# What is the origin of the name "degeneracy" pressure and "degenerate" Fermi gas?

What is the origin of the name "degeneracy" pressure and "degenerate" Fermi gas? I was trying to find the first paper that used the term "degenerate/degeneracy" to describe either "degeneracy" pressure or "degenerate" Fermi gas. But I failed to do so. Any help is highly appreciated!

• Presumably from the math usages. I’m not able to pull it up in the OED to get the time period of first usage of the various senses of the word. Sep 23, 2022 at 15:30

Just an informed hunch, seconding @Jon Custer's suggestion. The physics usage follows the older math usage, which signifies degeneracy as a limiting simplification which stretches a definition to an absurd, qualitatively simpler limit, such as a parabola being the degenerate limit of an ellipse with a focal point sent to infinity. Normally, the term associates to reduction of dimension or cardinality.

In physics, the term appears to originate in the degeneracy of matrix eigenvalues, the limit when a vanishing parameter of a matrix induces two or more distinct eigenvalues to merge to a common one, shared by two or more different eigenvectors (eigenstates). The dominant operator is the QM hamiltonian, thus involving degenerate energy eigenvalues. The converse limit "breaks the degeneracy", when introduction of a non-vanishing parameter (or perturbation) distinguishes two degenerate eigenstates and assigns unequal energy eigenvalues (levels, spectral lines, etc...) to them.

By the time the Dirac distribution for fermions was discovered, the concept was broadly used in physics, and the limiting case of all states of a system allowed by the Pauli exclusion principle being full up to the Fermi surface was naturally termed a "degenerate" gas: maximally ordered and quite un-gas-like, with a Pauli exclusion "pressure" keeping fermions from overlapping and condensing.

Fermi's original (1926) paper, Zur Quantelung des idealen einatomigen Gases, ZfPhys 36 902–912 thus utilizes the term Entartung (degeneracy).

Note added @Consigliere ZARF 's appreciated comment reminds us the term was already current in the old quantum theory (Sommerfeld 1921) to indicate "limiting case", and, before him, K Schwarzschild p550 §3 , (1916) "Zur quantenhypothese" Sitz ber Preuss Akad Wiss (Berlin), 548-568.

• Yeah, I looked at the English translation of Fermi's 1926 paper. There is a sentence: "giving the so-called degeneracy phenomena of the ideal gas for low temperatures." I thought maybe the usage of the term "degeneracy" to describe the low-temperature gas was already prevalent at that time. Perhaps I'm wrong, and he is indeed the first one to use the word "degeneracy" in this context.
– Cory
Sep 23, 2022 at 23:15
• I am not that knowledgeable about usage on low-entropy systems at that time... Sep 24, 2022 at 0:27
• This earlier discussion (to which you took part) points to Schwarzschild 1916 as perhaps the original use, much prior to Fermi 1926. Sep 29, 2022 at 18:57
• The earlier discussion left open the question whether the "math usage" was earlier, as you imply. This seems confirmed by finding ausgeartet in Plücker (1835), dégénérée in Bourdon (1825), degenerare in Scherffer (1771), degenerate in Wallis (1685)... Sep 30, 2022 at 3:38
• Wallis, p 272, indeed! Thanks so much: exceeds the OP in interest, for sure. Sep 30, 2022 at 19:18