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\section*{order-theoretic structure in quantum mechanics}

The following surveys how basic [[theorems]] about the standard foundation of [[quantum mechanics]] imply an accurate [[geometry|geometric]] incarnation of the ``[[phase space]] in [[quantum mechanics]]'' by an [[order theory|order-theoretic structure]] that combines with an [[algebra|algebraic]] [[structure]] to a [[ringed topos]], the ``[[Bohr topos]]''. While the notion of [[Bohr topos]] has been \emph{motivated} by the [[Kochen-Specker theorem]], the point here is to highlight that taking into account further theorems about the standard foundations of [[quantum mechanics]], the notion effectively follows automatically and provides an accurate and useful description of the geometry of ``quantum phase space'' also in [[quantum field theory]] formulated in the style of [[AQFT]].


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{QuantumPhaseSpaceAsJordanGeometry}{The quantum phase space as a Jordan-geometry given by a ringed topos}\dotfill \pageref*{QuantumPhaseSpaceAsJordanGeometry} \linebreak
\noindent\hyperlink{RelationToTheNonCommutativePhaseSpace}{Relation to the traditional non-commutative geometry}\dotfill \pageref*{RelationToTheNonCommutativePhaseSpace} \linebreak
\noindent\hyperlink{ListOfTheoremsInvoked}{List of theorems invoked}\dotfill \pageref*{ListOfTheoremsInvoked} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{QuantumPhaseSpaceAsJordanGeometry}{}\subsection*{{The quantum phase space as a Jordan-geometry given by a ringed topos}}\label{QuantumPhaseSpaceAsJordanGeometry}

Back when [[quantum mechanics]] was discovered in the first half of the 20th century, it was eventually formalized as [[mathematical physics]] and the traditional modern formulation emerged, where, in [[AQFT]] perspective, [[quantum observables]] are represented by suitable [[linear operators]] on a [[Hilbert space]] and more generally by elements of a [[C\emph{-algebra]] [[algebra of observables|of observables]], and where [[quantum states]] are certain (namely positive and normalized) linear functionals on these [[C}-algebra|C\emph{]]-[[algebras of observables]] ([[states on star-algebras]]).}

But one may still ask if the [[axioms]] in the definition of \emph{[[C\emph{-algebra]]\_ accurately capture the intended [[physics]]. This or similar questions were discussed back in the middle of the 20th century, when these notions were still in flux. Specifically in the 1930s [[Pascual Jordan]] argued (\hyperlink{Jordan32}{Jordan 32}) that the [[associative algebra]] [[structure]] on the observables is more [[structure]] than supported by the physics of states and observation, that instead only the underlying structure of what is now called the \emph{[[Jordan algebra]]} should matter.}}

Much later in 1978 this idea was formally validated by the [[Alfsen-Shultz theorem]] (\hyperlink{AlfsenShultz78}{Alfsen-Shultz 78}). This states that the [[space of quantum states]] for given [[quantum observables]] depends indeed only on the underlying [[Jordan algebra]] structure. This is not too surprising: the definition of a [[state on an operator algebra]] does not even mention the [[associative algebra]] structure but mentions only the [[positive operator|positivity]] structure, which is what the [[Jordan algebra]] captures.

Despite these insights, [[Jordan algebras]] found and find only marginal attention in [[mathematical physics]]. In a \hyperlink{ASReview04}{review from 2004} of the book of Alfsen and Shultz it says that back then Jordan algebras were hoped to shed light on conceptual problems of genuine [[quantum field theory]], but that these hopes never materialized. However, more recent developments change this picture a bit, we come to that below.

Much more recently in 2010, the [[Harding-Döring-Hamhalter theorem]] sheds a new light on the role of [[Jordan algebra]] structure. This theorem states mild conditions under which a [[Jordan algebra]] structure on [[quantum observables]] is equivalently encoded in the [[poset of commutative subalgebras]] of the full [[C\emph{-algebra]].}

These commutative subalgebras themselves are of course of old fame in quantum mechanics, they are the ``[[classical contexts]]'' given by tuples of [[quantum observables]] that all commute with each other and hence which can all be [[measurement|measured]] simultaneously without the [[uncertainty principle]] interfering. These [[classical contexts]] played a crucial role in the discussion of the foundation of quantum mechanics in the first half of the 20th century: back then people argued that for $A$ and $B$ two [[quantum observables]] which do not commute with each other it is unclear what it means physically to form their sum $A + B$ or their product $A B$, hence that a [[quantum state]] should be demanded to be a linear (and positive normed) [[linear functional|functional]] on all [[poset of commutative subalgebras|commutative subalgebras]], but not necessarily on the whole non-commutative algebra. Such a notion of ``quantum state on all [[classical contexts]]'' was called a \emph{[[quasi-state]]}.

The issue of whether quasi-states are a more accurate description of quantum [[measurement]] was settled in 1957 by [[Gleason's theorem]] (\hyperlink{Gleason57}{Gleason 57}), which says that given a [[Hilbert space]] of [[dimension]] greater than 2, then the [[quasi-states]] are automatically [[quantum states]] also on the full non-commutative [[algebra of observables]]. Typically this is viewed as making the notion of [[quasi-state]] obsolete, but since that is a formally weaker notion the opposite attitude makes sense: what is traditionally taken as the definition of [[quantum state]] is more accurately thought of as being a [[quasi-state]], hence something that is intrinsically related not to a non-commutative algebra of observables, but to the ``[[classical contexts]]'' of its [[poset of commutative subalgebras]].

Indeed, by combining the [[Alfsen-Shultz theorem]] with the [[Harding-Döring-Hamhalter theorem]], we have (under the pertinent mild assumptions) that two algebras of observables have the same [[space of quantum states]] already when they have the same [[poset of commutative subalgebras]]. Notice that where [[Gleason's theorem]] only involves the commutative subalgebras themselves, this Jordan-Alfsen-Shultz-Harding-D\"o{}ring-Hamhalter theorem crucially involves their [[order]] of inclusions, hence the actual [[poset]] structure of their inclusions.

Therefore there is an intrinsic [[order theory|order theoretic]] aspect in the standard foundations of [[quantum mechanics]]. But it is not \emph{just} order theory, for it is not the [[poset]] structure of inclusions alone that matters in the Jordan-Alfsen-Shultz-Harding-D\"o{}ring-Hamhalter theorem, but that poset structure together with the actual [[commutative C\emph{-algebra]] structure of each [[classical context]].}

There is an elegant way to combine these two aspects: a system of [[commutative ring|commutative algebras]] together with the [[order]] of their inclusions is equivalently a single [[algebra object]] [[internalization|internal to]] the [[sheaf topos]] over the [[poset]]. With some basics of [[topos theory]] in hand this is a trivial statement, but in view of the above it is worthwhile to make explicit: the Jordan-Alfsen-Shultz-Harding-D\"o{}ring-Hamhalter theorem (\hyperlink{HardingDoering10}{Harding-D\"o{}ring 10}, \hyperlink{Hamhalter11}{Hamhalter 11}) says that the collection of [[quantum observables]] in [[quantum mechanics]] is accurately formalized by a single [[commutative C\emph{-algebra]] [[internalization|internal]] to a [[sheaf topos]] over a [[poset]].}

It has been argued that this serves as a formalization of the views on [[quantum mechanics]] that in the middle of the 20th century [[Niels Bohr]] expressed in extensive but informal writing (see \hyperlink{Scheibe73}{Scheibe 73}): he said roughly that whatever [[quantum mechanics]] is, it must be expressible and must be expressed through classical contexts. In honor of this intuition, the above [[toposes]] deserve to be called \emph{[[Bohr toposes]]}, following the term ``Bohrification'' in (\hyperlink{HeunenLandsmanSpitters09}{Heunen-Landsman-Spitters 09}).

In fact, a [[Bohr topos]] is fairly trivial as far as [[toposes]] go, since, by the above, it is just a reflection -- precisely: the ``[[localic reflection]]'' -- of a purely [[order theory|order theoretic structure]]. But the key is that passing to the topos over the [[poset]] provides a home for the context-wise [[commutative C\emph{-algebra]] structure which makes the [[Bohr topos]] have the additional structure of a \emph{[[ringed topos]]}. This is the additional [[algebra|algebraic]] datum on top of the purely [[order theory|order theoretic]] datum in the [[Jordan algebra]] structure of [[quantum observables]].}

Ringed toposes have of course a long tradition in [[geometry]] (most famously in [[algebraic geometry]]). By [[Grothendieck]]`s foundational work, laid out in the \hyperlink{Hakim72}{thesis} of [[Monique Hakim]], ringed toposes form a general foundation for structured [[geometry]]. More recently this was further strengthened and refined by [[Jacob Lurie]] by the notion of [[structured (infinity,1)-topos|higher ringed toposes]], which we will see appear in [[quantum field theory]] below. In as far as the [[quantum observables]] in [[quantum mechanics]] are supposed to be the [[Isbell duality|dual]] of the \emph{[[phase space]]}, it is therefore natural to have this ``quantum phase space'' be realized as a [[ringed topos]].

Notice that this is a different geometric interpretation of ``quantum phase space'' than the traditional idea that quantum phase space is an object in [[non-commutative geometry]]! Here by the Jordan-Alfsen-Shultz-Harding-D\"o{}ring-Hamhalter theorem we see that it is not actually accurate to say that quantum phase space is dually given by a non-commutative [[C\emph{-algebra]], as in fact it is given dually just by a [[Jordan algebra]]. The [[ringed topos|ringed]] [[Bohr topos]] provides a natural [[geometry|geometric]] interpretation of this, one might call it ``Jordan geometry''.}

This geometric nature of the Bohr topos becomes more manifest as we consider its [[opposite category]]. By [[Gelfand duality]] this carries not an internal [[commutative C\emph{-algebra]] but its [[Gelfand spectrum]]: an actual internal [[topological space]]. This internal topological space has been called the \emph{[[spectral presheaf]]} (\hyperlink{ButterfieldHamiltonIsham98}{Butterfield-Hamilton-Isham 98}).}

While here we find this [[spectral presheaf]] as an accurate dual description of the space of [[quantum observables]] in [[quantum mechanics]] based on the Jordan-Alfsen-Shultz-Harding-D\"o{}ring-Hamhalter theorem, interest in the spectral presheaf originally came from the observation that it provides a clear geometric formulation of yet another theorem about the foundations of [[quantum mechanics]], namely of the \emph{[[Kochen-Specker theorem]]} (\hyperlink{KochenSpecker67}{Kochen-Specker 67}).

This again amplifies the role of the ``classical contexts'' of commutative subalgebras: one may ask if there is a [[hidden variable]] description of [[quantum mechanics]] that allows to assign actual values to all [[quantum observables]] such that in all [[classical contexts]] this assignment behaves as an actual [[classical observable]] in that it provides a (star-)[[homomorphism]] of [[associative algebras]] from the [[commutative C\emph{-algebra]] of the classical context to the real numbers. The [[Kochen-Specker theorem]] rules out such a [[hidden variable theory]] by stating that when the [[algebra of observables]] is that of [[bounded operators]] on a [[Hilbert space]] of [[dimension]] greater than 2, then such a ``hidden variable'' assignment cannot exist.}

In (\hyperlink{IshamButterfield98}{Butterfield-Isham 98}) it was observed, that this statement has a natural geometric interpretation in the [[Bohr topos]]: it simply says that the [[spectral presheaf]], hence the ``Jordan-algebraic geometry'' incarnation of quantum phase space, has no [[global element]]. This statement in turn is a characterization of how the quantum phase space is ``exotic'' as far as [[spaces]] go. It behaves like a non-trivial space, and yet there is no way to map a point into it as a whole, maps of points into it exist only locally.

In summary, the [[Bohr topos]] incarnation of the Jordan-Alfsen-Shultz-Harding-D\"o{}ring-Hamhalter characterization of quantum observables not only accurately captures the nature of quantum observables, but also makes other subtle nature of [[quantum mechanics]] becomes more explicitly evident than in other formulations.

To see how one can get more out of the [[Bohr topos]] incarnation of the quantum phase space, it serves to pass from plain [[quantum mechanics]] to the more general context of [[quantum field theory]]. Here the original [[Haag-Kastler axioms]] of [[AQFT]] demand that to a region of [[spacetime]] is to be assigned the [[quantum observables]] as a [[C\emph{-algebra]]/[[von Neumann algebra]]. But by the above discussion it is rather only the underlying [[Jordan algebra]] structure that matters, hence the [[Bohr topos]]. In light of this a [[local net of observables]] in [[AQFT]] is naturally regarded as a [[presheaf]] of [[ringed toposes]] on [[spacetime]], assigning the respective [[Bohr topos]] of local observables to each local region of spacetime.}

Such ``Bohrification of local nets of observables'' were analyzed in (\hyperlink{Nuiten12}{Nuiten 12}). There it was found that the natural structure of the transition functions of local nets of Bohr toposes of observables by [[geometric morphisms]] automatically capture the [[causal locality]] condition of [[local quantum field theory]]. This is now called ``Nuiten's lemma'' in (\hyperlink{WoltersHalvorson13}{Wolters-Halvorson 13}). Moreover, (\hyperlink{Nuiten12}{Nuiten 12}) shows that a natural [[descent]] condition on spacetime nets of [[Bohr toposes]] is equivalent to \emph{\href{local+net#StrongLocality}{strong locality}} of the [[quantum field theory]], something slightly weaker than \emph{\href{local+net#EinsteinLocality}{Einstein causality}}, which implies it. Since, by the above, the [[Bohr topos]] is the geometric incarnation of the [[Jordan algebra]] structure on [[quantum observables]], one might see this as a reply to the alleged lack of implications (in \hyperlink{ASReview04}{the AS review, 04}) of [[Pascual Jordan|Pascual Jordan's]] ideas from the 1930s to quantum field theory.

Be that as it may, notice that generally local systems of [[ringed toposes]] are to be expected to naturally encode [[quantum field theory]] on general grounds: the modern [[AQFT]]-style formulation of [[classical field theory]]/[[prequantum field theory]] via [[factorization algebras]] or similar assigns to subsets of [[spacetime]] the [[derived critical locus]] of local [[field (physics)|fields]] extremizing the given [[local action functional]], hence the [[derived geometry|derived space]] of solutions to the [[Euler-Lagrange equations]] of motion: the [[covariant phase space]] (pre-quantum). In [[physics]] this [[derived critical locus]] is modeled explicitly as a [[BV-BRST formalism|BV-complex]], but when realized in the full technical beauty of [[derived geometry]] it is in fact a [[structured (infinity,1)-topos|higher ringed topos]], as explained by [[Jacob Lurie]]. In view of this incarnation of [[classical field theory]] in [[AQFT]]-perspective as a net of [[structured (infinity,1)-topos|higher ringed topos]], it seems rather natural that under [[quantization]] it remains a net of ringed toposes, sending ringed toposes incarnating classical [[covariant phase spaces]] to ringed toposes incarnating their quantized version as quantum phase spaces.

\hypertarget{RelationToTheNonCommutativePhaseSpace}{}\subsection*{{Relation to the traditional non-commutative geometry}}\label{RelationToTheNonCommutativePhaseSpace}

In the above we highlighted that by \hyperlink{ListOfTheoremsInvoked}{the fundamental theorems} on the foundations of [[quantum mechanics]] -- going back to insights of [[Pascual Jordan]] way back in 1930 and formally affirmed by more recent results by Alfsen-Shultz and Harding-D\"o{}ring -- it follows that, accurately speaking, quantum phase space is not really an object in [[noncommutative geometry]], but rather in a kind of \emph{non-associative} ``Jordan geometry'' which is naturally captured by the [[ringed topos|ringed]] [[Bohr topos]] over the [[poset of commutative subalgebras]].

This observation indeed puts doubt on the long and widely held believe that the quantum phase space is an object in [[noncommutative geometry]], a belief that in fact motivated much of the development of [[noncommutative geometry]] in the first place.

But that this is not really true was ``well known'' all along: it is pointed out for instance in the foundational text (\hyperlink{BatesWeinstein97}{Bates-Weinstein 97}) on [[geometric quantization]]. On page 80 there is highlighted how given a classical [[phase space]] represented by a [[Poisson manifold]] $(X, \{-,-\})$, hence of a [[manifold]] that carries

\begin{enumerate}%
\item a commutative algebra of classical observables


\item equipped with a compatible non-commutative [[Lie bracket]] $\{-,-\}$ (the [[Poisson bracket]])



\end{enumerate}
the [[quantization]] of this data is to be thought of as applying to these two items separately:

\begin{enumerate}%
\item a non-associative but commutative [[Jordan algebra]] structure of quantum observables is the [[deformation quantization]] of the commutative algebra structure of classical observables;


\item a non-commutative [[Lie bracket]] structure is the deformation quantization of the Poisson bracket



\end{enumerate}
and that if the quantization of both items is given by a single [[C\emph{-algebra]] structure, then the non-associative commutative Jordan algebra structure is the one induced by the anticommutator}

\begin{displaymath}
A \circ B = \tfrac{1}{2}(A B + B A)
\end{displaymath}
while the non-commutative algebra structure is that given by the commutator

\begin{displaymath}
[A,B] = \tfrac{1}{2}(A B - B A)
\end{displaymath}
See also at \emph{\href{Jordan+algebra#OriginInQuantumPhysics}{Jordan algebra -- Origin in quantum physics}}. This splitting of the notion of quantization into a Lie-algebraic and a Jordan algebraic aspect is formalized in the notion of \emph{[[Jordan-Lie-Banach algebra]]}.

To understand that this makes good sense notice that this decomposition is that into [[kinematics]] and [[dynamics]] of [[quantum mechanics]]:

\begin{enumerate}%
\item [[kinematics]] --- the construction of the [[quantum observables]] and of the [[space of quantum states]] alone indeed does \emph{not} involve the associative product of the [[C\emph{-algebra]];}


\item [[dynamics]] --- the commutator [[Lie bracket]] structure is used to impose quantum [[Hamiltonian flows]] $\exp([A,-])$, hence (time) propagation along the [[trajectories]] generated by the observables $A$.



\end{enumerate}
But then observe in addition that when we pass from [[quantum mechanics]] to [[quantum field theory]] axiomatized as [[AQFT]], then in fact time propagation is no longer implemented by [[inner automorphisms]] of the form $\exp([A,-])$. Indeed it is impossible for any realistic [[physical system]] with infinitely many degrees of freedom (such as a [[field (physics)|field]]) to have time evolution given by an [[inner automorphism]]. That this does work for [[quantum mechanics]] is really an artifact of the degenerate case of finitely many degrees of freedom considered there.

Instead, in [[AQFT]] the [[spacetime]]-evolution of the quantum fields is all encoded in the transition functions of the [[local net of observables]]. By the \hyperlink{QuantumPhaseSpaceAsJordanGeometry}{above} discussion, this is however, accurately speaking, not actually to be thought of as a net of [[C\emph{-algebras]], but rather as a net of [[Jordan algebras]], hence as a net of [[Bohr toposes]]. This way the need for the non-commutative algebraic structure disappears and only the non-associative commutative [[Jordan algebra]] structure remains.}

In this context it may be noteworthy to recall what is a well-kept secret: despite much work on [[AQFT]] with the traditional [[Haag-Kastler axioms]] that demand to assign [[C\emph{-algebras]] to regions of spacetime, there is to date not a single non-[[free field theory|free field]] example in [[spacetime]] [[dimension]] greater than 3 of these axioms. (There are plenty of interesting example in dimension 2, though, see at \emph{[[conformal net]]} and pointers given there.)}

On the other hand, more recently the variant of the [[AQFT]] axioms known as \emph{[[factorization algebras]]} has been shown to admit plenty of interesting examples of [[quantum field theory]]. Comparison of the axioms is not straightforward and should be taken with a grain of salt, but it is maybe noteworthy that a [[factorization algebra]] is indeed a net that assigns to a region of spacetime \emph{not} an algebra structure. The algebra structure there is instead all encoded into the way that spacetime regions are included into each other.

\hypertarget{ListOfTheoremsInvoked}{}\subsection*{{List of theorems invoked}}\label{ListOfTheoremsInvoked}

For reference, the following lists that theorems about the standard foundations of quantum mechanics that are being referred to \href{QuantumPhaseSpaceAsJordanGeometry}{above}:

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


\item [[Alfsen-Shultz theorem]]


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


\item [[Kochen-Specker theorem]]


\item [[Nuiten's lemma]]



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[interpretation of quantum mechanics]]

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

\begin{itemize}%
\item [[Pascual Jordan]], \"U{}ber eine Klasse nichtassociativer hyperkomplexer Algebren, \emph{Nachr. Ges. Wiss. G\"o{}ttingen (1932), 569--575.}

\end{itemize}
\begin{itemize}%
\item [[Pascual Jordan]], [[John von Neumann]] and [[Eugene Wigner]], On an algebraic generalization of the quantum mechanical formalism, \emph{Ann. Math.} \textbf{35} (1934), 29--64.

\end{itemize}
\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})


\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})


\item [[Monique Hakim]], \emph{Topos annel\'e{}s et sch\'e{}mas relatifs}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).


\item Erhard Scheibe, \emph{The logical analysis of quantum mechanics} . Oxford: Pergamon Press, 1973.



\end{itemize}
\begin{itemize}%
\item [[Erik Alfsen]], [[Frederic Shultz]], \emph{A Gelfand Neumark theorem for Jordan algebras}, Advances in Math., 28 (1978), 11-56.


\item [[Erik Alfsen]], H. Hanche-Olsen, [[Frederic Shultz]], \emph{State spaces of $C^\ast$-algebras}, Acta Math., 144 (1980), 267-305.


\item Sean Bates, [[Alan Weinstein]], \emph{Lectures on the geometry of quantization} American Mathematical Society in the Berkeley Mathematics Lecture Notes series, 1997 (\href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf})


\item [[Jeremy Butterfield]], [[Chris Isham]], \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})


\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}


\item review of Alfsen-Shultz, 2004 (\href{http://www.ams.org/journals/bull/2004-41-04/S0273-0979-04-01019-5/S0273-0979-04-01019-5.pdf}{pdf})



\end{itemize}
\begin{itemize}%
\item [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], \emph{Bohrification of operator algebras and quantum logic}, in \emph{Deep Beauty} Cambridge University Press (2009) (\href{http://arxiv.org/abs/0909.3468}{arXiv:0909.3468}, \href{http://arxiv.org/abs/0905.2275}{arXiv:0905.2275})


\item [[John Harding]], [[Andreas Döring]], \emph{Abelian subalgebras and the Jordan structure of a von Neumann algebra} (\href{http://arxiv.org/abs/1009.4945}{arXiv:1009.4945})



\end{itemize}
\begin{itemize}%
\item [[Jan Hamhalter]], \emph{Isomorphisms of ordered structures of abelian $C^\ast$-subalgebras of $C^\ast$-algebras}, J. Math. Anal. Appl. 383 (2011) 391--399 (\href{dx.doi.org/10.1016/j.jmaa.2011.05.035}{journal})

\end{itemize}
\begin{itemize}%
\item [[Joost Nuiten]], \emph{[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]}, \href{http://qpl.science.ru.nl/}{Proceedings of QPL 2011} \href{http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011}{EPTCS 95}, 2012, pp. 211-218 (\href{http://arxiv.org/abs/1109.1397}{arXiv:1109.1397})


\item Sander Wolters, [[Hans Halvorson]], \emph{Independence Conditions for Nets of Local Algebras as Sheaf Conditions} (\href{http://arxiv.org/abs/1309.5639}{arXiv.1309.5639})



\end{itemize}


\end{document}