# The Many Classical Faces of Quantum Structures

## Abstract

**:**

## 1. Introduction

- known obstructions to hidden variable interpretations merely say that states cannot be located with exact precision in the state space, and are circumvented via open regions of states;
- the uncertainty principle cannot be expressed and therefore poses no interpretational problem;
- the measurement problem is obviated because the new notion of state space incorporates all classical data resulting from possible measurements.

- yields the same predictions as traditional quantum mechanics.

#### 1.1. Algebraic Quantum Theory

#### 1.2. Gelfand Duality

#### 1.3. Bohr’s Doctrine of Classical Concepts

#### 1.4. The Kadison–Singer Problem

#### 1.5. The Kochen–Specker Theorem

#### 1.6. Overview of This Article

## 2. Invariants

**Definition**

**1.**

**Definition**

**2.**

- a reflexive and symmetric binary (commeasurability) relation $\odot \subseteq A\times A$;
- elements $0,1\in A$;
- a (total) involution $\ast :A\to A$;
- a (total) function $\xb7:\mathbb{C}\times A\to A$;
- a (total) function $\parallel -\parallel :A\to \mathbb{R}$;
- (partial) binary operations $+,\xb7:\odot \to A$;

**Theorem**

**1**

**.**Every piecewise C*-algebra is the colimit of its commutative C*-subalgebras in the category of piecewise C*-algebras.

**Theorem**

**2**

**Theorem**

**3**

**.**Suppose there exist a category conservatively extending that of compact Hausdorff spaces and a functor F completing the following square.

**Theorem**

**4**

**.**Let A and B be C*-algebras. If $\mathcal{C}\left(A\right)$ and $\mathcal{C}\left(B\right)$ are isomorphic partially ordered sets, then A and B are isomorphic as piecewise Jordan algebras.

**Corollary**

**1**

## 3. Toposes

**Theorem**

**5**

**.**Let A be a C*-algebra. In the topos of contextual sets over $\mathcal{C}\left(A\right)$, the canonical contextual set $C\mapsto C$ is a commutative C*-algebra.

- Start with a quantum system A.
- Change the logical rules of set theory by moving to the topos of contextual sets over $\mathcal{C}\left(A\right)$.
- The quantum system A turns into a classical one given by the canonical contextual set $C\mapsto C$.

**Corollary**

**2**

**.**Let A be a C*-algebra. In the topos of contextual sets over $\mathcal{C}\left(A\right)$, there is a compact Hausdorff locale X such that the canonical contextual set is of the form $C\left(X\right)$.

**Proposition**

**1**

**.**Let A be a C*-algebra satisfying the Kochen–Specker Theorem 2. In the topos of contextual sets over $\mathcal{C}\left(A\right)$, the spectral contextual set has no points.

**Proposition**

**2**

**.**For any C*-algebra A, the internal locale X is determined by a continuous function from some locale $Spec\left(A\right)$ to $\mathcal{C}\left(A\right)$.

**Theorem**

**6**

**Corollary**

**3**

## 4. Domains

**Definition**

**3.**

**Lemma**

**1**

**Proposition**

**3**

**.**A C*-algebra A is finite-dimensional if and only if $\mathcal{C}\left(A\right)$ is Artinian, if and only if $\mathcal{C}\left(A\right)$ is Noetherian.

**Proposition**

**4**

**.**If $A={\u2a01}_{i=1}^{n}{\mathbb{M}}_{{n}_{i}}\left(\mathbb{C}\right)$, then the C*-subalgebras ${\mathbb{M}}_{{n}_{i}}\left(\mathbb{C}\right)$ correspond to directly indecomposable partially ordered subsets ${\mathcal{C}}_{i}$ of $\mathcal{C}\left(A\right)$, and furthermore ${n}_{i}$ is the length of a maximal chain in ${\mathcal{C}}_{i}$.

**Theorem**

**7**

**.**A C*-algebra A is scattered if and only if $\mathcal{C}\left(A\right)$ is a continuous domain if and only if $\mathcal{C}\left(A\right)$ is an algebraic domain.

**Corollary**

**4**

**.**For a scattered C*-algebra A, the Lawson topology makes $X=\mathcal{C}\left(A\right)$ compact Hausdorff. Hence to each scattered C*-algebra A we may assign a commutative C*-algebra $C\left(X\right)$.

## 5. Dynamics

**Definition**

**4.**

**Theorem**

**8**

**Lemma**

**2**

**.**The category of piecewise complete Boolean algebras and the category of piecewise AW*-algebras are equivalent.

**Proposition**

**5**

**.**Let A and B be typical AW*-algebras, and suppose that $f:Proj\left(A\right)\to Proj\left(B\right)$ preserve least upper bounds and orthocomplements. Then f extends to a Jordan homomorphism $A\to B$. It extends to a homomorphism if additionally $f\left((1-2p)(1-2q)\right)=\left(1-2f\left(p\right)\right)\left(1-2f\left(q\right)\right)$.

**Theorem**

**9**

**.**The functor that assigns to an AW*-algebra A its active lattice $AProj\left(A\right)$ is fully faithful.

**Corollary**

**5**

**.**Any normal piecewise Jordan homomorphism between typical AW*-algebras is a Jordan homomorphism.

## 6. Characterization

**Lemma**

**3**

**.**A partially ordered set is isomorphic to $\mathcal{C}\left(C\right(X\left)\right)$ for a compact Hausdorff space X if and only if it is opposite to a partition lattice whose points are in bijection with X.

**Theorem**

**10**

**.**A partially ordered set is isomorphic to $\mathcal{C}\left(B\right)$ for a piecewise Boolean algebra B if and only if:

- it is an algebraic domain;
- any nonempty subset has a greatest lower bound;
- a set of atoms has an upper bound whenever each pair of its elements does;
- the downset of each compact element is isomorphic to the opposite of a finite partition lattice.

**Theorem**

**11**

**.**Suppose that a C*-algebra A has a weakly terminal commutative C*-subalgebra $C\left(X\right)$. A category is equivalent to ${\mathcal{C}}_{\rightarrowtail}\left(A\right)$ if and only if it is equivalent to a semidirect product of $\mathcal{C}\left(C\right(X\left)\right)$ and $S\left(X\right)$.

## 7. Generalizations

**Theorem**

**12**

**Theorem**

**13**

## Acknowledgments

## Conflicts of Interest

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The Many Classical Faces of Quantum Structures. *Entropy* **2017**, *19*, 144.
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The Many Classical Faces of Quantum Structures. *Entropy*. 2017; 19(4):144.
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Heunen, Chris.
2017. "The Many Classical Faces of Quantum Structures" *Entropy* 19, no. 4: 144.
https://doi.org/10.3390/e19040144