Abstracts for Maths/Physics
seminars
Seeking topological
information in knot homologies
Yi Ni, Caltech
Knot homologies are generalizations of the polynomial invariants for
knots and links. The most famous
knot homologies are Khovanov homology and Knot Floer homology. I
will discuss the topological
information one can find in knot homologies, including genus, 4-genus,
fibration and triviality.
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Asymptotic Quantum
Electrodynamics and Representation Theory
Giuseppe Dito, Universite
de Bourgogne, France
It has been known for a long time that asymptotic fields in
QED are not free
fields and that Lorentz invariance is broken on the physical space.
In this talk, I will present a construction of an asymptotic
representation of the
Poincar\'e group defined on a Fock space $\mathcal F ^{\rho}$, $1/2
< \rho < 1$, with indefinite metric
built on a separable Hilbert space $E^\rho$. This Hilbert space appears
in the scattering theory of
the classical Maxwell-Dirac equations and contains electromagnetic
potentials decreasing as slowly as
$(1/|x|)^{3/2 - \rho + \epsilon}$ for some $\epsilon >0$. The
translation generators of this asymptotic
representation are given by quartic expressions with the feature that
the mass-square operator,
when restricted to the one-electron space, admits a sharp eigenvalue.
However, the standard indefinite metric is an
unbounded sesquilinear
form in $E^\rho$ and, as a consequence, Gupta-Bleuler methods
permit to construct a physical subspace $V_{GB}$ of $\mathcal F
^{\rho}$ which is only
invariant under $\mathbb{R}^4 \ltimes SU(2)$, but not under Lorentz
transformations.
This is a work in progress with Erik Taflin.