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Stochastic theory of emission and absorption of quanta

Michele Cini, Maurizio Serva

Published Apr 1, 1985• 1 citation
OpenAlex

Abstract

The attempt to reformulate quantum theory in the form of a theory of stochastic processes, which has recently met with considerable success, has failed up to now to describe the most typical quantum effect: the emission and absorption of field quanta. The authors explore the possibility of constructing a stochastic field theory based on the assumption that for any normal mode of a given classical field, the corresponding action variable becomes a stochastic variable assuming only positive integer values in units of h. The proposed method reproduces the known properties of simple quantum field theoretical models without recourse to probability amplitudes, which are quantities extraneous to classical probability theory. The probability transition rates, for simple states of the field, turn out to be those of simple Poisson processes.

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Stochastic theory of emission and absorption of quanta

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The attempt to reformulate quantum theory in the form of a theory of stochastic processes, which has recently met with considerable success, has failed up to now to describe the most typical quantum effect: the emission and absorption of field quanta. The authors explore the possibility of constructing a stochastic field theory based on the assumption that for any normal mode of a given classical field, the corresponding action variable becomes a stochastic variable assuming only positive integer values in units of h. The proposed method reproduces the known properties of simple quantum field theoretical models without recourse to probability amplitudes, which are quantities extraneous to classical probability theory. The probability transition rates, for simple states of the field, turn out to be those of simple Poisson processes.

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The attempt to reformulate quantum theory in the form of a theory of stochastic processes, which has recently met with considerable success, has failed up to now to describe the most typical quantum effect: the emission and absorption of field quanta. The authors explore the possibility of constructing a stochastic field theory based on the assumption that for any normal mode of a given classical field, the corresponding action variable becomes a stochastic variable assuming only positive integer values in units of h. The proposed method reproduces the known properties of simple quantum field theoretical models without recourse to probability amplitudes, which are quantities extraneous to classical probability theory. The probability transition rates, for simple states of the field, turn out to be those of simple Poisson processes.
The attempt to reformulate quantum theory in the form of a theory of stochastic processes, which has recently met with considerable success, has failed up to now to describe the most typical quantum effect: the emission and absorption of field quanta. The authors explore the possibility of constructing a stochastic field theory based on the assumption that for any normal mode of a given classical field, the corresponding action variable becomes a stochastic variable assuming only positive integer values in units of h. The proposed method reproduces the known properties of simple quantum field theoretical models without recourse to probability amplitudes, which are quantities extraneous to classical probability theory. The probability transition rates, for simple states of the field, turn out to be those of simple Poisson processes.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.
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.
A new class of stochastic variables, governed by a specifice set of rules, is introduced. These rules force them to loose some properties usually assumed for this kind of variables. We demonstrate that stochastic processes driven by these random sources must be described using an probability amplitude formalism in a close resemblance to Quantum Theory. This fact shows, for the first time, that probability amplitudes are a general concept and is not exclusive to the formalism of Quantum Theory. Application of these rules to a noisy, one-dimensional motion, leds to a probability structure homomorphic to Quantum Mechanics.
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.