Note that the variational bicomplex is related to what physicists call the “covariant phase-space” approach to classical field theory, see also [nL2].

Part II: Scalar quantum field theory

Date Venue Speaker Description References
Nov 20 MI 610 Rob Basics of quantum mechanics. Motivation. C*-algebras in physics. Uncertainty principle, Heisenberg algebra, Weyl algebra. Schrödinger equation, wave functions. Harmonic oscillator. [St1], [Gr]
Nov 27 MI 610 Shan Quantization of free bosonic field theories. Example: free real scalar field. (Skip superstuff and ignore superscript 0s.) [Fr1] §5.1–5.2
[De2] §1–3
Dec 8 BBG 069 Joey Canonical quantization of free scalar revisited. Locality (= microcausality).
Note the unusual date and time (13:15–15:00).		 [PS] §2; [Ze] §I.8
Dec 11 BBG 069 Jules Quantum fields as operator-valued distributions. Wightman axioms. [Sc] §8
[Ka] §1.0–1.2
Dec 18 MI 610 Peter Feynman path integral. Classical limit. [LB] §23; [Ze] §I.2
Jan 8 MI 610 Davide Scalar propagators as Green’s functions for Klein-Gordon. Feynman iε-prescription. Time ordering. [LB] §16–17; [Ze] §I.3–I.4
Jan 15 BBG 308 Shan Physical interpretation of particles and propagators. Higher correlation functions and Wick’s theorem. [PS] §4.2–4.3;
[LB] Ex 17.3, §18.5
Jan 22 MI 610 Michael Scattering. The S-matrix. Connection to particle physics. [PS] §4.5; [LB] §18, 20.4;
[St2] §6; [Ka] §4?
Jan 29 BBG 071 Jules Perturbative expansion of the S-matrix. Feynman diagrams. Example: φ4 theory. [LB] §19, 20.3; [Ze] §I.7; [PS] §4.4, 4.6; [Ka] §5?
Feb 5 MI 610 Joey LSZ reduction formula. Gell-Mann–Low formula. More about Feynman diagrams. Quantum-corrected propagator? [PS] §4.4, 7.2; [LB] §22; [Ha2] §7; [LB] §31.1–31.3?
Feb 12 MI 610 Ralph Wick rotation. Euclidean time. Statistical field theory. Schwinger functions? [DF1] §7.1–7.2; [LB] §21, 25; [PS] §9.3; [Ka] §2.1–2.2?
Feb 19 MI 610 Shan Renormalization of Feynman diagrams:
a first look at dimensional regularization.		 [Wi1] §1.4, 1.2–1.6
Feb 26 MI 610 Jules Perturbative renormalizability: generalities. [Wi1] §2.1, 2.2–2.3
Mar 2   extra   session MI 610 Jules Perturbative renormalization for φ3 theory. Comments on critical case.
Note the unusual date and time (13:15–15:00).		 [Wi1] §1–2; [PS] §10; [Sr] §12–18, 27
Mar 5 MI 610 Davide Composite operators. OPE for free theories. OPE and interactions? [Wi1] §3.1–3.5, 3.6–3.8?
[PS] §18.3 (first page)
Mar 11 MI 611 Joey Renormalization group. Wilsonian effective theory. More about dimensional regularization?
Note the unusual date and time (13:15–15:00).		 [Ze] §III.1; [Co] §1; [Gr] §1; [Po] §2; [PS] §12.1; [LB] §34.1–34.3, 35
Mar 19 MI 610 Michael More about the renormalization group. [Gr] §1.3
… … … Plan: Regularization revisited? Riemann ζ-function regularization? [Fr2] §V.3.5–V.4; [Et]; …
… … Jules Plan: Recap of renormalization group. Overview of renormalization-group equation. [Gr] §2–3, 5; [Po]; …; [PS] §12.2; [Ze] VI.8?
… … … Plan: Renormalization-group equation. [Gr] §3; [PS] §12.3–12.5; [Gr] §5

 

Part III: Fermionic quantum field theory

Date Venue Speaker Description References
Mar 26 MI 610 Davide Linear superalgebra. [DM] §1.1–1.6 (¬ 1.3.7); [Ma]
Apr 2 MI 610 Ralph More about linear superalgebra: Berezinian, change of variables. [DM] §1.10(C);
[Va] §3.6, 4.6
Apr 9 MI 610 Shan Supermanifolds. Batchelor’s theorem. Taylor expansion of sections. [DM] §2.1–2.3;
[Va] p.132, Lem 4.3.2
April 16 MI 610 Shan, Joey More about supermanifolds. Lie supergroups. Analysis on supermanifolds. [DM] §2.8–2.10; [Va] Thm 4.3.1; [Le] §2.3.1–2.3.7 (¬ 2.3.5), 3.2.6
April 22 MI 610 Joey Differential geometry on supermanifolds. Vector fields. Differential forms. Lie superalgebras. Integration. [Le] Lem 2.4.6, 2.4.8;
[DM] §3.2–3.3, Prop 3.10.5, 3.12.3
April 29 MI 610 Michael Clifford algebras. Spin, Pin as its subgroups. [LM] §I.1–I.2; [De1]; [Fi1]
May 6 MI 610 Michael Classification of Clifford algebras. Representation theory. [LM] §I.3–I.5; [De1]; [Fi1]
May 13 MI 610 Ralph Clifford modules (cont’d). Spin manifolds. Dirac operator. [LM] §II.1–II.5; [De1]
[Fi2] §2–3
May 20 No seminar.
May 27 No seminar.
Jun 3 No seminar; GQT school and colloquium.
Jun 15 MG 401 Jules Clifford algebras for Minkowski spacetime. Majorana, Dirac, Weyl (s)pinors.
Note the unusual date.		 [Fi1] §1, 5.4
[DF2] §1.1
Jun 17 MG 401 Jules, Davide Recap of free fermionic classical field theory. Canonical quantization. Explicit free classical solutions. [Fr1] §5.4, 5.2; [DF1] §3.4; [W+] §2 FP2, FP16
[PS] §3.2–3.3; [LB] §36
Jun 24 MI 420 Davide Spin-statistics theorem. [PS] §3.5; [Ka] §1.4
Jul 1 MG 401 Joey Fermions from supergeometry.
Invariant bilinear forms for (s)pinors [handout].
Note the unusual time (10:00–13:00).		 [W+] §2.FP2
[Ha1] §…; [De1] §…; [Fi1] §4

 

Part IV: Quantum Yang-Mills theory

In Fall 2015 the seminar is held biweekly, with three-hour talks on Wednesdays at 13:15–16:00.

Date Venue Speaker Description References
Sep 23 MG 401 Shan Recap of classical Yang-Mills. Wilson loop functionals. [Fr1] §1.5, 3.3–3.5, [Wi3] §7.6 (begin); [nL3]
Oct 14 MG 401 Joey Aharonov-Bohm experiment. Quantizing Yang-Mills. Faddeev-Popov quantization. Constraints in classical mechanics. [Ze] §I.1.5; [Ha2] §5, 15; [To] §6; [FM] §7.5; [vH]; [HT] §1.1
Oct 21 MI 610 Joey Dirac bracket. Longitudinal derivative. Koszul-Tate resolution. BRST differential. Quantization. [HT] §1.2–1.5, 2, 5.3, 8–10, 11.1–11.2, 14.1–14.2
Nov 4 MI 610 Mike Example: BRST for (classical) Maxwell theory. [HT] §19–19.1.5
Nov 25 MI 610 Davide Ghost dynamics. BV formalism. [HT] §11.2, 17
… … … Plan: Quantization in BV formalism. QED. Relation with Faddeev-Popov? [Ha2] §5; [To] §5; [Sr] §55; [LB] §14; [HT] §13–14

 

Part V: Supersymmetry

Date Venue Speaker Description References
Dec 9 16 23 Jan 6 MI Spr Ralph Introduction to supersymmetry.
Note: the Springer room is on the 7th floor

 

References

Our main references are the two volumes of Fields and Strings: A Course for Mathematicians (AMS, 1999), which are the written records of a one-year program held at the IAS in Princeton.

In particular we may use the following texts contained in those two volumes:

  • [De1] Deligne, Notes on Spinors (vol 1, pp 99–135)
  • [De2] Deligne, Note on Quantization (vol 1, pp 367–375)
  • [DF1] Deligne and Freed, Classical Field Theory (vol 1, pp 137–225)
  • [DF2] Deligne and Freed, Supersolutions (vol 1, pp 227–355)
  • [DF3] Deligne and Freed, Sign Manifesto (vol 1, pp 357–363)
  • [DM] Deligne and Morgan, Notes on Supersymmetry (vol 1, pp 41–97)
  • [Et] Etingof, Note on Dimensional Regularization (vol 1, pp 597–607)
  • [Fa] Faddeev, Elementary Introduction to Quantum Field Theory (vol 1, pp 513–550)
  • [Gr] Gross, Renormalization Groups (vol 1, pp 551–596)
  • [Ka] Kazhdan, Introduction to QFT (vol 1, pp 377–418)
  • [Wi1] Witten, Perturbative Quantum Field Theory (vol 1, pp 419–473)
  • [Wi2] Witten, Index of Dirac Operators (vol 1, pp 475–511)
  • [W+] Witten, Deligne, Freed, Jeffrey and Wu, Homework (vol 1, pp 609–717)
  • [Wi3] Witten, Dynamics of Quantum Field Theory (vol 2, pp 1119–1424)

Other references that might be useful are

  • [Ba] Baez, Torsors made easy (blog post, 2010)
  • [DB] Dragon, Brandt, BRST Symmetry and Cohomology (arXiv:1205.3293)
  • [EPZ] Esposito, Pelliccia and Zaccaria, Gribov Problem for Gauge Theories: a Pedagogical Introduction (arXiv:hep-th/0404240)
  • [FM] De Faria and De Melo Mathematical Aspects of Quantum Field Theory (Cambridge University Press, 2010) [ebook]
  • [Fi1] Figueroa-O’Farrill, Majorana Spinors, a modern approach (lecture notes)
  • [Fi2] Figueroa-O’Farrill, Spin geometry (lecture notes, 2010)
  • [Fo] Folland, Quantum Field Theory: A Tourist Guide for Mathematicians (AMS, 2008)
  • [Fr1] Freed, Classical Field Theory and Supersymmetry (IAS/Park City Lectures, 2001) [pdf] *
  • [Fr2] Fredenhagen, Quantum Field Theory (lecture notes, 2010) [pdf]
  • [Gr] Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2004)
  • [Ha1] Harvey, Spinors and Calibrations (Academic Press, 1990)
  • [Ha2] Hatfield, Quantum Field Theory of Point Particles and Strings (Westview Press, 1998)
  • [HT] Henneaux and Teitelboim, Quantization of Gauge Systems (Princeton University Press, 1994)
  • [Jo] Jost, Geometry and Physics (Springer, 2009) [pdf]
  • [LB] Lancaster and Blundell, Quantum Field Theory for the Gifted Amateur (Oxford University Press, 2014)
  • [Le] Leites, Introduction to the theory of supermanifolds (Rus. Math. Surv. 35:1, 1980)
  • [LM] Lawson and Michelsohn, Spin Geometry (Princeton University Press, 1990)
  • [Ma] Manin, Gauge Field Theory and Complex Geometry (Springer, 1997)
  • [nC1] Dimensional Analysis and Coordinate Systems (n-Category cafe, 2006)
  • [nC2] Unitary Representations of the Poincaré Group (n-Category cafe, 2009–2013)
  • [nL1] Gauge group and gauge transformation (entries on nLab)
  • [nL2] Covariant phase space (entry on nLab)
  • [nL3] Categorification (entry on nLab)
  • [nL4] Deligne’s theorem on tensor categories for supersymmetry (entry on nLab), see also this blog post
  • [Po] Polchinski, Renormalization and effective lagrangians (Nucl Phys B 231, 1984) [pdf]
  • [PS] Peskin and Schröder, An Introduction To Quantum Field Theory (Westview Press, 1995)
  • [Sc] Schottenloher, A Mathematical Introduction to Conformal Field Theory (Springer, 2008)
  • [Sr] Srednicki, Quantum Field Theory (Cambridge University Press, 2007) [pdf]
  • [St1] Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics: A Short Course for Mathematicians (World Scientific, 2008)
  • [St2] Strocchi, An Introduction to the Non-Perturbative Foundations of Quantum Field Theory (Oxford University Press, 2013)
  • [St3] Strassler, Virtual Particles: What are they? (blog post, 2011)
  • [To] Tong, Lectures on Quantum Field Theory (lecture notes, 2012)
  • [vdB] Van den Ban, Applications of representation theory in classical quantum mechanics (lecture notes, 2004)
  • [vH] Van Holten, Aspects of BRST Quantization (arXiv:hep-th/0201124)
  • [Va] Varadarajan, Supersymmetry for Mathematicians: An Introduction (AMS, 2004)
  • [Ze] Zee, Quantum Field Theory in a Nutshell (Princeton University Press, 2010)

and perhaps, at some point,

  • [Co] Costello, Renormalization and Effective Field Theory (AMS, 2011) [pdf]
  • [CG] Costello and Gwilliam, Factorization algebras in quantum field theory (draft, 2012) [pdf]
  • [Ma1] Mallios, Modern Differential Geometry in Gauge Theories: Maxwell Fields (Birkhäuser, 2006) [pdf]
  • [Ma2] Mallios, Modern Differential Geometry in Gauge Theories: Yang-Mills Fields (Birkhäuser, 2010) [pdf]
  • [Pa] Paugam, Towards the Mathematics of Quantum Field Theory (Springer, 2014) [pdf]

*) For convenience let’s number the subsections of [Fr]:

Lecture 1 (Classical mechanics): 1.1 Particle motion, 1.2 Some differential geometry, 1.3 Hamiltonian mechanics, 1.4 Lagrangian mechanics, 1.5 Classical electromagnetism, 1.6 Minkowski spacetime
Lecture 2 (Lagrangian field theory and symmetries): 2.1 The differential geometry of function spaces, 2.2 Basic notions, 2.3 Symmetries and Noether’s theorem, 2.4 Hamiltonian structures
Lecture 3 (Classical bosonic theories on Minkowski spacetime): 3.1 Physical lagrangians and scalar field theories, 3.2 Hamiltonian field theory, 3.3 Lagrangian formulation of Maxwell’s equations, 3.4 Principal bundles and connections, 3.5 Gauge theory, 3.6 Gauged σ-models
Lecture 4 (Fermions and the supersymmetric particle): 4.1 The supersymmetric particle, 4.2 A brief word about supersymmetric quantum mechanics, 4.3 Superspacetime approach
Lecture 5 (Free theories, quantization, and approximation): 5.1 Quantization of free theories: general theory, 5.2 Quantization of free theories: free fields, 5.3 Representations of the Poincaré group, 5.4 Free fermionic fields, 5.5 The general free theory, 5.6 General theory, 5.7 Perturbation theory
Lecture 6 (Supersymmetric field theories): 6.1 Introductory remarks and overview, 6.2 Super Minkowski spacetime and the super Poincaré group, 6.3 Examples of super Poincaré groups, 6.4 Representations of the super Poincaré group
Lecture 7 (Supersymmetric σ-models): 7.1 Three-dimensional theory, 7.2 A supersymmetric potential, 7.3 Dimensional reduction to n = 2 dimensions, 7.4 Dimensional reduction to n = 1

 

Last updated:  January 6, 2016