76 research outputs found
Derivation of the total twist from Chern-Simons theory
The total twist number, which represents the first non-trivial Vassiliev knot
invariant, is derived from the second order expression of the Wilson loop
expectation value in the Chern-Simons theory. Using the well-known fact that
the analytical expression is an invariant, a non-recursive formulation of the
total twist based on the evaluation of knot diagrams is constructed by an
appropriate deformation of the knot line in the three-dimensional Euclidian
space. The relation to the original definition of the total twist is
elucidated.Comment: 26 page
Cliffordization, Spin and Fermionic Star Products
Deformation quantization is a powerful tool for quantizing theories with
bosonic and fermionic degrees of freedom. The star products involved generate
the mathematical structures which have recently been used in attempts to
analyze the algebraic properties of quantum field theory. In the context of
quantum mechanics they provide a canonical quantization procedure for systems
with either bosonic of fermionic degrees of freedom. We illustrate this
procedure for a number a physical examples, including bosonic, fermionic and
supersymmetric oscillators. We show how non-relativistic and relativistic
particles with spin can be naturally described in this framework.Comment: 21 page
Star Products and Geometric Algebra
The formalism of geometric algebra can be described as deformed super
analysis. The deformation is done with a fermionic star product, that arises
from deformation quantization of pseudoclassical mechanics. If one then extends
the deformation to the bosonic coefficient part of superanalysis one obtains
quantum mechanics for systems with spin. This approach clarifies on the one
hand the relation between Grassmann and Clifford structures in geometric
algebra and on the other hand the relation between classical mechanics and
quantum mechanics. Moreover it gives a formalism that allows to handle
classical and quantum mechanics in a consistent manner.Comment: 21 page
Star products and perturbative quantum field theory
We discuss the application of the deformation quantization approach to
perturbative quantum field theory. We show that the various forms of Wick's
theorem are a direct consequence of the structure of the star products. We
derive the scattering function for a free scalar field in interaction with a
spacetime-dependent source. We show that the translation to operator formalism
reproduces the known relations which lead to the derivation of the Feynman
rules.Comment: 12 page
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