Quantum Dynamics

  The purpose of this section is to show that the Quantum Chance formulation is mathematically equivalent to standard quantum mechanics in both the Schrödinger and Heisenberg pictures.

Now that it has been shown that the symmetries of \(Q(\mathcal{H})\) are also implemented by symmetries of \(\mathcal{H}\), time symmetry is used to introduce a dynamics for quantum systems. To define dynamical evolution, consider systems that are invariant under time translation. For such systems, there is no absolute time, only time differences. The change from time \(0\) to time \(t\) is given by the symmetry transform \(\alpha_t : Q \rightarrow Q\), since the structure of the system of property values is indistinguishable at two values of time. If the state evolves first for a time \(t\) and then the resulting state for a time \(t' \), then this yields the same result as the original state evolving for a time \(t + t'\). It is assumed that a small time-period results in slight changes in the probability of property values occurring.

  The passage of time is represented by a continuous additive group $\mathbb{R}$ of real numbers into the group ${\mathop{\rm Aut}\nolimits} (Q)$ of automorphisms of $Q$ under composition. That is, a map $\alpha :\mathbb{R} \to {\mathop{\rm Aut}\nolimits} (Q)$, such that

 \[ \alpha_{t+t' }= \alpha_t \circ \ \alpha_{t'} \]

and the state $p_{\alpha_t}(x) $ is a continuous function of $t$. 

 The image of $\alpha$ is then a continuous one-parameter group of automorphisms on $Q$.\footnote{The group ${\mathop{\rm Aut}\nolimits} (Q)$ may be taken to be a topological group by defining, for each $\epsilon>0$,  an $\epsilon$-neighbourhood of the identity to be $\{ \alpha\mid |p_\alpha(x)- p(x)| < \epsilon$  for all $x$ and $p \}$. It is possible to directly speak of the continuity of the map $\alpha$, in place of the condition that $p_{\alpha_t}(x)$ is continuous in $t$.}

It has been shown that an automorphism $\alpha$ corresponds to a unitary operator. Therefore, the time evolution of the state $p_{\alpha_t}$ corresponds to that of the density operator $\rho_t = u_t  \rho u_t^{-1}$ and by Stone's Theorem (see Hall [1], section, 10.15.)

 \[u_t = e^{-\frac{i}{\hbar}  Ht},\] 

 where $\hbar$ is a the reduced Plank constant, the value of which is determined by experiment and \(H\) is a self-adjoint operator with units of energy; so

 \[  \rho_t = e^{-\frac{i}{\hbar} Ht} \rho \ e^{\frac{i}{\hbar} Ht}.\]

 Differentiating both side by time, \(t\),          

 \[  \partial_t \rho_t = -\frac{i}{\hbar} [ H, \rho_t ].\]

 This is the Liouville-von Neumann Equation and by correspondence with the classical Liouville Equation, \(H\) is the Hamiltonian of the quantum object.

  Conversely, this equation yields a continuous representation of $\mathbb{R}$ into ${\mathop{\rm Aut}\nolimits} (Q(\mathcal{H}))$.

  For $\rho = \Pi_\psi $, a pure state, $\rho_t  = \Pi_{\psi(t)}$ and this equation reduces to the Schr\"{o}dinger Equation:

 \[ \partial_t \psi(t) =-\frac{i}{\hbar} H \psi(t).\]                                    

This equivalence with standard quantum mechanics shows that the theory of quantum chance has the same predictive power. What the formulation of in terms of quantum chance does is make clear is that although the time evolution of probabilities of potential values of quantum properties is well defined, the actualisation of the values is not covered by the theory.

[1] Hall, B. C., Quantum Theory for Mathematicians, Springer, 2013

Modal categories and quantum chance

 The motivation for examining modal categories is to provide a more refined account of the ontology of dispositions and chance in quantum mechanics. Therefore, the emphasis will be on the real sphere although the modes apply also in the ideal sphere but in a way appropriate to that sphere. The modes of being are [1]

Necessity Not being able to be otherwise

Actuality Being this way and not otherwise

Possibility Being able to be one way or not

Contingency Not being necessary (also being able to be different)

Non actuality Not being so

Impossibility Not able to be so.

Intuitively necessity is more than actuality, actuality is more than possibility. This provides sense of direction because the lower mode is contained in the higher: what is actual must at least be possible, and what is necessary must at least be actual.  

With the negative modes impossibility is a minimum of being, extreme non-being.  Contingency is but the non-existence of necessity. The definiteness of the way of being is less even in non-actuality than in contingency, and even less in impossibility than in non-actuality?

 Of the three negative modes it is contingency that will be the most significant for our needs in quantum mechanics because it is a border case with a trace of positivity and provides a minimum opening to being-so. From the discussion of spin in the post on Quantum Objects it is contingent (not necessary) for a spin component to take only one of two values \(\pm \hbar/2 \). 

The language of uncertainty is inappropriate in the case of a world described by standard quantum mechanics. The values are not merely hidden but have no actual being until they come into existence.  Standard quantum mechanics does not provide a mechanism to describe the transition from contingency to actuality. It goes no further than to describe possibilities and the probability of them becoming actual.

In a world described by Bohmian Mechanics, the spin wavefunction guides the electron to take one of the two values after following a determined trajectory. For every exact starting point of the electron the end spin state is determined. It is only due to the practical impossibility of determining the starting point that makes the spin value seem contingent. It can be said to be epistemically contingent but not in reality. In the exactly specified situation the value that become actual become so necessarily.

If the world is as describe by Ghirardi, Rimini and Weber theory with the wavefunction still providing probabilities of outcome, the ontological situation is like standard quantum mechanics. This may seem puzzling because this theory is intended to dispel the mystery of quantum measurement (or, more generally, actuality), but even after the spontaneous wavepacket reduction, given by the GWR mechanism, some contingency remains or if the spin component value is actualised due to the spontaneous reduction nothing is explained by it. It is merely posited.  There is distinct version of this theory in which the wavefunction provides a mass density of the particle rather than a probability density. In typical situations the mass density will spontaneously reduce to a situation consistent with either \(+\hbar/2\) or \(-\hbar/2\). 

In the "may-worlds" description, the world is infinitely larger than can be observed. Only one of an infinite set of sub-worlds is observable. The spin component takes both values on two sub-worlds that are empirically separate although through corelation its known that if in one the spin takes value \(+\hbar/2\) then in the other it is \(-\hbar/2\). In this description of the world all values are actualised by necessity.

There are at least five descriptions of the world, including standard quantum mechanics, which are modally distinct in the real sphere.

Quantum chance

In an earlier post the mathematical formulation of quantum theory was presented in a way that brought probabilities rather than wavefunctions to the fore. In discussions of the meaning of probability the term chance is usually used when the probability does not describe a state of uncertainty or partial belief but objective random events. Before the event occurs the complete description of the state of the system is described by a probability measure.

Using the ontological terms adopted in this blog the probability measure over the \(\sigma\)-complex of the quantum system, an object in the inorganic level of real being, describes the objective chance that its properties, in the same inorganic level and represented by self-adjoint operators in the theory, will take one of a spectrum of possibilities that will become actual in that same inorganic level of reality.

The probability measure describes a contingent mode of being for the quantum system with a spectrum of valued that are possible and become actual. What is missing is an understanding of the timing of the actualisation. In all the versions of quantum theory considered so far time behaves the same way as in classical physics.

The timing of the actualisation of the possible values of the observable and the physical cause (at the inorganic level of reality) of this actualisation is the major open problem. 

The analysis of events in quantum tunnelling may provide some insight into this problem.

[1]   Hartmann, N. (2010) Möglichkeit und Wirklichkeit. 3rd edn. De Gruyter. Available at: https://www.perlego.com/book/1159964/mglichkeit-und-wirklichkeit-pdf (Accessed: 16 August 2022).

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).