For a long time I had been wondering who was the first to use the term “ghost” into quantum field theory. My first encounter with the term was in the context of Faddeev-Popov’s approach to quantization of non-Abelian gauge theories. But Faddeev and Popov, in their first articles, did not use “ghost”; instead, they used a more innocuous term, something like “a fictitious scalar field.” From what I could find out, almost immediately after Faddeev and Popov that strange field, scalar but with fermionic statistics, was renamed “ghost.” The post-Faddeev-Popov literature, however, does not contain any indication on who came up with this term. It also seems that the Russian equivalent “духи” was imported from English, rather than the other way around.
Who was that person who has managed to introduce into the vocabulary of particle physics, the science of the 20th century, a word that has origin in the superstitious beliefs at the dawn of human history?
Several days ago I ran onto David Derbes, a retired physics teacher from the University of Chicago Laboratory High School, who has helped the publication of a number of historical documents, including Dyson’s and Coleman’s lectures in quantum field theory. He told me about his latest object of study—the first preprint of Faddeev and Popov, in Russian and never published. I asked him if he knew the origin of the term “ghost.” David said did not know, but he told me he would try to find out.
With David’s help, now I think I know who has introduced the term “ghost” and when.
It turned out that the term was introduced by Wolfgang Pauli, in a different context. Pauli, a giant in physics, was also responsible for the introduction of the particle now called “neutrino” with a famous letter which started with the words „Liebe radioaktive Damen und Herren“, “Dear radioactive Ladies and Gentlemen.” And as with the neutrino, Pauli introduced the term “ghost” in a letter. In fact, in two letters. The first letter, sent to Källén on December 9, 1954, contains Pauli’s announcement of his intention to send a letter to TD Lee, where he would propose the word “ghost.” (The letter to Källén is letter [1942] in the book edited by Karl von Meyenn, see the end of this posting). Pauli predicted that once the term is proposed, its use will spread epidemically in the literature. But Pauli wrote that he did not think “ghosts” are physical, citing a quote, allegedly by Lichtenberg, “There are more things in the compendiums of physics, than are dreamt of in heaven and Earth.”
The letter to TD Lee was sent five days later on December 14, 1954, written in English, and copied to Dyson. Here are the beginning and the end of that letter. Note the way Pauli signed the letter. (Text taken from the Karl von Mayenn’s book; this is letter [1946] in that volume.)
“Dear Lee!
It is already some time that I started to study your paper seriously. Days became weeks, weeks two months and my file „Lee-model“ is still increasing – a proof of its importance. It is true that in my way of looking at it, most of this importance is concentrated in your footnote 4, p. 1331 and the rest of the paper seems to me, at least in first approximation, negligible in comparison to this small printed note.
(… a lot of technical discussions follow …)
The essential occurrence of negative probabilities in your example makes it, of course, extremely unphysical. But this is not my whole story, and in some other respects, I may have good consolation for you, too. Until now I only told you results which are proved. In the following concluding part of this letter I shall formulate guesses or conjectures of a more general kind, which should be merely considered as the outline of a program of further mathematical investigations.
Let us call a new energy-state with negative probability (negative in comparison with the other states of normal behaviour for small coupling constant), whose energy tends to ±∞ for (renormalized) coupling constant g going to 0, a ‚ghost‘. It is my opinion that the occurrence of ‚ghosts‘ will soon turn out to be a general feature of coupling constant renormalization. This feature has been revealed first by your example, the importance of which should therefore not be underrated. If the unrenormalized theory diverges logarithmically, the energy of the ghosts will behave for small g as 
. This seems to offer an explanation for the fact that in the examples, which could be really investigated until now, Dyson’s power series have the convergence radius zero. I suggest that this result (Thirring and others), which is presumably general, be brought in connection with the fact that the ghost energy is of the mentioned essentially singular type at the point g = 0. If my conjecture is right, the renormalized field-theory should have a mathematically rigorous solution, which, however, is unphysical because of the occurrence of negative probabilities in it. The ghosts have no physical reality whatsoever, they are the formal reaction of mathematics to the tricks played on her by the method of renormalization.
If my conjecture is right, this should also hold for quantum electrodynamics, the only case where we are certain that renormalization has anything to do with nature. The ghosts would then be situated very roughly, at the extreme high energy of e137 times electron mass (factors like l/π in the exponent not excluded). These ghost-states will then be only very seldom excited and one can understand the possibility that quantum-electrodynamics including renormalization, can give good approximations. But, in principle, it would have this defect too.
The situation is too new for us to think about the therapy now, we have first to think about good mathematical methods, to check the diagnosis and make the bacillus in the renormalization method generally visible. There are great difficulties in this problem. If, for instance, a ‚ghost‘ would be discovered in a Tamm-Dancoff approximation, how could we be sure that it is not only a result of the insufficiency of this approximation?
Nevertheless I think that this mathematical problem can be attacked and I suggested to Källén, who is at present in Copenhagen, that he resumes his old work of 1952 in the light of this new aspect. Our results will certainly be published in due time in one form or another but I think they have first to be put on a broader basis.
Meanwhile I hope that the end of this long letter will be the beginning of something else and I conclude with all good wishes for Xmas and for a really new year in physics to yourself and to all friends at Columbia University.
Sincerely yours,
W.Pauli
The society of ghost hunters
The president
TD Lee’s paper mentioned by Pauli is Phys. Rev. 95, 1329 (1954). Pauli would continue to refer to particles with negative norm as “ghosts.” For example, in a letter sent to to Heisenberg on May 18, 1955, Pauli expressed doubt that modes with negative norm in Heisenberg’s model can be quarantined from the rest of the Hilbert space. He asked: „Wieso bleibt der Geist in der Flasche?“ “Why does the ghost stay in the bottle?” (it appears that in German the word Geist, meaning ghost, is also used for a genie in a bottle).
Pauli’s ghost is, in the modern language, related to the Landau pole. With the invention of asymptotic freedom, Landau pole is no longer inevitable in quantum field theory. Pauli’s prediction that the term “ghost” would be widely used remains correct. “Ghost” lives on, most notably as the colorful name for Faddeev and Popov’s “fictitious scalar field.”
Reference:
W. Pauli, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a. Band IV, Teil II: 1953-1954 (Scientific Correspondence with Bohr, Einstein, Heisenberg, a.o. Volume IV, Part II: 1953-1954), edited by Karl von Meyenn, Springer, 1999.
Addendum (July 6, 2019): Although the English word “ghost” was first proposed by Pauli in his December 9, 1954 letter to Källén, Pauli conceived the use of the German word “Geist” for the same purpose a few days earlier, in his December 6, 1954 letter to Fierz.
của Einstein.
là tần số cơ bản của dao động của dây đàn,
với 
.
nay tương đương với một dao động tử điều hoà, và dây đàn là một tổ hợp các dao động tử điểu hoà với tần số 
, v.v. Các mức năng lượng của dao động tử điều hoà với tần số 
là 
. Như vậy để mô tả trạng thái lượng tử của dây đàn, ta cần một số vô hạn các số lượng tử 
trong đó 
là số lượng tử của dao động tử với tần số 
là năng lượng của trạng thái cơ bản. Để đơn giản từ nay ta sẽ đo năng lượng của dây đàn từ 
.
có năm trạng thái:
.
chính là entropy khi năng lượng bằng 
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