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entropy.6 To derive it he used several important properties ofthe entropy. But he did not use the following result: For anadiabatically isolated system the entropy increases with time

or stays constant . This is an important property of the en-tropy, and, because it was not used in its derivation, it ap-pears now as a theorem. Indeed, it is essentially the con-tent of the famous H-theorem, the history of which goes backto Boltzmann.2

IV. THE DENOUEMENT

You will not find the old arguments of von Neumann,7

Pauli and Fierz,8 and later expositions, for example, byGeorge Uhlenbeck9 in modern books. These scientists arevery famous, and you might wonder why their expositionsare now neglected. The answer is surprising: they all at-tempted to expound a result that is essentially wrong. How isit that a wrong or misleading result had had such a good andlong life 19291957 ? I am afraid that the answer must bethat the various authors thought the result was reasonable. Inaddition, the people who established it had a great reputation.So their result was just accepted, but apparently neverchecked in detailpossibly because the arguments were justtoo involved.

What does one expect is needed for ergodicity? I could but wont here give you a short plausibility argument thatsuggests that for ergodicity the system Hamiltonian musthave eigenvalues which are all different, that is, the Hamil-tonian must be nondegenerate. My argument is much simplerthan that employed by von Neumann7 and by Pauli andFierz8 and is also misleading! But at least it gives a hint atthe main and erroneous! conclusion of these early authorsthat ergodicity requires nondegeneracy of the Hamiltonian.Indeed their argument required more, which my simple argu-ment does not, namely that not only the energies but also theenergy differences must all be different.

This whole matter arose, as I have noted, at the beginningof my academic career which was at the University of Aber-

deen. I had been asked to lecture on statistical mechanics,and my friend Dr. now Professor Charles McCombieclaimed that you could not do statistical mechanics unlessyou first discuss the ergodic theorem. This claim made senseto me and took me to the long and complex paper by vonNeumann as well as to the paper by Pauli and Fierz. VonNeumanns paper was published in 1929,7 when he was only26 years old, three years before his famous book on themathematical foundations of quantum mechanics.10 ThePauliFierz paper8 was published when Pauli was 37 yearsold.

As I have mentioned, these papers were seriously flawed,but I did not discover this upon first reading of them. What Idid discover was that because surely a system may still be

ergodic if it exhibits a small number of degeneracies, onemight have wanted to know how many degeneracies wouldstill be acceptable. None of these papers dealt with this ques-tion and nobody writing in this area had raised it, let alonetried to find an answer to it. It seemed to me an obviousproblem for a Ph.D student, and this is how Ian Farquhar gotinvolved.

On allowing degeneracies of the Hamiltonian we foundthat the earlier arguments still went through. It is clear thatno one had looked at this question, for it suggested using theideas of Pauli and Fierz and von Neumann that degeneraciesof the Hamiltonian had nothing whatever to do with ergod-

icity. Essentially the old theorems were dead. I will not gointo the mathematics of this finding, as it would not help ushere. What to do? von Neumann was already ill he diedearly in 1957 , and I felt that we could not simply publishour results. We had to get Paulis reaction first. So we wrotea polite letter to Pauli asking for his opinion. Because I hadpublished in the area of statistical mechanics before, I imag-ine my name was not completely unknown to him. Anyway,nothing was said about that, and indeed nothing happenedfor what seemed a long time, but in fact it was only abouttwo or three months. Then we received a letter from Pauli,

saying that both he and Fierz agreed with us, and he wouldbe glad to communicate our paper to the Royal Society ofwhich he was of course a foreign member . In those dayspublications in the Royal Society were always preceded bythe name of the person who had communicated the paper. Ido not know of any other paper that had Paulis name at-tached in that manner. Anyway, that was that!11 We turnedour minds now to other things. Unanswered questions, how-ever, were how Pauli and Fierz discussed this matter, whatwas said, etc., apparently now unobservable. We will, how-ever, come to them next.12

V. THE PAULI LETTERS

In February 2001 there appeared in the American physicsjournal Physics Today an article about Pauli by Karl vonMeyenn and Engelbert Schucking.12 The former is the editorof Paulis scientific correpondence. Some time later, when Ivisited my cousin in Bern, it proved possible to have a pleas-ant meeting with von Meyenn. A little while later he told methat it appeared from a letter he found that Abraham Pais hadasked Pauli about me with a view of possibly inviting me toPrinceton.

Returning to the edition of Pauli letters, there seem to be astaggering 3500 of these, of which von Meyenn estimatesthat, about two thirds were written by Pauli himself. In anycase von Meyenn drew my attention to a Pauli letter of 9August 1956 no. 2320 in Vol. IV, Part III addressed to

Professor Fierz, in which he says my translation from theoriginal German As far as assumption B is concerned ...

This was a key assumption in the PauliFierz paper I con-sider it now not only as lacking in plausibility, but nonsense

his emphases . The view of the gentlemen Farquhar andLandsberg about this assumption is not expressed as sharplyas that, but I have the impression that they also do not be-lieve in it. It apears to me almost that they are laughing at us

Neumann, me, and you as the great masters ... . Here,then, I had at last a small hint at the nature of the PauliFierzdiscussion. It was Pauli who now realized the enormity ofthe error in the original von Neumann paper, and carried overinto their own PauliFierz . I had no idea that such letterswere exchanged in the several months during which we

waited for Paulis reply to the letter which we sent to him inAugust 1956.

VI. MORE RECENT IDEAS

The subject of irreversibility made great strides in the sec-ond half of the last century.14,15 To trace this subject in anydetail here is not really possible. Nonergodic systems exist ofcourse. Thus the matrix of transition probabilities might de-compose into blocks such that there are no transitions be-tween different blocks. Such a system is clearly nonergodic.In an ergodic flow system the phase space volume changes in

120 120Am. J. Phys., Vol. 73, No. 2, February 2005 Peter T. Landsberg


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