CONMEX - Sistema Carretero del Oriente del Estado de México

In the present article I discuss some features of the use of collective coordinates in two systems with a small number of degrees of freedom. In both cases the utility of collective coordinates is connected with the validity of an adiabatic approximation. The two examples are pedagogical in nature, but are perhaps appropriate in a volume honoring David Bohm, who has made major contributions to the theory of collective coordinates. First, however, I present some personal reminiscences of the days when I was a student of David Bohm.

My first recollection is of a seminar given in 1946 by David Bohm, shortly after he arrived in Princeton as an assistant professor. The subject was plasma physics. The talk was divided into three parts. The first part dealt with the plasma as a distinct state of matter, with an organization different from the solid, liquid and gas. The charge screening and lack of velocity locking was emphasized. The second part had to do with the widespread occurrence of plasmas in discharge tube physics, in astrophysics and in chemistry. He touched on the connection with microwave space-charge devices. Particular attention was paid to plasma oscillations and the frequent occurrence of instabilities. The third part dealt with metals viewed as quantum plasmas. The main tool of analysis was the linearization of the equations of motion for products of creation and annihilation operators by means of the random phase approximation.

I was then looking for a thesis advisor. In his low key fashion, Dave Bohm had opened up a vast panorama. It was clear that an enormous range of problems had to be explored. The intertwining of conceptual and practical problems was very appealing and exciting. How lucky to have the possibility of doing a thesis which was much more than doing the simple next step in an ongoing research program. I worked hard on my notes, and wrote up the lecture very carefully. I gave them to Dave and was taken on as a student.

We spent a tremendous amount of time together. There are advantages to having a bachelor as a mentor. We spent some time at the blackboard, but mainly talked. Then Dave wrote things down on paper. But even more vivid in my memory are the very long walks

through Princeton, with numerous stops for coffee. Dave developed ideas and responded to questions and criticisms. What was remarkable was that one could do theoretical physics without a blackboard and without pencil and paper. After returning to the office it seemed that the mathematics just settled into place, with significant results coming quickly.

I slowly realized that there was a hidden background. Dave had been an assistant to W.R.Smythe and had done essentially all of the difficult problems in Static and Dynamic Electricity. He had written papers on the theory of high energy accelerators and on the velocity distribution of electrons in nebulae. There was several years of total involvement with the behavior of plasmas in magnetic fields, in connection with his work for the Manhattan Project. These papers were written with his characteristic lucidity. The physical descriptions stand out and control the mathematical analysis. All of this work was germane to our research on the classical kinetic theory of plasmas and beams. There is undoubtedly more background that I am not aware of.

Dave was also deeply concerned with quantum problems. He thought about many-electron problems and superconductivity. He was at work on his text on quantum theory. He reformulated the theory of measurement in quantum mechanics. Most of all, he was fascinated by Bohr’s ideas on the role of complementarity in describing nature.

The problem of the divergences in quantum electrodynamics was very much on people’s minds. Dave worked on self-oscillations of finite-sized particles and on Kramer’s theory of non-relativistic electrodynamics. His lectures on advanced quantum mechanics dealt with electrodynamics. He gave much thought to the hypothesis of a minimum length in physics and to reconciling relativity and quantum mechanics for structures of finite extent. Along with other physicists such as Bohr, he felt that a really radical change in physical ideas was needed. Instead, Schwinger’s program of starting with the full Maxwell-Dirac Lagrangian, insisting on manifest covariance, and quarantining the divergences, represented a different kind of radical change. Feynman’s introduction of diagrams freed the imaginations of theoretical physicists to deal with what had been depressingly complicated formalisms in quantum field theory and many-body physics.

In the light of this rich background, it is now less surprising to me that one could do physics by talking. The use of analogy is very powerful when there is a well-defined mathematical basis for the analogies. Still, I continue to marvel at Dave’s extraordinary manner of expressing ideas and of constructing coherent intellectual structures. It made possible communication with non-physicists, who appreciated Dave’s ability to explain the fundamental ideas of physics. Indeed, this sometimes became ludicrous. I recall a social evening where, tongue in cheek, he constructed an elaborate and ‘convincing’ theory of the existence of ghosts and devils.

in many-body theory, and of wave and particle aspects in quantum theory, moved to the forefront of Dave’s thoughts. He was not satisfied with the equations-of-motion approach to many-body theory. He felt that a description was needed that dealt with collective and individual aspects simultaneously. This led to the successful auxiliary variable theory of Bohm and Pines.1 _{A beautiful account of the ideas is found in Dave’s later (1956) Les Houches}

lectures.2

He continued this work with T.Staver and D.Salt, and extended the description in his papers with G.Carmi.3 _{The ideas of these later papers have not}

been sufficiently appreciated by physicists.

On the problem of the foundations of quantum theory, his analysis of the Einstein, Podolsky, Rosen thought experiment gave rise to his causal reformulations. He continued this work in Brazil, Israel and England. The role of non-locality emerged. This culminated in the celebrated and startling Aharonov-Bohm analysis of electromagnetic potentials in quantum theory. It now occupies an important position in the physical foundations of gauge theories.

There is another aspect. It was the non-competitive atmosphere in which Dave did his work. I recall an incident. In our first paper on plasma

oscillations we had independently discovered the phenomenon of Landau damping. We used an orbit-tracing approach and considered the low collision frequency limit. It was used to understand particle trapping and the limitations of the linear theory. I came upon a copy of the Journal of Physics containing Landau’s solution of the linearized Vlasov equation.4 Due to wartime dislocations it arrived in Princeton after a delay of a year. I rushed to show it to Dave, who was in the shop constructing a frame for a hi-fi set. He was not at all perturbed at being scooped and simply admired the elegance and incisiveness of Landau’s paper.

Broader philosophical questions were often discussed in the numerous conversations that Dave had with me and others. I had taken courses in

mathematical logic from A.Church and pressed its claims on Dave. We talked about the relation to dialectical modes of thinking. I recall a conversation in the graduate student library at Princeton. It had a copy of the Catholic Encyclopedia. We looked at the article on the Holy Trinity, and noted how similar the language was to that of Bohr. However, nothing came of an attempt to generalize Bohr’s notion of wave-particle duality. Dave was always concerned with the philosophical problem of obtaining adequate concepts and modes of thinking to make sense of our experience. He has continued this with his explorations of the notions of implicate and explicate order.

Finally, I can only use old-fashioned language to describe his impact on me and others. Dave’s essential being was then, and still is, totally engaged in the calm but passionate search into the nature of things. He can only be characterized as a secular saint. He is totally free of

guile and competitiveness, and it would be easy to take advantage of him. Indeed, his students and friends, mostly younger than he is, felt a powerful urge to protect such a precious being. Perhaps the deep affection of his many friends helped to sustain him in the difficult years of the early 1950s.