The impact of quantum theory Till quantum mechanics came along, it would not have occurred to many that the structure of a physical theory itself may be constrained by limitations on the precision of measurements. In classical mechanics, dynamical variables were implicitly assumed to be precisely measurable at all times, from which it followed that any mathematically well-defined function of the dynamical variables was precisely measurable at any time. Quantum mechanics severed the ‘natural’ link between the two concepts. Two observables, well-defined at all times, would be simultaneously measurable only if they commuted with each other. However, a single observable could always be measured precisely. Although the details are well known, it may be worth recalling that while it was the uncertainty principle that unsettled the theorists, what the experimentalists were unearthing were effects of the superposition principle , with measurement errors that far exceeded the uncertainty principle constraints. What may be less well known (except to quantum field theorists) is that relativistic quantum field theory demands a severance of the link between being well defined and being measurable at yet another level. The classical electromagnetic field is well defined – and therefore measurable – at every point of space-time. However, no component of the quantized electric or magnetic field is precisely measurable at any point. The last fact was discovered by Landau and Peierls, but it remained for Bohr to point the way to the correct conclusion to be drawn.
The impact of quantum theory Till quantum mechanics came along, it would not have occurred to many that the structure of a physical theory itself may be constrained by limitations on the precision of measurements. In classical mechanics, dynamical variables were implicitly assumed to be precisely measurable at all times, from which it followed that any mathematically well-defined function of the dynamical variables was precisely measurable at any time. Quantum mechanics severed the ‘natural’ link between the two concepts. Two observables, well-defined at all times, would be simultaneously measurable only if they commuted with each other. However, a single observable could always be measured precisely. Although the details are well known, it may be worth recalling that while it was the uncertainty principle that unsettled the theorists, what the experimentalists were unearthing were effects of the superposition principle , with measurement errors that far exceeded the uncertainty principle constraints. What may be less well known (except to quantum field theorists) is that relativistic quantum field theory demands a severance of the link between being well defined and being measurable at yet another level. The classical electromagnetic field is well defined – and therefore measurable – at every point of space-time. However, no component of the quantized electric or magnetic field is precisely measurable at any point. The last fact was discovered by Landau and Peierls, but it remained for Bohr to point the way to the correct conclusion to be drawn.
In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of "hidden" variables, developed in a previous paper. We find that in principle, these "hidden" variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.Finally, we show that von Neumann's proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann's assumptions.In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.
Quantum mechanical uncertainty After reviewing the main characteristics of the classical description of light, let us discuss those aspects of the quantization of the electromagnetic field which are of relevance for the analysis of the phenomena we are interested in. It seems a reasonable place to start to make clear the fundamental difference between the classical and the quantum mechanical description of nature; we will come across this difference many times when discussing experiments, and it will often give us a headache. We have to deal with the physical meaning of what is called uncertainty. The starting point of the classical description is the conviction that natural processes have a “factual” character. This means that physical variables such as the position or momentum of a particle have, in each single case, a well defined (in general, time dependent) value. However, it will not always be possible to measure all the appropriate variables (for instance, the instantaneous electric field strength of a radiation field); furthermore under normal circumstances we are able to measure only with a finite precision. Hence the basic credo of classical physics should be given in the following form: we are justified in imagining a world with variables possessing well defined values which are not known precisely (or not known at all). In doing this we are not forming any conclusions that contradict our everyday experiences. This is the fundamental concept of classical statistics: we are satisfied with the use of probability distributions for the variables we are interested in, not from fundamental but purely practical reasons.
THE PROBLEM Within classical physics the relation between physical theory and ordinary experience was conceived in the following way: ordinary experience is something that can be described and understood in terms of physics. Such a description involves, apart from physical theory, certain approximations. But the conditons under which those approximations apply (they correspond to the initial conditions of, say, celestial mechanics), taken together with the theory, are supposed to be sufficient for giving a full account of ordinary experience. Consequently, classical theory of measurement (which, like any other theory of measurement, links together terms of ordinary experience and theoretical terms), is a piece of applied physics and all processes which happen during measurement can be analysed on the basis of the equations of motion only. When we enter quantum mechanics (QM), we are apparently presented with a completely different picture. For according to the current interpretation of elementary quantum mechanics, ordinary experience – and this now means classical physics – and physical theory (and this now means QM), belong to two completely different levels and it is impossible to give an account of the first in terms of the second. Any transition from the quantum level to the level of classical physics must be taken, not as a transition within QM from the general to the particular, but as an essentially new element which is incapable of further analysis.
Abstract We first present a summary of the quantization of the electromagnetic field in position space representation, using two main approaches: the Landau-Peierls approach in the Coulomb gauge and the Białynicki-Birula (BB) approach, based on the Riemann–Silberstein vector. We describe both in a framework that starts with a classical Hamiltonian structure and builds the quantum model in a bosonic Fock space by a precisely defined principle of correspondence. We show that the two approaches are completely equivalent. This is formulated by showing that there is a unitary map between the Fock spaces that makes them isomorphic. Since all the physically measurable quantities can be expressed in terms of scalar products, this implies that the two quantizations lead to exactly the same physical properties. We show furthemore that the isomorphism is preserved in the time evolutions. To show the equivalence, we use the concepts of helicity and frequency operators. The combination of these two operators provides a formulation that allows one to make the link between these two methods of quantization in a precise way. We also show that the construction in the BB quantization that avoids the presence of negative eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation for electrons and positrons, can be performed through an alternative choice of the canonical variables for Maxwell’s equations.
Introduction and overview Much that is studied in quantum measurement theory, and thus in this book, is very different from that studied in its classical counterpart, Bayesian inference. While quantum measurement theory is in one sense a minimal extension of the latter to quantum states, quantum measurements cause dynamical changes that never need appear in a classical theory. This has a multitude of ramifications, many of which you will find in this book. One such consequence is a limit to the information that can be obtained about the state of a quantum system, and this results in relationships between information gained and dynamics induced that do not exist in classical systems. Because of the limits to information extraction, choosing the right kinds of measurements becomes important in a way that it never was for classical systems. Finding the best measurements for a given purpose, and working out how to realize these measurements, is often nontrivial. The relationship between information and disturbance impacts the control of quantum systems using measurements and feedback, and we discuss this application in Chapter 5. The ability to extract information about quantum systems is also important in using quantum systems to make precise measurements of classical quantities such as time, acceleration, and the strengths of electric and magnetic fields. This subject is referred to as quantum metrology, and we discuss it in Chapter 6.
Measurement in classical physics Measurements in classical physics are the means by which we come to know the current state of a mechanical system without appreciably disturbing it. The laws of classical mechanics imply that we may interact with the system in such a way that this ideal may be approached arbitrarily closely, at least in principle (i.e., in theory, ignoring practical problems). The aim of this chapter is to examine how far we must revise this programme in the quantum theory of measurement. Did the classical physicist believe in the objective existence of material systems with well-defined position and momentum just because these quantities could be empirically determined with arbitrary precision at the same time? It seems she made a stronger assumption than this; that two successive observations of an apple, first in a tree and then on the ground, are connected to one another by virtue of the fact that the apple actually has a definite position and momentum at each instant, whether it is observed or not. Thus, the programme of classical physics was not merely to predict the results of future experiments by inserting data from previous ones into the laws of motion, but to show how the observations are a function of a causally connected sequence of unique processes undergone by the system of interest, which possesses properties independently of measurements. Measurement was then something of a secondary affair and indeed, prior to quantum mechanics, the theory of measurement as a distinct branch of theoretical physics did not exist.