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Open Systems & Information Dynamics

Geometric Factors in Electromagnetic Field Commutators and the Quantum Information Control of Vacuum Fluctuations

Luis Gerardo Pedraza Saavedra

Published Sep 1, 2007• 0 citations
Publisher

Abstract

The electromagnetic field commutation relations are defined in terms of geometric factors that are double averages over two finite four-dimensional space-time regions. The square root of any of the uncertainty relations derived from the aforementioned commutators is taken as a critical field, in the sense that any electromagnetic field much larger than it can be treated as classical. Another critical electromagnetic field associated with the quantum information control of vacuum fluctuations can be chosen as the square root of the mean quadratic fluctuation of each quantity of electromagnetic field, when the number of photons is defined and is equal to zero. Any electromagnetic field expectation value could be measured if it is much greater than the last critical field. This article covers a magnitude order comparison between the critical fields and its consequences for measuring the electromagnetic field information.

DOI
10.1007/s11080-007-9055-1

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The electromagnetic field commutation relations are defined in terms of geometric factors that are double averages over two finite four-dimensional space-time regions. The square root of any of the uncertainty relations derived from the aforementioned commutators is taken as a critical field, in the sense that any electromagnetic field much larger than it can be treated as classical. Another critical electromagnetic field associated with the quantum information control of vacuum fluctuations can be chosen as the square root of the mean quadratic fluctuation of each quantity of electromagnetic field, when the number of photons is defined and is equal to zero. Any electromagnetic field expectation value could be measured if it is much greater than the last critical field. This article covers a magnitude order comparison between the critical fields and its consequences for measuring the electromagnetic field information.
Quantisation of the electromagnetic field leads us simplistically to think of the field as fluctuating around some average value that could be predicted by a classical theory, whereas all statistical properties of the fluctuations must be derived from a quantum theory. The character of the fluctuations depends on the quantum state of the field. Even in a vacuum, where the average value of the field is zero, its mean square value due to fluctuations is positive. This quantum noise associated with these fluctuations is inherent and cannot be removed no matter how carefully we get rid of various environmental disturbances and fluctuations. But even though the quantum noise is here to stay, we can still control and manipulate it.
The geometric factors in the field commutators and spring constants of the measurement devices in the famous analysis of the measurability of the electromagnetic field by Bohr and Rosenfeld are calculated using a Fourier-Bessel method for the evaluation of folding integrals, which enables one to obtain the general geometric factors as a Fourier-Bessel series. When the space regions over which the factors are defined are spherical, the Fourier-Bessel series terms are given by elementary functions, and using the standard Fourier-integral method of calculating folding integrals, the geometric factors can be evaluated in terms of manageable closed-form expressions.
Several authors have used the Heisenberg picture to show that the atomic transitions, the stability of the ground state and the position-momentum commutation relation [x,p]=ih, can only be explained by introducing radiation reaction and vacuum electromagnetic fluctuation forces. Here we consider the simple case of a nonrelativistic charged harmonic oscillator, in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrodinger picture. We consider the effects of both classical zero-point and thermal electromagnetic vacuum fields. We show that the zero-point electromagnetic fluctuations are dynamically related to the momentum operator p=-ih d/dx used in the Schrodinger picture. Consequently, the introduction of the zero-point electromagnetic fields in the vector potential A_x(t) used in the Schrodinger equation, generates ``double counting'', as was shown recently by A.J. Faria et al. (Physics Letters A 305 (2002) 322). We explain, in details, how to avoid the ``double counting'' by introducing only the radiation reaction and the thermal electromagnetic fields into the Schrodinger equation.
We treat three standard problems of the nonrelativistic quantum theory by a Brownian-motion method devised by Sokolov and Tumanov. In all three cases, the commutation (Poisson bracket) relations for the particle's dynamical variables are deduced from the corresponding relations for the components of the fluctuating electromagnetic field. In addition, the particle ground-state energy is determined by the zero-temperature vacuum fluctuations of the field. The ground-state energy includes the lowest-order radiative correction, although the calculation makes no use of traditional perturbation theory or mass renormalization. If the vacuum radiation temperature is high, the energy of the particle goes to its equipartition value.
In this chapter, the authors prove that the free electromagnetic field is equivalent to an infinite set of uncoupled one-dimensional harmonic oscillators, one oscillator being associated with each value of the propagation vector k and polarization mode α. In order to quantize the electromagnetic field the authors shall have to express the field equations in Hamiltonian form and impose the canonical commutation rules on the canonically conjugate variables. Because the electromagnetic field possesses an infinite number of degrees of freedom it is also possible to formulate the theory in terms of a very different set of dynamical variables, the field operators themselves. To be more precise, the field operators only have off-diagonal matrix elements in the representation in which the photon occupation numbers are diagonal. In a state with a definite number of photons the expectation value of any field strength vanishes, and it is only meaningful to discuss fluctuations in the field strengths.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.