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 YangMills theory, based on ….
 Part V: Supersymmetry and SeibergWitten 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 Gbundles; associated bundle construction; connections and parallel transport.  [Fr1] §1.1, 3.4 
Jun 12  BBL 007  Ori  EulerLagrange 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  Energymomentum tensor. Examples: massive scalar field; general relativity.  [Fr1] Exercise 2.6 [DF1] §2.9 
Oct 16  MG 025  Shan  Electromagnetism revisited. Pure YangMills theory.  [Fr1] §1.5, 3.3–3.5 [DF1] §3.5, 4; [nL1]? 
Oct 23  MI 610  Davide  Classical vacua. Nonlinear σmodels. Gauged σmodels.  [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 phasespace” 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 operatorvalued 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 KleinGordon. Feynman iεprescription. 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 Smatrix. Connection to particle physics.  [PS] §4.5; [LB] §18, 20.4; [St2] §6; [Ka] §4? 
Jan 29  BBG 071  Jules  Perturbative expansion of the Smatrix. Feynman diagrams. Example: φ^{4} theory.  [LB] §19, 20.3; [Ze] §I.7; [PS] §4.4, 4.6; [Ka] §5? 
Feb 5  MI 610  Joey  LSZ reduction formula. GellMann–Low formula. More about Feynman diagrams. Quantumcorrected 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 φ^{3} 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 ζfunction regularization?  [Fr2] §V.3.5–V.4; [Et]; … 
…  …  Jules  Plan: Recap of renormalization group. Overview of renormalizationgroup equation.  [Gr] §2–3, 5; [Po]; …; [PS] §12.2; [Ze] VI.8? 
…  …  …  Plan: Renormalizationgroup 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  Spinstatistics 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 YangMills theory
In Fall 2015 the seminar is held biweekly, with threehour talks on Wednesdays at 13:15–16:00.
Date  Venue  Speaker  Description  References 
Sep 23  MG 401  Shan  Recap of classical YangMills. Wilson loop functionals.  [Fr1] §1.5, 3.3–3.5, [Wi3] §7.6 (begin); [nL3] 
Oct 14  MG 401  Joey  AharonovBohm experiment. Quantizing YangMills. FaddeevPopov 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. KoszulTate 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 FaddeevPopov?  [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 oneyear 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:hepth/0404240)
 [FM] De Faria and De Melo Mathematical Aspects of Quantum Field Theory (Cambridge University Press, 2010) [ebook]
 [Fi1] FigueroaO’Farrill, Majorana Spinors, a modern approach (lecture notes)
 [Fi2] FigueroaO’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 (nCategory cafe, 2006)
 [nC2] Unitary Representations of the Poincaré Group (nCategory 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 NonPerturbative 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:hepth/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: YangMills 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 σmodels
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 σmodels): 7.1 Threedimensional theory, 7.2 A supersymmetric potential, 7.3 Dimensional reduction to n = 2 dimensions, 7.4 Dimensional reduction to n = 1