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Foundations of Physics

The Radiation Field, at the Origin of the Quantum Canonical Operators

Ana Marı́a Cetto, Luis de la Peña

Published Jul 13, 2024• 4 citations
Publisher

Abstract

Abstract We show that the electromagnetic radiation field, conventionally introduced as a perturbation in quantum mechanics, is actually at the basis of the operator formalism. We first analyze the linear resonant response of the (continuous) variables x(t), p(t) of a harmonic oscillator to the full radiation field, i.e. the zero-point field plus an applied field playing the role of the driving force, and then extend the analysis to the response of a charged particle bound by a non-linear force, typically an atomic electron. This leads to the establishment of a one-to-one correspondence between the response functions and the respective quantum operators, and to the identification of the quantum commutator with the Poisson bracket of the response functions with respect to the normalized variables of the driving field. To complete the quantum description, a similar procedure is used to obtain the field operators as the response functions to the same normalized variables. The results allow us to draw important conclusions about the physical content of the quantum formalism, in particular about the meaning of the quantum expectation values and the coarse-grained nature of the quantum-mechanical description.

DOI
10.1007/s10701-024-00775-5

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Abstract We show that the electromagnetic radiation field, conventionally introduced as a perturbation in quantum mechanics, is actually at the basis of the operator formalism. We first analyze the linear resonant response of the (continuous) variables x(t), p(t) of a harmonic oscillator to the full radiation field, i.e. the zero-point field plus an applied field playing the role of the driving force, and then extend the analysis to the response of a charged particle bound by a non-linear force, typically an atomic electron. This leads to the establishment of a one-to-one correspondence between the response functions and the respective quantum operators, and to the identification of the quantum commutator with the Poisson bracket of the response functions with respect to the normalized variables of the driving field. To complete the quantum description, a similar procedure is used to obtain the field operators as the response functions to the same normalized variables. The results allow us to draw important conclusions about the physical content of the quantum formalism, in particular about the meaning of the quantum expectation values and the coarse-grained nature of the quantum-mechanical description.
Abstract Linear response theory is concerned with the way in which a physical system reacts to a small change in the applied forces. Here we show that quantum mechanics in the Heisenberg representation can be understood as a linear response theory. To this effect, we first address the question of the physical origin of the quantum operator formalism by considering the interaction of a bound electron with the radiation field, including the zero-point component, following the approach of stochastic electrodynamics. Once the electron has reached a stationary state, it responds linearly and resonantly to a set of modes of the driving radiation field. Such a response can lead the system to a new stationary state. Identifying a one-to-one relationship between the response variables and the corresponding operators results in the ( x,p ) commutator as the Poisson bracket of these variables with respect to the driving field amplitudes. To account for the order of the response variables, which is reflected in the non-commutativity of the operators, we introduce the concept of ordered covariance. The results obtained allow to establish a natural contact with linear response theory at the fundamental quantum level.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
Abstract The new quantum mechanics could at first be used to answer questions concerning radiation only through analogies with the classical theory. In Heisenberg’s original matrix theory, for instance, it is assumed that the matrix elements of the polarisation of an atom determine the emission and absorption of radiation analogously to the Fourier components in the classical theory. In more recent theories a certain expression for the electric density obtained from the quantum mechanics is used to determine the emitted radiation by the same formulæ as in the classical theory. These methods give satisfactory results in many cases, but cannot even be applied to problems where the classical analogies are obscure or non-existent, such as resonance radiation and the breadths of spectral lines. A theory of radiation has been given by the author which rests on a more definite basis. It appears that one can treat a field of radiation as a dynamical system, whose interaction with an ordinary atomic system may be described by a Hamiltonian function. The dynamical variables specifying the field are the energies and phases of its various harmonic conrponents, each of which is effectively a simple harmonic oscillator. One must, of course, in the quantum theory take these variables to be q-numbers satisfying the proper quantum conditions. One finds then that the Hamiltonian for the interaction of the field with an atom is of the same form as that for the interaction of an assembly of light-quanta with the atom. There is thus a complete formal reconciliation between the wave and light-quantum points of view.
This is the first in a series of two articles, the second of which provides an exact electro-magnetic field description of photon emission, absorption, and radiation pattern. Photon energy exchanges are analyzed and shown to be the triggered, regenerative response of a non-local eigenstate electron. This first article presents a model-based, hidden variable analysis of quantum theory that provides the statistical nature of wave functions. The analysis uses the equations of classical electro-magnetism and conservation of energy while modeling an eigenstate electron as a nonlocal entity. Essential to the analysis are physical properties that were discovered and analyzed only after the historical interpretation of quantum mechanics was established: electron non-locality and the standing electro-magnetic energy that accompanies and encompasses an active, electrically small volume. The standing energy produces a driving radiation reaction force that, under certain circumstances, is many orders of magnitude larger than currently accepted values. These properties provide a sufficient basis for the Schrödinger equation as a descriptor of non-relativistic eigenstate electrons in or near equilibrium. The uncertainty principle follows, as does the exclusion principle. The analysis leads to atomic stability and causality in the sense that the status of physical phenomena at any instant specifies the status an instant later.
The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.
Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant ℏ) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any values for physical quantities. Rather, one must go outside nonrelativistic classical mechanics in order to obtain a fundamental phase space distribution for classical physics. We assume that the values of physical quantities in classical theory at any temperature depend on the phase space probability distribution that arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant ℏ. All mechanical systems in thermal radiation will inherit the constant ℏ from thermal radiation. Here, we note the connections between classical and quantum theories (agreement and contrasts) at all temperatures for the harmonic oscillator in one and three spatial dimensions.