A Mathematical Formulation of Quantum Mechanics

The mathematical formulation to be discussed here is a modification of Kochen's Reconstruction of Quantum Mechanics [1]. It concentrates on a mathematical formulation using a union of σ-algebras that he calls a complex. In Kochen's reformulation it is through the interaction of a quantum system under consideration with another system that properties gain physical values. Such properties are said by Kochen to be relational or extrinsic, as opposed to the fixed intrinsic properties such as the charge of an electron.

While the mathematics depends heavily on the reconstruction by Kochen it makes use of von Neumann ∗-algebras (see Thirring [2]) rather than the Boolean algebras used by Kochen. This choice is made to avoid any hint of a logical interpretation. This blog also differs from Kochen's in the treatment and assignment of both intrinsic and extrinsic properties. The extrinsic properties will be treated here as an actualisation of potential, dispositional properties of the quantum entity that are shaped by its context.

In the mathematical formulation of quantum mechanics, a physical system, \(\mathbf{S}\), is characterized by a list of properties (called observables in the standard quantum mechanics), that can be represented by abstract self-adjoint operators, 

\[\mathcal{O}_\mathbf{S}=\{ \hat{X}_i = \hat{X}^*_i  | i \in \mathcal{I}_\mathbf{S}\} ,  \]

 with \(\mathcal{I}_\mathbf{S}\)  a set of indices depending on \(\mathbf{S}\), where every operator \(\hat{X} \in \mathcal{O}_\mathbf{S}\) represents a physical property characteristic of \(\mathbf{S}\), such as the total momentum, energy or spin of all particles localised in some bounded region of physical space and belonging to an ensemble of (possibly infinitely many) particles constituting the system \(\mathbf{S}\).

At every time \(t\), there is a representation of \(\mathcal{O}_\mathbf{S}\)  by self-adjoint operators acting on a separable Hilbert space \(\mathcal{H}_\mathbf{S}\): 

\[ \mathcal{O}_\mathbf{S} \ni  \hat{X}_i  \mapsto X_i =X^*_i \in \mathcal{A}(\mathcal{H}_\mathbf{S}) ,     \]

where \(\mathcal{A}(\mathcal{H}_\mathbf{S})\) is the \(*\)-algebra of all bounded operators acting on \(\mathcal{H}_\mathbf{S}\).

In general, the operators in the \(*\)-algebra do not all commute. This has deep physical consequences: 

• Properties that cannot appear together are represented in quantum mechanics by self-adjoint operators that do not all commute. 

• The sets of operators that do commute provide a set of properties for a measurable mathematical structure which is a \(\sigma\)-algebra.

The \(\sigma\)-algebra is essential to constructing mathematical probability models using the Kolmogorov axiomisation [3]. Therefore, each collection of commuting observables of the object has an associated \(\sigma\)-algebra.  So, a collection of \(\sigma\)-algebras that together capture all the property values associated with the object covers all possible events for that object in itself. This structure is the minimal one which contains all the $\sigma$-algebras arising from all the properties of the particle. 
Each dispositional property of an object, such as spin or position, has a \(\sigma\)-algebra of possible values that appear in interaction with other objects. All these \(\sigma\)-algebras form a complex property structure for the object under consideration. The formal definition of this notion is as follows:

 Let $F$ be a family of $\sigma$-algebras. The $\sigma$-complex $Q_F$ based on $F$ is the union, $\cup B$, of all $\sigma$-algebras $B$ lying in $F$. 

Generally, the family $F$ is left implicit, and reference is simply to a $\sigma$-complex $Q$ however the family \(F\) does tie the complex to an object that has properties that take values depending on the set of possible interactions that it can have with other objects. Usually $\sigma$-complexes that are closed under the formation of sub-$\sigma$-algebras are discussed. It is possible, in any case, to always close a $\sigma$-complex by adding all its sub-$\sigma$-algebras. This complex will include all the possible values that the dispositional properties of a particle can take in all possible interaction contexts.  \(B\) will denotes the \(\sigma\)-algebra relating to the property values of a particular (commuting) set of observables of the particle or system. 

In the mathematical formulation of Quantum Mechanics $\mathcal{H}$ is a Hilbert space. For the quantum object under consideration all properties are represented by self-adjoint operators and possible events are represented projection operators. For pair-wise commuting projection operators closed under the operation of orthogonal complement and countable product \(\prod_i \pi_i\) forms a \(\sigma\)-algebra. The family of all such \(\sigma\)-algebras form a \(\sigma\)-complex, \(Q(\mathcal{H})\), for the object in its possible interaction contexts. 

Proposition

In Quantum Mechanics, the operators representing the possible values of the properties of a system form the $\sigma$-complex $Q(\mathcal{H})$ of projections of the Hilbert space $\mathcal{H}$ of the system.       

It has been shown [1] that this proposition holds and the mathematical formalism using \(Q(\mathcal{H})\) is equivalent to the standard quantum formulation of the theory on \(\mathcal{H}\). However, the interpretation of the formalism here is that while a quantum object has a complete set of properties they appear only when the situation invokes the \(\sigma\)-algebra in the complex that allows that property to take a specific set of values.

[1] Simon Kochen. A Reconstruction of Quantum Mechanics. In: ArXiv e-prints (June 2015).
[2] WaIter Thirring. A Course in Mathematical Physics 3: Quantum Mechanics of Atoms and Molecules. Springer-Verlag, 1979.
[3] Andrey N. Kolmogorov. Foundations of Probability Theory. New York, 1950.