THE PUZZLE CONCERNING THE CONCEPT OF PROBABILITY IN SM Throughout the book we have been preoccupied with a single simple scenario. We consider now an n -particle system M that is in a state s t at t . We want to know whether s t has a certain property A . The question that we are asking has a perfectly clear physical meaning. Nevertheless, the information that the answer requires may not always be available to us. It may be the case that the experiment needed to answer the question is too expensive or difficult to perform. It is also possible that there are theoretical reasons why we cannot perform the experiment, or, simply, the experiment has not been performed yet. In all of these cases the need for probabilities arises. We need to calculate the probability P ( s t ∈ A ) and use our results to derive predictions concerning M . But, even if there is little doubt that we constantly use probabilities, there is a basic problem concerning the interpretation of the probabilities. It is difficult to define probabilities as observable parameters that depend only on the physical properties of M . This difficulty gives rise to a philosophical puzzle: If probabilities are not physical parameters that we discover by methods of observation and measurement only, how can we justify our willingness to be guided by them? How can we explain the utility of probabilities? In the previous chapters, we reviewed several attempts to solve this problem.
THE PUZZLE CONCERNING THE CONCEPT OF PROBABILITY IN SM Throughout the book we have been preoccupied with a single simple scenario. We consider now an n -particle system M that is in a state s t at t . We want to know whether s t has a certain property A . The question that we are asking has a perfectly clear physical meaning. Nevertheless, the information that the answer requires may not always be available to us. It may be the case that the experiment needed to answer the question is too expensive or difficult to perform. It is also possible that there are theoretical reasons why we cannot perform the experiment, or, simply, the experiment has not been performed yet. In all of these cases the need for probabilities arises. We need to calculate the probability P ( s t ∈ A ) and use our results to derive predictions concerning M . But, even if there is little doubt that we constantly use probabilities, there is a basic problem concerning the interpretation of the probabilities. It is difficult to define probabilities as observable parameters that depend only on the physical properties of M . This difficulty gives rise to a philosophical puzzle: If probabilities are not physical parameters that we discover by methods of observation and measurement only, how can we justify our willingness to be guided by them? How can we explain the utility of probabilities? In the previous chapters, we reviewed several attempts to solve this problem.
I interpreted the repeated measurement experiments of the previous chapter by saying that quantum mechanics can find probabilities only, not certainties (that is, that quantum mechanics is “probabilistic”, not “deterministic”). You may object, maintaining that the world is deterministic, but that my particular deterministic scheme (the “magnetic arrow”) is incorrect. The next chapter presents an ingenious argument, invented by Einstein, which shows that no local deterministic scheme could give the results observed by experiment. In order to understand that argument you need some background in probability. But in fact, a knowledge of probability is generally useful in day-to-day life as well as in physics. You walk across a street — what is the probability of your being hit by a car? You are advised to undergo elective surgery — what is the probability that the surgery will extend your life, and what is the probability that the surgery will go wrong and injure you? You breath some asbestos or smoke a cigarette — what is the probability of contracting cancer? Misconceptions about probability abound and can lead to disastrous public policy decisions. A knowledge of quantum mechanics is good for your soul, but it is of practical importance only to the designers of lasers, transistors, and superconductors. A knowledge of probability is of practical importance to everyone.
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.
Questions and Answers What Is Physics?Throughout this book we shall be concerned greatly with questions and answers.The subject that an observer asks a question of will be denoted by a bold symbol such as Ψ.On the other hand, the question that is asked of a subject will be denoted by a barred bold symbol such as Q.Then the answer A that is obtained when that question is asked of that subject will be denoted by A ≡ QΨ.This notation will be used to explain what quantized detector networks (QDN) is all about and in its mathematical implementation.Note that the left-right ordering here is significant: ΨQ will mean something very different.If I, the author, were asked the question Q ≡ What is physics?, my answer QI would be QI = Physics is the process whereby observers first formulate, then ask, and then answer, empirical questions.We need to clarify some points about this answer.By empirical, we mean according to the established principles of science, whereby theorists make propositions or assertions and then establish the truth values of those propositions relative to experiments carried out by unbiased experimentalists.This is done to acquire information about systems under observation (SUOs) in order to add to our knowledge about the relative physical Universe (that part of the Universe empirically accessible to us).We are motivated to do all this for numerous reasons, not the least of which is basic curiosity.This is the view of physics taken in this book.Physics is emphatically not a discipline that philosophers and metaphysicists can meaningfully contribute to, because without empirical testing, any proposition has to be regarded as scientifically vacuous (empirically meaningless).The questions relevant to physicists are posed through experiments and consequently will be referred to as empirical questions.Other sorts of questions will be referred to as theoretical questions.
The previous chapter addressed the challenge of making sense of the notion of a quantum state assignment being performed correctly without thereby acknowledging the notion of a quantum state a quantum system ‘is in’. The present chapter discusses what interpretation of quantum probabilities goes best with the account that I have called the ‘Rule Perspective’ and started to develop in the previous chapter. The first section considers what quantum probabilities should be taken to be probabilities of in the Rule Perspective; the second investigates what philosophical interpretation of probabilities applies best to quantum probabilities thus conceived. The third section considers two objections to the resulting account of quantum probabilities. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Motivated by some recent news, a journalist asks a group of physicists: "What's the meaning of the violation of Bell's inequality?" One physicist answers: "It means that non-locality is an established fact". Another says: "There is no non-locality; the message is that measurement outcomes are irreducibly random". A third one says: "It cannot be answered simply on purely physical grounds, the answer requires an act of metaphysical judgement". Puzzled by the answers, the journalist keeps asking questions about quantum theory: "What is teleported in quantum teleportation?" "How does a quantum computer really work?" Shockingly, for each of these questions, the journalist obtains a variety of answers which, in many cases, are mutually exclusive. At the end of the day, the journalist asks: "How do you plan to make progress if, after 90 years of quantum theory, you still don't know what it means? How can you possibly identify the physical principles of quantum theory or expand quantum theory into gravity if you don't agree on what quantum theory is about?" Here we argue that it is becoming urgent to solve this too long lasting problem. For that, we point out that the interpretations of quantum theory are, essentially, of two types and that these two types are so radically different that there must be experiments that, when analyzed outside the framework of quantum theory, lead to different empirically testable predictions. Arguably, even if these experiments do not end the discussion, they will add new elements to the list of strange properties that some interpretations must have, therefore they will indirectly support those interpretations that do not need to have all these strange properties.