nLab
Kochen-Specker theorem

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 randomness of quantum mechanics in a causal way. It roughly asserts that it is impossible to assign values to all physical quantities 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, developed in 1957 by Andrew Gleason (who incidentally is the person who communicated the original Kochen-Specker paper to the Journal of Mathematics and Mechanics ). Christopher Isham has recently shown that the Kochen-Specker theorem is equivalent to the statement that the spectral presheaf has no global elements.

Kochen-Specker theorem

Definition

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

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

$$\lambda :B(\mathscr{H})\to ℝ$$\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 $\lambda (\hat{B})=h(\lambda (\hat{A}))$.

Note that is ${\hat{A}}_1$ and ${\hat{A}}_2$ commute, it follows from the spectral theorem that there exists an operator $\hat{C}$ and functions $h_1$ and $h_2$ such that ${\hat{A}}_1=h_1(\hat{C})$ and ${\hat{A}}_2=h_2(\hat{C})$. It follows from FUNC that

$$\lambda ({\hat{A}}_1+{\hat{A}}_2)=\lambda ({\hat{A}}_1)+\lambda ({\hat{A}}_2)$$\lambda(\hat{A}_{1} + \hat{A}_{2}) = \lambda(\hat{A}_{1}) + \lambda(\hat{A}_{2})

and

$$\lambda ({\hat{A}}_1{\hat{A}}_2)=\lambda ({\hat{A}}_1)\lambda ({\hat{A}}_2).$$\lambda(\hat{A}_{1}\hat{A}_{2})=\lambda(\hat{A}_{1})\lambda(\hat{A}_{2}).

Now we have:

Theorem

(Kochen-Specker)

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

Consequences

If a valuation did exist and was restricted to a commutative sub-algebra of operators, it would be an element of the spectrum of the algebra. Since such elements do exist, valuations must exist on any commutative sub-algebra of operators but not on the non-commutative algebra, $ℬ(\mathscr{H})$, of all bounded operators. Isham calls these valuations local.

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

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

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

Isham calls this the category of contexts of $B(\mathscr{H})$. 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).

Definition

(spectral presheaf)

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

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

• 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 $V_2\stackrel{\kappa }{\to }ℝ$ to its restriction $V_1\hookrightarrow V_2\stackrel{\kappa }{\to }ℝ$.

Recall that the terminal object, $*=1_{{\mathrm{Set}}^{𝒱(\mathscr{H}{)}^{\mathrm{op}}}}$ in the category of presheaves on $𝒱(\mathscr{H})$ 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 $\lambda :1_{{\mathrm{Set}}^{𝒱(\mathscr{H}{)}^{\mathrm{op}}}}\to \underline{\Sigma }$ of the spectral presheaf, $\underline{\Sigma }$ would associate an element of the 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 notice that a multiplicative linear functional $\kappa :V\to ℝ$ ssatisfies the axioms of a valuation 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.

Observation

(Hamilton, Isham, Butterfield)

The Kochen-Specker theorem is equivalent to the statement that in the presheaf topos $[𝒱(\mathscr{H}{)}^{\mathrm{op}},\mathrm{Set}]$ the spectral presheaf $\Sigma$ has no global elements.

Related concepts

• Bohr topos

References

The original article is

• Simon Kochen, Ernst Specker, The problem of hidden variables in quantum mechanics , Journal of Mathematics and Mechanics, pdf.

The sheaf-theoretic interpretation of the theorem was proposed in

• Chris Isham, Jeremy Butterfield, A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations (arXiv:quant-ph/9803055)

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

• Jeremy Butterfield, John Hamilton?, Chris Isham, A topos perspective on the Kochen-Specker theorem, I. quantum states as generalized valuations, Internat. J. Theoret. Phys. 37(11):2669–2733, 1998, MR2000c:81027, doi; II. conceptual aspects and classical analogues Int. J. of Theor. Phys. 38(3):827–859, 1999, MR2000f:81012, doi; III. Von Neumann algebras as the base category, Int. J. of Theor. Phys. 39(6):1413–1436, 2000, arXiv:quant-ph/9911020, MR2001k:81016,doi; IV. Interval valuations, Internat. J. Theoret. Phys. 41 (2002), no. 4, 613–639, MR2003g:81009, doi

The original paper outlining Bell's theorem is

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

The original paper outlining Gleason's theorem is

• A.M. Gleason, Measures on the closed subspaces of a Hilbert space , Journal of Mathematics and Mechanics, pdf.

Discussion