Mathematical Physics Seminar on
Geometry of Quantum Fields
Fall 2018

Content: The goal of this semester is to give an introduction to Mathematical Foundations of Quantum Field Theory.
Organizers: Emanuel Knill and Markus Pflaum
Hours and Venue: Wednesday 3:00 p.m. - 3:50 p.m., MATH 350
Topics and talks:
  • Introduction to Wightman axcioms of QFT (Markus Pflaum, Sep 4)
  • Introduction to Euclidean Field Theory à la Osterwalder-Schrader (Emanuel Knill, Sep 11 & Sep 18)
  • The Wightman axioms, continued (Markus Pflaum, Sep 25)
  • Spin Statistics Theorem
  • Causal fermion systems (Magdalena Lottner, Oct 17)
  • Symmetries in QFT and Bargmann's theorem (Daniel Spiegel)
  • Distributional approach to renormalization à la Epstein-Glaser
  • Axiomatic field theory in curved space-time
  • Peierls bracket
  • Deformation quantization of field theories à la Fredenhagen et. al.
Literature
(with annotations):
  • Online Resources on mathematical QFT and foundations:
  • Pflaum, M.J., The Geometry of Classical and Quantum Fields, Lecture Notes.

    The FANCY Project on Functional Analysis and Noncommutative Geometry.

    The CRing Project on Commutative Algebra and related topics.

  • Distribution Theory and Pseudodifferential Operators:
  • Brouder, Ch., Nguyen Viet Dang, and Hélein, F., A smooth introduction to the wavefront set, Journal of Physics A: Mathematical and Theoretical 47 (2014).

    Shubin, M.A., Pseudodifferential Operators and Spectral Theory, Second Edition, Springer, Berlin, Heidelberg, New York 2001.

    Markus: A classic on the topic.

    Trèves, F.,Topological Vector Spaces, Distributions and Kernels, Academic Press Inc., New York, 1967.

    Markus: This is a comprehensive monograph on the functional analytic foundations of distribution theory. It also contains an exposition of nuclear spaces and the Schwartz kernel theorem.

  • Mathematics of Quantum Mechanics and Quantum Field Theory:
  • Folland, G.B., Quantum Field Theory : A Tourist Guide for Mathematicians, AMS (2008).

    Manny: Honest mathematician's overview of traditional QFT as commonly taught and presented in typical QFT textbooks. I found this to be a very instructive read.

    Takhtajan, L.A., Quantum Mechanics for Mathematicians, AMS (2008).

    Manny: Advanced mathematical treatment, not self-contained. Sometimes not as explanatory as I would have liked. I couldn't determine the general formulation of classical graded (super) mechanics from the chapter on this topic.

    Borcherds, R.E. and Barnard, A., Lectures on Quantum Field Theory, arXiv:0204014 (2002).

    Leonard: These are Borcherd's UC Berkeley notes on quantum field theory, which contain some details that his paper Renormalization and Quantum Field Theory glosses over. À la Borcherds, Lagrangians are defined as the symmetrization of the sheaf of derivatives of classical fields, Feynman measures are defined as linear functionals on a rank-1 free module over the symmetrization of the algebra of compactly supported sections of the sheaf of local actions, and renormalizations are defined as automorphisms of the symmetrization of the sheaf of local actions, where the automorphisms are of a sheaf of Hopf algebras. The group of renormalizations is shown to act freely and transitively on the set of Feynman measures, so there is no canonical Feynman measure, but there is a canonical orbit of such measures under renormalization.

  • Axiomatic Quantum Field Theory:
  • Streater, R.F. and Wightman, A.S., PCT, Spin, Statistics, and All That, Basic Books.

    Manny: A classic mathematical introduction to axiomatic QFT based on Wightman's axioms.

    R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer 1996.

    Manny: The most explanatory treatese on QFT that I have found. It has detailed discussions and describes the necessary background nicely, but for detailed physics arguments and proofs (when available) you need to follow the references. It addresses foundational issues that are rarely clarified elsewhere.

  • Perturbative Algebraic Quantum Field Theory:
  • K. Fredenhagen and K. Rejzner, Perturbative Algebraic Quantum Field Theory, arXiv:1208.1428.

    R. Brunetti, M. Dütsch, K. Fredenhagen, Perturbative algebraic quantum field theory and the renormalization groups, Advances in Theoretical and Mathematical Physics, Volume 13, Number 5 (2009), 1541-1599.

    K. Rejzner, Perturbative Algebraic Quantum Field Theory, Springer 2016.

    Manny: The most complete compilation of definitions and results relevant to perturbative AQFT. Its subtitle is "An introduction for mathematicians", but it is not an introduction, you have to be comfortable in an exceptionally broad range of mathematics to make sense of it, and you better know what physics motivates this. TMarch 21o read and understand what's going on here you need ready access to a massive amount of other literature for the details, including proofs of many important results. The referenced proofs are too often not mathematically satisfactory or complete without backtracking through further literature. Annoyingly, there are multiple references to unpublished results of the author. On the other hand, the field has progressed a lot in the last decade, and while it is far from settled, this is the most complete overview of what has been accomplished. It is a commendable if frustrating book.

    R. Brunetti, K. Fredenhagen, R. Verch, The Generally Covariant Locality Principle -- A New Paradigm for Local Quantum Field Theory, Communications in Mathematical Physics 237, 31-68 (2003) arXiv:0112041.

    Manny: This describes the modern AQFT framework for field theory on a general curved space-time background. The covariance is implemented with a category-theoretical formalism. Pay close attention to "relative Cauchy evolution", I found the underlying idea to be universally useful.

    K. Fredenhagen and K. Rejzner, Quantum field theory on curved spacetimes: Axiomatic framework and examples, Journal of Mathematical Physics 57 (2016).

  • Mathematical Renormalization:
  • Borcherds, R., Renormalization and quantum field theory, Algebra & Number Theory, Vol. 5, Nr 5 (2011), 627-658.

Last modified: Sun, January 28, 2018