Bub-Clifton theorem


Algebraic Quantum Field Theory

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



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT




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:


A broader textbook discussion is in

An interpretation in topos theory is proposed in

See also