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Physical Review

On the Measurement of Observables in Relativistic Quantum Mechanics

Otto Halpern, M. H. Johnson

Published Jun 1, 1941• 4 citations
Publisher

Abstract

The paper discusses the question how far observables in quantum theory permit accurate measurement if no assumptions are made about the values of conjugate variables. The physical connection between the uncertainty principle and the commutation relations is briefly discussed. A treatment of the $\ensuremath{\gamma}\ensuremath{-}\mathrm{r}\mathrm{a}\mathrm{y}$ microscope makes it appear that even with small-angle diffraction no great improvement beyond the Compton wave-length can be obtained. An analysis of an ideal arrangement to measure electric field strengths leads to the result that, admitting the existence of arbitrarily constituted test bodies, the accuracy of the measurement still cannot exceed certain limits which are mainly defined by the wave-length of the field and the spatial and temporal domain of measurement. These restrictions are due to the properties of the "vacuum," which are changed as a consequence of the possibility of pair production.

DOI
10.1103/physrev.59.896

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The paper discusses the question how far observables in quantum theory permit accurate measurement if no assumptions are made about the values of conjugate variables. The physical connection between the uncertainty principle and the commutation relations is briefly discussed. A treatment of the \ensuremathγ\ensuremath−ray\ensuremath{\gamma}\ensuremath{-}\mathrm{r}\mathrm{a}\mathrm{y}\ensuremathγ\ensuremath−ray microscope makes it appear that even with small-angle diffraction no great improvement beyond the Compton wave-length can be obtained. An analysis of an ideal arrangement to measure electric field strengths leads to the result that, admitting the existence of arbitrarily constituted test bodies, the accuracy of the measurement still cannot exceed certain limits which are mainly defined by the wave-length of the field and the spatial and temporal domain of measurement. These restrictions are due to the properties of the "vacuum," which are changed as a consequence of the possibility of pair production.
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 accuracies of measurements of almost all fundamental physical constants have increased by factors of about 10 000 during the past 60 years. Although some of the improvements are due to greater care, most are due to new techniques based on quantum mechanics. Although the Heisenberg Uncertainty Principle often limits measurement accuracies, in many cases the validity of quantum mechanics makes possible the vastly improved measurement accuracies. Seven quantum features that have a profound influence on the science of measurements are: (1) Existence of discrete quantum states of energy. (2) Energy conservation in transitions between two states. (3) Electromagnetic radiation of frequency v is quantized with energy hv per quantum. (4) The identity principle. (5) The Heisenberg Uncertainty Principle. (6) Addition of probability amplitudes (not probabilities). (7) Wave and coherent phase phenomena. Of these seven quantum features, only the Heisenberg Uncertainty Principle limits the accuracy of measurements, and its affect is often negligibly small. The other six features make possible much more accurate measurements of quantum systems than with almost all classical systems. These effects are discussed and illustrated.
The accuracies of measurements of almost all fundamental physical constants have increased by factors of about 10,000 during the past 60 years. Although some of the improvements are due to greater care, most are due to new techniques based on quantum mechanics. In popular accounts of quantum mechanics, such great emphases is placed on the Heisenberg Uncertainty Principle that it often appears that the primary effect of quantum mechanics should be to diminish measurement accuracy whereas in most cases it is the validity of quantum mechanics that makes possible the vastly improved measurement accuracies. Seven quantum features that have a profound influence on the science of measurements are: (1) Existence of discrete quantum states of energy Wi. (2) Energy conservation in transitions between two states. (3) Electromagnetic radiation of frequency ν is quantized with energy hν per quantum. (4) The identity principle. (5) The Heisenberg Uncertainty Principle. (6) Addition of probability amplitudes (not probabilities) so P=|ψ1 +ψ2 |2 ≠|ψ1|2 + |ψ2 |2. (7) Wave and coherent phase phenomena. Of these seven quantum features, only the Heisenberg Uncertainty Principle limits the accuracy of measurements, and its affect is often negligibly small. The other six features make possible much more accurate measurements of quantum systems than with almost all classical systems and the identity principle provides meaning and significance to highly precise measurements with quantized systems. These effects are discussed and illustrated.
We discuss two related issues pertaining to the foundations of relativistic quantum field theories. First: the question of observables; more particularly the question of what constraints, if any, are imposed upon the set of observables by the requirement of relativistic microcausality. It turns out that this requirement is in fact considerably less restrictive than has been supposed in the past. Second: the issue of physical states. We find that, in contrast to the nonrelativistic case, it is not possible to define the quantum state of a system in relativistic quantum field theories, because in this latter case no consistent description of how the state changes as the result of a measurement can be developed.
In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.
In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.