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\begin{document}

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\section*{Kochen-Specker theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{kochenspecker_theorem}{Kochen-Specker theorem}\dotfill \pageref*{kochenspecker_theorem} \linebreak
\noindent\hyperlink{sheaftheoretic_interpretation}{Sheaf-theoretic interpretation}\dotfill \pageref*{sheaftheoretic_interpretation} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[quantum mechanics]], the \emph{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) [[probability theory|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 (\hyperlink{Gleason57}{Gleason (1957)}) (who incidentally is the person who communicated the original Kochen-Specker paper to the \emph{Journal of Mathematics and Mechanics} ). Later in (\hyperlink{ButterfieldHamiltonIsham}{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|ringed]]) [[topos]], the ``[[Bohr topos]]''.

\hypertarget{kochenspecker_theorem}{}\subsection*{{Kochen-Specker theorem}}\label{kochenspecker_theorem}

\begin{defn}
\label{Valuation}\hypertarget{Valuation}{}
Let $B(\mathcal{H})$ be the algebra of [[bounded operator]]s on some [[Hilbert space]] $\mathcal{H}$. (In physics $\mathcal{H}$ is the space of states of a [[quantum mechanics|quantum mechanical]] system, and the elements $\hat A \in B(\mathcal{H})$ represent quantum observables.)

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

\begin{displaymath}
\lambda : B(\mathcal{H}) \to \mathbb{R}
\end{displaymath}
to the [[real number]]s, satisfying two conditions:

\begin{enumerate}%
\item \emph{value rule} -- the value $\lambda(\hat{A})$ belongs to the spectrum of $\hat{A}$;


\item \emph{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 $\lambda(\hat{B})=h(\lambda(\hat{A}))$.



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This has the following equivalent reformulationo, 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. \ref{Valuation} implies that a valuation satisfies

\begin{displaymath}
\lambda(\hat{A}_{1} + \hat{A}_{2}) = \lambda(\hat{A}_{1}) + \lambda(\hat{A}_{2})
\end{displaymath}
and

\begin{displaymath}
\lambda(\hat{A}_{1}\hat{A}_{2})=\lambda(\hat{A}_{1})\lambda(\hat{A}_{2})
  \,,
\end{displaymath}
hence that on [[poset of commutative subalgebras|commutative subalgebras]] it is a [[ring]] [[homomorphism]].

Therefore a valuation as in def. \ref{Valuation} is equivalently a function on the algebra which is an algebra homomorphism on each [[poset of commutative subalgebras|commutative subalgebra]].

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

\end{remark}
Now we have:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Kochen-Specker)}

No valuations on $B(\mathcal{H})$ exist if dim($\mathcal{H}$){\tt \symbol{62}}2.

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

\end{remark}
\hypertarget{sheaftheoretic_interpretation}{}\subsection*{{Sheaf-theoretic interpretation}}\label{sheaftheoretic_interpretation}

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

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(category of contexts)}

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

\begin{itemize}%
\item [[object]]s are \emph{commutative} [[von Neumann algebra|von Neumann subalgebras]] $V \subset B(\mathcal{H})$;


\item [[morphism]]s $V_1 \to V_2$ are inclusions $V_1 \subset V_2$.



\end{itemize}
\end{defn}
Isham calls this the \textbf{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.)

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(spectral presheaf)}

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

\begin{itemize}%
\item to $V \subset B(\mathcal{H})$ it assigned the set underlying the spectrum of $V$: the set of multiplicative [[linear functional]]s $\kappa : V \to \mathbb{R}$;


\item 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 $V_1 \hookrightarrow V_2 \stackrel{\kappa}{\to} \mathbb{R}$.



\end{itemize}
This is called the \emph{[[spectral presheaf]]}.

\end{defn}
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. \ref{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. Hence we have:

\begin{prop}
\label{}\hypertarget{}{}
(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.

\end{prop}
(\hyperlink{ButterfieldHamiltonIsham}{Butterfield-Hamilton-Isham 98})

For more see at \emph{[[Bohr topos]]}.

\hypertarget{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

\begin{itemize}%
\item [[beable]]

\end{itemize}
Other theorems about the foundations of [[quantum mechanics]] include:

\begin{itemize}%
\item [[order-theoretic structure in quantum mechanics]]

\begin{itemize}%
\item [[Gleason's theorem]]


\item [[Alfsen-Shultz theorem]]


\item [[Harding-Döring-Hamhalter theorem]]



\end{itemize}

\item [[Bell's theorem]]


\item [[Wigner theorem]]


\item [[Bub-Clifton theorem]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Bohr topos]]


\item [[interpretation of quantum mechanics]]

\begin{itemize}%
\item [[hidden variable theory]]

\end{itemize}

\item [[Bell's inequalities]]



\end{itemize}
Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

\begin{itemize}%
\item [[order-theoretic structure in quantum mechanics]]

\begin{itemize}%
\item [[Alfsen-Shultz theorem]]


\item [[Harding-Döring-Hamhalter theorem]]



\end{itemize}

\item [[Fell's theorem]]


\item [[Gleason's theorem]]


\item [[Wigner theorem]]


\item [[Bell's theorem]]


\item [[Bub-Clifton theorem]]


\item [[Kadison-Singer problem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original article is

\begin{itemize}%
\item [[Simon Kochen]], [[Ernst Specker]], \emph{The problem of hidden variables in quantum mechanics} , Journal of Mathematics and Mechanics 17, 59--87 (1967), (\href{http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf}{pdf})

\end{itemize}
Alternative proofs are in

\begin{itemize}%
\item Del Rajan, Matt Visser, \emph{Kochen-Specker theorem revisited} (\href{https://arxiv.org/abs/1708.01380}{arXiv:1708.01380})

\end{itemize}
The sheaf-theoretic interpretation of the theorem was proposed in

\begin{itemize}%
\item [[Chris Isham]], [[Jeremy Butterfield]], \emph{A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations} (\href{http://arxiv.org/abs/quant-ph/9803055}{arXiv:quant-ph/9803055})

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

\begin{itemize}%
\item [[Jeremy Butterfield]], John Hamilton, [[Chris Isham]], \emph{A topos perspective on the Kochen-Specker theorem}, \emph{I. quantum states as generalized valuations}, Internat. J. Theoret. Phys. 37(11):2669--2733, 1998, \href{http://www.ams.org/mathscinet-getitem?mr=1669557}{MR2000c:81027}, \href{http://dx.doi.org/10.1023/A:1026680806775}{doi}; \emph{II. conceptual aspects and classical analogues} Int. J. of Theor. Phys. 38(3):827--859, 1999, \href{http://www.ams.org/mathscinet-getitem?mr=1697983}{MR2000f:81012}, \href{http://dx.doi.org/10.1023/A:1026652817988}{doi}; \emph{III. Von Neumann algebras as the base category}, Int. J. of Theor. Phys. 39(6):1413--1436, 2000, \href{http://arxiv.org/abs/quant-ph/9911020}{arXiv:quant-ph/9911020}, \href{http://www.ams.org/mathscinet-getitem?mr=1788498}{MR2001k:81016},\href{http://dx.doi.org/10.1023/A:1003667607842}{doi}; \emph{IV. Interval valuations}, Internat. J. Theoret. Phys. \textbf{41} (2002), no. 4, 613--639, \href{http://www.ams.org/mathscinet-getitem?mr=1902067}{MR2003g:81009}, \href{http://dx.doi.org/10.1023/A:1015276209768}{doi}

\end{itemize}
The original paper outlining [[Bell's theorem]] is

\begin{itemize}%
\item J. S. Bell, \emph{On the Einstein Podolsky Rosen Paradox} , Physics, \href{http://www.drchinese.com/David/Bell_Compact.pdf}{pdf}.

\end{itemize}
The original paper outlining [[Gleason's theorem]] is

\begin{itemize}%
\item A. M. Gleason, \emph{Measures on the closed subspaces of a Hilbert space}, Journal of Mathematics and Mechanics, Indiana Univ. Math. J. 6 No. 4 (1957), 885--893 (\href{http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050}{web})

\end{itemize}
A technical discussion on the interplay of Gleason's and Kochen-Specker theorems and various issues regarding non-contextuality, locality and Bell inequality appears in Chapter 5 of

\begin{itemize}%
\item [[Valter Moretti]], Fundamental Mathematical Structures of Quantum Theory: Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation Springer (2019)

\end{itemize}
The hierarchy of strengths of no-go theorems in quantum context is studied from sheaf theoretic perspective in

\begin{itemize}%
\item [[Samson Abramsky]], Adam Brandenburger, \emph{The sheaf-theoretic structure of non-locality and contextuality}, \href{https://arxiv.org/abs/1102.0264}{arxiv/1102.0264}

\end{itemize}


\end{document}