## Abstract

The main focus is on the Hamilton–Jacobi techniques in classical general relativity that were pursued by Peter Bergmann and Arthur Komar in the 1960s and 1970s. They placed special emphasis on the ability to construct the factor group of canonical transformations, where the four-dimensional diffeomorphism phase space transformations were factored out. Equivalence classes were identified by a set of phase space functions that were invariant under the action of the four-dimensional diffeomorphism group. This is contrasted and compared with approaches of Paul Weiss, Julian Schwinger, Richard Arnowitt, Stanley Deser, Charles Misner, Karel Kuchař—and especially the geometrodynamical program of John Wheeler and Bryce DeWitt where diffeomorphism symmetry is replaced by a notion of multifingered time. The origins of all of these approaches are traced to Elie Cartan’s invariant integral formulation of classical dynamics. A related correspondence concerning the thin sandwich dispute is also documented.

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## Notes

Quoting a remark that Deser made to myself and Dean Rickles in an interview that we conducted with him in 2011 regarding general coordinate covariance “There is a fundamental conflict between the classical formalism and blindly pushing it to the quantum level.” Upon which I commented that the position that he had represented historically is that there is no benefit to be gained from having pursued the classical theory of general covariance. He replied that that was indeed the case.

This lecture is, beginning on page 19, a translation into German of the Battelle Rencontres lectures. Dieter Brill assisted with the translation, and as Brill communicated to me in April, 2018, this is consistent with “his famous cutting and pasting (literally) of other publications,”

[42], p. 315, following equation (7), Dirac gave these relations as between “operators or dynamical variables” provided the derivatives are “well-ordered.”

See [131] for more detail on the historical origins.

Cartan used the letter \(\alpha \) rather than

*s*.Cartan dealt explicitly with the standard particle Lagrangian \(L = \frac{m}{2}\left( (v^1)^2 +(v^2)^2 +(v^3)^2 \right) - U(v)\).

Cartan represented these variations by \(\delta \) rather than

*d*. I employ the modern notation that corresponds to the introduction of one forms.[33], p. 15.

[33], p. 14.

I have translated into more familiar notation, closely related to Schwinger’s own in his original 1951 presentation of his quantum action principle [108]. Weiss represented his generic fields by

*z*rather than \(\phi \). Also, initially Weiss conceived of the fields \(\phi _\alpha \) as operators, but then for the purposes of the 1938 paper reverted to the classical theory so as to recover Poisson brackets.[119], p. 105.

He cited [33] for proof, stating that he had learned this method from Born in Cambridge in 1933.

Weiss used the notation

*w*rather than \(u^0\).I have altered Weiss’ notation to conform to modern usage. He represented my \(\pi ^\alpha \) by \(P_\alpha \) and \({{\mathcal {P}}}_\mu \) by \(X_\mu \).

Strangely, Weiss did not refer to Léon Rosenfeld’s 1930 opus [96, 98] in which he presented a constrained Hamiltonian analysis of not only electromagnetic fields in interaction with evolving charged spinorial sources, but also with evolving curved metrics. For a discussion of the application of Rosenfeld’s methods to quantum electrodynamics in flat spacetime and Dirac’s awareness of his formalism, see [105]. See [104] for a detailed analysis of Rosenfeld’s groundbreaking work in constrained Hamiltonian dynamics. Rosenfeld followed up with an overview of the formalism in his 1932 review of quantum electrodynamics [97]. Born cited this work in his initial 1934 Hamilton–Jacobi approach to quantum electrodynamics [26]. As Weiss was Born’s student at this time it is hard to imagine that Born would not have brought Rosenfeld’s work to Weiss’ attention. Even more mysterious is Dirac’s evident failure to notify Weiss after Dirac took over Weiss’ thesis direction.

“During the 25 year period of quantum electrodynamical development, there was great formal progress in the manner of presenting the laws of quantum mechanics, all of which had its inspiration in a paper of Dirac. This paper (which is No. 26 in the collection, Selected Papers on Quantum Electrodynamics, Dover, 1958) discussed for the first time the significance of the Lagrangian in quantum mechanics.” [112], p. 420. Reference is to [111].

Schwinger represents the action by

*W*rather than*S*.Schwinger wrote this expression in terms of exact differentials.

See [109] for a compact discussion of the more general case in which anti-commutation relations arise.

see [100] for further details.

Joshua Goldberg informed me that Dirac was the direct cause.

See [100].

The article contains a note added in proof referring to a work [20] that they submitted for publication on March 31, 1960.

[4], p. 232.

Indeed, in remarks made in response to Bergmann’s talk at the Royaumont meeting in June, 1959 [21], they offered objections to the use of Weyl scalars with reasoning that would seem to apply also to their own coordinate conditions.

[48], p. 1161.

[89], p. 60.

They were later provided by Komar [71].

As Komar observed, one must assume either the spatial manifold is closed, or appropriate asymptotic conditions.

The square root was missing in Bergmann’s account, and was added by Komar in 1967 [71].

I have changed the symbol in an attempt to avoid some confusion that could result from Komar’s notation. Komar called this vanishing invariant \(\alpha _A\left( \vec {x}; g_{ab}(\vec {x}) , \frac{\delta S}{\delta g_{cd}}(\vec {x}) \right) \), [72], equation (1.8).

Dean Rickles and I learned this in an interview we conducted with Misner in 2011.

Komar completed his own doctoral thesis [70] under Wheeler’s direction in 1956.

[86], p. 499.

Syracuse University Bergmann Archive (SUBA), correspondence folder.

The final version was published in 1961 [125].

SUBA, correspondence folder.

SUBA, correspondence folder.

SUBA, correspondence folder.

Bergmann wrote

*N*instead of \(N^{-1}\), but this did not affect his argument.SUBA, correspondence folder.

SUBA, correspondence folder.

SUBA, correspondence folder.

SUBA, correspondence folder.

[15], pp. 51–52.

[68], p. 821.

[35], p. 480.

The definitive status of the conjecture seems to be that enunciated by Bartnick and Francaviglia [7], namely that the conjecture is valid provided certain geometrical conditions are assumed, and they are “not generically satisfied.” This result was generalized by Giulini [52] to include specific material sources.

Bryce DeWitt Papers (BDP), Briscoe Center for American History.

BDP.

[41], p. 60.

[74], p. 924.

My direct sources here include Jim Anderson, Dieter Brill, Joshua Goldberg, Karel Kuchař, Charles Misner, Ted Newman, Ralph Schiller, and Louis Witten.

Private communication.

Interview that I and Charles Torre conducted with Kuchař in April, 2016.

Interview that Dean Rickles and I conducted with Deser in March, 2011.

Private communication.

[99], p. 48.

[99], p. 52.

“The only remnants of this group on the instantaneous level are, in the ADM (Arnowitt, Deser and Misner [1962]) language, the super-hamiltonian H and supermomenta J which are interpreted as the generators of temporal and spatial deformations of a Cauchy surface, respectively. But these deformations do not form a group, nor are H and J components of a momentum map,” [55], p. 4.

[34], Chap. IV, Sec. C.

Bergmann makes this specific assertion in [16].

Bergmann did undertake measurability research involving the Weyl tensor with his student Gerrit Smith, but no use was made of Hamilton–Jacobi techniques. See for example [116].

An initial discussion appears in [101, 102], with further corrections in arXiv:1508.01277.

This is also essentially the prepublication summary that Carathéodory communicated to Weyl in 1935 in response to Weyl’s article. It is to be found on page 210 in the excellent Carathéodory biography by Maria Georgiadou [49].

de Donder called this a “généralisation du théorème direct de Jacobi,” [36], p. 116.

See [104] for an analysis of this work.

An initial analysis parameterized generally covariant theory was first undertaken by Bergmann [10] and Bergmann and Brunings [18]. The mystery deepens with the recognition that Rosenfeld had collaborated with de Donder in Paris in 1927, and had returned to Paris in 1931 to give a four-week course at the Institut Poincaré (see [29], p. 58) following the completion of his groundbreaking paper on constrained Hamiltonian dynamics. Rosenfeld would then presumably have discussed his work with de Donder before the latter published his book on the calculus of variations.

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## Acknowledgements

Thanks to Alex Blum for his critical reading of an earlier incomplete draft of this paper. And 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 both to him and Kurt Sundermeyer for some valuable comments on an earlier draft.

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## Appendices

### Appendix A: Generalizations of the Hamilton–Jacobi methods

De Donder, Weyl, and Carathéodory established the foundations for a generalization of the Hamilton–Jacobi equations, and I will in particular briefly describe the Carathéodory theory as nicely summarized by von Rieth [117].^{Footnote 71} It is basically an extension of the Poincaré-Cartan form \(p_i \delta q^i - H(q,p) \delta t\) to field theory. We wish to minimize the action \(A = \int d^4\!x {{\mathcal {L}}}(x^\mu ,\phi _A(x), \partial _\mu \phi _A(x)\). There exists an equivalent action obtained by subtracting a total divergence \(\Phi (x, \phi , \partial \phi )= \frac{dS^\mu (x,\phi )}{dx^\mu }=\partial _\mu S^\mu (x,\phi ) + \frac{\partial S^\mu (x,\phi )}{\partial \phi _A} \partial _\mu \phi _A \) from this Lagrangian. The fundamental idea is to place restrictions on unknown functions \(S^\mu (x,\phi )\) and \(\psi _{A\mu }(x,\phi )\) with \(\bar{{\mathcal {L}}}(x,\phi ,\partial \phi ) := \mathcal{L}(x,\phi ,\partial \phi ) - \Phi (x, \phi ,\partial \phi )\), such that

and \(\bar{{\mathcal {L}}}(x,\phi ,v(x,\phi )) > 0\) when \(v_{A \mu }(x,\phi ) \ne \psi _{A\mu }(x,\phi )\). Since by these assumptions the variation of \(\hat{{\mathcal {L}}}(x,\phi ,v)\) about \(v_{A \mu }(x,\phi ) = \psi _{A\mu }(x,\phi )\) vanishes, we have when varying only \(v_{A\mu }\), \(0 = \frac{\partial {{\mathcal {L}}}(x,\phi ,v)}{\partial v_{A\mu }}\bigg | _{v = \psi } \delta v_{A\mu }- \frac{\partial S^\mu (x,\phi )}{\partial \phi _A} \delta v_{A\mu }\). Therefore we conclude that

and (9.1) becomes

Finally we can derive Carathéodory’s first main result: Consider a new action

Substituting from (9.3) and employing (9.2) this becomes

where in the last step we apply Stoke’s theorem. Thus when we assume that the values of \(\phi _A\) are fixed on the 3-dimensional closed spacetime boundary the integral is independent of \(\phi _A\) in the interior! This is an extension of David Hilbert’s “independence theorem” to which Weiss’ thesis advisor Born referred in his 1934 [26] article.^{Footnote 72} As observed by Born, in a model in which there is only one independent variable *t* and *L* has no explicit *t* dependence we have

where \(p^A(\phi ,\psi ) := \frac{\partial L(\phi ,\psi )}{\partial \psi _A}\) and \(H(\phi ,\psi ) := -L(\phi ,\psi )+ \frac{\partial L(\phi ,\psi )}{\partial \psi _A} \psi _A \). The fact that (9.6) is an exact differential leads to a set of differential equations that must be satisfied by the unknown “geodesic fields” \(\psi _A(\phi )\). Because \(\frac{\partial {\hat{A}}}{\partial \phi _A} = p^A\) and \(\frac{\partial {\hat{A}}}{\partial t} = -H\) we have as a consequence that \(\frac{\partial p^A}{\partial \phi _B} - \frac{\partial p^B}{\partial \phi _A} = 0\) and \(\frac{\partial H}{\partial \phi _A} - \frac{\partial p^A}{\partial t} = 0\). These are differential equations to be satisfied by the \(\psi _A(t,\phi )\). But as Born points out the most efficient way to solve them is to solve first for \({\hat{A}}\) by replacing the \(p^A\) argument in *H* by \(\frac{\partial {\hat{A}}}{\partial \phi _A}\) - where it is assumed, of course, that one is dealing with a non-singular theory for which one can solve for the velocities \(\psi _A\) in terms of the momenta \(p^B\). The result is the standard Hamilton–Jacobi equation

Carathéodory and de Donder did extend these methods to field theory, but as far as I can tell their treatments did not include models in which arbitrary gauge symmetries were present. In the non-singular case where one can solve for \(\psi _{A\mu }\) in terms of the generalized canonical momenta \(p^{A\mu } := \frac{\partial \mathcal{L}(x,\phi ,\psi )}{\partial \psi _{A\mu }}\), and \({{\mathcal {H}}} := p^{A\mu } \psi _{A\mu } - {{\mathcal {L}}} = {{\mathcal {H}}}(x,\phi ,p)\). One then obtains generalized canonical equations of the form \(\partial _\mu \phi _A = \frac{\partial {{\mathcal {H}}}}{\partial p^{A\mu }}\) and \(\partial _\mu p^{A\mu } = - \frac{\partial \mathcal{H}}{\partial \phi _A}\). And according to (9.3) we obtain the generalization of the Hamilton–Jacobi equation^{Footnote 73}

There is a mystery regarding de Donder’s incorporation of a special kind of covariance symmetry in his formalism. He did consider so-called parameterized finite dimensional and field theoretic models. He showed that a necessary and sufficient condition for reparameterization covariance was that the Lagrangian be homogeneous of degree one in derivatives with respect to the parameters. The corresponding generalized Hamiltonian was therefore of degree zero, with the consequence that what we now call a constraint would arise. One particularly relevant finite dimensional case is the relativistic free particle, and in this case the analysis is equivalent to that first developed by Rosenfeld in 1930 [96, 98],^{Footnote 74} and further amplified by Bergmann and Dirac. The puzzle for me is that he did succeed in deriving the generalized Hamiltonian for Einsteinian gravity with the corresponding generalized Hamilton and Hamilton–Jacobi equations, yet he did not address the underlying general covariance.^{Footnote 75}

### Appendix B: The free relativistic particle

I will illustrate the Bergmann Komar Hamilton–Jacobi analysis with the free relativistic particle with spacetime position \(q^\mu \) as a function of the parameter \(\theta \). Consider the reparameterization covariant Lagrangian \(L_p = - m \left( -{\dot{q}}^2\right) ^{1/2}\), where \({\dot{q}}^\mu := \frac{dq^\mu (\theta )}{d \theta }\). We have \(p^\mu = m {\dot{q}}^\mu /\left( -{\dot{q}}^2\right) ^{1/2}\) and the Hamiltonian constraint \(H = p^2 + m^2 = 0\), with the Hamilton–Jacobi equation \(\eta ^{\mu \nu }\frac{\partial S}{\partial q^\mu } \frac{\partial S}{\partial q^\nu } + m^2 = 0\). First I address Bergmann’s 1966 argument that *S* cannot depend on \(\theta \). It follows here from the identity

On the other hand,

so we conclude that \( \frac{\partial ^2 S}{\partial q^\mu \partial \theta } = 0\), and therefore \(\frac{\partial S}{\partial \theta } = f(\theta )\), which does not affect the dynamics and can be taken to be zero.

Next I consider Komar’s selection of invariants in identifying equivalence classes of solutions. This is really a selection of gauge conditions, in a manner that will be discussed in detail in [103]. As Komar notes, without these conditions it is not possible to solve the Hamilton–Jacobi equation for the momenta in terms of the configuration variables. The simple choice I will make here is

where the \(\alpha ^a\) are numerical constants, but there are of course many other possibilities. The \(p^a\) is the analogue of \(\alpha ^0_A\) in (6.7). These are vanishing invariants under evolution in \(\theta \) since as we know \({\dot{p}}^a = 0\). Thus we now have a set of four Hamilton–Jacobi equations which are

and consequently, from \(H\left( p\right) = 0\),

We conclude that \(S = \alpha _a q^a - \left( {\vec {\alpha }}^2 + m^2 \right) ^{1/2} q^0\).

Continuing with the extension of Komar’s analysis to this model, we have

This is indeed invariant under infinitesimal reparameterizations \(\theta ' = \theta - \epsilon (\theta )\) on the constraint surface \(p^0 =\left( {\vec {p}}^2 + m^2 \right) ^{1/2}\), whereby \(\delta q^\mu = {\dot{q}}^\mu \epsilon \), so

Continuing with the notation used in representing the numerical value of the invariant, I will represent this numerical value by \(\beta ^a\) and the corresponding phase space function by \(\beta _0^a := q^a - \frac{\alpha ^a}{\left( {\vec {\alpha }}^2 + m^2 \right) ^{1/2} } q^0\). Then we have the vanishing invariant \({\bar{\beta }}^a = \beta _0^a - \beta ^a = 0\). Note also, as in Komar’s vacuum gravitational case, the invariant phase space functions satisfy the canonical Poisson bracket \(\left\{ {\bar{\beta }}_a, {\bar{\alpha }}^b \right\} = \delta ^b_a\).

Of course, from (9.13) we obtain the free relativistic particle solution in the chosen gauge, namely \(\theta = q^0\),

Notice also that just as was noted by Komar in the case of general relativity, the action *S* actually undergoes a change under the action of the reparameterization group,

Komar connected this change with a change under time evolution. One must however distinguish between evolution and transformations of solution trajectories under diffeomorphisms. This is spelled out in detail in [93] and will be discussed further in [103]. A specific field dependence is required to obtain variations in configuration-velocity space that are projectable under the Legendre map to phase space. In this model the infinitesimal descriptor of projectable reparameterizations take the form \(\epsilon = \left( - {\dot{q}}^2 \right) ^{-1/2} \xi (\theta )\). And under these reparameterizations the action undergo the \(q^0 = \theta \) dependent variations \(-\xi (q^0)m^2\).

I observe finally that the numerical value of \(p^a =\alpha ^a \) fixes an equivalence class under reparameterizations. The invariant \({\bar{\beta }}^a\) alters this value.

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Salisbury, D. A history of observables and Hamilton–Jacobi approaches to general relativity.
*EPJ H* **47**, 7 (2022). https://doi.org/10.1140/epjh/s13129-022-00039-8

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DOI: https://doi.org/10.1140/epjh/s13129-022-00039-8