15,410 research outputs found
Quantum stochastic convolution cocycles II
Schuermann's theory of quantum Levy processes, and more generally the theory
of quantum stochastic convolution cocycles, is extended to the topological
context of compact quantum groups and operator space coalgebras. Quantum
stochastic convolution cocycles on a C*-hyperbialgebra, which are
Markov-regular, completely positive and contractive, are shown to satisfy
coalgebraic quantum stochastic differential equations with completely bounded
coefficients, and the structure of their stochastic generators is obtained.
Automatic complete boundedness of a class of derivations is established,
leading to a characterisation of the stochastic generators of *-homomorphic
convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum
Levy process on a compact quantum group are given and, with respect to both of
these, it is shown that an equivalent process on Fock space may be
reconstructed from the generator of the quantum Levy process. In the examples
presented, connection to the algebraic theory is emphasised by a focus on full
compact quantum groups.Comment: 32 pages, expanded introduction and updated references. The revised
version will appear in Communications in Mathematical Physic
Algebraic constructive quantum field theory: Integrable models and deformation techniques
Several related operator-algebraic constructions for quantum field theory
models on Minkowski spacetime are reviewed. The common theme of these
constructions is that of a Borchers triple, capturing the structure of
observables localized in a Rindler wedge. After reviewing the abstract setting,
we discuss in this framework i) the construction of free field theories from
standard pairs, ii) the inverse scattering construction of integrable QFT
models on two-dimensional Minkowski space, and iii) the warped convolution
deformation of QFT models in arbitrary dimension, inspired from non-commutative
Minkowski space.Comment: Review article, 57 pages, 3 figure
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