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Register a new userCallias-type operators associated to spectral triples
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Added by
Hermann Schulz-Baldes
Publication date
2021
fields
Physics
and research's language is
English
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No Arabic abstract
Callias-type (or Dirac-Schrodinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.
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We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyahs $L^2$-index theorem, which states that the index of a Callias-type operator on a non-compact manifold $M$ is equal to the $Gamma$-index of its lift to a Galois cover of $M$. We also prove the cobordism invariance of the index of Callias-type operators. In particular, we give a new proof of the cobordism invariance of the von Neumann index of operators on compact manifolds.
The notion of spectral localizer is extended to pairings with semifinite spectral triples. By a spectral flow argument, any semifinite index pairing is shown to be equal to the signature of the spectral localizer. As an application, a formula for the weak invariants of topological insulators is derived. This provides a new approach to their numerical evaluation.
The asymptotic expansion of the heat-kernel for small values of its argument has been studied in many different cases and has been applied to 1-loop calculations in Quantum Field Theory. In this thesis we consider this asymptotic behavior for certain singular differential operators which can be related to quantum fields on manifolds with conical singularities. Our main result is that, due to the existence of this singularity and of infinitely many boundary conditions of physical relevance related to the admissible behavior of the fields on the singular point, the heat-kernel has an unusual asymptotic expansion. We describe examples where the heat-kernel admits an asymptotic expansion in powers of its argument whose exponents depend on external parameters. As far as we know, this kind of asymptotics had not been found and therefore its physical consequences are still unexplored.
We review the relationship between positive operator-valued measures (POVMs) in quantum measurement theory and asymptotic morphisms in the C*-algebra E-theory of Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout (CMP 226), is integrally related to positive asymptotic morphisms on locally compact spaces via an asymptotic Riesz Representation Theorem. Examples and applications to quantum physics, including quantum noise models, semiclassical limits, pure spin one-half systems and quantum information processing will also be discussed.
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.
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