The $\dagger$ symbol is often used in quantum mechanics,and also often in general mathematics to represent the conjugate transpose operation.For Hermitian matrices we can write $$A^\dagger=A$$Who introduced this notation?

  • $\begingroup$ Who introduced this in QM literature, or generally in the mathematical literature? $\endgroup$
    – DanielC
    Feb 2 '20 at 16:24
  • $\begingroup$ Actually interested to know about both! $\endgroup$ Feb 2 '20 at 19:06
  • $\begingroup$ I would have thought that Dirac's 1939 paper "A new notation for quantum mechanics" was the answer, but I was wrong. In fact, a cursory search of my library's bookshelves shows no QM books by big-name authors using this notation (Pauli, Dirac, v Neumann, Landau and Lifshitz). I'm sure there's examples, but it would be nice to have a particular instance. $\endgroup$ Feb 2 '20 at 19:53
  • 2
    $\begingroup$ Someone who shouldn't have (editorializing here) since it's difficult to tell from a plain old "t" to those of us with aging eyesight. $\endgroup$ Feb 3 '20 at 12:48
  • $\begingroup$ @ConsigliereZARF Very good! $\endgroup$ Feb 6 '20 at 22:48

In a now-deleted comment, Consigliere ZARF listed a number of papers published in Zeitschrift für Physik in the late 1920's that used this notation. The earliest was Pascual Jordan's 1927 "Über eine neue Begründung der Quantenmechanik", using the notation on pp.816-817; with about 10 other papers published in the following few years, all in the ZfP, all behind a paywall. According to Duncan and Janssen "(Never) Mind your p’s and q’s: Von Neumann versus Jordan on the Foundations of Quantum Theory", p.57, the Jordan paper was where the notation was introduced. The paper "Über die Grundlagen der Quantenmechanik" by D. Hilbert, J. von Neumann, and L. Nordheim, Mathematische Annalen (1928) uses (on page 19) $A^+$ for the adjoint of $A$.

Up to the point reading Consigliere ZARF's comment I had focused on English-language literature, which seemed to lag about a decade behind the German. The rest of this post describes what I found there.

An approximate answer: Google books search for "dagger+Hermitian" led to these early uses of the notation.

Page 59 of W. H. Furry's 1938 "Note on the Theory of the Neutral Particle", Phys. Rev. vol 54, p.56 has this:

In dealing with such matrices we shall use an asterisk (*) to indicate the complex conjugate matrix, a prime (') to indicate the transposed matrix, and a dagger ($\dagger$) to indicate the Hermitian adjoint matrix.

which seems to me to be an indication that this notation needed explaining. A few years later, Booth and Wilson, in "Radiative processes involving fast mesons" (Proc. Roy Soc A, v175, July 1940) wrote this in a footnote on p.487:

Note that in order to conform to the usual custom we modify Kemmer’s notation and use a dagger to denote the Hermitian conjugate.

indicating that the notation had by then become standard (although not universal, or else Kemmer's notation from the year before would not have needed adjusting).

For all I know, it is possible that Furry was the true original introducer of the dagger notation. It is certain that the notation was in use in the late 1930's.

Added: Furry cites an earlier paper of his (Phys Rev v51, p.125 (1937) and another by Pauli ("Contributions mathématiques à la théorie des matrices de Dirac", Annales de l'institut Henri Poincaré, Volume 6 (1936) no. 2, p. 109-136) , that use a superscript plus sign $A^+$ for the same thing. Page 119 of Pauli's paper has

En général, nous désignerons par $\gamma^{+\mu}$ les matrices conjuguées hermitiques (transposées et conjuguées) des $\gamma^\mu$

and so on, with what I think is a typo in the next bit: "(c’est-à-dire telles que $\gamma^{+\mu}_{\rho\sigma}=\gamma^{*\mu}_{\rho\sigma}$)".

The leap from superscript $+$ to superscript $\dagger$ is understandable. It was taken, for instance, by Pauli in his 1943 paper, "On Dirac's New Method of Field Quantization", if not earlier. (See equation (4) on p.176.) But not, evidently, by Schwinger in his 1948 paper "Quantum Electrodynamics. I. A Covariant Formulation", who uses $C^+$ for the Hermitian conjugate of $C$, on p.1441.


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