1. Preface to the translation. According to the philosopher of science Dean Rickles, “Léon Rosenfeld was the first to attempt a direct quantization of the gravitational field (using both then-available methods: covariant and canonical) in 1930” ([56], p. 12). Rosenfeld’s first work on canonical approach [58] has been analyzed and translated by Donald Salisbury in [60–62]. The paper on covariant method [59] has been briefly reviewed by Salisbury [62], Alessio Rocci [57] and recently reproduced and reviewed in [48]. Both Rosenfeld’s works have been contextualized by Rickles in his book on the history of quantum gravity. We have translated Rosenfeld’s paper [59] making only minor changes in the notation. Like in the original German version, the references quoted by Rosenfeld are in the footnotes. By following Salisbury [60], we used this last section to add some comments to the text. They are intended to help the reader to contextualize Rosenfeld’s paper. Unlike Salisbury, our comments required us to add some references not included in the original paper and listed at the end of this section. For this reason, this section is entitled Comments with References.
2. In the abstract, we maintained Rosenfeld’s term gravitational quanta, the literal translation of Gravitationsquanten, for historical consistency. The term graviton will be coined after Rosenfeld’s work (in 1934, see [56], p. 31; footnote 29, and in 1939, see [56], p. 139; footnote 55).
3. In the original paper, the footnotes are unnumbered and only the symbols * and ** are used. To avoid confusion caused by the different numbering of the pages, we used Arabic numerals.
4. In his brief introduction, Rosenfeld quoted three papers (see footnote 1). In the first, Werner Heisenberg and Wolfgang Pauli described the Lagrangian and Hamiltonian formalism for the fields. Like in Rosenfeld’s work (see Sect. 2), the symbol $$ \cal H\it $$ indicated the Hamiltonian density, while $$ \overline{\cal H\it } $$ denoted the spatial integral of the Hamiltonian density, i.e., the Hamiltonian. In his previously published paper on the canonical approach [58], Rosenfeld had specified that the integration domain defining $$ \overline{\cal H\it } $$ “must be chosen in such a manner that field quantities assume a constant value on the boundary, indeed, such values that [the Lagrangian] vanishes there” ([60], p. 72). In their paper, Heisenberg and Pauli introduced also the variational derivative for fields, the energy-momentum tensor and Dirac’s expression of the creation and annihilation operators in terms of $$N $$ and $$\Theta $$, the conjugated number and phase variables, respectively; see Eq. (16). In the other two papers quoted by Rosenfeld, both Robert Oppenheimer and Ivar Waller pointed out the serious difficulties for quantum electrodynamics claimed by Rosenfeld, i.e., the emergence of divergent integrals leading to the nonsense prediction that the spectral lines would be infinitely displaced.
5. At the classical level, the gravito-electromagnetic system does not produce divergent quantities. As noticed by Rickles “That it was Heisenberg’s idea makes sense, since he was then searching for ways of bypassing the divergences (including through his ‘lattice worlds,’)” ([56], p. 122). But according to Blum, when and where Rosenfeld discussed this idea with Heisenberg remains uncertain ([46], p. 259; footnote 14).