Review of Lewis, Peter J. Quantum Ontology: A Guide to the Metaphysics of Quantum Mechanics, 2016
Lewis presents an overview of quantum mechanics and phenomena related to the measurement problem: interference, entanglement, matrix and wave mechanics, and the interpretation of states. In Chapter 2 consideration is given to non-commuting observables and the EPR argument. The measurement problem is explained and GRW, Bohm’s theory and MWI are examined.
The book is very well written and the physics is interspersed with history. The explanations of many concepts are reasonably good for the general public. In this book, Lewis initially describes the measurement problem well:
.. if measuring devices are just physical systems obeying the linear Schrödinger dynamics, then the application of a measuring device to a system cannot cause an exception to the linear Schrödinger dynamics. Measuring devices must obey the Schrödinger dynamics, since every physical system does, but they must also violate the Schrödinger dynamics if they are to enact the measurement postulate. Quantum mechanics, understood as including the measurement postulate, is not just incomplete; it is inconsistent. (P.50 Kindle edition)
Although Lewis initially describes the problem well as quoted above, he does not appear to apply this description particularly well when analyzing Bohm’s theory. In fact, it appears to us that his understanding of the measurement problem is still similar to his definition of the problem previously given in a 2007 paper , which we refer to as the philosophers’ measurement problem:
The measurement problem, in a nutshell, is the problem that at the end of a measurement, there is nothing in standard quantum mechanics that represents the determinate outcome of the measurement—rather, every possible measurement outcome is represented in some branch of the final wavefunction. So if we take the standard theory seriously as a description of physical systems, then we have no explanation of the fact that measurements have outcomes. A minimal condition, then, for solving the measurement problem, then, is that a theory provides an explanation of our determinate measurement results.
This definition is significantly different from Lewis’s initial description given above in his book. Understanding the distinction between these two potential definitions is crucial toward a full understand of the quantum measurement problem. Lewis states in his book:
... the measurement problem does not arise for Bohm’s theory: The wave function always obeys the Schrödinger equation and never undergoes a collapse, and it is the particle positions rather than the wave function that generate the results of the measurement. (P. 57 Kindle edition)
which indicates that Lewis believes his initial description is equivalent to his definition given in his 2007 paper . While this is correct for the philosophers’ measurement problem, this is not correct for the physical measurement problem that we have defined in our book, which demands that a solution to the measurement problem provide the conditions under which a measurement occurs and the theoretical basis for such conditions. To see the problem, consider if a measurement device consisting of particles (as originally suggested by Lewis in his first definition) is included in the Bohmian analysis. That is, suppose one has a particle that splits into two paths and for which each path interacts with a bonafide measurement device. Now, after the interaction, consider another device that is used to test the state of the two devices. In the spirit of Bohm’s theory, a test particle could move to the left if the state of the two devices is entangled in the manner that is predicted by the Schrödinger equation (this state is shown in our book, in Chapter 3). The particle moves to the right for the subspace spanned by all states orthogonal to the predicted entangled state. Now, according to Bohm’s theory, the test particle will always be found to move to the left, because this is the prediction of Schrödinger’s equation, and the theory is set-up to agree with Schrödinger’s equation. However, from the measurement postulate, we know that the state of the two devices are in a product state. In such a case, the test particle could also move to the right. Hence in an actual experiment that happened to include a bonafide detector, Bohm’s theory will produce the wrong answer!
The measurement problem, as Lewis initially described it, “Quantum mechanics, understood as including the measurement postulate, is not just incomplete; it is inconsistent” is the correct deduction in this instance, and not that the measurement problem does not arise for Bohm’s theory. One might argue that the fact that the detector is bonafide needs to be included in the analysis of Bohm to get the correct answer. But then this can only be done given that we know what constitutes a measurement device and given that we know that a measurement will be done in a given basis. But then we have only reverted back to the main problem of the physical measurement problem, which is to determine the specific conditions under which a set of particles constitutes a bonafide measurement device. On this matter, both Bohm’s theory and Many-Worlds Theory are silent. It appears to us that the issues raised above lead Lewis later to several problematic statements, for example:
Bohr is almost certainly wrong when he says that quantum mechanics requires us to renounce causation at the microscopic level. Causation is a varied and flexible notion, and it is hard to imagine a physical theory that isn’t causal in any sense. All of the theories canvassed here provide causal explanations of quantum phenomena.
In terms of Bohr’s original model of an electron rotating around an atom with precise classical position and momentum, it is true that later Bohr renounced this causal model, and for good reason. Additionally, other than early in his career, nearly all of Bohr’s statements regarding the measurement process points to when there is an amplification of quantum phenomenon, and not at the microscopic level. Neither Bohm’s theory nor Many-Worlds Theory canvassed in his book provide any credible scientific resolution of the physical measurement problem. And as we have mentioned, this will give the incorrect result when a bonafide measurement device is included in the analysis. We strongly disagree that Bohr’s argument is “almost certainly wrong” and believe several of Lewis’s conclusions are based on application of the rather incomplete philosophers’ definition of the problem. We believe that when the full physical definition of the problem is considered, non-deterministic phenomenon is much more difficult to rule out.
 P. Lewis, How Bohm’s Theory Solves the Measurement Problem, Philosophy of Science, vol. 74, No. 5, December 2007