Review of S. Gao, The Meaning of the Wave Function: In Search of the Ontology of Quantum Mechanics, 2017.
Gao initially reviews the mathematical formalism and the Born rule. Gao then examines whether the wave function is real and complete or whether it represents an epistemic state about the system. He argues that Aharonov’s protective measurement provides a direct experience with the wavefunction, as the entire wavefunction can be measured when it is a nondegenerate eigenstate of the Hamiltonian. He argues in Chapter 3 that this supports a version of the reality of the wave function in terms of a property of the system. He reviews Bohm’s theory and derives the Schrödinger’s equation assuming linear time evolution and using spacetime translational invariance and the nonrelativistic version of Einstein’s energy-momentum relationship. The ontology of quantum mechanics is examined, with emphasis on the charge distribution of a single electron. Analysis is done using protective measurement for which he argues that a quantum system has a well-defined charge distribution in space, in exactly the same sense that a classical system has a well-defined charge distribution in space. In Chapter 8 Gao suggests a reformulation of the measurement problem in terms of a conflict between 1) the mental state of an observer supervenes on the wave function and 2) the evolution of the wavefunction via Schrödinger’s equation. He then examines Bohm’s theory and Everett’s theory in light of this new formulation of the mental state of an observer. Gao presents his own ideas relating to the collapse theories in terms of the mental states of observers. As well, he presents a collapse theory based on the eigenstates of the Hamiltonian. Later in the book, the issue of the lack of a Lorentz invariant formulation of collapse is reviewed and whether or not a preferred frame can be detected.
Gao’s book presents several new ways of looking at the wave function and quantum mechanics. His use of protective measurement to aid in philosophical issues regarding the wavefunction is interesting and novel. His derivation of Schrödinger’s equation using spacetime invariance assuming linear state evolution and the energy-momentum relationship appears to be correct though the relation between spacetime invariance and Schrödinger’s equation has been extensively discussed previously in the literature . Gao utilizes protective measurement to argue that a quantum system has a well-defined charge distribution in space, in exactly the same sense that a classical system has a well-defined charge distribution in space. This has not been obvious to us in all situations, and if Gao is correct, a rather interesting result. He seems to understand that a physical solution to the measurement problem is needed rather than an interpretation, but does not make this as clear in his book as we feel is needed. The book goes into some detail in putting forward a solution for a role of conscious observers, which is a sufficient condition for measurement. It seems to be reasonable as a source of inspiration for those looking toward consciousness as a possible explanation of measurement. Gao goes out on a limb to propose his own ideas in this regard, which on one hand is a strength, but on the other hand can be a weakness if one treads too far into the realm of speculation.
Gao looks at the possibility that the mental state of an observer intervenes with certain branches of the wave function. While interesting, more background here would have been useful, including a full explanation of the Albert-Loewer single-mind theory. As well, the argument as explained in [2, p. 189] that two observers in the Albert-Loewer single-mind theory looking at a quantum phenomenon would not be able to properly communicate what they see. This leads apparently to Gao’s desire to have mental states intervening with physical states.
Gao then examines the possibility of a mental state intervening with physical states, which he calls psychophysical supervenience. He considers Everett’s theory and Bohm’s theory in terms of the mind intervening in wave function evolution. He finds that one will meet some serious difficulties if assuming the mental state of an observer supervenes either on certain branches of the observer's wave function or on other additional variables.
Gao does seem to understand that the measurement problem cannot be solved by an interpretation, but he does not appear to have identified the specific problems with either Bohm’s theory or Many-Worlds Interpretation, for which a physical theory (such as GRW) that differs from an interpretation is needed to explain measurement. Gao also does not fully define the measurement problem, except quoting Maudlin’s formulation as being an incompatibility of:
(C1). the wave function of a physical system is a complete description of the system;
(C2). the wave function always evolves in accord with a linear dynamical equation, e.g. the Schrödinger equation;
(C3). each measurement has a definite result (which is one of the possible measurement results whose probability distribution satisfies the Born rule)
However, (C1), (C2), and (C3) do not constitute the physical measurement problem as defined in our book . Gao then states that Maudlin’s formulation must be supplemented to include the experience of conscious observers:
It has been realized that the measurement problem in fact has two levels: the physical level and the mental level, and it is essentially the determinate-experience problem (Barrett, 1999, 2005). The problem is not only to explain how the linear dynamics can be compatible with the existence of definite measurement results obtained by physical devices, but also, and more importantly, to explain how the linear dynamics can be compatible with the existence of definite experience of conscious observers.
In terms of collapse theories, he states, “collapse theories are still plagued by a few serious problems such as the tails problem (Albert and Loewer, 1996).” Moreover:
I will give a few speculations here. I conjecture that the mental content of an observer being in a post-measurement superposition like (8.2) is composed of the mental content corresponding to every branch of the superposition, and in particular, the modulus squared of the amplitude of each branch determines the vividness of the mental content corresponding to the branch (Gao, 2016). Under this assumption, when the modulus squared of the amplitude of a branch is close to zero, the mental content corresponding to the branch will be the least vivid.
While we agree that consciousness is a sufficient condition for measurement, it is not known whether or not consciousness is a necessary condition. The precise quantum physics of consciousness we expect will need to be developed from the ground-up slowly in a methodical manner. Gao continues:
Under this assumption, when the modulus squared of the amplitude of a branch is close to zero, the mental content corresponding to the branch will be the least vivid. It is conceivable that below a certain threshold of vividness an ordinary observer or even an ideal observer will not be consciously aware of the corresponding mental content. Then even though in collapse theories the post-measurement state of an observer is still a superposition of different outcome branches with similar structure, the observer can only be consciously aware of the mental content corresponding to the branch with very high amplitude, and the mental content corresponding to the branches with very low amplitude will not appear in the whole mental content of the observer. This may solve the structured tails problem of collapse theories.
Proposing such a detailed solution and contending “this may solve the structured tails problem of collapse theories” is perhaps within the realm of metaphysical possibility, but is hardly scientific certitude. It very may well be that consciousness is related to measurement, but little would currently suggest to us that the fine details are correct in Gao’s description.
In Sec. 8.4 Gao presents his own model of wave function reduction:
In this subsection, I will present a concrete model of energy-conserved wavefunction collapse based on the above analysis (see also Gao, 2013a). I assume that this superposition of energy eigenstates will collapse to one of the eigenstates after a discrete dynamical process, and the collapse evolution satisfies the conservation of energy at the ensemble level...In the end, the probability of one branch will be close to one, and the probabilities of other branches will be close to zero. In other words, the initial superposition will randomly collapse to one of the energy branches in the superposition respectively.
The dynamics of the model appears to have each superposition term either remain (or removed) with probability of the squared coefficient corresponding to each term at every Planck time. Hence in the limit of a large number of Planck times, the superposition will settle on one of the eigenstates of the Hamiltonian with probabilities that correspond to the Born rule. There is a parameter “k” that is defined that can control the rate of the collapse and so the mechanism could be tested. This later model is a reasonably clear-cut mathematical attempt that is easily analyzed. His model is a physical model unlike an interpretation and hence addresses the physical measurement problem that we have defined, which is a positive feature of it. An issue is that Gao’s model conserves energy only on-average and not on every-trial during measurement. Hence during measurement occurrences, one could expect to see jumps of energy, i.e. energy is not strictly being conserved. This was also a major issue behind the argument of Bohr-Kramers-Slater (BKS), and Bohr conceded to Einstein that this was incorrect after experiments were conducted by Compton & Simon and Bothe & Geiger in the 1920’s and later confirmed by others in the 1950’s. Interestingly this issue is discussed in our book, and has been raised anew by Aharonov, and we recommend further experiments to test this. However, we predict in our book that energy and momentum are both conserved on the ensemble level and also on every trial in the measurement process. Hence if we are correct, then Gao’s model is wrong. And, we do expect that any new experiments will confirm the early experiments, and no experiments that we are aware of since have reported a violation of every-trial energy or momentum. On the other hand, violations could be argued to occur during very short times such as in Gao’s model at Planck times, for which specialized experimentation would be needed.
 M. Hamermesh, Galilean Invariance and the Schrödinger Equation, Annals of Physics 9, 518-521 (1960); J.M. Levy-Leblond, Galilei Group and Nonrelativistic Quantum Mechanics, J. Math. Phys. 4, 776 (1963).
 J. Barrett, The Quantum Mechanics of Minds and Worlds, Oxford University Press, 1999.
 M. Steiner, R. Rendell, The Quantum Measurement Problem, Inspire Institute, 2018