Quantum chance

In contrast to other formulations of quantum mechanics the theory proposed here describes a world of objective chance events.

Quantum States

As presented in an earlier post, the properties of a quantum object are represented by self-adjoint operators, and these operators act on the elements of a Hilbert space.  The concept of the state of a quantum system or object will now be presented.  This concept of state is closely aligned with that in statistical mechanics, but the probabilistic quantum state is a property of individual object rather than in a statistical ensemble. As quantum states generalise classical probabilities, the discussion starts with a brief overview of probability measures.

Probability Measures

Classical probability [1] provides a probability measure, a countably additive, [0,1]-valued measure, that is a function
\[p:  B \to [0,1].\]
with domain $B$, a $\sigma$-algebra of subsets of the state space \(\Omega\), such that $p$($\mathbf{1}$) = $1$, where \(\mathbf{1}\) is the identity in the algebra \(B\), and

\[p(\bigcup_i e_i) = \sum_i  p(e_i) \hbox{ for pair-wise disjoint elements } e_1, e_2, \dots \hbox{ in }B.\]
In the case of a system $\mathbf{S}$, an object more complex than a single particle, the probability function $p$ gives the probabilities over the $\sigma$-algebra of events for that system as a whole.  A physical theory predicts the probabilities of outcomes of any possible situation given the complete initial state. 

States of a system \(\mathbf{S}\) in quantum mechanics are, in general, represented by density matrices acting on a Hilbert space, \(\mathcal{H}_\mathbf{S}\). Density matrices are non-negative, trace-class operators, \(\rho\),  of trace \(1\), that is:
$$\rho =  \rho^* \ge 0,  \mathsf{with } \, \text{tr}(\rho)=1.$$
The expectation of a physical quantity, \(\hat{X} \in \mathcal{O}_\mathbf{S}\), the set of properties for the system, at time \(t\) in a state given by a density matrix \(\rho\) is defined by
$$ p(X(t)) := \text{tr}(\rho X(t))$$
where \(X(t)\) is the operator representing the physical property \(\hat{X(t)}\). This can be extended to all bounded operators in \(\mathcal{A}(\mathcal{H}_\mathbf{S})\), the \(*\)-algebra of all bounded operators acting on \(\mathcal{H}_\mathbf{S}\)  .

From the mathematical formulation by Kochen [2], adapted here, there is a one-to-one correspondence between states $p$ on the \(\sigma\)-complex \(Q(\mathcal{H})\) and density operators.
The use of the same symbol as for the classical event probability \(p(x)\) for the quantum state is because this mathematical description of the quantum state is a generalisation of the classical probability model from a single \(\sigma\)-algebra to a \(\sigma\)-complex.  A density operator $\rho$ defines a probability measure $p$ on $Q(\mathcal{H})$, the \(\sigma\)-complex of projections. The converse, that a state $p$ defines a unique density operator $\rho$ on $\mathcal{H}$, follows from a theorem of Gleason [3]. A state on the lattice of projections on \(\mathcal{H}\) defines a unique density operator. A lattice consists of a partially ordered set in which every pair of elements has a unique least upper bound and a unique greatest lower bound.

Pure and mixed states

In Quantum Mechanics there is a one-one correspondence between the pure states of a system and rays of unit vectors $\psi$ in $\mathcal{H}$, such that $p(X) = (\psi, X \psi)$. The pure states correspond to one-dimensional projections $ \pi_\psi$ (with $\psi$ in the image of $\pi_\psi$) and $p(X) = \text{tr}(\pi_\psi X) = (\psi, X \psi)$.  

That even the pure states predict probabilities that are not \(0\) or \(1\) is what is to be expected of projections onto properties values. A pure state simply predicts the probabilities of property values that can become actual in interactions. Mixed states are mixtures of the pure states and in general there is no unique decomposition of a mixed into pure states. 

Properties or observables 

An observable is nothing more than a property of a quantum object that has a set of possible values that become actual values in interaction with other objects and can appear in various contexts. It need not actually be "observed" in an experiment to do so.  In the theory, each such property is represented by a self-adjoint operator that has a spectrum that corresponds to the set of possible property values. 

This concept of property can, as argued above, be treated within the \(\sigma\)-complex formulation [2]. Kochen shows that there is a one-one correspondence between observables \(\omega\) such that
\[ \omega: B(\mathbb{R}) \to Q(\mathcal{H}),\] 
where \(B(\mathbb{R})\) is the \(\sigma\)-algebra of Borel sets (Any set in a topological space that can be formed from open sets, or from closed sets, through the operations of countable union, countable intersection, and relative complement is a {\em Borel set}.) generated by the open intervals of the real numbers, \(\mathbb{R}\).

Given $\pi_\lambda = \omega((-\infty,\lambda])$, there is a Hermitian operator \(A\) such that  

\[A=\int \lambda d\pi_\lambda.\]

Conversely, given a Hermitian operator $A$ on $\mathcal{H}$, the spectral decomposition above defines the representation of a property $\omega$ as the spectral measure $\omega(s)= \int_s d\pi_\lambda$, for $s\in B(\mathbb{R})$.  This establishes the one-one correspondence and the equivalence of a key element of the reconstruction to that of standard quantum mechanics.      
                                                                                                      
It follows that if $\omega: B(\mathbb{R}) \to Q(\Omega) $ is a representation of a property with corresponding self-adjoint operator $X$, then, for the state $p$ with corresponding density operator $\rho$, the expectation of $\omega$

\[{\mathop{\rm Exp}\nolimits}_p(\omega) = \text{tr}(X \rho).\]

The result shows the close connection between the spectrum of an operator and the \(\sigma\)-algebra of property values. For instance, for the case of a discrete operator \(X\), the spectral decomposition \(X =
\sum a_i \pi_i\) defines the \(\sigma\)-algebra of property values generated by the set \(\{\pi_{i}\}_i\). Conversely, given the \(\sigma\)-algebra of property values, the elements of the set \(\{\pi_i\}_i\) allow the definition, for each sequence of real numbers \(a_i\), the Hermitian operator \( \sum a_i \pi_i\). 

Symmetries

In the standard formulation of quantum physics symmetries appear as unitary transformations of Hilbert space vectors or Hermitian operators. Here they also appear naturally as symmetries   of a $\sigma$-complex.

An automorphism of a $\sigma$-complex $Q$ is a  one-one transformation     $\alpha:  Q\to Q$  of  $Q$ on to $Q$ such that for every $\sigma$-algebra $B$ in $Q$ and all $ e, e_1, e_2, \cdots$     in every $B$

 \[  \alpha(e^\bot)=\alpha(e)^\bot \hbox{ and }\alpha (\sum_i e_i)=\sum_i \alpha(e_i),\]

where \(\bot\) denote the orthogonal complement.

There is a one-one correspondence between symmetries $\alpha: Q(\mathcal{H}) \to Q(\mathcal{H})$ and unitary operators $u$ on $\mathcal{H}$ such that $\alpha(X) = uXu^{-1}$, for all $X \in Q(\mathcal{H})$.

If a state $p$ corresponds to the density operator $\rho$, then      

\[p_\alpha(X) = p(\alpha^{-1} (X)) = \text{tr}(\rho u^{-1} Xu) = \text{tr}(u \rho u^{-1}X), \kern 4pc \text{(*)} \]

so that the state $p_\alpha$ corresponds to the density operator $u \rho u^{-1} $.  This shows the mathematical equivalence between the \(\sigma\)-complex and the density matrix formulations under symmetry transformation.

Now that the mathematical formalism in terms of \(\sigma\)-complexes has been presented it is possible to develop a theory that describes the micro-physical world as governed by quantum chance. By chance is meant objective probability that is a physical property of objects in the real sphere rather than a description of uncertainty in the epistemic sphere. This will, among other attributes, allow the derivation of the quantum conditional probability that provide an explanation of quantum state reduction. 

In standard quantum mechanics the term `reduction' usually means the reduction of the wavefunction and so is conceptually tied to the Schrödinger picture. Although there is a mathematical equivalence between the Schrödinger picture and the formalism developed in this paper, the state as a set of probability measures on a \(\sigma\)-complex of projection operators describes a physics of dispositions rather than waves in either real space or some more abstract space. As such the position is closer to that of matrix mechanics of early quantum physics, but with more ontological detail on the role of probabilities.  

Conditional states

State reduction is a phenomenon that may seem unique to quantum mechanics and has no counterpart in classical mechanics, but this not the case. Conditional probability plays that role classically but the objective ensemble or relative frequency interpretations of probability are available, which means that no conceptual problem is posed by the `reduction' of the probability distribution. With this in mind, conditional probability will now be generalised to quantum mechanics.                                                         

If $p$ is a state on $Q(\mathcal{H})$ and $Y$ a projection operator in $ Q(\mathcal{H}) $ such that $p(Y) \ne  0$, then it will be shown that there exists a unique state $p(\, \cdot\!\mid Y)$ conditional on $Y$. If $\rho$ is the density operator corresponding to $p$, then  to see that the operator $Y\rho Y/\text{tr}(Y \rho Y)$ corresponds to the state $p(\, \cdot\!\mid Y)$, note that if $X$ lies in the same $\sigma$-algebra as $Y$, then $X$ and $Y$ commute, so

$$\begin{eqnarray}\label{eq:reduction}
 \text{tr}(Y \rho YX)/ \text{tr}(Y \rho Y) &=&  \text{tr}(\rho XY)/ \text{tr}(\rho Y)  \\ 
&=& p(X \cdot Y)/p(Y)   \\ 
&=& p(X\mid Y)   ,
\end{eqnarray}$$

which is the standard form for conditional probability. It is proposed that the expression

$$\begin{eqnarray}\label{eq2:reduction}
p(X\mid Y) =  \text{tr}(Y \rho YX)/ \text{tr}(Y \rho Y)  \kern 4pc \text{(**)}
\end{eqnarray}$$

generalises the conditional probability state to the case when \(X\) and \(Y\) do not commute.

The mapping of \(\rho\) to \(Y \rho Y/  \text{tr}( \rho Y)  \) is the formula for the reduction of state given by the von Neumann-Lüders Projection Rule [4]. In the standard quantum mechanics this rule is an additional principle appended to quantum mechanics. Here it appears as the unique answer to conditioning a state to a specific projection operator. 

From the symmetry transformation of the state, equation (*),

$$\begin{eqnarray}
p_\alpha (X |Y) &=& \frac{p_\alpha (X Y)}{p_\alpha(Y)}  \\
 &=& \frac{p (\alpha^{-1}(X Y))}{p(\alpha^{-1}(Y))}  \\
 &=& \frac{p (\alpha^{-1}(X) \alpha^{-1}(Y))}{p(\alpha^{-1}(Y))}   \\ 
&=& p(\alpha^{-1}( X) | \alpha^{-1}(Y))
\end{eqnarray}$$

The results in this subsection hold when \(X\) and \(Y\) are in the same commutative von Neumann sub-algebra corresponding to a sub-\(\sigma\)-algebra of \(Q(\mathcal{H})\).The next subsection covers the more interesting case of when they are in different sub-algebras and there is a correction to the form of the conditional probability that is well known from classical probability. 

Total probability in classical and quantum conditional probability

The Law of Total probability (or Law of Alternatives in the discrete case, as here) provides a useful method to partition and analyse conditional probabilities. Firstly, in the classical case, let $Y_1, Y_2, \cdots$ lie in a $*$-algebra of commuting projection operators with $Y_i Y_j= 0$ for $i  \ne  j$, and let $Y=\sum_i Y_i$, then:

$$\begin{eqnarray} \label{eq:LoA}
p(X \mid Y) &=& p(\sum_i (X Y_i))/p(Y)  \\
             &=& \sum_i (p(X  Y_i)/p(Y_i))  (p(Y_i)/p(Y))  \\
             &=& \sum_i   p(X \mid Y_i)p(Y_i \mid Y) ,
\end{eqnarray}$$                                                                                                  

Not only is this partition useful but also displays clearly how the quantum case differs from the classical.

From equation (**) , above, 

$$\begin{eqnarray}\label{eq:QLoA}
p(X \mid Y)& =&  \text{tr}(Y\rho Y X)/ \text{tr}(\rho Y)  \\
&=& \text{tr}(\sum_{i,j} Y_i \rho Y_j  X) /  \text{tr}(\rho Y)  \\
&=&\sum_i  \text{tr}(Y_i \rho Y_i X)/  \text{tr}(\rho Y) + \sum_{i\ne j} \text{tr}(Y_i \rho Y_j X)/ \text{tr}(\rho Y)  \\
&=&\sum_i p(X \mid Y_i)p(Y_i\mid Y)    \\
& &+\sum_{i\ne j} \text{tr}(Y_i \rho Y_j X)/ \text{tr}(\rho Y).
\end{eqnarray}$$

This shows that for conditional properties an interference term must be added to the classical law of total probability.   When \(X\) commutes with each \(Y_i\) the interference term is zero.

 A measurement of an property is the most familiar example of conditioning with respect to several properties. If a property is represented by an operator \(A\) has a spectral decomposition $\sum_i a_i \pi_{i} $, then measuring the observable amounts to registering the values of the properties given by the $\pi_{i} $. The interaction algebra $B_A$ is that generated by the $\pi_{i}$. 

Conditioning can be used to define the context for the dynamics of a quantum object. In general, this requires the notion of conditioning with respect to several conditions. 

Given a system with $\sigma$-complex $Q$ and disjoint elements $Y_1, Y_2 , \dots$  in a  common $\sigma$-algebra in $Q$ with $\sum_i y_i=\mathbb{1}$, and a state $p$, the state conditional on $Y_1, Y_2 , \dots$ is defined to be   $p(\, \cdot\!\mid   Y_1,Y_2 , \dots  ) =  \sum p(Y_i)p(\, \cdot\!\mid   Y_i)$. This can be written more compactly as $p(\, \cdot\!\mid B_A) $, the state conditional on the interaction algebra $B_A$.

For a quantum system, with $\sigma$-complex $Q(\mathcal{H})$ and if $\rho$ is the density operator corresponding to the state $p$:        

 \[ p( \cdot \mid  B_A )  =  \sum  \text{tr}(\rho Y_i)(Y_i\rho Y_i/ \text{tr}(\rho Y_i)) =  \sum Y_i \rho Y_i,\] 

so that for each $X$ the probability $p(X \mid B_A)  =   \sum_i  \text{tr}(Y_i\rho Y_i X)$. 

The natural definition for applying a symmetry to the conditioned state is given by 

\[p_\alpha (X \mid B_A) = p(\alpha^{-1}(X)\mid \alpha^{-1}(B_A)).\]

The $\sigma$-algebra $B_A$ generated by the $Y_1, Y_2, \dots$ is simply a sub-algebra of the $\sigma$-complex.   

Summary

There is an equivalence between quantum states and the probabilities describing objective chance. That we are dealing with a generalisation of probability theory is made evident in the equation for the conditional probability that introduces extra terms due to quantum interference.

References

[1] Andrey N. Kolmogorov. Foundations of Probability Theory. New York, 1950.

[2] Simon Kochen. A Reconstruction of Quantum Mechanics. In: ArXiv e-prints (June 2015).

[3] A. M. Gleason. Measures on the Closed Subspaces of a Hilbert Space. In: J. Math. & Mech. 6 (1957), p. 883.

[4] Gerhardt Lüders. Über die Zustandsänderung durch den Messprozess. In: Annalen der Physik 8 (1951), pp. 322-328.

Strata of Real Being

To discuss ontology in general and to compare competing ontological proposals some structure is useful. This structure will help to place candidate beings in relation to others, understand their way of being and, for example, avoid the category mistake of identifying elements of physical reality with their mathematical representation.  Here we will explore the proposal that reality is indeed stratified and secondly what that stratification may be. In critical ontology it will be important to distinguish whether these levels are levels of reality per se or levels in a description of reality. 

The starting point cannot just be comprehensive scepticism about whether anything exists or not. That will get us nowhere. We will start by taking the phenomena around us as the they appear. There appear to be things that take up space and are stable. There appear to be things that take space and grow. There are appear to be things that have some purpose and make choices. There appear to be things that are made up of other things.   In addition, there are beings with emotional states and awareness of making choices. There is also awareness of resistance or pushback from other things. There is awareness of being the capacity of some beings to change the state of other things. We may be mistaken about particular things and their relationship to each other but the appearances are the appearances.  In talking about things and beings, we have implicitly used categories such a space, time, choice, emotion, awareness and substance. The categories will play an important role is characterising the levels in which beings exist. There will be separate post on categories.

Hartmann proposes a pluralistic view of reality, that covers all the things and beings listed above, that appear in four dynamically interrelated strata [1]:

1. Inorganic 
2. Organic 
3. Psychic 
4. Spiritual. 

The layers are presented in the order of decreasing ontological strength. That is, in each layer the beings depend on those in stronger layers to exist. However, the beings in the less strong layers cannot be reduced to beings in stronger layers. They have their specific categories of existence. There are also multi strata beings as will be illustrated at the end of this post. The question of whether these levels exist and the nature of their existence will not be examined here other than to recognise them as structural proposals and so will be seen to at least exist at the spiritual level. 

The name given to each level is not problematic except for the fourth.  Spiritual being is a translation from the German geistige Sein and the use of the term Geist follows the tradition influenced by Hegel and Dilthey—as it is used, for example, in Geisteswissenschaft—and its meaning covers both individual mind as well as superindividual culture, which is why it lacks an adequate English translation. No religious sense of the term is intended here. Beings at the spiritual level cannot exist without the other levels. 

Although the listing of the four strata suggests a hierarchy, to provide this thought with some substance something needs to be said about the relationship between the strata. The levels of reality are characterised (and therefore distinguished) by their categories. With regard to the levels of reality, it is obvious that the categories in question are ontological rather than epistemological. There are categories that pertain to reality as whole - such as time, whole, part. At the inorganic level the categories of physics are different from those of chemistry. Some examples follow.

1. Inorganic stratum, with categories such as space, time, process, substance, is organised through causality.
2. Organic stratum, with categories such as metabolism, assimilation, self-regulation, self-reproduction and adaptation, is organically organised - the details of the organising principles are unknown for the time being.
3. The psychic stratum, with categories such as consciousness, pleasure, act and content, has its own psychic network of relations. Equally unknown is the inner essence of the form of determination of mental acts.
4. Spirit includes categories of fear, hope, will, freedom, thought, personality, but also society, historicity, or intersubjectivity. It is organised as three modes of spirit;
• Personal - thought, freedom, conscious action and moral will
• Objective - superindividual but living as culture, collective or society
• Objectivated - superindividually existing but non-living. As in cultural artifacts and institutions

All categories at the inorganic level, such as space and time, recur at the organic level.  This recurrence of categories is called superinformation. There is also autonomy off levels and this is known as superposition - building above. For example, consciousness, a category of the psyche, is not spatial.

In addition there are the “fundamental” categories (common for all strata of real being) that come in pairs: 

principle–concretum; structure–modus; form–matter; inner–outer; determination–dependence; quality–quantity; unity–manifoldness; harmony–conflict; contrast–dimension; discretion–continuity; substratum–relation; element–structure.

The above only provides a sketch. For more detail on the categories of real being see Cicovacki [1] who provides a bridge to the detailed work by Hartmann.

The basic structure can be summarised in a diagram.

Hartmann's model of reality includes mind and spirit superposed on an ontologically stronger foundation of life and matter. There is, however,  a feedback loop that is not explicit in the diagram. The objectivated spirit gives rise to artifacts, such as books and architecture, that are also actualised as things at the material level. For example, a person, James Joyce say, has some thoughts (personal spirit) about Dublin society (objective spirit) and creates a novel, Ulysses (objectivated spirit), that is published as book (still objectivated spirit) and printed in several editions (inorganic objects).

The being James Joyce was made of inorganic matter organised as an inorganic life form with a mind and emotional life on which was superposed a spirit that could choose to work for years on intricate works of literature. He would also often choose to drink heavily after a day's work. He loved his immediate family.

A further example. This blog exists as objectivated spirit. It came into being because of my choice and was enabled by the infrastructure supplied by Google. Making and maintaining the blog are choices made by me at the level of my personal spirit. The blog infrastructure is dependent on organised matter (inorganic level) the organisation of it is objectivated spirit because it was designed. The blog and its posts are captured and encoded in the blog infrastructure. The posts are instantiated on the screen of a reader's device. The readers use their sense organs and brain (organic level) to access the information that is processed through their psychic level and, hopefully, understood at the level of their personal spirit.  The blog readers and I form a loose community that exists as objective spirit.

Simple beings such as electrons may only inhabit the inorganic level but the existence of complex beings may be across many or all strata. This complex existence is a process rather than a hierarchy.

[1] Cicovacki, Predrag, The Analysis of Wonder. Bloomsbury USA. 2014

The combination of quantum objects

 Quantum objects have been introduced in an earlier post. These objects, if not the most fundament entities, are quite basic to the constitution of the inorganic layer of reality. In contributing to the complexity of the real world these objects must combine. 

Many calculations in quantum mechanics can be carried out by considering a single isolated system and its the operator representing the total energy of the system - the Hamiltonian. That is how the spectrum of the Hydrogen atom is calculated and how tunnelling is analysed. However, it is especially important in examining contextual issues to consider mathematical representation of the combination and interaction of objects. Having introduced the concept of the σ-complex for an isolated system, the σ-complex of two systems \(S_1\) and \(S_2\) will now be constructed.

As is standard in the von Neumann formulation [1], if the Hilbert spaces \(\mathcal{H}_1\) associated with object \(S_1\)  and \(\mathcal{H}_2\) associated with object \(S_2\), the Hilbert space of  the combined object \(S_1 + S_2\) is the tensor product \(\mathcal{H}_1 \otimes  \mathcal{H}_2\).

 Given the combined system $S_1 + S_2$ with the $\sigma$-complex  $Q(\mathcal{H}_1)\oplus Q(\mathcal{H}_2)$, there is a unique Hilbert space $\mathcal{H}_1\otimes \mathcal{H}_2$ such that $Q(\mathcal{H}_1) \oplus Q(\mathcal{H}_2) \cong Q(\mathcal{H}_1\otimes\mathcal{H}_2)$, where \(\cong \) means equivalence unique up to isomorphism. For the proof, in finite dimensional Hilbert spaces, see Kochen [2]. 

The combination of two quantum objects gives a quantum object. This would build up a world consisting only of quantum objects. Is this the case or do the quantum charactertics weaken with many combinations? We will return to this question.

[1] John von Neumann. Mathematische Grundlagen der Quantenmechanik. German. Springer, Berlin, Heidelberg, 1932

[2] Simon Kochen. A Reconstruction of Quantum Mechanics. In: ArXiv e-prints (June 2015).

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.

Quantum objects

Quantum mechanics is part of physics and experimentation is an activity in physics. Therefore, the entities of interest in physics must be observable to some extent. The must make an appearance. That is, entities of interest must be objects; they appear in at least one field of sense. 

The ontological framework for the discussion will be that quantum objects are physical objects that have some properties that can appear in one of a collection of fields of sense. To move from ontology to physical theory an explanatory structure is needed and, when possible, a mathematical formulation is adopted that describes the behaviour of the quantum objects in a precise way.  

There are puzzling and  paradoxal effects of quantum mechanics, be they the thought experiments of Einstein, Podolsky and Rosen (EPR) or Wigner's friend, Schrödinger's cat or the actual double slit experiment. These point to the need to re-examine what is meant by a physical property of an object.  In quantum mechanics it is known that all properties need a specific context in which to appear as quantified values. This is known to be the case in general, from the work of Kochen and Specker [1] but has been a common understanding among physicist based on the insights obtained from experiments such as that of Stern and Gerlach [2].  This experiment measures some consequences of the electron having spin. In quantum mechanics spin, while having the same dimensions as classical spin, is quite a different phenomenon and will be discussed below.

Before moving on to spin consider the Heisenberg Uncertainty Principle, which means an electron, for example, cannot have a precise value of its position and its momentum at the same time. Does this mean that it sometimes has position as a property and sometimes momentum but never both together? That understanding of the theory would mean that these properties are of the context rather than the object of the experiment. According to the ontology of objects and their properties being developed in this blog a particle, such as an electron, always has the properties of position and momentum but these properties take values by appearing in distinct Fields of Sense. This means that the context in which the particle position takes a value is different from that in which its momentum takes a value.

We now consider a particle with spin \(\frac{1}{2}\), an electron,  situated and free to move in three dimensions. Following Faddeev and Yakubovskii [3], the electron combines its spin Hilbert space (\(\mathbb{C}^2\)) with that associated with the electron's movement in three dimensions (\(\mathbb{R}^3\)). The Hilbert space \(\mathcal{H}\) for the electron's position and momentum is the space \(L^2(\mathbb{R}^3)\) of square-integrable complex-valued functions. This Hilbert space is the state space that can represent the electron in position coordinates (neglecting spin). In mathematical notation, the state space with spin becomes the enlarged Hilbert space

\[\mathcal{H}_S = L^2(\mathbb{R}^3)\otimes\mathbb{C}^2.\] This captures the intuition that the state function becomes a pair of complex valued \(L^2(\mathbb{R}^3)\) functions. For a pure state \({\Psi(x)}\) in \(\mathcal{H}\) there are two orthogonal states in \(\mathcal{H}_S\):

\[\Psi_{\uparrow} = \binom{\Psi(x)}{0} , \, \, \, \Psi_{\downarrow} =  \binom{0}{\Psi(x)}.\]

Focussing on the spin property, \(\hat{S}\), a self-adjoint operator \(S\) on \(\mathbb{C}^2\), representing \(\hat{S}\) can be represented as the linear combination of four linearly independent matrices. A convenient set of such matrices are the identity matrix and the well-known Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\)

\[I = \begin{pmatrix} 1 & 0\\0 & 1\end{pmatrix}, \sigma_1 = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix},\sigma_2 = \begin{pmatrix}0 & -i\\i & 0\end{pmatrix} \sigma_3 = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}\]

Quantum physics is brought into this mathematics by introducing the Plank constant, \(\hbar\), and defining the self-adjoint spin-\(\frac{1}{2}\) operators as  

\[ S_j = \frac{\hbar}{2} \sigma_j, j =  1, 2, 3 \label{eq:Sj} ~~~~~~~~~~~~~(1)\]

and these have the angular momentum commutation relations

\[[S_1,S_2]= iS_3, \,  \, [S_2,S_3]= i S_1, \,  \, [S_3,S_1]= iS_2.\]

The operators \(S_j\) have eigenvalues \(\pm \hbar/2\), which are the admissible values of the projections of the spin on some direction. The vectors \( \Psi_{\uparrow} = \binom{\Psi(x)}{0}\) and \(\Psi_{\downarrow} = \binom{0}{\Psi(x)}\) are eigenvectors of the operator \(S_3\) with eigenvalues \(+\hbar/2\) and \(-\hbar/2\). The eigenvectors of \(S_1\) and \(S_2\) are linear combinations of \(\Psi_{\uparrow}\) and \(\Psi_{\downarrow} \).  Therefore, these vectors describe states with a definite value of the third projection of the spin. Similar constructions can be made for the other spin components. But as the components do not commute, they cannot simultaneously take precise values in all components. The spin will appear to interact with some other object, involving a magnetic interaction, to take a value of \(+\hbar/2\) or \(-\hbar/2\). Independent of the direction defined by the interaction, the possible values the spin property can take are \(\pm \hbar/2 \), with probabilities determined by the state of the electron. The probabilities of \(\pm \hbar/2 \) appearing sum to one.  

The property values appear when the appropriate Field of Sense is available. However, in relation to this point it should be noted, that the isolated system that has properties represented by Hermitian operators, such as the spin-\(\frac{1}{2}\) \( S_j\) , equation (1), that have eigenvalues \(\pm \frac{1}{2}\hbar\) in the absence of any interaction term coupling it to another object. The spin values are quantitative characteristics of the spin property, but the values will only actually appear once the appropriate Field of Sense is available to provide a context for an interaction with another object.

A Field of Sense affords a context for the appearance of properties of the quantum object, whether, position, momentum or spin components.

[1] Simon Kochen and Ernst P. Specker. The Problem of Hidden Variables in Quantum Mechanics. In: J. Math. & Mech. 17 (1967), p. 59.

[2] W. Gerlach and O. Stern. Der experimentelle Nachweis der Richtungsquantelung. In: Zeitschrift für Physik, (1922), pp. 349-352.

[3] L D Faddeev and O A Yakubovskii. Lectures on Quantum Mechanics for Mathematics Students. Student Mathematical Library. 47 vols. American Mathematical Society, 2000.