Contents

# Contents

## Idea

In quantum mechanics, the Kochen-Specker theorem – developed in 1967 by Simon Kochen and Ernst Specker – is a no-go theorem that places limits on the types of hidden variable theories that may be used to explain the (apparent) probabilistic nature of quantum mechanics in a causal way. It roughly asserts that it is impossible to assign values to all physical observables while simultaneously preserving the functional relations between them. It is a complement to Bell's theorem, developed by John Bell in 1964, and is related to Gleason's theorem, proven by (Gleason (1957)) (who incidentally is the person who communicated the original Kochen-Specker paper to the Journal of Mathematics and Mechanics ). Later in (Butterfield-Hamilton-Isham 98) it was observed that the Kochen-Specker theorem is equivalent to the statement that the spectral presheaf has no global elements, which led to the proposal that the phase space in quantum mechanics is naturally to be understood as a (ringed) topos, the “Bohr topos”.

## Kochen-Specker theorem

###### Definition

Let $B(\mathcal{H})$ be the algebra of bounded operators on some Hilbert space $\mathcal{H}$. (In physics $\mathcal{H}$ is the space of states of a quantum mechanical system, and the elements $\hat A \in B(\mathcal{H})$ represent quantum observables.)

A valuation on $B(\mathcal{H})$ is a function

$\lambda : B(\mathcal{H}) \to \mathbb{R}$

to the real numbers, satisfying two conditions:

1. value rule – the value $\lambda(\hat{A})$ belongs to the spectrum of $\hat{A}$;

2. functional composition principle (FUNC) – for any pair of self-adjoint operators $\hat{A}$, $\hat{B}$ such that $\hat{B}=h(\hat{A})$ for some real-valued function $h$ we have $\lambda(\hat{B})=h(\lambda(\hat{A}))$.

###### Remark

This has the following equivalent reformulation, which is crucial for the sheaf-theoretic interpretation discussed below.

Observed that if $\hat{A}_{1}$ and $\hat{A}_{2}$ commute, then it follows from the spectral theorem that there exists an operator $\hat{C}$ and continuous functions $h_{1}$ and $h_{2}$ such that $\hat{A}_{1}=h_{1}(\hat{C})$ and $\hat{A}_{2}=h_{2}(\hat{C})$. Then the axiom FUNC in def. implies that a valuation satisfies

$\lambda(\hat{A}_{1} + \hat{A}_{2}) = \lambda(\hat{A}_{1}) + \lambda(\hat{A}_{2})$

and

$\lambda(\hat{A}_{1}\hat{A}_{2})=\lambda(\hat{A}_{1})\lambda(\hat{A}_{2}) \,,$

hence that on commutative subalgebras it is a ring homomorphism.

Therefore a valuation as in def. is equivalently a function on the algebra which is an algebra homomorphism on each commutative subalgebra.

(Observe the difference to quasi-states (quantum states), which are positive linear functions on commutative subalgebras, not necessarily respecting the ring structure.)

Now we have:

###### Theorem

(Kochen-Specker)

No valuations on $B(\mathcal{H})$ exist if dim($\mathcal{H}$)>2.

###### Remark

If a valuation did exist and was restricted to a commutative sub-algebra of operators, it would be an element of the Gelfand spectrum of the commutative sub-algebra. Since such elements do exist, valuations do exist on any commutative sub-algebra of operators even if not on the whole non-commutative algebra, $\mathcal{B}(\mathcal{H})$, of all bounded operators. Isham calls these valuations on commutative subalgebras local. In the Bohr topos of the algebra of observables (see there for more), the local valuations are just the internal valuations.

## Sheaf-theoretic interpretation

Chris Isham and Jeremy Butterfield gave a topos theoretic reformulation of the Kochen-Specker theorem as follows.

###### Definition

(category of contexts)

Let $\mathcal{V}(\mathcal{H})$ be a category (the poset of commutative subalgebras of the algebra $B(\mathcal{H})$ of bounded operators) whose

• objects are commutative von Neumann subalgebras $V \subset B(\mathcal{H})$;

• morphisms $V_1 \to V_2$ are inclusions $V_1 \subset V_2$.

Isham calls this the category of (classical) contexts of $B(\mathcal{H})$. Each commutative algebra is viewed as a context within which to view a quantum system in an essentially classical way in the sense that the physical quantities in any such algebra can be given consistent values (as they can in a classical context). These classical contexts were maybe first amplified by Niels Bohr as being the contexts through which all of quantum mechanics is to be perceived. (Therefore the word “Bohr topos” for the concept that is meant to formalize this perspective of Bohr.)

###### Definition

(spectral presheaf)

Let $\Sigma : \mathcal{V}(\mathcal{B})^{op} \to Set$ be the presheaf on the category of contexts such that

• to $V \subset B(\mathcal{H})$ it assigned the set underlying the spectrum of $V$: the set of multiplicative linear functionals $\kappa : V \to \mathbb{R}$;

• to an inclusion $i : V_1 \hookrightarrow V_2$ it assigns the corresponding function $i^* : \Sigma(V_2) \to \Sigma(V_1)$ that sends a functional $V_2 \stackrel{\kappa}{\to} \mathbb{R}$ to its restriction $V_1 \hookrightarrow V_2 \stackrel{\kappa}{\to} \mathbb{R}$.

This is called the spectral presheaf.

Recall that the terminal object, $* = 1_{Set^{\mathcal{V}(\mathcal{H})^{op}}}$ in the category of presheaves on $\mathcal{V}(\mathcal{H})$ is the presheaf that assigns the singleton set $*$ (the terminal object in Set) to each commutative algebra.

A global element of the spectral presheaf $\Sigma$ is a morphism $e : * \to \Sigma$ in the presheaf topos. Being a natural transformation of functors, such a global element $\lambda : 1_{Set^{\mathcal{V}(\mathcal{H})^{op}}} \to \underline{\Sigma}$ of the spectral presheaf, would associate an element of the Gelfand spectrum of an algebra $V$ to that algebra such that all local valuations are global, i.e. for $V \subseteq W$ valuations on $V$ are local valuations on $W$ but global on $V$.

Because a multiplicative linear functional $\kappa : V \to\mathbb{R}$ satisfies the axioms of a valuation, def. , when restricted to the self-adjoint elements of $V$.

By the Kochen-Specker theorem these cannot exist, hence a global element of $\Sigma$ cannot exist. Hence we have:

###### Proposition

(Hamilton, Isham, Butterfield)

The Kochen-Specker theorem is equivalent to the statement that the spectral presheaf $\Sigma$ of the algebra of bounded operators has no global elements if the dimension of the Hilbert space is greater than 2.

For more see at Bohr topos.

Other theorems about the foundations of quantum mechanics include:

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

The original article:

Alternative proofs:

Review:

• Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, Jan-Åke Larsson, Kochen-Specker Contextuality [arXiv:2102.13036]

The sheaf-theoretic interpretation of the theorem was proposed in

The formulation in terms of presheaves on the category of commutative sub-algebra of $B(\mathcal{H})$ was proposed in part III of

The original paper outlining Bell's theorem:

• John S. Bell?, On the Einstein Podolsky Rosen Paradox, Physics, pdf.

The original paper outlining Gleason's theorem is

• Andrew M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, Indiana Univ. Math. J. 6 No. 4 (1957), 885–893 (web)

A technical discussion on the interplay of Gleason's and Kochen-Specker theorems and various issues regarding non-contextuality, locality and Bell's inequality:

The hierarchy of strengths of no-go theorems in quantum context is studied from sheaf theoretic perspective in

Strengthening of the Kochen-Specker theorem and relation to the “free will theorem”:

• Simon Kochen, On the Free Will Theorem [arXiv:2207.06295]

The new theorem […] does require any use of free will on the experimenter’s part. The theorem also strengthens the Kochen-Specker Theorem

Relation of Kochen-Specker to Bell's inequalities: