Abstract
Peter Bergmann and his students embarked in 1949 on a mainly canonical quantization program whose aim was to take into account the underlying four-dimensional diffeomorphism symmetry in the transition from a Lagrangian to a Hamiltonian formulation of Einstein’s theory. Along the way they developed techniques for dealing in phase space with the arbitrariness that arises in classical solutions given data at a given coordinate time. They argued that even though one seemed to destroy the full covariance through the focus on a temporal foliation of spacetime, this loss was illusory. In work undertaken with Anderson and I. Goldberg in the early 1950s, they constructed explicit expressions for both the generator of transformations between solutions which could be physically distinct, i.e., did not necessarily correspond to changes in spacetime coordinates, and invariant transformations that did correspond to four-dimensional diffeomorphisms. They argued that the resulting factor algebra represented diffeomorphism invariants that were the correct candidates for promotion to quantum operators. Early on they convinced themselves that only the corresponding construction of classical invariants could adequately reflect the fully relativistic absence of physical meaning of spacetime coordinates. Efforts were made by Bergmann’s students Newman and Janis to construct these classical invariants. Then in the late 1950s, Bergmann and Komar proposed a comprehensive program in which classical invariants could be constructed using the spacetime geometry itself to fix intrinsic spacetime landmarks. At roughly the same time, Dirac formulated a new criterion for identifying initial phase space variables, one of whose consequences was that Bergmann himself abandoned the gravitational lapse and shift as canonical variables. Furthermore, Bergmann in 1962 interpreted the Dirac formalism as altering the very nature of diffeomorphism symmetry. One class of infinitesimal diffeomorphism was to be understood as depending on the perpendicular to the given temporal foliation. Thus even within the Bergmann school program, the preservation of the full four-dimensional symmetry in the Hamiltonian program became problematic. Indeed, the ADM and associated Wheeler-DeWitt program that gained and has retained prominence since this time abandoned the full symmetry. There do remain dissenters – raising the question whether the field of quantum gravity has witnessed a Renaissance in the ensuing decades – or might the full four-dimensional symmetry yet be reborn?
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Notes
- 1.
“Wie Sie ersehen können, ist meine Ausbildung noch keineswegs in irgendeiner Richtung als abgeschlossen zu betrachten. Ich bin mir darüber klar, dass ich besonders auf dem Gebiet der Relativitätstheorie einerseits, der Quantentheorie anderseits, noch sehr viel zu lernen habe. Wenn es mir möglich wäre, würde ich sehr gern in der Richtung weiterarbeiten, die Verbindung zwischen diesen beiden Gebieten zu suchen.” The Albert Einstein Archives at the Hebrew University of Jerusalem (AEA), 6–222, Letter from Bergmann to Einstein dated March 14, 1936.
- 2.
AEA, 6–282, Letter from Bergmann to Einstein dated January 24, 1949.
- 3.
“Sie suchen einen selbständigen und neuen Weg zur Lösung der prizipiellen Schwierigkeiten. Bei diesem Bestreben kann einem Niemand helfen, am wenigsten Einer, der einigermassen fixierte Ideen hat. Sie wissen z. B., dass ich auf Grund gewisser Ueberlegungen fest glaube, dass der Wahrscheinlichkeitsbegriff nicht primär in die Realitätsbeschreibung eingehen darf, während Sie daran zu glauben scheinen, dass man zuerst eine Feldtheorie aufzustellen und diese dann nachträglich zu ‘quantisieren’ hat …Ihr Versuch, eine Feldtheorie abstrakt durchzuführen, ohne von vornherein über die formale Natur der Feldgrössen zu verfügen, erscheint mir nicht glücklich, weil formal zu arm und unbestimmt.” AEA, 6–282, Letter from Einstein to Bergmann dated January 26, 1949.
- 4.
AEA, 6313, Recommendation dated April 18, 1954.
- 5.
Explicitly, \(M^\alpha { }_\mu = \frac {1}{3!}\epsilon ^{\alpha \beta \gamma \delta }\epsilon _{\mu \nu \rho \sigma }\frac {\partial x^\nu }{\partial u^\beta } \frac {\partial x^\rho }{\partial u^\gamma }\frac {\partial x^\sigma }{\partial u^\delta }\).
- 6.
Goldberg has expressed to me the opinion that the parameterization idea was inspired by Weiss. Weiss had introduced a parameterized version of electromagnetism in flat spacetime, but he did not conceive of the x μ(u) as dynamical variables. See Rickles and Blum (2015) for more background and also Salisbury (2020b) for Weiss’ influence on Hamilton-Jacobi techniques in general relativity.
- 7.
- 8.
Bergmann (1949, 680).
- 9.
Equations (3.1) in BPSZ50 have incorrect right-hand sides. The typos were corrected in Anderson and Bergmann (1951), equation (4.3).
- 10.
- 11.
Bergmann et al. (1950, 88). Curiously, even though Joshua Goldberg had just begun his own thesis work (Goldberg 1952) under Bergmann’s direction at this time, and it was devoted both to the covariance foundations of the EIH approach and the Hamiltonian form of general relativity, he has communicated to me that he never shared this view.
- 12.
“I am sorry that we did not get together in New York …In the meantime, I have received the reprints you were kind enough to send me. I shall reciprocate in kind and send you one I have here at the moment and regularly send you my output …We are having a project at Syracuse in which we are attempting to investigate the general properties of covariant field theories (whether or not they be impressed on a Riemannian geometry), with the special purpose of learning how to quantize covariant theories. If you should have any papers relating to this general subject, I should certainly appreciate knowing about them.” Alfred Schild Papers at the University of Texas at Austin (ASP), Box 86-27/2.
- 13.
Private communication.
- 14.
Syracuse University Bergmann Archive, Syracuse, NY, USA (SUBA), Correspondence folder.
- 15.
ASP, Box 86-27/2. See also Blum and Salisbury (2018).
- 16.
ASP, Box 86-27/2.
- 17.
SUBA, Correspondence folder.
- 18.
SUBA, Correspondence folder.
- 19.
Private communication, and cf. also Salisbury (2020a).
- 20.
See Salisbury and Sundermeyer (2017) for a detailed analysis of this paper.
- 21.
This is a direct expression of the transformation property of a scalar density \(\mathfrak {S}\left (y(x)\right )\) of weight one under a coordinate transformation x ′μ(x), \(\mathfrak {S}'\left (y'(x')\right ) = \mathfrak {S}\left (y(x)\right ) \left |\frac {\partial x}{\partial x'}\right |\).
- 22.
See Salisbury and Sundermeyer (2017).
- 23.
This two-index object should not be confused with the parameters introduced in BPSZ50.
- 24.
See Salisbury and Sundermeyer (2017) for details.
- 25.
See Salisbury (2009) for more context and his correspondence with Dirac!
- 26.
- 27.
Bergmann (1979, 175).
- 28.
Anderson and Bergmann (1951, 1023).
- 29.
Anderson and Bergmann (1951, 1023).
- 30.
See Salisbury and Sundermeyer (2017) and in particular equation (37).
- 31.
Josh Goldberg has informed me that Bergmann himself took some time to come to this realization.
- 32.
I should add parenthetically that the evidence suggests that Rosenfeld was probably aware of the daunting challenge one faced in finding a constraint algebra that corresponded to the conventional diffeomorphism Lie algebra. Regarding the algebra he confined his attention to spatial diffeomorphisms whose generators did satisfy a closed Poisson bracket algebra. This discussion is in his Sect. 6 (Rosenfeld 1930, 2017). See also (Salisbury and Sundermeyer 2017, 43–44).
- 33.
Anderson explained to the author and Rickles in 2011 that his exposure to Dyson did not come from Syracuse, but rather from a visit to Mexico in the summer of 1951 where he worked with Alejandro Medina and “tried to understand Dyson’s paper, and the renormalization program. So then I got very much involved in quantum field theory.”
- 34.
For vacuum general relativity, we read off from \(\bar \delta g_{\mu \nu } = - g_{\mu \nu , \alpha } \delta \xi ^\alpha + 2 g_{\alpha (\mu } \delta \xi ^\alpha _{, \nu )} =: - g_{\mu \nu , \alpha } \delta \xi ^\alpha + F_{(\mu \nu ) \rho }{ }^{(\alpha \beta ) \sigma } \delta \xi ^\rho _{, \sigma }\) that \( F_{(\mu \nu ) \rho }{ }^{(\alpha \beta ) \sigma } = 4 \delta ^\sigma _{(\mu } \delta ^{(\beta }_{\nu )} \delta ^{\alpha )}_\rho \). The proposed field operator equations are then \(\left \{\hat L^{\sigma \alpha } \cdot \hat g_{\alpha \rho }\right \} = 0\).
- 35.
A January 1957 APS abstract with Bergmann (Janis and Bergmann 1957) is similarly limited in scope.
- 36.
It is not clear when Bergmann became aware of Komar’s work. Géhéniau summarized his joint work with Debever at the July 1955 Bern meeting. They proved that there existed at most 14 independent second-order spacetime scalars. Bergmann posed a question, following the presentation, relating to the existence of only four second differential order scalars that existed for vacuum spacetimes that possessed no symmetry. There is an edited proof (which does not appear to be in Bergmann’s handwriting) of the Géhéniau article that was to appear in the Bern proceedings. The proof is in the Syracuse Bergmann archives (SUBA) in a folder labeled Bern Correspondence and does not mention Komar. The document states that Bergmann later had a private discussion with Wigner, and Wigner’s response to Bergmann’s inquiry follows. However, the ultimate published version contains the comment “The question that must be decided (and that Komar in Princeton has also addressed) concerned the characterization of definitively distinct solutions of Einstein’s gravitational equations.” “Die Frage, die entschieden werden sollte (und die auch Herr Komar in Princeton angegriffen hatte) betraf die Charakterisierung wesentlich verschiedener Lösungen der EINSTEINschen Gravitationsgleichungen.” It is likely that Bergmann heard the basis of his question directly from Komar at the April 1955 Washington meeting of the American Physical Society, where they were both present on the same day. Komar likely referred in his report (Komar 1955) to an observation that would appear in his Ph.D. thesis (Komar 1956), citing the as yet unpublished (Géhéniau and Debever 1956) result that in a generic vacuum spacetime, there exist four independent scalar invariants of second differential order. We know that Komar did go to Syracuse as a postdoctoral researcher, presumably at the beginning of the 1957 academic year. He stayed at Syracuse until 1963, promoted eventually to Associate Professor (Goldberg 2013).
- 37.
A. Janis (1958). True observables and the generalized equivalence problem. SUBA, Correspondence folder.
- 38.
Letter from Bergmann to Janis dated September 15, 1958. SUBA, Correspondence folder.
- 39.
Letter from Bergmann to Dirac dated October 9, 1959. SUBA, Correspondence folder.
- 40.
Letter from Bergmann to Dirac dated October 9, 1959. SUBA, Correspondence folder.
- 41.
Letter from Dirac to Bergmann dated November 11, 1959. SUBA, Correspondence folder.
- 42.
Letter from Anderson to Bergmann, undated. SUBA, Correspondence folder.
- 43.
SUBA, Correspondence folder. There is a related undated letter. Anderson writes “Enclosed is what I hope is a corrected and correct version of the paper we discussed over the phone. As I mentioned then, I agree in the main with your comments and in fact appreciated them.” Unfortunately we are not in possession of the earlier draft.
- 44.
See Pons et al. (1997) for a proof.
- 45.
Letter from Bergmann to Rosen dated September 26, 1973. SUBA, Correspondence folder.
- 46.
In 1989, Bergmann (1989, 298), characterized Dirac’s procedure as having “first appeared by magic.” It is possible that he might have been inspired to employ the normal by Weiss’s work (Weiss 1936) – which he nominally supervised. But I now doubt this. The Weiss construction employs the covariant normal. It does not depend on the metric. But most significantly his canonical momenta are simply the conjugates of the field derivatives with respect to the parameter time, and his formalism did not contemplate reparameterization covariance – the context in which a notion of D-invariance would arise.
- 47.
Bergmann (1962), equation (27.11).
- 48.
He and Komar give the derivation later in Bergmann and Komar (1972).
- 49.
In Pons et al. (1997) we call it the “diffeomorphism-related group.”
- 50.
- 51.
Bergmann and Komar (1972, 175).
- 52.
Bergmann and Komar (1972, 176).
- 53.
See Salisbury (2020b) for a detailed analysis.
- 54.
Some authors do dispute this equivalence. See in particular (Kiriushcheva and Kuzmin 2011).
- 55.
Research status report for period December 1, 1964 to May 31, 1965, Bryce S. DeWitt Papers, Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin, Box 4RM235.
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Acknowledgements
I would like to thank Jürgen Renn and the Max Planck Institute for the History of Science for support offered me as a Visiting Scholar. Thanks also to Alexander Blum for his critical reading and many constructive suggestions.
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Salisbury, D. (2020). Toward a Quantum Theory of Gravity: Syracuse 1949–1962. In: Blum, A.S., Lalli, R., Renn, J. (eds) The Renaissance of General Relativity in Context. Einstein Studies, vol 16. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-50754-1_7
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