Content: |
The goal of this semester is to give an introduction to the topological and geometric descriptions of quantum matter and quantum fields. |
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Organizer: |
Markus Pflaum |

Hours and Venue: |
Wednesday 3:00 p.m. - 3:50 p.m., MATH 350 |

Topics and talks: |
- Recapitulation of Hilbert spaces and their uses in QM and QFT ( Jan 22)
- An Introduction to C*-Algebras (Robin Deeley, Jan 29)
- Algebraic Quantum Mechanics (Markus Pflaum, Feb 5
- Pseudodifferential Operators (Alex Nita, Feb 12)
- The Standard Model and Beyond (Andrew Hamilton, Feb 19)
- Founding Quantum Field Theory on Spinors (Andrew Hamilton, Feb 26)
- Wightman Axioms of QFT (March 4)
- Euclidean Field Theory à la Osterwalder-Schrader and Wick Rotation (Robert Maier, March 11)
- The Haag-Kastler Axioms (Daniel Spiegel, March 22)
- The Spectral Theory in Rigged Hilbert Space (Daniel Spiegel)
- Foliations (Juan Moreno, April 22, via zoom)
- Mathematical Theory of Fractons (Mike Hermele, April 29, via zoom)
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? |

Literature(with annotations): |
- Online Resources on mathematical QFT and foundations:
- Distribution Theory and Pseudodifferential Operators:
- Mathematics of Quantum Mechanics and Quantum Field Theory:
- Axiomatic Quantum Field Theory:
- Perturbative Algebraic Quantum Field Theory:
- Mathematical Renormalization:
Pflaum, M.J.,
The The CRing Project on Commutative Algebra and related topics. Brouder, Ch., Nguyen Viet Dang, and Hélein, F.,
Shubin, M.A., Markus: Trèves, F., Markus: Folland, G.B., Manny: Takhtajan, L.A., Manny: Borcherds, R.E. and Barnard, A., Leonard: Streater, R.F. and Wightman, A.S., Manny: R. Haag, Manny: K. Fredenhagen and K. Rejzner, R. Brunetti, M. Dütsch, K. Fredenhagen,
K. Rejzner, Manny: R. Brunetti, K. Fredenhagen, R. Verch, Manny: K. Fredenhagen and K. Rejzner, Borcherds, R., |