Kadison-Singer problem

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

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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

The *Kadison-Singer problem* is the question:

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

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

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

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

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.