L\'eon Rosenfeld's general theory of constrained Hamiltonian dynamics

This commentary reflects on the 1930 discoveries of L\'eon Rosenfeld in the domain of phase-space constraints. We start with a short biography of Rosenfeld and his motivation for this article in the context of ideas pursued by W. Pauli, F. Klein, E. Noether. We then comment on Rosenfeld's General Theory dealing with symmetries and constraints, symmetry generators, conservation laws and the construction of a Hamiltonian in the case of phase-space constraints. It is remarkable that he was able to derive expressions for all phase space symmetry generators without making explicit reference to the generator of time evolution. In his Applications, Rosenfeld treated the general relativistic example of Einstein-Maxwell-Dirac theory. We show, that although Rosenfeld refrained from fully applying his general findings to this example, he could have obtained the Hamiltonian. Many of Rosenfeld's discoveries were re-developed or re-discovered by others two decades later, yet as we show there remain additional firsts that are still not recognized in the community.


Introduction
Léon Rosenfeld's 1930 Annalen der Physik [53] 1 paper developed a comprehensive Hamiltonian theory to deal with local symmetries that arise in Lagrangian field theory. Indeed, to a surprising degree he established the foundational principles that would later be rediscovered and in some respects extended by the individuals who until recently have been recognized as the inventors of the methods of constrained Hamiltonian dynamics, Peter Bergmann and Paul Dirac. Not only did he provide the tools to deal with the only local gauge symmetries that were known at the time, namely local U(1) and local Lorentz covariance, but he also established the technique for translating into a Hamiltonian description the general covariance under arbitrary spacetime coordinate transformations of Einstein's general theory of relativity. Some of this pioneering work either became known or was independently rediscovered over two decades later. But for unknown reasons Rosenfeld never claimed ownership, nor did he join later efforts to exploit his techniques in pursuing canonical approaches to quantum gravity. 2 He was brought to Zurich in 1929 by Wolfgang Pauli with the express purpose of helping to justify procedures that had been employed by Heisenberg and Pauli in their groundbreaking papers on quantum electrodynamical field theory. With the understanding that second quantization should naturally include all known fundamental interactions, Rosenfeld and Pauli apparently jointly decided that a new procedure was needed that would also take into account the dynamics of Einstein's gravitational field in interaction with electromagnetism and charged spinorial source fields. Among Rosenfeld's achievements are the following: He was the first to (1) Show that primary phase space constraints always arise as a consequence of local Lagrangian symmetries, (2) Show that local symmetries always involve singular Lagrangians, (3) Exploit the identities that result from the symmetry transformation properties of the Lagrangian 3 to construct the constrained Hamiltonian that contains arbitrary spacetime functions, (4) Translate the vanishing conserved charge 2 He did present an unpublished seminar entitled "Conservation theorems and invariance properties of the Lagrangian" in Dublin in May of 1946 where he repeated the invariance arguments but did not relate the discussion to phase space. Niels Bohr Archive, Rosenfeld Papers. 3 These identities were first exploited by Felix Klein in the context of general relativity, as we shall discuss below. that arises as a consequence of symmetry transformations of the Lagrangian into a phase space expression, (5) Show explicitly that this symmetry generator, which we call the Rosenfeld-Noether generator, generates the correct infinitesimal transformations of all of the phase space variables, (6) Derive secondary and higher constraints through the requirement that primary constraints be preserved in time, (7) Show how to construct the constrained Hamiltonian and general covariance generator for general relativity -both for vacuum relativity and gravitation in dynamical interaction with the electromagnetic field and charged spinorial sources. Most of the advances listed here a now accepted wisdom -yet none have until recently been attributed to Rosenfeld. Following a brief introduction to Rosenfeld in section 2 we will illustrate the seven accomplishments using two familiar simple models, the free electromagnetic field and the relativistic free particle. Then in section 4 we will present a detailed analysis of the first six of these achievements, referring to the general theory in Part 1 of his article. Where possible we employ Rosenfeld's notation. Section 5 is devoted to a description of the seventh achievement as it is related to Rosenfeld's general relativistic application. In section 6 we will take Rosenfeld's general findings and apply them to his example. Here we revert to modern notation and construct in detail the Hamiltonian and symmetry generators for Rosenfeld's tetrad gravity in interaction with the electromagnetic and spinorial fields. It is curious that he did not give the explicit expressions for the Hamiltonian in either the 1930 paper or the 1932 follow-up in which he reviewed the then current status of quantum electrodynamics.
In an Appendix we will give a capsule history of the later, better-known development of constrained Hamiltonian dynamics.
Before proceeding, it is clear from the title of Rosenfeld's article that he aimed at quantizing the Einstein-Maxwell-Dirac field. From the modern perspective he could perhaps be accused of a certain naivete in supposing that his fields could be promoted to quantum mechanical q-numbers through the simple expedient of forming a self-adjoint Hermitian operator by taking one half of the sum of the field operator and its Hermitian adjoint. But this is what he did in his equation preceding (R10). (Henceforth we will refer to his equations by adding the prefix R). The corresponding self-adjoint operators are expressed with an underline. This notation tends to make his text harder to read than necessary. And since, unless otherwise noted, we will be discussing the classical theory we will omit these underlines. This was a period of intense debate and evolving views regarding the recently established theories of wave and matrix mechanics, and Rosenfeld was ideally placed amidst the contenders. His position was somewhat unique given his working knowledge of general relativity and his previous efforts in unifying relativity with the incipient quantum wave theory. In addition to seeking an assistantship with Niels Bohr he also wrote to Albert Einstein, proposing that if he were successful with Einstein's aid in obtaining a research fellowship from the International Education Board he work under Einstein's supervision "on the relations between quantum mechanics 4 For more on Rosenfeld's life and collaborations see [32] and relativity." 5 Einstein replied almost immediately from Berlin, endorsing the project. 6 Rosenfeld also sought at the same time an arrangement with Niels Bohr who wrote back, advising according to Rosenfeld that coming to Copenhagen at the moment " was not convenient and I had better postpone it". 7 Finally he wrote to Wolfgang Pauli who invited him to come to Zurich. Surprisingly, given Rosenfeld's general relativistic background, when asked by Pauli what he wished to do in Zurich, he replied that he intended to work on a problem involving the optical properties of metals. But he "got provoked by Pauli to tackle this problem of the quantization of gravitation and the gravitation effects of light quanta." 8 In an autobiographical note written in 1972 Rosenfeld says that in Zurich, where he arrived in the Spring of 1929, he "participated in the elaboration of the theory of quantum electrodynamics just started by Pauli and Heisenberg, and he pursued these studies during the following decade; his main contributions being a general method of representation of quantized fields taking explicit account of the symmetry properties of these fields, a general method for constructing the energy-momentum tensor of any field, a discussion of the implications of quantization for the gravitational field . . ." 9 10 tor potential A µ with associated field tensor F µν = A ν,µ − A µ,ν where A ν,µ := ∂Aν ∂x µ . (We take the metric to have diagonal elements (1, −1, −1, −1)). The flat space free electromagnetic field Lagrangian is The statement that the Lagrangian is invariant under the local symmetry (3.2) is the identity (Note that ∂Lem ∂Aµ,ν = F µν = −F νµ and the identity results as a consequence of this anti-symmetry.) In particular, the coefficient of each distinct ξ ,αβ vanishes identically when these coefficients are understood as functions of A µ,ν . But now we introduce momenta p α conjugate to the A β . DefiningȦ ν := A ν,0 , the momenta are defined to be p α := ∂Lem ∂Ȧα . The seven Rosenfeld results applied to this model, numbered in parentheses, are (1) There is a primary constraint expressing the vanishing of the coefficient of ξ ,00 .
(2) In making the transition to a Hamiltonian version of free electromagnetism we would like to be able to solve the defining equations for the momentum for the velocitiesȦ µ in terms of the A ν , A ν,a ( where a is a spatial index), and p α . This is clearly not possible in this case sinceȦ 0 does not even appear in these relations. Another way of viewing this problem is to note that the defining relations are linear in the velocities and take the form then to solve for the velocities in terms of the momenta we would need to find the reciprocal of the Hessian matrix ∂ 2 Lem ∂Ȧα∂Ȧ β . But this matrix is singular since It is singular as a consequence of the invariance of the Lagrangian under the gauge transformation (3.2).
(3) Since the time derivative of the nought component of the potential does not appear in the momenta, we can choose any value we wish for it without violating these relations. So let us takeȦ 0 = λ where λ is an arbitrary spacetime dependent function. The remaining velocities can be solved, yieldingȦ a = p a + V ,a .
Substituting into the canonical Hamiltonian we find The field B a = ǫ abc A b,c is the magnetic field.
(4) The identity (3.3) can be conveniently rewritten in terms of the Euler-Lagrange equations, We deduce that when the Euler-Lagrange equations are satisfied we have a conserved charge where we have assumed that the arbitrary ξ go to zero at spatial infinity. Since ξ also has arbitrary time dependence it is clear that in addition to the primary constraint p 0 = 0 we must also have a secondary constraint p a ,a = 0.
(5) The constraint M em generates the infinitesimal symmetry transformations and δp a = 0.
(6) The deduction (4) may be understood as a derivation of a higher order(secondary) constraint in the sense that if we write M em = d 3 x p 0ξ + N ξ , then we have d dt M em = d 3 x ṗ 0ẋ i + p 0ξ +Ṅ ξ + Nξ = 0. The vanishing of the coefficient ofξ then yieldsṗ 0 = −N = 0.
(7) Since this model is not generally covariant the achievement number seven is not relevant.
Our next model is generally covariant, and it will serve to display some important differences with models that obey internal gauge symmetries like the U(1) symmetry.
We consider the parameterized free relativistic particle. Let x µ (θ) represent the particle spacetime trajectory parameterized by θ. Under a reparameterization θ ′ = f (θ), where f is an arbitrary positive definite function, x µ transforms as a scalar, We introduce an auxiliary variable N(θ) and we assume that it transforms as a scalar density of weight one, Then the particle Lagrangian takes the form whereẋ µ := dx µ dθ . It is quadratic in the velocities and Rosenfeld's general theory is therefore directly applicable. The Lagrangian transforms as a scalar density of weight one under parameterizations, i.e., Consequently, the equations of motion are covariant under reparameterizations.
Now consider an infinitesimal reparameterization θ ′ = θ+ξ(θ) with corresponding variations δx µ (θ) := x ′µ (θ + ξ(θ)) − x µ (θ) = 0, Then (3.6) yields the identity Again it will be convenient to express this identity in terms of the Euler-Lagrange equations. For this purpose we introduce the δ * variation associated with the infinitesimal reparameterization. (It is actually minus the Lie derivative.) To save writing we will represent the variables x µ and N by a generic Q α . For an arbitrary function of variables Φ we define This has the property that δ * Φ = d dθ (δ * Φ). In terms of the Q α the identity (3.7) is Thus we may rewrite (3.8) as where δLp δQα = ∂Lp ∂Qα − d dθ ∂Lp ∂Qα = 0 are the Euler-Lagrange equations. One final rewriting of this identity yields a conserved charge. Substituting δQ α = δ * Q α +Q α ξ we find Proceeding with Rosenfeld's achievements applied to this model we have (1) The second derivativeξ could arise in (3.7) only ifṄ were to appear in the Lagrangian, and this would spoil to reparameterization covariance. Thus we must have p N := ∂Lp ∂Ṅ = 0.
(3) We can takeṄ = λ where λ is positive-definite but otherwise an arbitrary function of θ. The remaining velocities follow from the definitions Solving forẋ µ we haveẋ Substituting these velocities into the Hamiltonian we have (4) According to (3.10) the conserved charge associated with the free relativistic particle is where we have used the same procedure described in item (3) to obtain a phase space function involving also the arbitrary function λ.
In the last equality we used the equation of motion. This is the correct δ * variation for a scalar. Also we have where again in the last equality we used the equation of motion. This is the correct δ * variation of a scalar density. (6) We deduce that in addition to the primary constraint p N = 0 we have a secondary constraint p 2 + m 2 = 0.
(7) This is a generally covariant model, and as we shall see, the construction of the generator of infinitesimal diffeomorphisms does also apply to general relativity. It is significant, however, that the charge we have obtained only works for infinitesimal variations. As we shall discuss in detail later, this deficiency is related to the fact that we need to apply the equations of motion in order to obtain the correct variations.

Rosenfeld's original contributions in the general theory
Concerning the invention of constrained Hamiltonian dynamics there is little in the work of Bergmann [5] , Bergmann and Brunings [8], Dirac [18,19], Bergmann, Penfield, Schiller, and Zatkis [10], Anderson and Bergmann [1], Heller and Bergmann [30], and Bergmann and Schiller [11] that was not already achieved or at least anticipated by Léon Rosenfeld over twenty years earlier. He also pioneered the field of phase space symmetry generators.
Rosenfeld assumed that the Lagrangian was quadratic in the field velocities, taking the form U(1) transformations with no index ξ, and local Lorentz transformations with a latin index ξ r . Rosenfeld does not make this distinction in his abstract formalism, and it is our hope that this notation will make his article more accessible.
Accordingly, the symmetry variations of the field variables are Rosenfeld lets a "prime" represent the transformed variable, and Rosenfeld also introduced δ * variations with the definition where Φ is any functional of x and Q(x) and ∂Φ(x) ∂x ν is the partial derivative with respect to the spacetime coordinate. The δ * variations are minus the Lie derivative in the direction δx ν . Utiyama [62] in 1947 followed Rosenfeld's lead in employing the δ * notation. E. Noether [44] in 1918 denoted these variations in the functional form byδ. P. G. Bergmann [5], beginning in 1949, continued Noether's use of theδ notation. These variations are now called "active" variations.
or equivalently where δL δQα − ∂L ∂Qα ,µ = 0 are the Euler-Lagrange equations and as stated above K µ varies only under internal symmetries with descriptors ξ r . Rosenfeld assumed it be linear in derivatives of the field, In fact, since only δQ α (x) = c αr (x, Q)ξ r (x) comes into play in the variation of K µ , it follows since the identity (4.5) cannot depend on second derivatives of the ξ r that the variation of K µ ,µ takes the form δK µ ,µ = (r αµ c αr ξ r ) ,µ =: (I µ r ξ r ) ,µ , (4.8) where according to (R73) Thus we will work with the identity (4.5) in the form This identity incorporates (R12), (R13) and (R75).
Since according to (4.4) we can equivalently write (4.6) in the form, This relation does not appear explicitly in Rosenfeld's work, but he then exploited the several identities that follow from these fundamental identities, namely the identical vanishing of the coefficient of each derivative of the arbitrary ξ µ , ξ, and ξ r . He was not the first to deduce these identities. This discovery can be traced to Felix Klein [35], and although Rosenfeld did not specifically identify Klein's procedure, he did cite one of his essential results, namely the appearance of the four field equations that did not involve accelerations when using Einstein's 1918 Lagrangian that was quadratic in the time derivatives of g µν [23]. 11 In any case, Rosenfeld was the first to project these relations to phase space. We think it likely that it was the Klein procedure that Rosenfeld refered to in his introduction when he noted that "in the especially instructive example of gravitation theory, Professor Pauli helpfully indicated to me a new method that allows one to construct a Hamiltonian procedure in a 11 See his remark preceding equation (R120).
definitely simpler and natural way when identities are present". Pauli had exploited one of these identities in his Encyclopedia of the Mathematical Sciences contribution on relativity [45], and had cited Klein. One might be justified in interpreting this sentence as a recognition by Rosenfeld that Pauli had communicated to him the fundamental ideas of the general theory presented in this paper. We will comment on this hypothesis in our concluding remarks. 12 Indeed, the series of volumes was Klein's creation, and Klein carefully read the article and offered constructive criticism. 13 (1) Primary constraints Substituting (4.3) into the identity (4.10) we find that the identically vanishing while the coefficients of ξ ,ρσ give With regard to the remaining transformations, the coefficient of ξ r ,µ gives ∂L ∂Q α,µ c αr + I µ r ≡ 0. (4.14) After introducing the momenta P α := ∂L ∂Qα Rosenfeld obtains the phase space constraints (R18c) and In fact, Pauli derived the contracted Bianchi identities in the same manner that was later employed by Bergmann for generally covariant theories [5]. He performed an integration by parts of the identity, and then let the ξ µ on the boundary vanish. Pauli did not offer a genuinely Klein inspired approach until his updated annotated relativity article appeared in 1958 [46]. He shared this derivation first in a letter dated 9 October 1957, addressed to Charles Misner [64]. 13 See the discussion of the article in [24] This last relation corresponds to (R79). 14 Looking at the vanishing coefficient of Q α,µν in the identity (4.11) under the variations δQ α = c αr ξ r , Rosenfeld showed in (R80) that I 0 r is independent ofQ α . Thus the three relations (4.15)-(4.17) are primary constraints, using the terminology introduced by Anderson and Bergmann in 1949 [1].

(2) Singular Lagrangians
For the quadratic Lagrangian the momenta take the form where A αβ and D are functions of Q γ and their spatial derivatives. Then (4.12), (4.13) and (4.14) deliver the additional identities, and corresponding to (R25) and (R26). The first is the statement that the c 0 αµ , c 0 α and c 0 αr are null vectors of the Hessian matrix 15 A αβ . As we shall see, Rosenfeld used all of these relations in his construction of the Hamiltonian.

(3) Construction of the Hamiltonian
In solving (4.18) for the velocities, Rosenfeld refered to "the theory of linear equations" but did not give an explicit reference. His procedure was unique as far as we can tell. Here we repeat Rosenfeld's argument in §3, filling in some additional details to make the argument more comprehensible. 14 Rosenfeld actually defines F := P α c 0 αr , in his Case 2. Thus in an effort to introduce a unified and hopefully more comprehensible notation, we are representing the actual constraint with a 'prime'. 15 Cecile DeWitt-Morette indicated to one of us several years ago that she denotes this the "Legendre matrix", but the Hessian terminology now seems to be widespread.
He first supposed that he had found, presumeably through a suitable linear combination of the linear equations (4.18), a non-singular submatrix of the Hessian matrix of rank N − r 0 , where N is the total number of Q α variables, and r 0 is the number of primary constraints. Label the indices of the non-singular matrix by α ′ and the remaining indices by α ′′ . Let A α ′ β ′ represent the inverse of the non-singular Explicitly, Now since these null vectors must be expressible as linear combinations of the c 0 α r , c 0 α µ and c 0 α , it follows that since c 0 and we have therefore solved for P α ′′ as a linear combination of the P α ′ , A similar relation holds for D α ′′ . It follows that a special solutionQ 0 α of (4.18) is The general solution is thereforė where the λ µ , λ and λ r are arbitrary functions.
This method for solving linear singular equations is to be contrasted with a procedure pursued by Bergmann and his collaborators, beginning in 1950 [6]. His group employed the so-called "quasi-inverses", but again without explicit references.
The procedure was first published by E. H. Moore in 1920 [39]. It was subsequently rediscovered and extended by R. Penrose in 1955 [47]. Dirac invented his own idiosyncratic method in 1950 [18]. 16 Rosenfeld then substituted the general solutions (4.22) into the standard Hamil- Explicitly, we have we have finally Rosenfeld's (R35), where The Hamilton equations follow as usual from the variation of the Hamiltonian density, where we used (R33) and the Euler-Lagrange equations. Rosenfeld did not posit a new variational principle. 17 He simply proved that the Hamiltonian equations, with the Hamiltonian containing the arbitrary functions λ µ , λ and λ r , are equivalent to the Euler-Lagrange equations.

(4) Diffeomorphism and gauge generators
For the purpose of constructing the phase space generators of infinitesimal symmetry transformations it is convenient to rewrite the identity (4.11) equivalently as as Rosenfeld does explicitly in (R56c) for his Case 1.
This form of the identity is actually the basis of Noether's second theorem.
She gives the scalar density form explicitly in her equation (13)  17 See [27], for example, where in the context of non-singular systems one speaks of the 'modified Hamilton's principle' δ t2 t1 p iq i − L(q, p, t) = 0. 18 "With this information it is possible to show that theC of (6.8) is actually the generator of thē δy A and theδπ A transformations. The calculation is straightforward and closely follows a similar calculation in Rosenfeld, 13 so that we shall not carry it out". The reference is to the paper we are analyzing here.
to a verbal communication from Fräulein Emmy Noether herself". 19 One might conclude that Noether was essentially involved in his work [36].
On-shell, that is on the solutions of the Euler-Lagrange equation δL δQα = ∂L ∂Qα − ∂ ∂x µ ∂L ∂Qα,µ = 0, the identity (4.25) implies that the current is conserved, from which by applying the Gauß integral theorem one obtains a conserved charge where we introduced the momenta, the Hamiltonian density (4.24) and the energy- Rosenfeld gives no reference for these constructions, but it is most likely that he learned of these objects from Pauli [45], who in turn refers to F. Klein [35]. The pseudo-tensor was in fact first written down by Einstein [23], and that publication stimulated the symmetry analysis of E. Noether [44] and Klein.
Rosenfeld was the first to promote the vanishing charge (4.26) to a phase space symmetry generator, and also the first to show that it is a linear combination of phase space constraints. Up to the time that Rosenfeld accomplished this feat, attention had been paid only to the nonvanishing conserved Noether charges that follow from global symmetries. Strangely, although it is manifestly evident in Rosenfeld's analysis, he never stated explicitly that this charge was constrained to vanish.

(5) Infinitesimal variations generated by the Rosenfeld-Noether generator
In a tour de force Rosenfeld proved that the charges (4.26) generated the correct δ * Q α and δ * p α variations of all of the canonical variables under all of the infinitesimal symmetry transformations.This is obvious for the configuration variables Q α since It is less obvious for variations of the momenta, but Rosenfeld gives an explicit proof. Bergmann and his collaborators, who did over twenty years later consider the realization of general coordinate transformations as canonical transformations, did not provide an analogous proof.
Rosenfeld showed in the equation preceding (R51) that for the generalized mo- Then he used the identity (4.10) to conclude that and thus This is indeed the variation generated by d 3 xM 0 .
It is obvious from the conservation of Noether charge that follows from (4.11) that the coefficients of each of the time derivatives of the arbitrary descriptors that appear in the charge density must vanish. Also, as Rosenfeld noted, the coefficients of the highest time derivative of the descriptor ξ are constraints, now called primary constraints. But then he noted that setting equal to zero the time derivative of the charge density yielded a recursion relation among the vanishing coefficients. In particular, employing an integration by parts we obtain his relations (R63)

The Einstein-Maxwell-Dirac theory
The gravitational action We will translate Rosenfeld's notation into conventional contemporary form. He His e k raises Minkowski indices. Also, to avoid confusion when considering specific components, we use a capital letter to represent contravariant coordinate objects.
So e k h k,µ= e K µ , and E ν K= e k h k,µ is the reciprical of kh k,µ and h ν k e k h k,µ= δ ν µ is the statement that E ν K e K µ = δ ν µ .
Expanding the Christoffel symbols in terms of the tetrads we find The curvature in terms of the spin connection is 20 3) 20 Rosenfeld never explicitly referred to the spin coefficients.
Then the scalar curvature density is Rosenfeld took the gravitational Lagrangian density to be where κ := 8πG/c 2 . This is his expression (R104). Thus Rosenfeld's gravitational Lagrangian is manifestly a scalar density under arbitrary coordinate transformations. It is also expressible as the sum of two manifest scalar densities, This is the content of (R105 Rosenfeld addressed this issue in his Case 2.

The electromagnetic action
We have the conventional electromagnetic action E in terms of the vector potential A µ and field tensor F µν = A ν,µ − A µ,ν , The matter action Rosenfeld's matter Lagrangian is where γ µ := E µ I Γ I and we denote the constant Dirac gamma matrices as Γ I . Also ψ := ψ † Γ 0 and is the spinor connection consistent with the Christoffel connection. It was first constructed independently by H. Weyl [65] and V. Fock [25]. Both authors were attempting a geometric unification of Dirac's electron theory with gravity. 21 We will use the properties and and hence

The momentae and the identities Case 1 -General covariance
Rosenfeld's Case 1 assumes that the Lagrangian transforms as a scalar density under arbitrary coordinate transformations. This property is satisfied separately by L g , E, and W under the transformations In particular, with L = L g + E + W =: The identically vanishing coefficient ofξ µ in the identity (5.15) then yields the four primary constraints  which is (R108). In the second line of (R109) Rosenfeld in principle displays two additional primary constraints, namely and which he, however, did not include among his "eigentliche Identitäten" (proper identities). We will return to this neglect at the end of this section. and we find that Referring to (4.10) we read off from this expression that as in (57a) in the foundational quantum field article by Heisenberg and Pauli [29] that served as Rosenfeld's inspiration for this paper. This interpretation is consis-

Symmetry generators
Next we construct the symmetry variations according to paragraphs §13 and §14. The secondary constraints ( where b.t. denotes a boundary term. Next, the generator for infinitesimal U(1) transformations is It may be rewritten further by using the relations (5.23) and (5.24) and then employing (as did Rosenfeld) the algebra (5.30), where the {φ A , ψ A } are all first-class constraints. The expressions (5. 35, 5.36, 5.38) have this generic structure, although the first class property need not hold for the coefficients of the ξ A andξ A in the Rosenfeld-Noether generators above. Indeed, we expect that one needs to work with modified transformations in order to respect the Legendre projectability of the transformations [51]. These questions will be addressed in a future publication. One can also realize canonical active finite 3-D diffeomorphism transformations.

Finite canonical transformations and the symmetry group algebra
Suppose, for example, that we wish to actively transform a scalar phase space function φ(x) under the diffeomorphism x ′a = x a + ξ a ( x). Its finite actively transformed change is then If we undertake the infinitesimal transformation x µ 1 = x µ + ξ µ 1 (x) followed by x µ 2 = x µ + ξ µ 2 (x) and then subtract the them in reverse order, the descriptor of the overall coordinate transformation is Thus higher order time derivatives appear with each commutation. The appearance of these time derivatives was an early concern of Bergmann -although not explicitly stated by him or his collaborators in the period prior to Dirac's gravitational Hamiltonian breakthrough in 1958 [20]. He did however refer to this challenge in a later recollection 23 We can surmise from Rosenfeld's discussion in his section §6 that he must have recognized this obstacle since as we noted above he confined his discussion of the group algebra to the spatial diffeomorphisms and the internal symmetry transformations. He concluded that the vanishing generators must satisfy a closed Lie algebra.
In the language that was introduced later by Dirac [18] (without ever mentioning the work of Rosenfeld with which he had been familiar since 1932) 24 , the constraints 22 The closure of this phase space algebra will later be the defining property of first class constraints [18]. Curiously, Dirac introduced this notion without ever mentioning the work of Rosenfeld with which he had been familiar since 1932. See [55] for a discussion of relevant correspondence in 1932 between Rosenfeld and Dirac.
23 "During the early Fifties those of us interested in a Hamiltonian formulation of general relativity were frustrated by a recognition that no possible canonical transformations of the field variables could mirror four-dimensional coordinate transformations and their commutators, not even at the infinitesimal level. That is because (infinitesimal or finite) canonical transformations deal with the dynamical variables on a three-dimensional hypersurface, a Cauchy surface, and the commutator of two such infinitesimal transformations must be an infinitesimal transformation of the same kind. However, the commutator of two infinitesimal diffeomorphisms involves not only the data on a three-dimensional hypersurface but their "time"-derivatives as well. And if these data be added to those drawn on initially, then, in order to obtain first-order "time" derivatives of the commutator, one requires second-order "time" derivatives of the two commutating diffeomorphisms, and so forth. The Lie algebra simply will not close." [7] must be first class.
The impossibility of generating finite canonical transformations corresponding to coordinate transformations for which δx 0 = 0 was a feature not only of Rosenfeld's Noether charge, but also the generators that were written down first by Anderson and Bergmann [1] in 1951, and later by Bergmann and Schiller [11] in 1953. Although Dirac never concerned himself with the question whether the full diffeomorphism group could be realized as a canonical transformation group, he is the one who unintentionally invented the framework in which this goal could be achieved. The key was the decomposition of infinitesimal coordinate transformations which were either tangent to a given foliation of spacetime into fixed time slices, or perpendicular to the foliation. Bergmann and Komar [9] subsequently gave a group-theoretical interpretation of this decomposition, pointing out that the relevant group was a phase space transformation group that possessed a compulsory dependence on the spacetime metric. In 1983 we [54] provided an explicit proof that this dependence was required in order to obtain a Lie algebra that did not involve higher time derivatives of the descriptors. More recently, Pons, Salisbury and Shepley [49] showed that this demand on the structure of the group algebra is equivalent to the demand that the permissible variations of configuration-velocity variables be projectable under the Legendre transformation from configuration-velocity space to phase space.

Expanding upon Rosenfeld's application: Construction of the Hamiltonian
Because of the existence of primary constraints it is not possible to solve uniquely for the momentae in terms of the velocities. As we noted earlier, Rosenfeld pioneered a method for obtaining general solutions that involved as many arbitrary functions as there were primary constraints, where the constraints arising in both Cases 1 and 2 must be taken into account. Rosenfeld then employed these general solutions in the construction of the Hamiltonian.
Rosenfeld did not display the explicit expression for the Hamiltonian for his general relativistic model. We do not know why. It is, however, straightforward to apply his method to construct it. We undertake the construction here.
We begin with the momentum conjugate to the tetrads e µI , (R118),  48) where is the velocity-independent term in L g .
The total Hamiltonian density is H g + H em , where the electromagnetic contribution H em can also be found applying Rosenfeld's method. See [55]. It has the structure H em = H c em + λF + λ i χ i where the χ i are the spinorial constraints from (5.23, 5.24). In the "usual" Dirac-Bergmann procedure one would require the stabilization of primary constraints, and/or find new constraints or fix the multipliers λ. Although the Hamiltonian (5.58) generates the correct field equations, some additional work needs to be done to be able to compare with later publications on canonical tetrad-spinor formulations, as for instance [42,43] 25 The present article may thus be seen as an atonement to Rosenfeld by one of the authors. 26 Gerne möchte ich Dich in dieser Verbindung auf die lange Arbeit von Rosenfeld, Annalen der Physik (4), 5, 113, 1930 aufmerksam machen. Er hat sie seinerzeit bei mir in Zürich gemacht und hiess hier dementsprechend der Mann, der das Vierbein quantelt (klingt wie der Titel eines became apparent in his Part 2 that the special cases that Rosenfeld identified in his Part 1 were chosen with the Einstein-Maxwell-Dirac theory in mind, and the article might have been more accessible had he simply addressed this model from the start rather than formally treating a wider class of theories. It is this lament by Pauli that leads us to suspect that Rosenfeld's general theory was indeed more general than the unidentified Pauli suggestion that Rosenfeld acknowledged in his introduction. Yet the paper was known, in particular already in 1932 by Dirac, as has been documented elsewhere [55], yet Dirac did not cite it in his papers on constrained Hamiltonian dynamics [18,19]. Strangely, in another paper of 1951 concerned with electromagnetism in flat spacetime Dirac did refer to Rosenfeld in addition to his foundational papers in declaring that "an old method of Rosenfeld (1930) is adequate in this case" in making the transition from a Lagrangian to the Hamiltonian. 27 With regard to the Syracuse group, the paper was only discovered following the publications by Bergmann [5] and Bergmann-Brunings [8] of their initial foundational papers on constrained Hamiltonian dynamics. 28 As we noted earlier, following this discovery Schiller made explicit use of the Rosenfeld paper in constructing the phase space generators of symmetry transformations that we have elected to call Rosenfeld-Noether generators. On the other hand, in the joint publication by Bergmann and Schiller [11] that focused on these charges Rosenfeld was not cited.
27 [22], p. 293 28 J. Anderson related to D. S. in 2006 that it was he who had found the paper and brought it to the attention of Bergmann. In this same conversation R. Schiller indicated that the paper was the inspiration for his Ph. D. thesis, conducted under Bergmann's direction.
invariance of a Lagrangian and uses them for obtaining phase-space constraints, but he also (2) proposes an expression for the generator of phase-space symmetry transformations, and (3) details a procedure to derive a Hamiltonian density from a singular Lagrangian in a manner more mathematically satisfying than later ones by Dirac and by Bergmann and his Syracuse group.
The history-of-science story of the Klein-Noether identities is another story of early discovery and later rediscovery. Felix Klein in 1918 derived a chain of identities for general relativity in his attempt to arrive at conservation laws in general relativity [35]. Similar chains of identities exist for arbitrary local symmetries; they shall not be derived here (for details see Sect The full set of Klein-Noether identities was investigated also by J. Goldberg [26], exhibited by R. Utiyama [63], mentioned by A. Trautman [61] -all of them not citing F. Klein. (It seems that the first reference to Klein is in [3].) The identities were called extended Noether identities in [37,38] , cascade equations in [33,34], Noether's third theorem in [13,14], and Klein identities in [48].
Another result concerning the Klein-Noether identities -already visible in the Rosenfeld article, and still widely unknown today -is the fact that these are entirely equivalent to the chain of primary, secondary, ... constraints in the Hamiltonian treatment [37,38].
And still another history-of-science story lays dormant under repeated efforts to find generators of phase-space symmetry transformations. After Rosenfeld, the investigations into the manner in which the constraints of a theory with local symmetries relate to the generators of these symmetries in phase space restarted with the work of Anderson and Bergmann [1], Dirac [21], and Mukunda [40,41]. It soon be-came clear that the phase space symmetry generator is a specific linear combination of the first-class constraints. In 1982, Castellani devised an algorithm to determine a symmetry generator [15]. This was completed by Pons/Salisbury/Shepley [49] by taking Legendre projectability into account and thereby extending the formalism to incorporate finite symmetry transformations.
It seems to have gone unnoticed that L. Rosenfeld already in 1930 showed that the vanishing charge associated with the conserved Noether symmetry current is the sought-after phase-space symmetry generator, called the Rosenfeld-Noether generator in this article. The figure who came the closest to affirming this fact was L.
Lusanna who indeed contemplated a wider scope of symmetry transformations including several specific pathological cases [37,38]. The proof by Rosenfeld, repeated in Section 4, is not easy to digest at first reading, but it is valid for infinitesimal transformation. One consequence that all examples suggest is that one can read off the first-class constraints of the theory in question from the Rosenfeld-Noether generator. Recall that in the "usual" handling of constrained systems, sometimes referred to as the Dirac-Bergmann algorithm, one needs to establish a Hamiltonian first in order to find all constraints beyond the primary constraints. Only then can first and second-class objects can be defined.
In a forthcoming article we will show how Rosenfeld's approach can be generalized so that Legendre projectability is respected. One significant result of this analysis is that whenever local symmetries beyond general covariance are present, such appropriately chosen symmetries must be added to the general coordinate transformations to achieve canonically realizable transformations. 29 With his attempt to quantize the Einstein-Maxwell-Dirac theory Rosenfeld made an ambitious effort that was "well before its time". Keep in mind that prior to Rosenfeld's article no results on the Hamiltonian formulation of pure Einstein gravity were known, that Weyl's ideas of electromagnetic gauge invariance were not generally ac-cepted, and that spinorial entities were still treated ad hoc. He can be forgiven for not having reached today's level of understanding. He did not derive explicitly all of the first-class constraints from the Klein-Noether identities although he did appreciate their importance as group generators. Nor did he display the full Hamiltonian for his model even though as we have seen he was certainly in position to do so in a straightforward application of his method. Thus he could have derived a tetrad formulation for general relativity with gauge fields nearly five decades before it appeared on the quantum gravitational research agenda.
As a matter of fact the canonical formulation of general relativity in terms of tetrads and spin connections became a hot topic only in the 1970's -even though Bryce DeWitt and Cecile DeWitt-Morette had addressed this issue already in 1952 [17]. The preponderance of articles on canonical general relativity around 1950 were formulated in terms of the metric and the Levi-Civita connection. Rosenfeld obviously was aware that this was possible in the case of vacuum general relativity.
In item (3) of his §15 he writes "The pure (vacuum) gravitational field could be described by the g µν instead of the h i,ν . Then we would be dealing with another variation of the 'second case' ". Indeed, he notes that as a consequence of the general covariance four primary constraints would arise (that first appeared explicitly in Bergmann and Anderson). It would be of interest to apply a modified version of Rosenfeld's program to both the Dirac [20] and to the ADM [2] Lagrangians. These differ by divergence terms. 30 The divergence terms do not however transform as scalar densities under general coordinate transformations, so their treatment would require a simple modification of Rosenfeld's Case two.
Of course, Ashtekar's invention of new gravitational variables initiated an interest in tetrad variables that form the basis of today's active research in loop quantum gravity. And we can thank Rosenfeld for not only setting down the first stones of the foundations for this canonical loop approach to quantum gravity. Remarkably, in ad- 30 See e.g. [16] dition he pioneered the development of the gauge theoretical phase space framework that undergirds all current efforts at unifying the fundamental physical interactions.

A.1 Singular Lagrangians
Assume a classical theory with a finite number of degrees of freedom q k (k = 1, ..., N) defined by its Lagrange function L(q,q) with the equations of motion For simplicity, it is assumed that the Lagrange function does not depend on time explicitly; all the following results can readily be extended. A crucial role is played by the matrix (sometimes called the "Hessian") If det W = 0, not only the Lagrangian but the system itself is termed 'singular', and 'regular' otherwise.
From the definition of momenta by one immediately observes that only in the regular case can the p k (q,q) be solved for all the velocities in the formq j (q, p) -at least locally.
In the singular case, det W = 0 implies that the N × N matrix W has a rank R smaller than N -or that there are P = N-R null eigenvectors ξ k ρ : This rank is independent of which generalized coordinates are chosen for the Lagrange function. The null eigenvectors serve to identify those of the equations of motion which are not of second order. By contracting these withq j one gets the P on-shell equations Being functions of (q,q) these are not genuine equations of motion but -if not fulfilled identically -they restrict the dynamics to a subspace within the configurationvelocity space (or in geometrical terms, the tangent bundle T Q). For reasons of consistency, the time derivative of these constraints must not lead outside this subspace. This condition possibly enforces further Lagrangian constraints and by this a smaller subspace of allowed dynamics, etc.
The previous considerations are carried over to a field theory with a generic Lagrangian density L(Q α , ∂ µ Q α ). Rewrite the field equations as With the choice of the time variable T = x 0 the Hessian is defined by 31 If the rank of this matrix is R < N , it has P = N-R null eigenvectors A.2 Klein-Noether identities and phase-space constraints

A.2.1 Klein-Noether identities
In 1918, Emmy Noether [44] wrote an article dealing with the consequences of symmetries of action functionals For symmetry transformationsδ S Q α , δ S x µ her central identity is with the Noether current

A.3 Dirac-Bergmann algorithm
Since the fields and the canonical momenta are not independent, they cannot be taken as coordinates in a phase space as one is accustomed in the unconstrained case.
This difficulty was known already by the end of the 1920's, and after unsatisfactory attempts by eminent physicists such as W. Pauli

A.3.2 Weak and strong equations
It will turn out that even in the singular case, one can write the dynamical equations in terms of Poisson brackets. But one must be careful in interpreting them in the presence of constraints. In order to support this precaution, Dirac contrived the concepts of "weak" and "strong" equality.
If a function F (p, q) which is defined in the neighborhood of Γ P becomes identically zero when restricted to Γ P it is called "weakly zero", denoted by F ≈ 0: (Since in the course of the algorithm the constraint surface is possibly narrowed down, a better notation would be F ≈ | Γ P 0 .) If the gradient of F is also identically zero on Γ P , F is called "strongly zero", denoted by F ≃ 0: It can be shown that Indeed, the subspace Γ P can itself be defined by the weak equations φ ρ ≈ 0.

A.3.3 Canonical and total Hamiltonian
Next introduce the "canonical" Hamiltonian by Its variation yields constraints. In order that these be respected, the variation of H C needs to be performed together with Lagrange multipliers. This gives rise to define the "total" Hamiltonian H T := H C + u ρ φ ρ (A. 16) with arbitrary multiplier functions u ρ in front of the primary constraint functions.
Varying the total Hamiltonian with respect to (u, q, p) one obtains the primary constraints and Since the multipliers u ρ are not phase-space functions, the Poisson brackets {F, u ρ } are not defined. However, these appear multiplied with constraints and thus the last term vanishes weakly. Therefore the dynamical equations for any phase-space function F (q, p) can be written aṡ

A.3.4 Stability of constraints
For consistency of a theory, one must require that the primary constraints are conserved during the dynamical evolution of the system: There are essentially two distinct situations, depending on whether the determinant of C ρσ vanishes (weakly) or not • det C = 0: In this case (A.19) constitutes an inhomogeneous system of linear equations with solutions u ρ ≈ −C ρσ h σ , where C is the inverse of the matrix C. Therefore, the Hamilton equations of motion (A.18) becomė which are free of any arbitrary multipliers. new constraints φρ ≈ 0ρ = P + 1, ..., P + S ′ called "secondary" constraints.
The primary and secondary constraints define a hypersurface Γ 2 ⊆ Γ P . In a further step one has to check that the original and the newly generated constraints are conserved on Γ 2 . This might imply another generation of constraints, defining a hypersurface Γ 3 ⊆ Γ 2 , etc., etc. In most physically relevant cases, the algorithm terminates with the secondary constraints.
The algorithm terminates when the following situation is attained: There is a hypersurface Γ C defined by the constraints hold. In the following, weak equality ≈ is always understood with respect to the "final" constraint hypersurface Γ C .

A.3.5 First-and second-class constraints
Curiosity about the fate of the multiplier functions leads to the notion of first-and second-class objects.
Some of the equations (A.21) may be fulfilled identically, others represent conditions on the u ρ . The details depend on the rank of the matrixĈ. If the rank ofĈ is P , all multipliers are fixed. If the rank ofĈ is K < P there are P -K solutions The most general solution of the linear inhomogeneous equations (A.21) is the sum of a particular solution U ρ and a linear combination of the solutions of the homogeneous part: with arbitrary coefficients v α . Together with φ ρ , also the linear combinations It was argued that a theory with local variational symmetries necessarily is described by a singular Lagrangian and that it acquires constraints in its Hamiltonian description. The previous section revealed the essential difference between regular and singular systems in that for the latter, there might remain arbitrary functions as multipliers of primary first-class constraints. An educated guess leads to suspect that these constraints are related to the local symmetries on the Lagrange level.
This guess points in the right direction, but things aren't that simple. Dirac, in his famous lectures [18,19] introduced an influential invariance argument by which he conjectured that also secondary first-class constraints lead to invariances. His argumentation gave rise to the widely-held view that "first-class constraints are gauge generators". Aside from the fact that Dirac did not use the term "gauge" anywhere in his lectures, later work on relating the constraints to variational symmetries revealed that a detailed investigation on the full constraint structure of the theory in question is needed; see J. Pons [52].

A.4.2 Relating Lagrangian and Hamiltonian symmetries
Why at all should the symmetry transformations as given by (A.10), that is δ ǫ Q α = A α r (Q) · ǫ r (x) + B αµ r (Q) · ǫ r ,µ (x) + ..., (A. 29) be related to canonical transformations Is there a mapping between the parameter functions ǫ r and ǫ I ? Can one specify an algorithm to calculate the generators of Noether symmetries in terms of constraints?
Ignoring Rosenfeld, it seemed that the very first people to address these questions were Anderson and Bergmann (1951) -even before the Hamiltonian procedure for constrained systems was fully developed. N. Mukunda [41] started off from the chain (A.12) of Klein-Noether identities and built symmetry generators as linear combinations of first class primary and secondary constraints from them, assuming that no tertiary constraints are present. L. Castellani [15] devised an algorithm for calculating symmetry generators for local symmetries, implicitly neglecting possible second-class constraints.

A.5 Second-class constraints and gauge conditions
The previous subsection dealt at length with first-class constraints because they are related to variational symmetries of the theory in question. Second-class constraints where now weak equality refers to the hypersurface Γ R defined by the weak vanishing of all previously found first-and second-class constraints, that is the hypersurface Γ C together with the constraints (A.33). The idea is that the quest for stability of these constraints, namely does not vanish 35 . In this case the multipliers are fixed to: Some remarks concerning the choice of gauge constraints Ω α : • The condition of a non-vanishing determinant (det(Λ αβ ) = 0) is only a sufficient condition for determining the arbitrary multipliers connected with the primary FC constraints.
• The gauge constraints must not only be such that the "gauge" freedom is removed (this is guaranteed by the non-vanishing of det Λ), but also the gauge constraints must be accessible: for any point in phase space with coordinates (q, p), there must exist a transformation (q, p) → (q ′ , p ′ ) such that Ω α (q ′ , p ′ ) ≈ 0 . This may be achievable only locally.
• In case of reparametrization invariance (at least one of) the gauge constraints must depend on the parameters explicitly -and not only on the phase-space variables.