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\begin{document}

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\section*{quantum mechanics}

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

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

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{npov}{nPOV}\dotfill \pageref*{npov} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{quantum_mechanical_systems}{Quantum mechanical systems}\dotfill \pageref*{quantum_mechanical_systems} \linebreak
\noindent\hyperlink{observables_and_states}{Observables and states}\dotfill \pageref*{observables_and_states} \linebreak
\noindent\hyperlink{spaces_of_states}{Spaces of states}\dotfill \pageref*{spaces_of_states} \linebreak
\noindent\hyperlink{flows_and_time_evolution}{Flows and time evolution}\dotfill \pageref*{flows_and_time_evolution} \linebreak
\noindent\hyperlink{Subsystems}{Quantum subsystems}\dotfill \pageref*{Subsystems} \linebreak
\noindent\hyperlink{quantum_physics_in_general}{Quantum physics in general}\dotfill \pageref*{quantum_physics_in_general} \linebreak
\noindent\hyperlink{formulations_and_formalization}{Formulations and formalization}\dotfill \pageref*{formulations_and_formalization} \linebreak
\noindent\hyperlink{ordertheoretic_structure_in_quantum_mechanics}{Order-theoretic structure in quantum mechanics}\dotfill \pageref*{ordertheoretic_structure_in_quantum_mechanics} \linebreak
\noindent\hyperlink{quantum_mechanics_in_terms_of_compact_categories}{Quantum mechanics in terms of $\dagger$-compact categories}\dotfill \pageref*{quantum_mechanics_in_terms_of_compact_categories} \linebreak
\noindent\hyperlink{foundational_theorems_of_quantum_mechanics}{Foundational theorems of quantum mechanics}\dotfill \pageref*{foundational_theorems_of_quantum_mechanics} \linebreak
\noindent\hyperlink{applications_of_quantum_mechanics}{Applications of quantum mechanics}\dotfill \pageref*{applications_of_quantum_mechanics} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

While [[classical mechanics]] considers deterministic evolution of particles and fields, quantum physics follows nondeterministic evolution where the probability of various outcomes of [[measurement]] may be predicted from the state in a Hilbert space representing the possible reality: that state undergoes a unitary evolution, what means that the generator of the evolution is $\sqrt{-1}$ times a Hermitean operator called the \emph{quantum Hamiltonian} or the [[Hamiltonian operator]] of the system. The theoretical framework for describing this precisely is the \textbf{quantum mechanics}. It involves a constant of nature, Planck constant $h$; some quantum systems with spatial interpretation in the limit $h\to 0$ lead to classical mechanical systems (not all: some phenomena including non-integer spin are purely quantum mechanical, but the properties depending on their existence survive in the ``classical'' limit); in limited generality, one can motivate and find the \emph{nonfunctorial} procedure to single out a [[right inverse]] to taking this \emph{classical limit} under the name [[quantization]].

While quantum mechanics may be formulated for a wide range of physical systems, interpreted as particles, extended particles and fields, the quantum mechanics of fields is often called the [[quantum field theory]] and the quantum mechanics of systems of a fixed finite number of particles is often viewed as the quantum mechanics in a narrow sense.

\hypertarget{npov}{}\paragraph*{{nPOV}}\label{npov}

Mathematically, despite the basic formalism of quantum mechanics which is sound and clear, there are two big areas which are yet not clear. One is to understand quantization, in all cases -- of particles, fields, strings and so on. The second and possibly more central to nLab is a problem how to define rigorously a wide range of quantum field theories and some related quantum mechanical systems like the hypothetical [[superstring theory]]. Regarding that this is a central goal, we also put emphasis on the [[interpretation of quantum mechanics]] via the picture which is a special case of a FQFT, and where the time evolution functorially leads to evolution operators.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We discuss some basic notions of quantum mechanics.

\hypertarget{quantum_mechanical_systems}{}\subsubsection*{{Quantum mechanical systems}}\label{quantum_mechanical_systems}

Recall the notion of a \emph{[[classical mechanical system]]}: the formal dual of a real \emph{commutative} [[Poisson algebra]].

\begin{defn}
\label{QuantumSystem}\hypertarget{QuantumSystem}{}
A \textbf{quantum mechanical system} is a [[star algebra]] $(A, (-)^\ast)$ over the [[complex number]]s. The [[category]] of of quantum mechanical systems is the [[opposite category]] of $\ast$-algebras:

\begin{displaymath}
QuantMechSys := {\ast}Alg_{\mathbb{C}}^{op}
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
It makes sense to think of this as a deformed version of a real [[Poisson algebra]] as follows:

\begin{itemize}%
\item the Poisson-[[Lie algebra|Lie bracket]] of a [[Poisson algebra]] corresponds to the [[commutator]] of the $\ast$-algebra:

\begin{displaymath}
[a,b] := a b - b a
  \,,
\end{displaymath}

\item the commutative algebra structure of the Poisson algebra coresponds to the [[Jordan algebra]] structure of the $\ast$-algebra, with commutative (but non-associative!) product

\begin{displaymath}
(a,b) := a b + b a
  \,.
\end{displaymath}


\end{itemize}
With this interpretation the [[derivation]]-property of the Poisson bracket over the other product is preserved: for all $a,b,c \in A$ we have

\begin{displaymath}
[a,(b,c)] = ([a,b],c) + (b,[a,c])
  \,.
\end{displaymath}
We thus may regard a non-commutative star-algebra as a \emph{[[nonassociative algebra|non-associative]] [[Poisson algebra]]} : a [[Jordan-Lie algebra]]. See there for more details.

\end{remark}
\hypertarget{observables_and_states}{}\subsubsection*{{Observables and states}}\label{observables_and_states}

\begin{defn}
\label{}\hypertarget{}{}
Given a quantum mechanical system in terms of a [[star algebra]] $A$, we say

\begin{itemize}%
\item an \textbf{[[observable]]} is an element $a \in A$ such that $a^\ast = a$;


\item a \textbf{[[state]]} is a [[linear function]] $\rho : A \to \mathbb{C}$ which is \emph{positive} in the sense that for all $a \in A$ we have $\rho(a a^\ast) \geq 0 \in \mathbb{R} \hookrightarrow \mathbb{C}$.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
One can formalize the idea that a quantum mechanical system is like a deformed [[classical mechanical system]] as follows:

To every ${}^\ast$-algebra $A$ is associated its [[poset of commutative subalgebras]] $Com(A)$. Then the corresponding quantum mechanical system is a [[classical mechanical system]] [[internalization|internal]] to the [[sheaf topos]] $Sh(Com(A))$:

\begin{itemize}%
\item The $\ast$-algebra canonically induces a commuative algebra $\underline A \in Sh(Com(A))$;


\item the (classical) [[state]]s of $\underline{A}$ in $Sh(Com(A))$ are in natural bijection with the quantum states externally on $A$;


\item the (classical) [[observable]]s of $\underline{A}$ in $Sh(Com(A))$ correspond to the external quantum observables on $A$.



\end{itemize}
(\ldots{}details\ldots{})

\end{remark}
One also says that the internal classical mechanical system $(Sh(Com(A)), \underline{A})$ is the ``[[Bohrification]]'' of the external quantum system $A$. See there for more details.

\hypertarget{spaces_of_states}{}\subsubsection*{{Spaces of states}}\label{spaces_of_states}

Given a $\ast$-algebra $A$ together with a [[state]] $\rho$ on it, the [[GNS construction]] provides an [[inner product space]] $H_\rho$ together with an [[action]] of $A$ on $H_\rho$ and a [[vector]] $\Omega = \sqrt(\rho)$ -- the [[vacuum vector]] -- such that for all $a \in A$ the value of the state $\rho : A \to \mathbb{C}$ is obtained by applying $a$ to $\sqrt{\rho}$ and then taking the inner product with $\sqrt \rho$:

\begin{displaymath}
\rho(A) = \langle \sqrt\rho, a \sqrt \rho\rangle
  \,.
\end{displaymath}
If the [[star algebra]] $A$ happens to be a [[C-star algebra]], then this inner product space is naturally a [[Hilbert space]].

Historically and still often in the literature, such a Hilbert space is taken as a fundamental input of the definition of quantum systems.

Traditionally, [[Dirac]]`s ``bra-ket'' notation is used to represent vectors in such Hilbert spaces of states, where $|\psi\rangle$ represents a state and $\langle\psi|$ represents its linear adjoint. State evolutions are expressed as unitary maps. Self-adjoint operators represent physical quantities such as position and [[momentum]] and are called [[observables]]. [[measurements|Measurements]] are expressed as sets of [[projectors]] onto the eigenvectors of an observable.

In [[mixed state]] quantum mechanics, physical states are represented as [[density operators]] $\rho$, state evolution as maps of the form $\rho \mapsto U^\dagger \rho U$ for unitary maps $U$, and [[measurements]] are positive operator-valued measures (POVM's). There is a natural embedding of pure states into the space of density matrices: $|\psi\rangle \mapsto |\psi\rangle\langle\psi|$. So, one way to think of mixed states is a probabilistic mixture of pure states.

\begin{displaymath}
\rho = \sum_i a_i |\psi_i\rangle\langle\psi_i|
\end{displaymath}
Composite systems are formed by taking the tensor product of Hilbert spaces. If a [[pure state]] $|\Psi\rangle \in H_1 \otimes H_2$ can be written as $|\psi_1\rangle \otimes |\psi_2\rangle$ for $|\psi_i\rangle \in H_i$ it is said to be \emph{[[separable]]}. If no such $|\psi_i\rangle$ exist, $|\Psi\rangle$ is said to be \emph{[[entangled]]}. If a mixed state is separable if it is the sum of separable pure states. Otherwise, it is entangled.

\hypertarget{flows_and_time_evolution}{}\subsubsection*{{Flows and time evolution}}\label{flows_and_time_evolution}

As for classical mechanics, 1-parameter families of flows in a quantum mechanical system are induced from observables $a \in A$ by

\begin{displaymath}
\frac{d}{d \lambda} b_\lambda = \frac{1}{i \hbar}[b_\lambda, a]
  \,.
\end{displaymath}
In a non-relativistic system one specifies an observable $H$ -- called the [[Hamiltonian]] -- whose flow represents the time evolution of the system. (This is the [[Heisenberg picture]].)

We comment on how to interpret this from the point of view of [[FQFT]]:

Quantum mechanics of point particles may be understood as a special case of the formalism of [[quantum field theory]]. It is interpreted as the quantum analog of the [[classical mechanics]] of point particles. Of course, we can take a configuration space of a system of particles looking like the configuration space of a single particle in a higher dimensional manifold.

Remark: related query on the relation between QFT and quantum mechanics (of particles and in general) can be found \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2573&Focus=22060#Comment_22060}{here}.

One may usefully think of the quantum mechanics of a point particle propagating on a [[manifold]] $X$ as being $(0+1)$-dimensional [[quantum field theory]]:

the fields of this system are maps $\Sigma \to X$ where $\Sigma \in Riem Bord_1$ are 1-dimensional [[Riemannian manifold]] [[cobordism]]s. These are the \emph{trajectories} of the particle.

After [[quantization]] this yields a 1-dimensional [[FQFT]] given by a [[functor]]

\begin{displaymath}
U(-) : Riem Bord_1 \to Hilb
\end{displaymath}
from [[cobordism]]s to [[Hilbert space]]s (or some other flavor of [[vector space]]s) that assigns

\begin{itemize}%
\item to the point the \emph{space of states} $\mathcal{H}$, typically the space of $L_2$-sections (with respect to a [[Riemannian metric]] on $X$) of the background [[gauge field]] on $X$ under which the particle in question is charged


\item to the cobordism of Riemannian length $t$ the operator

\begin{displaymath}
U(t) := \exp\left(\frac{t}{i \hbar } H \right)
  : \mathcal{H} \to \mathcal{H}
  \,,
\end{displaymath}
where $H$ is the [[Hamiltonian operator]], typically of the form $H = \nabla^\dagger \circ \nabla$ for $\nabla$ the [[covariant derivative]] of the given background [[gauge field]].



\end{itemize}
Such a setup describes the quantum mechanics of a particle that feels forces of backgound [[gravity]] encoded in the [[Riemannian metric]] on $X$ and forces of background gauge fields (such as the [[electromagnetic field]]) encoded in the covariant derivative $\nabla$.

(This is the [[Schrödinger picture]].)

\hypertarget{Subsystems}{}\subsubsection*{{Quantum subsystems}}\label{Subsystems}

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{A}$ an algebra describing a quantum system, def. \ref{QuantumSystem}, a \textbf{subsystem} is a subalgebra (a [[subobject]]) $B \hookrightarrow \mathcal{A}$.

Two subsystems $B_1, B_2 \hookrightarrow \mathcal{A}$ are called \textbf{independent subsystems} if the [[linear map]]

\begin{displaymath}
B_1 \otimes B_2 \to \mathcal{A}
\end{displaymath}
\begin{displaymath}
(b_1, b_2) \mapsto b_1 \cdot b_2
\end{displaymath}
from the [[tensor product]] of algebras (the [[composite system]]) factors as an [[isomorphism]]

\begin{displaymath}
B_1 \otimes B_2 \stackrel{\simeq}{\to} B_1 \vee B_2 \hookrightarrow \mathcal{A}
\end{displaymath}
through the algebra $B_1 \vee B_2$ that is generated by $B_1$ and $B_2$ inside $\mathcal{A}$ (the smallest subalgebra containing both).

\end{defn}
See for instance (\hyperlink{BrunettiFredenhagen}{BrunettiFredenhagen, section 5.2.2}).

\begin{defn}
\label{}\hypertarget{}{}
Given two independent subsystems $B_1, B_2 \hookrightarrow \mathcal{A}$, and two [[states]] $\rho_1 : B_1 \to \mathbb{C}$ and $\rho_2 : B_2 \to \mathbb{C}$, then the corresponding \textbf{product state} $\rho_1 \otimes \rho_2$ on $B_1 \vee B_2$ is defined to be

\begin{displaymath}
(\rho_1 \otimes \rho_2) : 
  (b_1 , b_2) \mapsto \rho_1(b_1) \rho_2(b_2)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
There exist states on $B_1 \vee B_2$ that are not (convex combinations of) product states. This phenomenon is called [[entanglement]].

\end{defn}
\hypertarget{quantum_physics_in_general}{}\subsection*{{Quantum physics in general}}\label{quantum_physics_in_general}

More generally, \emph{quantum physics} is all the known physics not including [[classical physics]] in wider sense; it includes relativistic and nonrelativistic phenomena. Quantum mechanics is the standard formalism with the Hilbert space, unitary evolution etc. which explains theoretically phenomena of quantum physics: in this generality of the formalism a la von Neumann, it includes the quantum field theory. Quantum statistical mechanics in fact uses some additional assumptions not in exact quantum mechanics, which are believed to be derivable eventually (like quantum ergodicity etc.). Thus quantum statistical mechanics may or may not be included within quantum mechanics.

Remark: Another way to look at quantum processes is via [[quantum channels]] which are completely positive trace-preserving maps.

For a previous query about quantum physics (includes experimental phenomena) and quantum mechanics (formalism for such, sometimes with or without statistical principles) see \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2573&Focus=22059#Comment_22059}{here}.

\hypertarget{formulations_and_formalization}{}\subsection*{{Formulations and formalization}}\label{formulations_and_formalization}

\hypertarget{ordertheoretic_structure_in_quantum_mechanics}{}\subsubsection*{{Order-theoretic structure in quantum mechanics}}\label{ordertheoretic_structure_in_quantum_mechanics}

See \emph{[[order-theoretic structure in quantum mechanics]]}.

\hypertarget{quantum_mechanics_in_terms_of_compact_categories}{}\subsubsection*{{Quantum mechanics in terms of $\dagger$-compact categories}}\label{quantum_mechanics_in_terms_of_compact_categories}

Many aspects of quantum mechanics and [[quantum computation]] depend only on the abstract properties of [[Hilb]] characterized by the fact that it is a [[†-compact category]].

For more on this see

\begin{itemize}%
\item [[quantum mechanics in terms of †-compact categories]].

\end{itemize}
\hypertarget{foundational_theorems_of_quantum_mechanics}{}\subsection*{{Foundational theorems of quantum mechanics}}\label{foundational_theorems_of_quantum_mechanics}

The following circle of [[theorems]]

\begin{itemize}%
\item [[Kochen-Specker theorem]]


\item [[Gleason's theorem]]


\item [[Alfsen-Shultz theorem]]


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



\end{itemize}
all revolve around the phenomenon that the ``[[phase space]]'' in quantum mechanics and hence the [[space of quantum states]] are all determined by the [[Jordan algebra]] structure on the [[algebra of observables]], which in turn is determined by the [[poset of commutative subalgebras]] of the algebra of observables. See at \emph{[[order-theoretic structure in quantum mechanics]]} for more on this.

There is also

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

\end{itemize}
which says roughly that linear maps between [[spaces of quantum states]] are [[unitary operators]] (or anti-unitary) already when they preserve [[norm]], hence preserve [[probability]].

\hypertarget{applications_of_quantum_mechanics}{}\subsection*{{Applications of quantum mechanics}}\label{applications_of_quantum_mechanics}

Quantum mechanics, as opposed to [[classical mechanics]], is necessary for an accurate description of reality whenever the characteristic scale is sufficiently small. For instance

\begin{itemize}%
\item In [[chemistry]] (``[[quantum chemistry]]'') the properties of [[atoms]] and [[molecules]] are derived from quantum mechanics.


\item In [[solid state physics]] the properties of [[metals]] etc. are described by quantum mechanics of electron gases.


\item In [[particle physics]] of course, [[quantum field theory]] is the appropriate description.



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[harmonic oscillator]]


\item [[hydrogen atom]]


\item [[quantum lattice system]]



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

\begin{itemize}%
\item [[classical mechanics]]

\begin{itemize}%
\item [[semiclassical approximation]]

\end{itemize}

\item [[quantization]]

\begin{itemize}%
\item [[deformation quantization]], [[geometric quantization]]

\end{itemize}

\item \textbf{quantum mechanics}

\begin{itemize}%
\item [[Planck's constant]]


\item [[Schrödinger picture]], [[Heisenberg picture]], [[Dirac picture]]


\item [[quantum probability]]


\item [[propagator]]


\item [[quantum superposition]], [[quantum fluctuation]], [[quantum entanglement]]


\item [[quantum measurement]]

\begin{itemize}%
\item [[positive-operator valued probability measure]]

\end{itemize}

\item [[hidden variable theory]],

\begin{itemize}%
\item [[EPR paradox]], [[Bell's inequalities]]


\item [[Kochen-Specker theorem]]



\end{itemize}

\item [[no-cloning theorem]]


\item [[Born-Oppenheimer approximation]]


\item [[quantum tunneling]]


\item [[supersymmetric quantum mechanics]]


\item [[quantum logic]], [[quantum computing]]

\begin{itemize}%
\item [[linear logic]], [[linear type theory]]

\end{itemize}

\item [[finite quantum mechanics in terms of dagger-compact categories]]


\item [[JBW-algebraic quantum mechanics]]


\item [[interpretation of quantum mechanics]]


\item [[quaternionic quantum mechanics]]



\end{itemize}

\item [[quantum field theory]]

\begin{itemize}%
\item [[FQFT]], [[AQFT]]

\end{itemize}

\item [[quantum chemistry]]


\item [[quantum biology]]



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

Classical textbooks (on the [[Hilbert space]] description) include

\begin{itemize}%
\item [[John von Neumann]], \emph{Mathematische Grundlagen der Quantenmechanik}. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.


\item [[George Mackey]], \emph{The Mathematical Foundations of Quamtum Mechanics} A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963


\item E. Prugovecki, \emph{Quantum mechanics in Hilbert Space}. Academic Press, 1971.


\item [[Paul Dirac]], \emph{[[The Principles of Quantum Mechanics]]}, International series of monographs on physics, Oxford University Press, 1978


\item Anthony Sudbery, Quantum mechanics and the particles of nature: an outline for mathematicians



\end{itemize}
Lecture notes include

\begin{itemize}%
\item Uni Bonn, \emph{\href{http://www-cip.physik.uni-bonn.de/~baehren/scripts/quantum.html}{Lecture scripts and Online courses -- Quantum mechanics}}


\item [[Valter Moretti]], \emph{Mathematical Foundations of Quantum Mechanics: An Advanced Short Course} (\href{http://arxiv.org/abs/1508.06951}{arXiv:1508.06951})



\end{itemize}
Some standard references include

\begin{itemize}%
\item Glimm and Jaffe, [[Glimm-Jaffe|Quantum physics - a functional integral point of view]]


\item Movshev's course has mathematically nice references: \href{http://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/syllabusfinal.html}{link}; and \href{http://www.math.columbia.edu/~woit/qftnotes1.pdf}{here} is a link to Woit's list of more physical tradition references.


\item [[Klaas Landsman]], \emph{[[Mathematical Topics Between Classical and Quantum Mechanics]]}


\item [[Klaas Landsman]], \emph{Fundations of Quantum Theory} Springer 2017


\item P. Cartier, C. DeWitt-Morette, \emph{Functional integration: action and symmetries}, Cambridge Monographs on Mathematical Physics, 2006.


\item Leon Takhtajan, \emph{[[Quantum mechanics for mathematicians]]} , Amer. Math. Soc. 2008.


\item Sean Bates, [[Alan Weinstein]], \emph{Lectures on the geometry of quantization}, \href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf}


\item F. Strocchi, \emph{An introduction to the mathematical structure of quantum mechanics}


\item [[Valter Moretti]], \emph{Spectral Theory and Quantum Mechanics} 2nd edition Springer 2018


\item [[Valter Moretti]], \emph{Fundamental Mathematical Structures of Quantum Theory} Springer 2019



\end{itemize}
Dicussion of quantum measurement is in

\begin{itemize}%
\item Jan Perina, Z. Hradil, [[Branislav Jurčo]], \emph{[[Quantum optics and fundamentals of physics]]}, Kluwer 1994

\end{itemize}
Introduction to mathematical foundations of quantum physics in [[quantum probability]], [[operator algebra]]:

\begin{itemize}%
\item Jonathan Gleason, \emph{The $C^*$-algebraic formalism of quantum mechanics}, 2009 (\href{http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf}{pdf}, [[GleasonAlgebraic.pdf:file]])


\item [[Jürg Fröhlich]], B. Schubnel, \emph{Quantum Probability Theory and the Foundations of Quantum Mechanics}. In: Blanchard P., Fröhlich J. (eds.) \emph{The Message of Quantum Science}. Lecture Notes in Physics, vol 899. Springer 2015 (\href{https://arxiv.org/abs/1310.1484}{arXiv:1310.1484}, \href{https://doi.org/10.1007/978-3-662-46422-9_7}{doi:10.1007/978-3-662-46422-9\_7})


\item [[Jürg Fröhlich]], \emph{The structure of quantum theory}, Chapter 6 in \emph{The quest for laws and structure}, EMS 2016 (\href{https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure}{doi}, \href{https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8}{doi:10.4171/164-1/8})


\item [[Klaas Landsman]], \emph{Foundations of quantum theory -- From classical concepts to Operator algebras}, Springer Open 2017 (\href{https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf}{pdf})



\end{itemize}
see also

\begin{itemize}%
\item [[Lucien Hardy]], \emph{Quantum Theory From Five Reasonable Axioms} (\href{https://arxiv.org/abs/quant-ph/0101012}{arXiv:quant-ph/0101012})

\end{itemize}
Generalization of the algebraic perspective to [[quantum field theory]] is discussed in

\begin{itemize}%
\item [[Rudolf Haag]], \emph{[[Local Quantum Physics -- Fields, Particles, Algebras]]}

\end{itemize}
for more on this see at \emph{[[AQFT]]} and at \emph{[[perturbative AQFT]]}

Different incarnations of this [[C\emph{-algebra|C}-algebraic]] locality condition are discussed in section 3 of

\begin{itemize}%
\item Sander Wolters, \emph{Quantum toposophy},

\end{itemize}
relating it to the [[topos theory|topos-theoretic]] formulation in

\begin{itemize}%
\item [[Joost Nuiten]], \emph{[[schreiber:bachelor thesis Nuiten|Bohrification of local nets of observables]]}

\end{itemize}
Aspects of quantum mechanics in [[category theory]] and [[topos theory]] are discussed in

\begin{itemize}%
\item [[Hans Halvorson]] (ed.) \emph{Deep Beauty -- Understanding the quantum world through mathematical innovation} Cambridge (2011) (\href{http://www.tacron.com/uploads/Deep_Beauty_-_Understg_the_Quantum_World_Through_Mathem._Innov._-_H._Halvorson__Cambridge__2011__WW.pdf}{pdf})

\end{itemize}
This discusses for instance [[higher category theory and physics]] and the [[Bohr topos]] of a quantum system.

[[!redirects quantum physics]] [[!redirects quantum theory]]

[[!redirects quantum mechanical system]] [[!redirects quantum mechanical systems]]



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