The double slit experiment

Having discussed the issues with quantum measurement in general, and shown that standard interpretation of quantum mechanics is incomplete, Young's double slit experiment with electrons will be discussed in two variants (for background see Chapter 1 of Hall [1]):

• The standard configuration, to be described below, Figure (1).
• A configuration with a pointer that acts behind the slits to point in the direction of the passing electron, Figure (2). 

Figure (1) The setup for the double slit experiment is shown. An electron source sends one particle at a time toward the screen with the slits. The slits are marked by \(\delta_1\) and \(\delta_2\) and a sample region \(\Delta\) is shown on the detector screen.

Standard configuration

The experiment, Figure (1), is as follows:

• There is a source of electrons that move towards a screen with two slits.
• The intensity of the beam is low and only one electron is moving towards the detector at any time.
• The slits are marked $\delta_1$ and $\delta_2$
• \(\Delta\) be some arbitrary region on the electron detector.

Let $Y_1$ and $Y_2$ be the projection operators for position in the regions of the two slits $\delta_1$ and $\delta_2$. Then $Y_1 + Y_2$ is the projection of position for the union $\delta_1 \cup \delta_2.$ Let $X$ be the operator for position in a local region $\Delta$ on the detection screen. Assume the electron is only constrained to pass through the slits without being constrained as to which, then under those conditions the conditional probability is given by the Law of Alternatives:

\[\begin{eqnarray}
p(X|Y_1 + Y_2) &=& p(X|Y_1)p(Y_1|Y_1 + Y_2) + p(X|Y_2)p(Y_2|Y_1 + Y_2)\nonumber \\
& &+ [\textbf{tr}(Y_1 \rho Y_2 X)+\textbf{tr}(Y_2 \rho Y_1 X)]/ \textbf{tr}(\rho (Y_1+ Y_2)).
\end{eqnarray} \, \, \, \, \, \, \, (1)\]

This can be written more compactly as
\[
p(X|Y_1 + Y_2) = p(X|Y_1)p(Y_1|Y_1 + Y_2) + p(X|Y_2)p(Y_2|Y_1 + Y_2) + p(X| Y_1+Y_2 )_I  \, \, \, \, (2)
\]
where \( p(X| Y_1+Y_2 )_I\) is the interference term. 

Note that if $X$ commutes with either $Y_1$ or $Y_2$, this interference term vanishes, because \(Y_i Y_j = 0\) for \( i \ne j \) . This can be observed if the detector is right next to the two-slit screen because \(X\) then coincides with either \(Y_1\) or \(Y_2\). If the detector is a distance from the two-slit screen (e.g. \(X= \Delta\)), then the state of the electron evolves unitarily via  $\alpha_t,$ so $\alpha_t(Y_i)=u_t Y_i u^{-1}_t$ no longer commutes with $\alpha(X)$, giving rise to the non-zero interference term.

 

The standard explanation of the interference effect is that the state of the particle is, or acts as, a coherent pair of waves emanating from the slits, which exhibit constructive and destructive interference effects. This was, of course, the explanation for Young's original experiment with light. For individual quantum particles, however, there is the unexplained local event observed at the detection screen.  This is a problem for the theory. While the Born interpretation of the wavefunction provides a probability distribution for the particle position it requires the detection screen, operating outside what is described by the mathematical theory, to act as a sampling mechanism for that distribution.

The explanation given in this blog is that the two-slit screen functions as a preparation of the state for the particle, by which the state is conditioned, or reduced, to pass through the region $\delta_1\cup\delta_2 $. This reduction is not a position measurement, since $\delta_1\cup\delta_2$ is not a localised region (as it would be for a single-slit screen). Once the particle reaches a detection screen then, in interaction with the screen, it appears in a random local region $\Delta$ and its position takes a value. Just as in the standard Born interpretation of the wavefunction, it is not explained in the theory how the electron takes the value that the detector detects other than invoking random sampling of the possible values.

So, the interference pattern on the detector is built up over time as more electrons arrive and are sampled by the detector. 

The introduction of an interaction with a pointer

This section is adapted from Bricmont [2], Appendix 5.A and Maudlin [3].

The experiment, illustrated in Figure (2), is now as follows

• There is again a source of electrons that move towards a screen with two slits. 
• The intensity of the beam is low and only one electron is moving towards the detector at any time.
• The slits are marked $\delta_1$ and $\delta_2$.
• \(\Delta\) be some arbitrary region on the electron detector.
• A pointer \(P\) is introduced. It is a quantum object with three states neutral, \(P_0\), points to slit \(1\), \(P_1\) and points to slit \(2\), \(P_2\). The interaction with the electron causes the pointer to move towards it.

Figure (2) The setup for the double slit experiment is as in Figure (1) but for the addition of a three state pointer that interacts with the electron as it passes through slit \(\delta_1\) or \(\delta_2\).

Again let $Y_1$ and $Y_2$ be the projection operators for position in the regions of the two slits $\delta_1$ and $\delta_2$. Then $Y_1 + Y_2$ is the projection of position for the union $\delta_1 \cup \delta_2.$ Let $X$ be the operator for position in a local region $\Delta$ on the detection screen. The operator representing the pointer has three eigenstates and therefore a three-dimensional Hilbert space \(\mathcal{H}_P\). Without the pointer the Hilbert space is \(\mathcal{H}_0\). The Hilbert space of the total system is \(\mathcal{H}_P \otimes \mathcal{H}_0\). The total system consists of a single electron and a pointer constrained by the screen with the two slits, and the detector.

The possible constituent states are: 

• \(\phi_1\) be the state of the pointer pointing towards the slit \(\delta_1\)
• \(\phi_2\) be the state of the pointer pointing towards the slit \(\delta_2\)
• \(\phi_0\) be the state of the pointer pointing in the neutral direction \(P_0\)
• \(\psi_1\) be the state of the electron passing through slit \(\delta_1\)
• \(\psi_2\) be the state of the electron passing through slit \(\delta_2\)
• \(\Psi_0\) be the state of the electron with the pointer in the neutral position \(P_0\).

where \(\phi_0\), \(\phi_1\) and \(\phi_2\) are eigenstates and therefore orthogonal. This is not the case for \(\psi_1\) and \(\psi_2\). 

Assuming the pointer starts in its neutral state, the initial wave function is
\[\begin{eqnarray}
\Psi_0 &=& \phi_0 \otimes (\psi_1 +\psi_2) \nonumber\\
&=&\phi_0 \otimes \psi_1 + \phi_0 \otimes \psi_2
\end{eqnarray}\]
Two treatments of the situation will now be discussed. In the first, the electron carries its charge through either the \(\delta_1\) or \(\delta_2\) and the pointer reacts and in the second the charge is not constrained to pass through only one slit at a time. The first treatment would be consistent with the ontology of Bohmian mechanics or stochastic mechanics. The second would be consistent with the electron with its charge passing through both slits or not physically existing at all at that point in the experiment. This is consistent with the ontology proposed by Bell [4] for the formulation of quantum mechanics proposed by GRW [5]. In their ontology there can be a local event only with extremely low probability in a run of the experiment. 

Treatment I: The pointer reacts to which slit the electron passes through

Here the situation is idealised to assume that the pointer reacts perfectly to the electron going through either slit 1 or slit 2. This is not a measurement because the reaction is neither registered nor signalled. At no point does anyone know which slit the electron has passed through.

Time unitary evolution in quantum mechanics is linear, therefore \(\Psi_0\) evolves to

\[
\Psi = \phi_1 \otimes \psi_1 + \phi_2 \otimes \psi_2.
\]
Inserting this for the state into equation (1), and using the notation for the interference term in equation (2), gives
\[
p(X| Y_1+Y_2 )_I=\frac{\textbf{tr}(Y_1 \rho_\Psi Y_2 X)+\textbf{tr} (Y_2 \rho_\Psi Y_1 X)}{\textbf{tr} (\rho_\Psi (Y_1+ Y_2 ))} 
\]
\[
p(X| Y_1+Y_2 )_I= \frac{\mathfrak{N}}{\mathfrak{D}},
\]
where
\[
\mathfrak{N} =(\phi_2 \otimes \psi_2 ,P \otimes X \phi_1 \otimes \psi_1) +(\phi_1 \otimes \psi_1, P \otimes X \phi_2 \otimes \psi_2 )
\]
\[
\mathfrak{D}=(\phi_1 \otimes \psi_1 , \phi_1 \otimes \psi_1)+(\phi_2 \otimes \psi_2, \phi_1 \otimes \psi_1)
\]
\[
+(\phi_1 \otimes \psi_1, \phi_2 \otimes \psi_2) +
(\phi_2 \otimes \psi_2, \phi_2 \otimes \psi_2)
\]
Using that the states of pointer are orthogonal
\[
p(X| Y_1+Y_2 )_I= 0
\]

The quantum interference term disappears. \(p(X| Y_1+Y_2 )\) is just a combination of the pattern for each slit on its own. So, even though no measurement is registered the presence of the pointer and its interaction with the electron is enough to eliminate the interference pattern. This is often explained (by Feynman [6] for example) by the electron being watched to determine which slit the electron passes through. The pointer is reacting to but not determining the outcome. The interference pattern disappears due to what is known as entanglement, not measurement.

Treatment II: The pointer does not react to which slit the electron passes through

In this treatment the assumption that the total charge is carried through only one of the two slits is not made or if it does the pointer cannot unambiguously react to it. This leads to a more general linear combination of the possibilities. Generally, the \(\Psi\) evolves to
\[\Psi = \sum_{i \in \{0,1,2\}} a_i \phi_i \otimes \psi_1 + \sum_{i \in \{0,1,2\}}b_i \phi_i \otimes \psi_2.
\]
\[
a_0=b_0, a_1 = b_2, a_2=b_1.
\]
The pattern to be observed on the detection screen in this treatment would now be 
\[
(\Psi, P \otimes X \Psi) = (\sum_{i \in \{0,1,2\}} a_i \phi_i \otimes \psi_1 + \sum_{i \in \{0,1,2\}}b_i \phi_i \otimes \psi_2, \]

\[
 P \otimes X [\sum_{i \in \{0,1,2\}} a_i \phi_i \otimes \psi_1 + \sum_{i \in \{0,1,2\}}b_i \phi_i \otimes \psi_2]).
\]
Using the orthogonality of the pointer states,
\[
(\Psi, P \otimes X \Psi) =\sum_{i \in \{0,1,2\}}|a_i|^2 (\psi_1, X \psi_1) + \sum_{i \in \{0,1,2\}}|a_i|^2 (\psi_2, X \psi_2)+\]
\[\sum_{i \in \{0,1,2\}} a^*_1 a_2 (\psi_2, X \psi_1) + \sum_{i \in \{0,1,2\}}a^*_2 a_1 (\psi_1, X \psi_2)
\]
where superscript \(*\) denotes the complex conjugate.
Using \(C= \sum_{i \in \{0,1,2\}}|a_i|^2\) to simplify to 
\[
(\Psi, P \otimes X \Psi) = C( (\psi_1, X \psi_1) + (\psi_2, X \psi_2))+
 \mathfrak{Re}\{2 a^*_1 a_2 (\psi_2, X \psi_1)\}.
\]

So, the interference pattern (\(\mathfrak{Re}\{2 a^*_1 a_2 (\psi_2, X \psi_1\}\)) persists. This behaviour is consistent with a physical situation where no charged particle exists in the region of the slits, as in the Bell ontology for the GRW collapse theory.

Experimental tests and ontological comparisons

The setup with the pointer, as described above, is an idealisation. This pointer is a quantum object that will react reliably to a passing charge particle but with no registration of the direction pointed. If there is no passing charged particle, then there would be nothing to react to. 

It is conceivable that the pointer could be realised by a molecule with an appropriate electrical dipole moment that can be fixed in position immediately behind the screen, between the two slits, but free to rotate. Maudlin [3] discusses the setup with a reacting proton trapped between the slits. Any practical experiment would implement the pointer in a way that would inevitably deviate for the ideal. This could lead to a situation where the interference pattern is weakened but not destroyed.

If quantum theories are constructed to be empirically equivalent but with distinctly different ontological models, then a discussion of how credible these ontological models are within different scenarios can provide a valid critical comparison. The result in Treatment I is consistent with an ontology in which the electron carries its charge on one continuous trajectory, such as in Bohmian mechanics or Nelson's stochastic mechanics. That is, each electron exists in the region of only one of the slits. Then the presence of a pointer reacting to the charge but not measuring it would be sufficient to destroy the interference pattern. This would give support to

• A Bohm or Nelson type theory in which the electron follows a continuous trajectory through the experimental setup. The trajectory is deterministic in the case of Bohm but stochastic in the case of Nelson.
• A quantum chance theory. The local appearance of the electron as a dispositional property that appears as a value locally in the region of only one of the slits due to the interaction of the electron with the pointer. However, the theory does not as it stands describe how this appearance is made actual. It would be an assumption that the pointer acts to sample the distribution.

By contrast, if pointer shows no reaction, as in Treatment II, then that would undermine the explanatory force of the Bohm or Nelson ontologies and indicate a quantum world in which one or more of the following is the case:

• A registering measurement is needed to destroy the interference pattern. This could be called the Copenhagen point of view.
• The charge is spread across possible positions (although this would have to be split equally across the two slits to give no pointer reaction)
• The charged particle may not actually exist in the region of pointer. Although the Bell ontology for GRW could be thought of as a mechanism for locally actualising the charge, the mechanism that they propose does not occur frequently enough to produce the effect in this experiment.
• The proposal for a theory of quantum chance in which dispositional property of the electron to appear at a locality does not entail the actual appearance due to the interaction with the pointer.

Treatments I and II both assume a behaviour of the pointer. There is a full mathematical formalism that would, in principle, provide the answer to whether the theory predicts that the interference pattern persists or disappears once the electron point interaction is included in the Hamiltonian of the system shown in Figure (2). 

The theory put forward in this blog is open to either the outcome where the pointer reacts and to that in which it does not. This is because it provides no quasi-classical insight into what the pointer or the electron may do. The quantum chance theory provides transition probabilities that must be calculated from first principles. They correspond to dispositional powers that do not appear as such in any Field of Sense. This is because the dispositional properties, although existing in the real sphere, only provide an effect when an interaction and context affords a Field of Sense, and the form of the appearance depends on the details of the interaction. What is clear is that there is no local appearance of the electron to which the pointer can react. The physical details of the pointer interaction with the quantum state my give rise to an actual local appearance of the electron but it may not. A treatment using the total system Hamiltonian will have no mechanism to break the symmetry between the two slits and so cannot be expected to eliminate the interference pattern.

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

[2] Bricmont, J., Quantum Sense and Nonsense, Springer Nature, 2017

[3] Maudlin, T., (2019) Philosophy of Physics: Quantum Theory, Princeton University Press., 2019

[4] Bell, J. S., Are there quantum jumps?, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge University Press, 2004, 201-212

[5] Ghirardi, G. C., Rimini, A. & Weber, T., Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D, American Physical Society, 1986, 34, 470-491

[6] Feynman, R., The Feynman Lectures on Physics, Volume III Quantum Mechanics, 

California Institute of Technology, 2013

The so-called measurement problem in quantum mechanics

 In some following posts specific experimental situations will be discussed. To prepare for these it is appropriate to start with a general discussion of measurement in quantum mechanics.       

Adapted from Kochen [1].                        

 The Measurement Problem refers to a postulate in standard quantum mechanics, which assumes that an isolated system undergoes unitary evolution via Schrödinger's equation and then an eigenvalue of the operator representing the observable being measured (an observable is a property of the system that the experimental setup is designed to measure) is randomly selected as the result of the measurement, as presented by Bohm [2], for example. However, if a property $\hat{A}$ of a system $S$ is measured by an apparatus $T$, the total system $S+T$, if assumed to be isolated, then undergoes unitary evolution. The random selection of an eigenvalue is an additional mechanism.

The mathematical formulation of an ideal measurement, in standard quantum mechanics, is as follows for system \(S\) in a pure state \(\phi_k\):

• Take the spectral decomposition of an operator representing an observable to be $A =\sum_i a_i \pi_{i}$.
• Each $\pi_{i}$ is a one-dimensional projection with eigenstate $\phi_{i}$ and \(\{a_i\}_i\) is the set of eigenvalues. 
• The apparatus $T$  is assumed to be sensitive to the different eigenstates of $A$. 
• Hence, if the initial state of $S$ is $\phi_{k}$ and the apparatus $T$ is in a neutral state $\psi_0$, so that the state of $S+T$ is $\phi_{k}\otimes \psi_0$
• The system evolves into the state $\phi_{k}\otimes \psi_k$, where the $\{\psi_i\}_i$ are the states of the apparatus operator corresponding to the states $\{\phi_{i}\}_i$ of the system, \(S\). 
• \(T\) and its interaction with \(S\) will have been chosen to achieve this 
• a perfectly designed measurement apparatus to be in \(\psi_l\) whenever \(S\) is in \(\phi_{l}\) for all \(l\). 

This all looks reasonable, and the key assumption is that the measuring apparatus does what it is supposed to. But now, for the case of a more general initial state, $\phi=\sum_i c_i \phi_{i}$:

• By linearity, if $S$ is in the initial state $\phi=\sum_i c_i \phi_{i}$, then 
• $S+T$ evolves into the state $\Gamma=\sum_i c_i \phi_{i} \otimes\psi_i$.  

A problem with this for standard quantum mechanics is that the completed measurement gives a particular apparatus state $\psi_k$, say,  indicating that the state of $S$ is $\phi_{k}$, so that the state of the total system is $\phi_{k}\otimes \psi_k$, in contradiction to the derived evolved state  $\sum_i c_i \phi_{i} \otimes \psi_i$. This evolution does not describe what happens in an experiment.

In contrast, the reduction can also be considered from the viewpoint of the conditioning of the states. If the state $p$ of $S+T$ just prior to measurement is $\rho_\Gamma$, corresponding to $\Gamma=\sum_i c_i \phi_{i} \otimes \psi_i$ then after the measurement it is in the conditioned state, by equation~(**) in the post Quantum chance, 

\[\begin{eqnarray}
p(\cdot |(\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k}))&\nonumber\\
=&\frac{(\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k}) \rho_\Gamma (\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k})}{ \textbf{tr}((\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k}) \rho_\Gamma )}\nonumber\\
=&\frac{(\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k}) (\sum_i c_i \phi_{i} \otimes \psi_i) (\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k})}{ \textbf{tr}((\pi_{\phi_k} \otimes I)( I \otimes \pi_{\psi_k}) (\sum_i c_i \phi_{i} \otimes \psi_i) )}\nonumber\\
=&\pi_{\phi_k \otimes \psi_k}. \nonumber
\end{eqnarray}\] 

Hence, the new conditioned state of $S+T$ is the reduced state $\phi_{k} \otimes \psi_k$ This is not surprising as it is conditioned on being in just the \(k\)th state of \(\rho_\Gamma\) and so it projects that element out.  This is not a resolution of the measurement problem but merely makes use of the probabilistic formulation of the theory to show conditioning forces the state reduction.

 Whereas standard quantum mechanics must add a means to reconcile the unitary evolution of $S+T$ with the measured reduced states of $S$ and $T$, the interpretation argued for in this paper take the opposite approach to the orthodox interpretation. The point of departure is not the unitary development of an isolated system, but rather the result of an interaction.  It is the conditions under which dynamical evolution occurs that must be further investigated, rather than the additional reduced state mechanism. Therefore, it should not be taken for granted, as assumed in standard quantum mechanics, that an isolated system evolves unitarily.  The question to be addressed is whether in a measurement the $\sigma$-complex structure of $S+T$ undergoes a symmetry transformation at separate times of the process. This is formalised as the condition for the existence of a representation $\alpha:\mathbb{R}\to {\mathop{\rm Aut}\nolimits} (Q)$. The outcome of a measurement cannot be given by a unitary process.

A completed measurement or a state preparation has two distinct elements of $Q(\mathcal{H}) (=Q(\mathcal{H}_S \otimes \mathcal{H}_T))$ at initial time 0 which end up being mapped to the same element at a later time $t$. One such element is an initial state \(\phi \otimes \psi_0\) results in a state \(\phi_k \otimes \psi_k\), for some $k$. However, a second such element \(\phi_k \otimes \psi_0\) also results in the state \(\phi_k \otimes \psi_k\). If the state $\phi$ is chosen to be distinct from $\phi_{k}$, then the two elements \(\pi_{\phi \otimes\psi_0}\)  and \(\pi_{\phi_k \otimes\psi_0}\) of $Q(\mathcal{H})$ both map to the same element \(\pi_{\phi_k \otimes\psi_k}\).  However, any automorphism $\alpha_t$ is a one-to-one map on $Q$, so the measurement process cannot be described by a representation $\alpha:\mathbb{R}\to {\mathop{\rm Aut}\nolimits} (Q)$, and hence a  \textit{unitary evolution cannot explain what is observed}.     

The Measurement Problem must be resolved by a theory that includes state reduction in its dynamics in addition to periods of unitary evolution. The GRW theory provides an example of a partial mechanism for this. Partial because it only reduces the wavefunction to one that is more localised rather than full transition from possibility to actuality.  Bohmian mechanics avoids this by proposing a particle trajectory dynamics that requires no more state reduction than in classical probability.  In Bohmian mechanics the particle always has an actual position.                                                                     

For a composite system it should not only be outside forces that can break symmetry, but internal interactions. In the state \(\Gamma=\sum_i c_i \phi_i \otimes \psi_i\) introduced above the total, but still isolated, system \(S + T\) has a set of \(i\) property values associated with the states \(\{\phi_i \otimes \psi_i\}_i \). However, the interacting object  \(T\) as part of the system \(S + T\) will have the state \(\phi_{k}\)  of \(S\) appear with probability \(| c_k |^2 \). This provides a matrix mechanics interpretation of reduction as a physical transition probability for the system \(S\) in the presence of the apparatus \(T\).  State reduction does take place in isolated compound systems with internal interactions and the reduction of the state is due to the combined system's properties but traceable to the dispositional power to take specific property values associated with \(S\).  

In an experiment the results are recorded at the time of the experiment. This experimental recording is not part of the formal theory. The theory provides transition probabilities but nothing to time the transition.

[1] Kochen, S., A Reconstruction of Quantum Mechanics, ArXiv e-prints, 2015

[2] Bohm, Arno, Quantum Mechanics, Springer, 2001

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.