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Kadison-Singer problem

Contents

Context

Functional analysis

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

Given a separable Hilbert space \mathcal{H}, such as the sequence space 𝓁 2\mathcal{l}_2, write B()B(\mathcal{H}) for its C*-algebra of bounded linear operators and D()D(\mathcal{H}) for the subalgebra of diagonal operators.

The Kadison-Singer problem is the question:

Does every pure state ψ\psi on D()D(\mathcal{H}) extend to a state ρ\rho on B()B(\mathcal{H})?

D() ψ ρ B() \array{ D(\mathcal{H}) &\overset{\psi}{\longrightarrow}& \mathbb{C} \\ \downarrow & \nearrow_{\mathrlap{\exists \rho}} \\ B(\mathcal{H}) }

This was proven to be the case in Marcus-Spielman-Srivastava 13.

Other theorems about the foundations and interpretation of quantum mechanics include:

References

The problem was stated in

  • Richard V. Kadison and Isadore Singer, Extensions of pure states Americal Journal of Mathematics, 81(2):383–400, 1959

and its solution was proven in

  • Adam Marcus, Daniel A. Spielman, and Nikhil Srivastava, Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem, 2013. (arXiv:1306.3969)

Review includes

  • Nicholas Harvey, An introduction to the Kadison-Singer conjecture and the paving conjecture (pdf)

Last revised on December 11, 2017 at 08:47:05. See the history of this page for a list of all contributions to it.