nLab
Bub-Clifton theorem

Context

Algebraic Quantum Field Theory

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

Philosophy

Contents

Idea

The two fundamental no-go theorems for hidden variable reconstructions of the quantum statistics, the Kochen-Specker theorem and Bell's theorem, may be formulated as results about the impossibility of associating a classical probability space with a quantum system, when certain constraints are placed on the probability measure.

The Bub-Clifton-Halvorson theorem (Bub-Clifton 96, Clifton-Bub-Halvorson 03) on the other hand is a positive result about the possibility of associating a classical probability space with a quantum system in a given state.

Given a Hilbert space HH and a quantum observable represented by a Hermitean operator on this space, and given a pure state, the Bub-Clifton theorem characterizes a maximal sub-lattice of the Birkhoff-vonNeumann Hilbert lattice of subspace of HH (the “quantum logic” of HH) such that there are sufficiently many yes-no questions on the elements in the lattice to recover all the probabilities induced by the given pure state to compatible sets of projections (classical contexts).

A useful summary is in Bub 09, pages 1-2.

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

References

A broader textbook discussion is in

An interpretation in topos theory is proposed in

See also