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,
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
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