Abstract
An editorial note by one of us in this journal in 2020, argued in favour of the name Schrödinger–Lichnerowicz formula for the formula, \(g^{\mu \nu }\nabla _\mu \nabla _\nu +m^2 + R/4\), for the ‘square’ of the Dirac operator in curved spacetime since it had been obtained by Schrödinger in 1932 and rediscovered by Lichnerowicz in 1962. However, unfortunately, it overlooked the rediscovery of the formula by Asher Peres in 1963. We briefly recall the context of each of these discoveries and reflect on the naming of mathematical formulae in general and of this formula in particular.
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Addendum to: Gen Relativ Gravit https://doi.org/10.1007/s10714-019-2625-z
In an editorial note [1] by one of us (BSK) to the recent (re-)publication [2] of an English translation (by Claus Kiefer) of the 1932 paper [3] of Erwin Schrödinger, it was mentioned that the latter paper of Schrödinger had been (by thirty years!) the first place where the formula \(g^{\mu \nu }\nabla _\mu \nabla _\nu +m^2 + R/4\) had been obtained for the ‘square’ \((-i\gamma ^\mu \nabla _\mu - m)(i\gamma ^\nu \nabla _\nu \psi - m)\) of the Dirac operator.
It was also mentioned in [1] that, around 30 years later, the corresponding formula was rediscovered by André Lichnerowicz. (As stated in [1], this was done in the special case of zero mass and on a Riemannian manifold rather than a spacetime in [4] but actually the formula for the square of the massless Dirac operator also appeared in the slightly earlier paper [5] in a Lorentzian context.) Lichnerowicz used the result in [4], to prove the vanishing of the index—i.e. the Hirzebruch \(\hat{A}\)-genus—for even-dimensional compact Riemannian manifolds admitting spin structures with non-negative (but not identically zero) scalar curvature. (This was an early application of the Atiyah Singer index theorem, when applied to the case of Riemannian manifolds with spin structures.) And it was argued in [1] that, of the several names (including “Bochner–Lichnerowicz formula”, “Lichnerowicz–Bochner–Weitzenböck formula” etc. as well as “Schrödinger–Lichnerowicz formula”) that had been used by other authors, the name Schrödinger–Lichnerowicz formula seemed the most appropriate.
However, it was unfortunately overlooked in [1] that, also in 1963, Asher Peres rediscovered the same formula in [6] in aid of making the point that the ‘squared’ Dirac equation in an external gravitational field doesn’t contain a gyro-gravitational term analogous to the gyro-magnetic term (see the top of Page 10 in [1]) \(\tfrac{1}{2}\sigma ^{ab}F_{ab}\) in the ‘squared’ Dirac equation in an external electromagnetic field. Also, Peres pointed out that it is not analogous to the equation for a classical spinning particle in a curved spacetime where there is a term of form \(\frac{1}{2}R_{\alpha \beta \gamma \delta }v^\beta S^{\gamma \delta }\) where \(v^\alpha \) is the velocity vector and \(S^{\alpha \beta }\) the angular momentum tensor of the classical spinning particle.Footnote 1 Actually, in obtaining the formula for the ‘square’ of the Dirac operator, [6] makes reference to the equations in Peres’s earlier paper [7] which also discusses the theory associated with the names of Einstein, Cartan, Kibble, Sciama and Weyl and others (see Footnote 11 in [1]) of gravity with torsion and thus, arguably, Peres deserves to have his name included in that list of names too. (Somewhat ironically though, [6] does not mention that the theory involving torsion discussed in [7] would lead to a gravitational counterpart of gyro-magnetic coupling.)
This unfortunate omission leads us to reflect on the often noted fact that the naming of theorems in mathematics and of results in physics is often the result of historical quirks and accidents and frequently unsatisfactory or unfair in one way or another. In the case of the formula for the ‘square’ of the Dirac operator, we feel that in an ideal world, in which Lichnerowicz and Peres and everyone else would have been aware of Schrödinger’s prior contribution, it would have deserved to be called, simply, the ‘Schrödinger formula’. Indeed not only did Schrödinger obtain it 30 years earlier but it is all the more creditable that, as explained in [1], he managed to do so without a formula for the spin connection. In contrast, both Lichnerowicz and Peres benefited from the fact that, by 1963, the notion of spin connection had (as also discussed in [1]) by then penetrated the realm of the routine in both physics and mathematics. However, there are other reasons for naming formulae than historical priority. And perhaps it is good to append some other names so as to distinguish this particular formula of Schrödinger from the several other formulae that bear his name. If so, though, perhaps it should more justly be known as the Schrödinger–Lichnerowicz–Peres formula.
Let us also take the opportunity of this addendum to correct Footnote 10 of [1] which suggested that the square-roots of g in Eq. (74) in [2, 3] were unnecessary. The square roots are in fact needed since, in [3], and unlike in modern usage, the symbol ‘\(\nabla _k\)’ was always taken to mean ([3, Eq. (74)]) \(\frac{\partial }{\partial _k} - \varGamma _k\) even when it acts on a spinor vector.
Notes
It is not really correct, though, to say, as Peres does that “Dirac particles \(\dots \) cannot be used to test general relativity.” since of course, say in a coordinate system, the term \(g^{\mu \nu }\nabla _\mu \nabla _\nu \) in the ‘squared’ Dirac equation is not the same as the scalar Laplacian.
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Acknowledgements
GWG thanks Mike Stone for pointing out to him Schrödinger’s priority over Lichnerowicz and for subsequent discussions.
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Gibbons, G.W., Kay, B.S. Addendum to: Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field I. Gen Relativ Gravit 54, 134 (2022). https://doi.org/10.1007/s10714-022-03026-y
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DOI: https://doi.org/10.1007/s10714-022-03026-y