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CRC Press eBooks

Measurement of an Electromagnetic Field

Michael B. Mensky

Published Oct 19, 2017• 0 citations
Publisher

Abstract

This chapter discusses quantum measurements of relativistic systems, such as quantum fields. The quantum theory of fields can be constructed with the help of a functional integral with integration over field configurations. It is natural that a description of measurements of quantum fields can be achieved by the restriction of such integrals. The main point is that different definitions of electromagnetic field strength lead to different conclusions about the existence or nonexistence of an absolute quantum restriction on the measurability of the field strength. To analyse the measurement process in quantum mechanics, one can use a simple model of the sensitive element of the instrument, sometimes called a 'meter'. The purely classical part of the instrument is not then explicitly considered. The coupling of the apparatus to the measured system is described by an interaction Lagrangian that is proportional to the product of the measured quantity and the coordinate of the meter.

DOI
10.1201/9781315140292-7

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CRC Press eBooks•Mar 5, 2018•0 citations
This chapter discusses quantum measurements of relativistic systems, such as quantum fields. The quantum theory of fields can be constructed with the help of a functional integral with integration over field configurations. It is natural that a description of measurements of quantum fields can be achieved by the restriction of such integrals. The main point is that different definitions of electromagnetic field strength lead to different conclusions about the existence or nonexistence of an absolute quantum restriction on the measurability of the field strength. To analyse the measurement process in quantum mechanics, one can use a simple model of the sensitive element of the instrument, sometimes called a 'meter'. The purely classical part of the instrument is not then explicitly considered. The coupling of the apparatus to the measured system is described by an interaction Lagrangian that is proportional to the product of the measured quantity and the coordinate of the meter.
The quantum mechanical measurement process is analysed in detail, especially from the group theoretical point of view. The first three chapters review the results obtained before briefly. In §1, the classical theory of measurement, originated by von Neumann, is birefly described. The importance of considering more general schemes of measurements than the original one is, however, emphasized. In §2, the role of conservation laws, in imposing limitations on the possibility of measurements, is discussed. Accurately measurable quantities are defined in this context. In §3, the role of conservation laws is further analysed. It is shown that the conserved quantity of the apparatus cannot be used for a “pointer reading.” The following two chapters discuss the nature of the exactly measurable quantities. In §4, exactly measurable quantities are explicitly determined. It is shown that the measurement of projection operators in the Hilbert space of the object which commute with the operators of the restricted Poincaré group give all the information on the states of the object that can be obtained by accurate measurements. In §5, the measurement on objects which are composed of more than one particle is discussed. The last chapter, §6, deals with approximately measurable quantities, but no complete estimate of the unavoidable inaccuracy of the measurement of these is obtained. It also contains some concluding remarks.
Measurements play an integral part in any scientific theory. Indeed, the observations revealing causal behaviour and symmetries in our world of experience lie at the heart of determining the truth or the falsity of scientific hypotheses. In this regard, quantum theory provides one of the most stringently tested and comprehensive description of the natural world. This thesis is an examination of the mathematical structures of quantum measurements with an emphasis on the measurement implementations of different operational devices: quantum observables, channels and instruments. The measurability of all the quantum properties a system bears is, in principle, guaranteed by quantum measurement theory. However, as physical processes, measurements are constrained by the laws of physics and it has been long known that certain conservation laws can inhibit the measurements of some of the quantum observables. In this work it is shown that such constraints stem from the fact that the measured observables inherit the symmetries of the measurement devices implementing them. A quantum measurement set-up can implement a multitude of operational devices as a function of the initial state of the apparatus. In other words, one can realize various maps – even incompatible ones – simply by controlling the apparatus’ state. Due its close resemblance with the classical analogue, this operation is often referred as quantum programming. In this thesis the capabilities and restrictions of quantum programming are explored using a measurement-theoretic approach. Moreover, quantitative methods are developed to address the mutual relations between the programming states of an arbitrary pair of observables or channels.
Quantum restrictions on the measurability of an electromagnetic field strength and their relevance to the energy-time uncertainty relation are considered. The minimum errors in measuring electromagnetic field strengths, as they were estimated by the author (1988) in the framework of the phenomenological method of restricted path integral (RPI), are compared with the analogous estimates found by Landau and Peierls (1931) and by Bohr and Rosenfeld (1933) with the help of certain measurement setups. RPI-based restrictions, including those of Landau and Peierls as a special case, hold for any measuring schemes meeting the strict definition of measurement. Their fundamental nature is confirmed by the fact that their associated field detectability condition has the form of the energy-time uncertainty relation. The weaker restrictions suggested by Bohr and Rosenfeld rely on an extended definition of measurement. The energy?time uncertainty relation, which is the condition for the electromagnetic field to be detectable, is applied to the analysis of how the near-field scanning microscope works.
This chapter considers a more realistic variation of the measuring procedure, taking into account the fact that the measured particles become entangled with the measuring apparatus at some finite time. The results are also generalized to quantum field theory, where quantum states are represented by wave functions depending on an infinite number of space-time coordinates. Thus, to make Bohmian mechanics covariant, one needs first to reformulate the standard quantum mechanics (QM) in a covariant way such that time is treated on an equal footing with space. Another physically interesting situation is when the entanglement with the environment takes the form, where are coherent states. In general, in the Bohmian interpretation all the other quantities may have a distribution totally different from those predicted by purely probabilistic QM. The space probability density should be generalized to the space-time probability density.
This chapter examines the reasons why measurement theory is a particularly controversial part of modern theoretical physics. The problem of measurement in quantum mechanics is considered as nonexistent or trivial by an impressive body of theoretical physicists and as presenting almost insurmountable difficulties by a somewhat lesser but steadily growing number of their colleagues. The law of evolution in time constitutes the essential part of the conventional measurement theory. In this theory, the existence of a Hamiltonian describing the instrument-system interaction is postulated, not derived. In other words, this assumption is considered as a legitimate idealization of the experimental facts. A wave packet reduction can be considered as a phenomenological procedure that, in many instances, can be applied in order to calculate the results of successive observations and the correlations between them without introducing into the formulas the complicated wave functions of the various instruments.