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In his book on quantum mechanics in the chapter on perturbation theory Dirac says in a footnote:

A system with only one stationary state belonging to each energy-level is often called non-degenerate and one with two or more stationary states belonging to an energy-level is called degenerate, although these words are not very appropriate from the modern point of view.

I have two questions about this:

1) Why did Dirac deem the terms (non-)degenerate inappropriate?

2) Why do we, with our even more modern point of view, still use them?

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    $\begingroup$ Maybe he didn't like the derogatory implication of "degenerate" ? $\endgroup$
    – Carl Witthoft
    Oct 25, 2018 at 13:14
  • $\begingroup$ Possibly. My first association was with the Nazis use of the term 'degenerate art'. $\endgroup$
    – blackholedynamite
    Oct 25, 2018 at 15:10
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    $\begingroup$ I doubt PAMD was referring to cultural connotations of the time---he was not the type to pay attention to such. He may have been deferring to category epimorphisms in Hilbert space. I doubt catastrophy theory was around at the time. Merging energy levels "degenerate" to a single one, and have their "degeneracy lifted" by running the relevant parameter backwards. He may have objected to the inappropriate final, irreversible slant of the term. $\endgroup$
    – Cosmas Zachos
    Oct 25, 2018 at 16:29
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    $\begingroup$ @CosmasZachos Dirac's book was published in 1930. It was a year before von Neumann embedded quantum mechanics into Hilbert spaces (inspired by Dirac's book) and they entered physicists' vernacular. Dirac may be alluding to the old quantum mechanics of Bohr, where energy was the only quantum number, so multiple states meant degenerate eigenvalues. In general, there are multiple quantum numbers and true degeneracy would mean multiple states even when all of them are equal. $\endgroup$
    – Conifold
    Oct 25, 2018 at 21:03
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    $\begingroup$ The footnote in question is actually absent from the first edition archive.org/details/in.ernet.dli.2015.177580/page/n171. The earliest point it could have been added was '35 as I was unable to find a scan of the second edition. It's present in the third from '47. $\endgroup$
    – blackholedynamite
    Oct 25, 2018 at 22:17

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Here is my interpretation of what Conifold’s unchronological comment may have been trying to say. The term “degeneracy” was first introduced by Schwarzschild (1916, p. 550) in the context of “old quantum theory”. (Credit by Sommerfeld (1921, p. 500).)

In (1927, p. 4) von Neumann “embedded quantum mechanics into Hilbert spaces” after participating in Hilbert’s 1926/27 seminar (and reading Dirac’s papers). Thereafter, degeneracy took the meaning of operator eigenvalue multiplicity, which is what Dirac must be referring to as “the modern point of view” in his book (1935 or 1947).

So Dirac may be objecting to the old QM word having been imported in this way — as opposed to using a “more appropriate” new one to mean “multiplicity > 1”. But no handy alternative emerged, so we still use degeneracy. (It would be interesting to see if similar uses in linear algebra, or algebraic geometry, pre- or post-date this episode.)

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