We consider theories in which the Schr\"odinger equation is modified so that the reduction of the state vector becomes a dynamical process characterized by a mean reduction time ${\ensuremath{\tau}}_{R}$. It is generally believed that such a theory and quantum theory differently predict the outcome of any two experiments performed in rapid enough succession (separated by a time interval $\ensuremath{\lesssim}{\ensuremath{\tau}}_{R}$), such as Papaliolios's successive measurements of a photon's polarization. It is shown that the predictions of a plausible class of dynamical reduction theories (for which the ensemble average of ${|〈{\ensuremath{\varphi}}_{n}|\ensuremath{\psi}〉|}^{2}$ remains constant during the reduction process, where $|\ensuremath{\psi}〉$ is the state vector, and $|{\ensuremath{\varphi}}_{n}〉$ is any one of the basis states to which it may reduce) and quantum theory do not differ for any Papaliolios-type experiment. However, there are deviations from the predictions of quantum theory if the second experiment measures interference between the superposed states created by the first measurement. As an example, the result of a recent two-slit neutron interference experiment by Zeilinger et al. is applied to the theory of Pearle, placing an upper limit ${\ensuremath{\tau}}_{R}\ensuremath{\gtrsim}5$ sec on the neutron self-reduction time.
We consider theories in which the Schr"odinger equation is modified so that the reduction of the state vector becomes a dynamical process characterized by a mean reduction time \ensuremathτR. It is generally believed that such a theory and quantum theory differently predict the outcome of any two experiments performed in rapid enough succession (separated by a time interval \ensuremath≲\ensuremathτR), such as Papaliolios's successive measurements of a photon's polarization. It is shown that the predictions of a plausible class of dynamical reduction theories (for which the ensemble average of ∣〈\ensuremathφn∣\ensuremathψ〉∣2 remains constant during the reduction process, where ∣\ensuremathψ〉 is the state vector, and ∣\ensuremathφn〉 is any one of the basis states to which it may reduce) and quantum theory do not differ for any Papaliolios-type experiment. However, there are deviations from the predictions of quantum theory if the second experiment measures interference between the superposed states created by the first measurement. As an example, the result of a recent two-slit neutron interference experiment by Zeilinger et al. is applied to the theory of Pearle, placing an upper limit \ensuremathτR\ensuremath≳5 sec on the neutron self-reduction time.
In theories which describe the reduction of the state vector as a physical process, the possibility exists, for certain experiments, of predictions which differ from those of quantum theory. These are ``interrupted reduction interference'' experiments, characterized by an interaction which triggers the reduction, followed rapidly (before the reduction is completed) by a measurement of interference between the superposed states that make up the state vector (possible examples: double Stern-Gerlach experiment, two-slit neutron interference). We consider the general class of stochastic reduction theories, and ask whether they allow superluminal communication by means of such experiments. We show that, if the state vector that precedes reduction is precisely reproducible, then superluminal communication can occur in certain circumstances, unless the off-diagonal elements of the density matrix decay exponentially, with a universal time constant. We also show, in that case, that no state vector ever reduces in a finite time, so such a theory is not satisfactory. However, superluminal communication can be avoided if reduction is triggered only in irreproducible state vectors, of such complexity that prior to reduction the ``effective'' density matrix, constructed from the ensemble of such state vectors and traced over the variables outside the experimenter's control, is diagonal. Then predictions are identical to those of quantum theory for ``interrupted reduction interference'' experiments and thus apparently for all experiments. The lesson of this paper is that the ``effective'' density matrix must always be used to make physical predictions in dynamical reduction theories. This supplies a resolution of the problem of reconciling state-vector reduction with relativity: even if the reduction dynamics is not relativistically invariant, its experimental predictions are. It also implies that the ``effective'' entropy increases during a measurement, but remains constant during reduction, which is the reverse of a common dictum.
Causal quantum theory is an umbrella term for ordinary quantum theory modified by two hypotheses: state vector reduction is a well-defined process, and strict local causality applies. The first of these holds in some versions of Copenhagen quantum theory and need not necessarily imply practically testable deviations from ordinary quantum theory. The second implies that measurement events which are spacelike separated have no nonlocal correlations. To test this prediction, which sharply differs from standard quantum theory, requires a precise definition of state vector reduction. Formally speaking, any precise version of causal quantum theory defines a local hidden variable theory. However, causal quantum theory is most naturally seen as a variant of standard quantum theory. For that reason it seems a more serious rival to standard quantum theory than local hidden variable models relying on the locality or detector efficiency loopholes. Some plausible versions of causal quantum theory are not refuted by any Bell experiments to date, nor is it evident that they are inconsistent with other experiments. They evade refutation via a neglected loophole in Bell experiments---the collapse locality loophole---which exists because of the possible time lag between a particle entering a measurement device and a collapse taking place. Fairly definitive tests of causal versus standard quantum theory could be made by observing entangled particles separated by $\ensuremath{\approx}0.1$ light seconds.
The reduction of the state vector is rightly held as one of the most puzzling aspects of quantum theory. Before adding a few more pages to the inflatory literature on the subject, I wish to voice a somewhat sobering remark. Namely, there is a striking contrast between the amount of theoretical discussions about the nature and significance of this notion, and its practical import. Indeed, the so-called “axiom” of reduction of the state vector has a rather limited use in the effective working of quantum theory. There are very few situations where we really need to assert that, after a measurement, the state vector has been reduced to the eigenvector of the physical property corresponding to the eigenvalue resulting from the measurement (“observed”). Most of the time, in effect, the system under investigation is forgotten or discarded after the measurement - or even destroyed by it (photons). We rarely make successive observations of the same system, so that we do not usually need to know what the state of a system is after a measurement. One might argue that the reduction of the state vector is operationally used mainly to describe the preparation of a system before acting upon it, that is, to ensure that we are indeed dealing with spin up electrons or velocity selected protons such as we wish to experiment about. Anyway, the statement of reduction is to be considered more as a kind of consistency condition to be required at the beginning or end of a theoretical analysis than as an effective tool of this analysis - such as, for instance, the statement of unitary evolution, which leads to the concrete construction of the evolution operator, hence the S-matrix, cross-sections, etc. In other words, the practical status of the state vector reduction, even if it is considered as an axiom of quantum theory, is different enough from that of the constructive axioms of the theory.
The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrodinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrodinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.
An experiment for checking the Dynamical Spacetime approach to wavefunction collapse is proposed. The Dynamical Spacetime approach predicts deviations from Born's rule, when a solid evolves into a three-state superposition, and when the displacement between the superposed states is, at the reduction point in time, significantly larger than the spatial variation of the solids nuclei, being typically on the order of a tenth of an Angstroem. The solid is brought into the three-state superposition by splitting a photon into three beams and by detecting it in each beam by avalanche photodiodes, which displace the solid at different distances with the help of a piezoactuator. The challenge of the experiment is the precise prediction of the setup's reduction point in time to ensure a sufficient separation between the states at this point in time. This is addressed by avoiding interactions of the setup with the environment during superposition, and by a precise calculation of the setup's reduction point in time with the help of a formulary for the Diosi-Penrose criterion for solids in quantum superpositions. Since the measurement of reduction probabilities is not disturbed by state decoherence, the experiment can be performed at room temperature. The quantitative analysis demonstrates that the predicted increase of the reduction probability of one state by a factor of 1.5 with respect to Born's rule can be measured by a few hundred statistically significant measurements.
An alternative calculational procedure for quantum theory is presented, employing three features which differ from the corresponding features of orthodox quantum theory. These are: (1) the state vector is never to be “reduced”; (2) quantum theory only makes predictions about ensembles of identical experiments (not about individual experiments); and (3) the measuring apparatus must always be included in the state vector. What is new about this alternative theory is the demonstration that Features 2 and 3 (which have hitherto been discussed separately) can be consistently combined to yield an interpretation of quantum theory which has the additional advantage of Feature 1. It is shown that alternative and orthodox theories yield identical predictions for any practical experiment. However, two classes of experiments (which are impractical) for which the orthodox and alternative theories yield different predictions are described. Therefore, it is, in principle, possible to decide which theory Nature has chosen to obey. It is concluded that there is no reason, based upon experimental evidence, for preferring one theory over the other.