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Physics-Uspekhi

Measurability of quantum fields and the energy-time uncertainty relation

Michael B. Mensky

Published May 31, 2011• 8 citations
Publisher

Abstract

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.

DOI
10.3367/ufne.0181.201105f.0543

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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 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.
It is shown in two examples that to measure the momentum of a particle with great accuracy, allowing for large uncertainties in the position, it is still necessary to consider additional quantum effects. For the case of measurements by electromagnetic means, the uncertainties in the measurability of field strengths and of charge-current densities should be considered. Within the framework of the Bohr-Rosenfeld analysis, uncertainties in the field measurements do not limit the accuracy of momentum measurements. What happens when the Bohr-Rosenfeld approximations are not made is an open question.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.