Trjitzinsky Memorial Lectures

presented by
Elliott Lieb
Princeton University

April 5-7, 2000

A reception will be held in the
Reading Room, Levis Faculty Center,
immediately following the lecture on Wednesday, April 5


Lectures to be presented:

Wednesday, April 5
4:00 p.m., Room 314 Altgeld Hall

The Quantum Mechanical World View:
A Highly Successful but Still Incomplete Theory

Ordinary matter is held together with electromagnetic forces, and the dynamical laws governing the constituents (electrons and nuclei) are those of quantum mechanics. These laws, found in the beginning of this century, were able to account for the fact that electrons do not fall into the nuclei and thus atoms are quite robust. It was only in 1967 that Dyson and Lenard were able to show that matter in bulk was also stable and that two stones had a volume twice that of one stone. Simple as this may sound, the conclusion is not at all obvious and hangs by a thread-- namely the Pauli exclusion principle. In the ensuing 3 decades much was accomplished to clarify, simplify and extend this result. We now understand that matter can, indeed, be unstable when relativistic effects and magnetic fields are taken into account -- unless the electron's charge is small enough (which it is, fortunately). These delicate and non-intuitive conclusions will be summarized. The requisite mathematical apparatus needed for these results is itself interesting. Finally, we can now hope to begin an analysis of the half-century old question about the ultimate theory of ordinary matter, called quantum electrodynamics (QED). This is an experimentally successful theory, but one without a decent mathematical foundation. Some recent, preliminary steps in resolving the infinities of QED will be presented.

Thursday, April 6
4:00 p.m., Room 314 Altgeld Hall

The Mathematics and Physics of the Second Law of Thermodynamics

The essence of the second law is the statement that all processes can be quantified by an entropy function whose increase is a necessary and sufficient condition for a process to occur. It is one of the few really fundamental physical laws (in the sense that no deviation, however tiny, is permitted) and its consequences are far reaching. Since the entropy principle is independent of any statistical mechanical model, it ought to be derivable from a few logical principles without recourse to Carnot cycles, ideal gases and other assumptions about such things as 'heat', 'hot' and 'cold', 'temperature', 'reversible processes', etc. In this lecture the foundations of the subject and the construction of entropy from a few simple axioms will be presented.

Friday, April 7
4:00 p.m., Room 314 Altgeld Hall

The Bose Gas: A Subtle Many-Body Problem

Now that the properties of the ground state of quantum-mechanical many-body systems (bosons) at low density, r, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. One of these is that the leading term in the energy/particle is 4par where a is the scattering length (which will be defined in the lecture). Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained jointly with J. Yngvason and this work will be presented. The reason behind the mathematical difficulties will be emphasized. Another problem of great interest is the existence of Bose-Einstein condensation, and what little is known about this rigorously will also be discussed. With the aid of the methodology developed to prove the lower bound, two other problems have been successfully addressed. One is the fact that the Gross-Pitaevskii equation correctly describes the ground state in the 'traps' actually used in the experiments. The other is a very recent proof that Foldy's 1961 theory of the high density gas of charged particles correctly describes their ground state.

 
History of Trjitzinsky Lectures

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