ABSTRACT. During the Spring of 1966, Ed Jaynes presented a seminar course on quantum electronics that included the now famous “Jaynes-Cummings Model” and his Neoclassical Theory (NCT). As part of this seminar series, the NCT description of a two-level atom in an applied field was formulated as a formidable set of coupled nonlinear differential equations. Undaunted, Ed posted the equations on the Washington University Physics Department bulletin board and offered a prize of “a steak dinner for two” at a restaurant of the choice of the person who solves the equations. Within days, Bill Mitchell was able to present an elegant solution at one of the quantum electronics seminars. This early success of a new approach to doing theoretical physics encouraged Ed to challenge the knowledge hungry Physics Department with Steak Dinner Problem II. This problem was a specific mathematical formulation of the exact (i.e. without the Rotating Wave Approximation) description of the interaction of a two-level atom with a single quantized electromagnetic field mode. Jaynes' formulation of the problem appears to have anticipated the use of Bargmann Hilbert space in QED. This problem has remained unsolved for 26 years in spite of the efforts of numerous researchers, most of whom were probably unaware of Jaynes' offered prize. Recent efforts to solve this problem will be described.

ABSTRACT. A reformulation of the Dirac theory reveals that iћ has a geometric meaning relating it to electron spin. This provides the basis for a coherent physical interpretation of the Dirac and Schödinger theories wherein the complex phase factor exp(–iφ / ћ) in the wave function describes electron zitterbewegung, a localized, circular motion generating the electron spin and magnetic moment. Zitterbewegung interactions also generate resonances which may explain quantization, diffraction, and the Pauli principle.

Introduction

Edwin T. Jaynes is one of the great thinkers of twentieth century science. More than anyone else he has deepened and clarified the role of statistical inference in science and engineering. To my mind, his greatest accomplishment has been to recognize that in the evolution of statistical mechanics the principles of physics had gotten confused with principles of statistical inference, and then to show how the two can be cleanly separated to produce a simpler yet more powerful theoretical system.

I share with Ed Jaynes the belief that quantum mechanics suffers from an analogous muddle of probability with physics, which is at the root of the perennial controversy over physical interpretation.

Though a Jaynesian revolution of the “quantum muddle” remains elusive, I will report here on a promising possibility that has been overlooked.

ABSTRACT. It is pointed out that the conclusions drawn from a recent quantum interference experiment with light suggest an operational resolution of the Schrödinger cat paradox.

On this occasion in honor of Prof. E.T. Jaynes, we recall that he devoted some of his research efforts to the interpretation of quantum mechanics, which led him to propose several experimental tests. Although the ‘neoclassical’ theory he developed was found not to be confirmed by experiment, it nevertheless played a role in encouraging us to think critically about quantum mechanics and to carry out new experiments. This short contribution is concerned with a well-known quantum problem of interpretation.

The quantum paradox known as Schrödinger's cat, in which the cat is cast in a linear superposition of the state of being alive and the state of being dead, has been debated since the beginnings of quantum mechanics. Whereas most physicists are ready to concede the existence of superposition states for a microscopic quantum system like an atom, such states appear to be ruled out for a macroscopic system like a cat by common experience. The question then arises at which level classical concepts take over from quantum mechanical ones.

In a very clear and readable discussion of this and other paradoxes Glauber has pointed out that the noise inevitably associated with the amplification process accompanying the measurement of a microscopic system leaves the cat in a mixed state rather than a pure one.

ABSTRACT. The Jaynes-Cummings Model (JCM), a soluble fully quantum mechanical model of an atom in a field, was first used (in 1963) to examine the classical aspects of spontaneous emission and to reveal the existence of Rabi oscillations in atomic excitation probability for fields with sharply defined energy (or photon number). For fields having a statistical distribution of photon numbers the oscillations collapse to an expected steady value. In 1980 it was discovered that with appropriate initial conditions (e.g. a near-classical field), the Rabi oscillations would eventually revive – only to collapse and revive repeatedly in a complicated pattern. The existence of these revivals, present in the analytic solutions of the JCM, provided direct evidence for discreteness of field excitation (photons) and hence for the truly quantum nature of radiation. Subsequent study revealed further nonclassical properties of the JCM field, such as a tendency of the photons to antibunch. Within the last two years it has been found that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat). This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) and to explore still further the relationship between classical and quantum physics.

ABSTRACT. In this paper it is shown how Jaynes' maximum entropy principle or, more generally, his maximum calibre principle can be cast in such a form that the stochastic process that underlies observed data can be determined under the assumption that the process is Markovian. Under suitable constraints it becomes possible to derive the Fokker-Planck equation and the Îto-Langevin equation of that process.

Introduction

Jaynes' maximum entropy principle has found wide-spread applications not only in systems in thermoequilibrium but also in systems far from it. In addition, Jaynes treated time-dependent processes by means of his maximum calibre principle. In the present paper we show how his principles can be cast in such a form that the underlying process that is observed by certain correlation functions can be represented by a Fokker-Planck equation and an Îto-Langevin equation. It will be assumed throughout our paper that the process is Markovian. At the end of this contribution explicit examples treated recently by Lisa Borland and the present author will be presented.

Derivation of the Fokker-Planck Equation

In order to express the definition of a Markov process in a rigorous mathematical form, we choose a time sequence t0, t1, …tM. We may attribute a probability distribution to the path taken by the state vectors at the corresponding times.

ABSTRACT. E.T. Jaynes has been the central figure for over three decades in showing how maximum entropy estimation (MEE) provides an extension of logic to cases where one cannot carry out Aristotelian deductive reasoning. Here I will review two early applications of MEE which I hope will provide some insight into how these ideas were being used in quantum and statistical mechanics around 1960.

In May of ‘61 I turned in my Ph.D. thesis (Scalapino, 1961) to Stanford University and, following a well-known dictum, went on to work on other problems. Now, three decades later, on this, the occasion of Ed Jaynes’ seventieth birthday, I decided to look back to see what we were doing when I was first getting to know Ed and his special approach to problems.

Not surprisingly, my thesis contains several applications of the principle of Maximum Entropy Estimation (MEE). In 1957 Ed had written two seminal articles (Jaynes, 1957(a), 1957(b)) showing how one could use Shannon's (1948) Information Theory to construct density matrices for a variety of different problems in equilibrium statistical mechanics. I had been very much taken with this work and wanted to understand how it could be applied to other systems.

Introduction

Humans have long been interested in devices which extend their senses, the telescope being an early example of such a device. We have since progressed to systems which operate in domains beyond the normal senses, as in the use of x-rays in medical imaging or seismic waves to probe the earth. Nevertheless, most existing remote sensing systems operate with considerable human input and interpretation; naturally, there is great interest in developing remote sensing systems which can with greater autonomy identify or recognize salient aspects of their environments. Potential applications of such systems abound:

1. Air-traffic control (identification of incoming aircraft).

2. Air defense (recognizing friendly or hostile aircraft).

3. Quality assurance (assembly line inspection).

4. Medical screening (identifying anomalies in radiographic or tomographic images).

5. Security and surveillance systems.

6. Robotics (navigation, carrying out tasks).

7. Exploration geophysics (interpreting seismic data).

8. Analytical chemistry (neutron activation analysis, nuclear magnetic resonance, other kinds of spectroscopy).

9. Detection of unexploded subsurface munitions.

All of these systems have in common the detection and processing of signals. In some cases (e.g., radar), the system must also emit a signal which interacts with the environment and is subsequently detected in modified form.

Though our understanding of human perception and information processing is incomplete, the very fact that recognition and identification are biologically possible demonstrates that the problem is not fundamentally intractable.

ABSTRACT. Can quantum probability theory be applied, beyond the microscopic scale of atoms and quarks, to the human problem of reasoning from incomplete and uncertain data? A unified theory of quantum statistical mechanics and Bayesian statistical inference is proposed. QSI is applied to ordinary data analysis problems such as the interpolation and deconvolution of continuous density functions from both exact and noisy data. The information measure has a classical limit of negative entropy and a quantum limit of Fisher information (kinetic energy). A smoothing parameter analogous to a de Broglie wavelength is determined by Bayesian methods. There is no statistical regularization parameter. A priori criteria are developed for good and bad measurements in an experimental design. The optimal image is estimated along with statistical and incompleteness errors. QSI yields significantly better images than the maximum entropy method, because it explicitly accounts for image continuity.

Introduction

Jaynes has been an eloquent advocate for a compelling hypothesis: Probability Theory as Logic. That is, probabilities represent degrees of belief; probability theory develops and applies universal principles of logical inference from incomplete information. Two of his primary interests have been Bayesian probability theory and the interpretation of quantum mechanics. Bayesian probability theory yields inferences by systematically and consistently combining new data with prior knowledge. Jaynes pioneered the maximum entropy (ME) class of Bayesian methods for density function estimation (Jaynes, 1983), which has been applied successfully to numerous data analysis problems.

ABSTRACT. It is now well established that effects traditionally associated with vacuum electromagnetic field fluctuations can be described equally well in terms of source fields (radiation reaction). This remarkable reconciliation of two previously unconnected points of view, which was stimulated by Jaynes' neoclassical theory, is reviewed and explained in a general and simple way.

Background

The Jaynes-Cummings paper seems to have been the first to employ dressed states of two-state atoms in fields (Jaynes 1963). One result found in that paper is that semiclassical radiation theory could serve as an excellent approximation even under certain conditions where the average number of photons in the field is small. The accuracy of semiclassical theory paved the way to Jaynes' “neoclassical theory,” where even spontaneous emission was treated without field quantization (Jaynes 1972, Milonni 1976).

The neoclassical theory was a subject of much debate and misunderstanding in the 1970s. It turned out not to be so easy to dismiss it on experimental grounds.

ABSTRACT. The picture of solo pulsars as usually presented is that they are born following a supernova explosion, coincident with the formation of a rapidly spinning neutron star. They emit astonishingly regular pulses of radiation whose periods gradually lengthen over time scales of one to ten million years, taken usually as a pulsar lifetime. The present work advances a substantially different picture than the above. In what follows, a model is presented which makes plausible an alternative view of pulsars which addresses a number of previously puzzling questions. In this new view, a neutron star typically goes through three stages. In the beginning, just after its formation, the magnetic field of the rapidly spinning neutron star tumbles erratically. Only later, on a time scale << 10 yrs., the tumbling motion gives way to pulsar behavior, an accurate limit cycle behavior whose period is given simply as proportional to the moment of inertia divided by the conserved angular momentum. In a time << 10 yrs. the pulsar dies, as the magnetic axis aligns itself with the rotation axis, a stable fixed point in the parameter space of the model. There are thus three time scales: the pulsar period, typically of the order of one second, an intermediate time determined principally by mechanical (or viscous) damping, and the third the damping time of the pulsar period, as angular momentum is radiated away over a million years or more.

ABSTRACT. A series of recent experiments and calculations are described that study the classical limit of an atom. The classical limit of an atom is defined to be the quantum mechanical state that most nearly approximates the ideal of a classical particle traveling in a Kepler orbit. It is found that correspondence between the quantum and classical descriptions is always in the form of ensembles of many realizations so long as the electron is taken to be a point particle. Even in the limit of a wave packet made up of states with all quantum numbers large there are distinctly quantum interference features in the evolution of the quantum ensemble. However, when the classical theory is modified by allowing within the ensemble only the Kepler orbits corresponding to energies and angular momenta allowed by Bohr's old quantum theory, most of the interference features are produced by the ensemble of classical particles.

Ed Jaynes has been characterized variously as a physicist, a statistician, an inventor, a free thinker, a teacher, a mentor, and a rabble rouser. It is hard to dispute his right to any of these mantles he might wish to claim, but it is equally hard to find a finite set of descriptors that do him justice. From my own selfish point of view his most important contribution has been to act as a source of ideas and new ways of looking at some long-standing and fundamental problems.

Introduction

In this paper, we shall consider (1) the derivation of prior distributions for use in Bayesian analysis, (2) Bayesian and non-Bayesian model selection procedures and (3) an application of Bayesian model selection procedures in the context of an applied forecasting problem. With respect to derivations of prior distributions, Jaynes (1980, p.618) has stated:

“It was from studying these frashes of intuition in Jeffreys that I became convinced that there must exist a general formal theory of determination of priors by logical analysis of the prior information–and that to develop it is today the top priority research problem of Bayesian theory.”

As is well known, many including Jeffreys (1967), Lindley (1956), Savage (1961), Hartigan (1964), Novick and Hall (1965), Jaynes (1968), Box and Tiao (1973), Bernardo (1979), and Zellner (1971, 1977, 1990) have worked to solve the “top priority research problem of Bayesian theory,” to use Jaynes’ words. Herein we shall consider further the maximal data information prior (MDIP) distribution approach put forward and discussed in Zellner (1971, 1977, 1990). A MDIP is a solution to a well-defined optimization problem and its use provides maximal data information relative to the information in a prior density. Also, as shown below, a MDIP maximizes the expectation of the logarithm of the ratio of the likelihood function, ℓ(θ|y), to the prior density, π(θ), i.e. ℓn[ℓ(θ|y)/π(θ)]. Several explicit MDIPs will be presented including those for parameters of time series processes–see Phillips (1991) for a lively discussion of this topic.

ABSTRACT. It is pointed out that the self-consistent treatment of radiation reaction is a common feature of many Ph.D. theses of E. T. Jaynes' students and “grandstudents” in the 30-year period 1962 - 1992. In this way they are all descended from Microwave Laboratory Report 502, written by Jaynes at Stanford in 1958. A number of examples are described briefly, and a new example that extends the framework laid down in M. L. 502 is presented.

Microwave Laboratory Report 502

In 1958 E. T. Jaynes wrote a paper entitled Some Aspects of Maser Theory that was issued as Microwave Laboratory Report No. 502 at Stanford University. The cover of the report is shown in Fig. 1.

In the Abstract of M. L. 502 Jaynes identified the work as an examination of the relation between quantum electrodynamics and the semiclassical theory of radiation. M. L. 502 was intended to serve as a preliminary step in a study of the ultimate limitations on noise figure and frequency stability in molecular-beam masers. The Abstract says, among other things, that “the semiclassical theory, as extended here, is a far more reliable means of calculating radiation processes than usually supposed.” In the present author's view, M. L. 502 reported the first examination of dynamical questions that are now associated with the field of cavity quantum electrodynamics.

ABSTRACT. Given two sets of data that are repeated measurements of the same physical quantity, one “control” and one “trial,” there are three problems of interest to the experimenter: (1) determine if something changed, (2) if something changed, what? and (3) estimate the magnitude of the change. These three problems are addressed using probability theory as extended logic. In the first section, the probability that the data sets differ is computed independent of what changed, i.e., independent of whether or not the means or standard deviations changed. In the second section, two probability distributions are computed: first, the probability that the means changed is computed independent of whether or not the standard deviations changed. Then second, the probability that the standard deviations changed is computed independent of whether or not the means changed. In the third section, the problem of estimating the magnitude of the changes is addressed. Here the probability density functions for both the difference in means and the ratio of standard deviations is computed. The probability for the ratio of standard deviations is computed independent of whether or not the means are the same, just as the probability for the difference in means is computed independent of whether or not the standard deviations are the same. This last calculation generalizes the solution of both the two-sample problem (different means and same but unknown standard deviations) and the Behrens-Fisher problem (different means and different unknown standard deviations).

Introduction

In 1963, E. T. Jaynes and F. W. Cummings published a paper entitled “Comparison of Quantum and Semiclassical Radiation Theories, with Application to the Beam Maser.” This paper was written at a time when the quantum theory of the laser had not yet been worked out, and represented an attempt at discussing the new behaviour that could be expected from treating the field quantum mechanically instead of classically. Jaynes and Cummings began by considering the interaction between a single two-level molecule and a single field mode, a simple situation which has become known as the Jaynes-Cummings model. Treating then the maser medium as a beam of two-level systems initially in a mixture of their upper and lower states characterized by a temperature T, they found that the field mode reached a thermal equilibrium with temperature Tf = ΩT/ω, where ω is the transition frequency of the two-level systems and Ω is the field mode frequency, provided that T > 0. For T < 0, the steady state field was found to contain an infinite amount of energy, an unphysical result resulting from the neglect of losses in the model. Jaynes and Cummings then proceeded to develop an alternative semi-classical theory of the beam maser, introducing their famous neo-classical theory of spontaneous emission.

An important follow-up paper was published by F. W. Cummings, who studied the long-time behavior of a two-level system interacting with a quantized single-mode field initially in a coherent state.