In Chapter 2, we will review basic facts of quantum measurement that are usually discussed in basic texts on quantum mechanics. These include a motivating experiment – the Stern–Gerlach effect – and discussions of measurement results, statistics, the Born rule, and wavefunction collapse.
This tome is a formal presentation of the unsharp observable approach to quantum mechanics using the positive operator valued (POV) concept of an observable. It is intended for philosophers and mathematicians as well as physicists. This is a very formalistic book. There are, however, portions that should be read by all experimentalists performing quantum mechanical studies as well as graduate students completing their first graduate year quantum mechanics course. If we interpret quantum mechanics as making statistical predictions on the results of states prepared in identical fashions, we will often find that the observed probabilities are nearly, but not exactly, the expected 1 or 0 for measurements that are expected to always give the same result. Hence, rather than demanding sharp measures with probabilities of exactly 1 or 0, quantum mechanics allows a reasonable interpretation of unsharp results of nearly 1 or nearly 0. As the authors point out in chapter II, POV measures and the associated unsharp measures are not essential for an interpretation of quantum mechanical results. However, more sharp variables, and hence a larger Hilbert space, will be needed than are required by the unsharp approach. Also, the theoretical interpretation will often not be as straightforward with only sharp observables. As I have previously stated, all graduate students, both those of a theoretical and those of an experimental bent, should read parts of this book (after they have completed a normal introduction using sharp variables). They should at least read the first chapter which explains the unsharp variable measure concept as well as giving some examples, one of which is the Stern - Gerlach experiment. In chapter VII, the last chapter, the Stern - Gerlach experiment photon polarization and other quantum optical experiments and wave-particle duality experiments for photons are examined in detail using the POV formalism. This would also be an excellent chapter for all graduate students to read. Some sections may send them back to some of the intermediate chapters, hopefully to peruse in greater detail. The intermediate chapters contain the formal development of the POV formalism and compare it to the normal method. There is also a brief introduction to measurement theory using unsharp observables. As the authors state, this is not intended to be a complete solution to the problem of measurement theory. The next two chapters deal with uncertainty and phase space formalism as considered by the unsharp observables. Philosophers of physics and mathematicians will find this an excellent introduction to the POV formalism and the distinction between sharp and unsharp measurements. Many texts will gloss over both the philosophical and mathematical consequences inherent in the full statistical formalism of quantum mechanics. This monograph is not, by the authors' admission, a full treatment of quantum measurement theory. Also, although in the introduction the authors discuss the importance of applying quantum mechanics to individual objects, this book mainly considers the interpretations as statistical measures. There is little attempt to deal with the quantum mechanical interpretations of the behaviour of a single particle. That is, questions about measurements on a single particle or the interpretation of the wavefunction for a single particle are not fully covered. The authors claim that the POV approach is also the best for individual systems, but do little in applications. The major example given is in the last chapter where they consider interference measurements for single photons. They might have referenced more of the current alternative approaches to measurement theory of single particle wavefunctions. The readers of this journal will be disappointed, but not surprised, that the covariant nature of observables is restricted almost entirely to the Galilei group. The only Poincare group material is the example of photon localization in the last chapter. This is a reflection of the great difficulty of applying the special or the general theories of relativity to the Hilbert space formalism.
This paper proposes a geometric selection framework for quantum measurement that reframes collapse as a contextual, basin-dependent selection process rather than a universal instantaneous postulate. Beginning with the Stern–Gerlach experiment, measurement outcomes are interpreted as symmetry-constrained selections in an internal SU(2) state space, recovering standard spin statistics without invoking hidden classical variables. The framework is extended to sequential measurements and interferometric configurations through the introduction of an explicit, computable collapse-rate functional. This functional interpolates continuously between unitary evolution and effective projection, depending on both branch distinguishability and a phenomenological measure of measurement-basin accessibility. The Born rule is recovered exactly at the moment of selection, ensuring full compatibility with standard quantum mechanics in appropriate limits. The model predicts small, testable deviations from standard quantum mechanics in near-threshold interferometric regimes, where coherence is only partially suppressed. These deviations are distinct from those predicted by objective-collapse models with universal rates and can be constrained using existing and near-term matter-wave interferometry platforms. The approach provides a minimal, empirically grounded extension of measurement theory that remains falsifiable and compatible with current experimental bounds.
The basic thesis of this book is that there is no quantum measurement problem. I do not mean by this that the apparent paradoxes of quantum mechanics arise because we fail to recognize ‘that quantum theory does not describe physical reality’ (Fuchs and Peres 2000a). Quantum theory describes reality just fine, like any other scientific theory worth taking seriously: describing (and explaining) reality is what the scientific enterprise is about. Nor do I mean that we have found solutions to the problem, ways to modify or supplement quantum mechanics so that the paradoxes are removed. I don’t actually think that we have found any very satisfactory solutions in this sense (at least not in the relativistic domain) but that isn’t my theme: even if we did have these modifications, they would be ways of changing from quantum physics, which supposedly does have a measurement problem, to some other theory, which supposedly doesn’t.
The Stern–Gerlach (SG) experiment (1922) was one of the earliest experiments for which the result could not be explained using classical physics by any means. Hence it demonstrated unambiguously the importance of quantum effects. The SG experiment can be explained using the quantum theory for the spin-½ particle, which is a quantum system that can be represented simply by 2 × 2 matrices. We will use the spin-½ particle to introduce some of the basic concepts of quantum mechanics, such as state space, operators, measurement, the uncertainty principle, and the evolution equation. Furthermore, the theory for the spin-½ particle can be readily generalized to finite dimensional quantum systems, i.e., systems that can be represented by finite dimensional matrices.
Since this book is closely connected to quantum mechanics, these introductory words will briefly and informally recall some postulates of this theory. Interested readers are urged to consult traditional books on quantum mechanics to complement the discussion ahead and for descriptions of experimental evidence that give rise to the postulates. In physics there are several “equivalent formulations” of quantum mechanics, and a catalogue of the most important appears in [StyO2]. The discussion here is restricted to the nonrelativistic case.
In this paper I present a thought experiment that gives different results depending on whether or not the wavefunction collapses. Since the wavefunction does not obey the Schrodinger equation during the collapse, conservation laws are violated. This is the reason why the results are different. Quantities that are conserved if the wavefunction does not collapse might change if it does. I also show that using the Born Rule to derive probabilities of states before a measurement given the state after it (rather than the other way round as it is usually used) leads to the conclusion that the memories that an observer has about making measurements of quantum systems have a significant probability of being false memories.