|Content:||The goal of this semester is to give an introduction to the
topological and geometric descriptions of quantum matter and quantum fields.
|Hours and Venue:||Wednesday 3:00 p.m. - 3:50 p.m., MATH 350|
|Topics and talks:|
Abstract: As first noticed in the 1970s, the charges of fermions follow patterns that allow the Standard Model U(1)xSU(2)xSU(3) to be embedded in Grand Unified groups SU(5) and SU(2)xSU(2)xSU(4), which in turn embed in SO(10), or equivalently in its covering group Spin(10). Spin(10) seems to know about Lorentz transformations: notably, the Dirac pseudoscalar miraculously equals the Spin(10) pseudoscalar. But Spin(10) lacks a time dimension. To accommodate time, 2 dimensions must be adjoined (the complex structure of spinors means that they live naturally in even dimensions). The main result is: The Dirac (Poincare) and SM algebras are commuting subalgebras of the spin algebra associated with the group Spin(11,1) of transformations of spinors in 11+1 spacetime dimensions. Each of the 3 spatial dimensions of 3+1 dimensional spacetime proves to be a 5-brane.
Abstract: Spinors -- spin 1/2 particles -- come already quantized. They come in two species, row spinors and column spinors, which already have the properties of fermion creation and annihilation operators: you can multiply a row spinor by a column spinor or vice versa, but you cannot multiply a row spinor by a row spinor or a column spinor by a column spinor. The anticommutation property of spinors can be traced to the fact that the spinor metric in 4 dimensions is antisymmetric. This seminar explores building quantum field theory on a foundation of the known properties of spinors, rather than on postulated rules. Spinors have properties beyond those invoked by qft; for example, the algebra of outer products of spinors yields (is isomorphic to) the multivector algebra of strings and branes of all dimensions. Might spinors point to a way forward for qft at unification scales?
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.
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.
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.
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.
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).
Borcherds, R., Renormalization and quantum field theory, Algebra & Number Theory, Vol. 5, Nr 5 (2011), 627-658.