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Purely geometric path integral for spin foams

Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for a path integral to be at least formally equivalent to the corresponding canonical quantization, at each point in the space of histories it is important that the integrand have not only the correct phase -- a topic of recent focus in spin-foams -- but also the correct modulus, usually referred to as the measure factor. The correct measure factor descends from the Liouville measure on the reduced phase space, and its calculation is a task of canonical analysis. The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst formulation, in which the basic variables are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the connection degrees of freedom have been integrated out. We call this a purely geometric Plebanski-Holst path integral. In prior work in which one of the authors was involved, the measure factor for the Plebanski-Holst path integral with both connection and two-form variables was calculated. Before one discretizes this measure and incorporates it into a spin-foam sum, however, one must integrate out the connection in order to obtain the purely geometric version of the path integral. To calculate this purely geometric path integral is the principal task of the present paper, and it is done in two independent ways. Gauge-fixing and the background independence of the resulting path integral are discussed in the appendices.


Introduction
In the path integral approach to constructing a quantum theory, the integrand of the path integral has two important parts: a phase part given by the exponential of i times the classical action, and a measure factor. The form of the phase part in terms of the classical action ensures that solutions to the classical equations of motion dominate the path integral in the classical limit so that one recovers classical physics in the appropriate regime. The measure factor, however, arises from careful canonical analysis, and is important for the path integral to be equivalent to the corresponding canonical quantum theory. In most theories, this means that it is important, in particular, in order for the path integral theory to have such elementary properties as yielding a unitary S-matrix that preserves probabilities. The importance of having the correct measure factor is thus quite high.
Spin-foams are a path integral approach to quantum gravity in which one does not sum over classical gravitational histories, but rather quantum histories arising from canonical quantization. Specifically, in spin-foams, one sums over histories of canonical states of loop quantum gravity. These histories possess a natural 4-dimensional space-time covariant interpretation, whence each can be thought of as a quantum space-time. This approach allows one to retain the understanding gained from loop quantum gravity, such as the discreteness of area and volume spectra, while at the same time formulating the dynamics in a way that makes space-time symmetries more manifest [1].
The starting point for the derivation of spin-foams is the Plebanski-Holst formulation of gravity [2][3][4][5][6], in which the basic variables are a connection ω and what is called the Plebanski two-form, Σ. However, in the final spin-foam sum, the connection ω is usually not present, and one sums over only spins and intertwiners, which determine certain eigenstates of Σ alone. Because of this, the continuum path integral most directly related to the spin-foam sum is a Plebanski-Holst path integral in which only Σ appears, and in which the connection has been integrated out. We call such a path integral purely geometric because Σ directly determines the geometry of space-time.
Because of the quantum mechanical nature of the histories used in spin-foams, ensuring that the summand has the required phase part and measure factor is not completely trivial. Only within the last couple years has the correct phase part been achieved [7,8]. Regarding the measure factor, a first step has been carried out in the work [5], where the correct measure factor is calculated for the Plebanski-Holst path integral with both ω and Σ present. However, until now, the measure factor for the path integral with Σ alone, necessary for spin-foams, had not yet been calculated. To carry out this calculation is the main purpose of the present paper. In order to be sure about all numerical factors, we do this in two different ways: (1.) by starting from the path integral in [5] and then integrating out the connection degrees of freedom, and (2.) by starting from the ADM path integral and introducing the necessary variables from there. We find that these two ways of calculating the measure factor exactly match, as must be the case, as the canonical measure factor ultimately descends from the Liouville measure on the reduced phase space, which is independent of the formulation of gravity used [5].
The path integral derived in this paper is ready to be discretized and translated into a spin-foam model, a task which will be carried out in forthcoming work. Furthermore, when this is accomplished, we would like to emphasize that, because both primary and secondary simplicity constraints are already incorporated in the continuum path integral [5,9], they will be automatically incorporated in the resulting spin-foam model as well.
It should also be kept in mind that the raison d'être of the canonical path integral is to ensure formal equivalence with canonical quantization, and it may be that one can use such equivalence as a more direct criterion for obtaining the correct 'measure factor' in a spin-foam model. The work [10] has begun to explore use of such alternative argumentation.
We begin the paper by reviewing some background on the path integral and how it is used. After that, we derive the purely geometric Plebanski-Holst path integral, first starting from [5], and then starting from the ADM path integral. Gauge-fixing and the background independence of the derived path integral are then discussed in appendices.

Path integral generalities
For an unconstrained field theory with canonically conjugate variables ϕ, π, the phase space path integral takes the form [11] where the integration is over histories of pairs (ϕ, π), S[ϕ, π] is the 'phase space action' d 4 x(φπ − H) with H the Hamiltonian density, and DϕDπ is a formal Cartesian product of Lesbesgue measures at each point in space-time -or, equivalently, a Cartesian product of Liouville measures on phase space at each moment of time. For a system with second class constraints, ifφ,π are taken to be canonically conjugate coordinates on the constrained phase space, then (1) still applies. However, if one uses coordinates on the unconstrained phase space, one obtains [12] where the combined factor δ n (C i )|{C i , C j }| 1 2 is independent of the way the constraint surface is represented by constraint functions C i . For a system with first class constraints C i in which the action takes the form S[ϕ, π, λ] = S o [ϕ, π] + λ i C i , with λ i Lagrange multipliers, one can write the path integral as [13] Alternatively, one may introduce gauge fixing functions ξ i so that ξ i and C i together form a second class set of constraints, in which case (2) again applies. If we assume that, e.g., the ξ i are pure momentum or pure configuration, so that they all Poisson commute, the path integral then takes the form [14] Z In fact, under very general assumptions (which, however, have yet to be fully proven in the case of gravity) (3) and (4) are equal upto an infinite constant equal to the gauge volume (see [13,15,16], and appendix C), so that they determine the same physics in the manner to be reviewed below. In the rest of this paper for simplicity we will use expression (3). However, all derivations in this paper can just as well be done starting from expression (4) -one need only reinsert the omitted factor δ n (ξ)| det{C i , ξ j }|. The path integral was originally discovered as a way to write transition amplitudes between states in quantum mechanics. Let {O i } denote a set of phase space functions whose quantum analogues { O i } form a complete commuting set. Let Ψ {(Oi,νi)} denote the simultaneous eigenstate of the operators { O i } with eigenvalues {ν i }. Then the path integral determines the transition amplitude between two such states via where U (T ′ − T ) is the time evolution operator from t = T to t = T ′ . For a theory with first class constraints of which the Hamiltonian is a linear combination -i.e., a time reparametrization invariant theory -there is no time evolution operator, and, instead of (5), the interpretation of the path integral involves a 'rigging map' or 'group averaging map' η [17] from kinematical (unconstrained) states to physical states (states annihilated by the constraints) [1,18,19]: This is the case relevant for us. Note the physical inner product, and therefore Z(O), is only physically meaningful up to rescaling by a constant. In the following we will use the notation = to denote "equality upto rescaling by a constant." By setting O = 1, one obtains the so-called partition function. Often in the literature one says that the partition function defines the quantum theory. Of course, what is meant is that the mathematical form -or more precisely, the integrand -of the partition function defines the quantum theory. One then uses the integrand to integrate against general functions O, the results of which are used as above. In this paper, we are explicit about this fact for clarity.
The transition amplitude (5) or (6) defines the dynamics of the theory. This, together with a kinematical quantum framework of states and relevant operators, is enough to allow one to calculate any quantity of physical interest. In the case of GR, canonical loop quantum gravity provides the kinematics. In this case, by choosing the observables O i appropriately, the canonical states Ψ {(Oi,νi)} can be made to be spin-network states [20], generalized spin-network states [21], or Livine-Speziale coherent states [22]. In all of these cases, the relevant observables O i are purely geometric, depending only on the pull-back of Σ IJ µν to the spatial slice.

Derivation of a purely geometric path integral from Plebanski-Holst
We will consider both Euclidean and Lorentzian gravity simultaneously, defining s := +1 and s := −1 respectively in these two cases. In additition to space-time manifold indices, denoted here by lower case Greek letters, the Plebanski and Plebanski-Holst formulations of gravity make use of mixed tensors with 'internal' indices I, J, K, · · · ∈ {0, 1, 2, 3} which are raised and lowered with a fixed 'internal' metric η IJ := diag(s, 1, 1, 1). The basic variables are an so(η)-valued connection ω IJ µ and an so(η)-valued two-form X IJ µν , the Plebanski two-form. In the theory, X IJ µν is constrained to satisfy the so-called simplicity constraint.
where ǫ IJKL denotes the 'internal' alternating tensor. This constraint has 20 independent components per point and restricts X IJ µν to belong to one of five sectors. The first is the degenerate sector, in which ǫ µνρσ ǫ IJKL X IJ µν X KL ρσ = 0. The other four sectors each correspond to X IJ µν taking one of the following four forms 2κ ǫ IJ KL e K ∧ e L (II±) for some non-degenerate tetrad e I µ (see [5,23]), where we have chosen to include the factor κ := 16πG here instead of explicitly in front of the action (given below). Given an element Y IJ ∈ so(η), it is useful to define (γ) IJ . Using these variables and notation, we start from the Plebanski-Holst path integral derived in [5] where the action is an integral over the space-time manifold M, and the "(II±)" subscript indicates that we are restricting the integration to the (II+) and (II−) sectors, in which The V appearing in (8) is the space-time volume density and V s is the spatial volume density determined by e I µ . For brevity, from now on we will omit the "(II±)" subscript on the path integral (8), leaving it understood. In the end, to ensure that solutions to Einstein's equations dominate in the classical limit of the spin foam sum, it is necessary to restrict to a particular combination of sectors (II+) and (II−) called the Einstein-Hilbert sector [6][7][8]. However, this is not relevant for the work of the present paper. Without the simplicity constraint, the action appearing in (8) X, is the so-called BF action. Restriction to the (II±) sectors reduces this action to the Holst action [24] The equations of motion derived by varying this action are , and e µ I e ν J F IJ µν = 0, the first being the torsion-free condition on ω, and the second being equivalent to the Einstein equations.
It should be noted that the path integral (8) is valid only when O is purely geometric (i.e., a function of X IJ µν only). For, in the derivation in [5], when the Henneaux-Slavnov trick [9] is used, a change of variables involving ω IJ µ is performed. If the presence of O is made explicit throughout the derivation, one sees that if O depends explicitly on ω IJ µ , then the argument leading to (8) will change, as will the final path integral expression 1 However, the restriction of O to be purely geometric is not a real restriction, as, on-shell, ω is uniquely determined by X. Furthermore, the principal application of (8) will be to computing the physical inner product between spin-network states, which are eigenstates of X only (see equation (6)).
The goal of this section is to integrate out the connection in the expression (8). To make this task easier, it is helpful to first show that the γ dependent term in the action can be dropped, which is done in the first of the following subsections. Using the fact that the integral over the connection is of Gaussian form, in the second subsection we integrate out the connection, giving the result in terms of the determinant of the appropriate matrix. The final subsection computes the determinant of this matrix.

Eliminating the γ-dependent term in the action
the part of the Lagrangian in (8) without the γ term, and with the γ term, respectively. For this subsection, introduce a global time coordinate t, and let Σ t ′ denote the hypersurface on which t = t ′ , providing us with a foliation of M into hypersurfaces, and fix a vector field t a such that t a ∂ a t = 1. We use lower case latin indices a, b, c, ... to denote indices associated with one or more of the spatial manifolds Σ t ′ . Let ǫ abc and ǫ abc denote the totally antisymmetric symbol (i.e., Levi-Civita symbol of density weight −1 and 1, respectively) on each hypersurface. Then define Note that, if we define π a IJ := ǫ abc X bcIJ , X bcIJ and let D a denote the derivative operator induced on each Σ t by the pull-back of ω, then one sees that G IJ = D a π a IJ and (γ) π a IJ are the usual Gauss constraint without and with γ term. Finally, let The important point about these latter expressions are that they are independent of ω IJ t .
To show that G IJ = 0 implies d ω X IJ = 0, we first decompose the latter into its pull-back d ω X IJ ← −−− − = 0 to each hypersurface and the pull-back of t a (d ω X IJ ) abc = 0 to each hypersurface. That so that G IJ = 0 implies all of the components of the equation d ω X IJ = 0, as desired.
Lemma 2. The restriction to sectors (II±) of the simplicity constraints and G IJ = 0 together imply G IJ = 0 and d ω X IJ = 0. In sectors (II±) of the simplicity constraints, X IJ = (const.)ǫ IJ KL e K ∧ e L , so that d ω X IJ = 0 becomes d ω e I = 0, whose covariant exterior derivative yields the Bianchi identity F I J ∧ e J = 0, which implies Proof.
Decomposing (γ) G and G into self dual and anti-self-dual parts, one is able to calculate the determinant on the right hand side, resulting in (11).
Proposition 4. The 1/γ term in the partition function (8) can be dropped, so that Proof.
Where, in step 4, lemma 3 has been used, and in step 5, lemma 2 has been used.

Integrating out the connection
The next step is to evaluate the Gaussian integral: Because, in the path integral (8), X is constrained to be of the form (9), we assume X to be of this form throughout the section. We begin by noting that if we define the action becomes This casts the integral in the explicit Gaussian form in equations (27)(28) in appendix A. Using the result (30), one has that, modulo an overall X-independent constant, where with ω[X] the unique connection determined by X via the equation of motion of BF theory found by varying ω, Because X is of the form (9), ω[X] is furthermore the unique Lorentz spin-connection determined by the tetrad e, i.e., such that d ω e I = 0, so that the action reduces to the Holst and Palatini actions where ≈ denotes equality when X is of the form (9), and the last equality holds because, when ω = ω[X], the extra 'topological' term added to the Palatini action to give the Holst action vanishes due to the Bianchi identity [24]. In order to use equation (13) in the path integral, it remains to calculate the determinant of a.

The determinant of a
For this calculation, it will be convenient to let Define the inverse volume 4-form V αβγδ := ǫ IJKL e α I e β J e γ K e δ L = V −1 ǫ αβγδ . Then Next, we decompose the source (ω IJ µ ) and target (ρ µIJ ), of the matrix of interest a, in a way that will aide in the calculation. First, we "internalize" the space-time indices: and define In terms of this decomposition of ω IJ µ and ρ µIJ , (15) becomes We are ready to calculate the determinant of a. From (14) and (16), The 24 by 24 middle matrix ∂PM IJ ∂WN KL , expressed in terms of the decompositions P IJ M = (P i 0 , P i ,P i j , P,P i 0 ,P i ,P i j ,P ) and (17), is seen to be where, recall, s = +1 for Euclidean gravity and s = −1 for Lorentzian gravity. This yields where the constant is just (4κ) −24 time the determinant of the numerical matrix in (19), which is non-zero, as one can check. From (16),

Final continuum path integral
Using this expression in (13) and substituting that into (12) yields the final pure geometric continuum path integral We wish to note that this expression derives from the full canonical path integral, with primary as well as secondary simplicity constraints imposed, and with the full requisite determinant factors. The only assumption necessary in the derivation is that O does not explicitly depend on ω. In spin foam sums, each spin foam is labeled by quantum numbers which are a discrete analogue of precisely the two-form X, making the above expression ideally suited for use with spin-foams.

ADM path integral
As a secondary derivation of (20) we start from the ADM formalism. Because one has no second class constraints in the ADM formalism, one doesn't need to use the Henneaux-Slavnov trick [9]. As a consequence, this derivation can also be thought of as a 'check' on the use of the Henneaux-Slavnov trick in this case.

Integrating out the momentum
The non-gauge-fixed canonical path integral in the ADM formalism in terms of the canonical variables (h ab , π ab ) is where recall a, b, c . . . denote spatial indices. Here N and N a are lapse and shift lagrange multipliers and H ADM is the Hamiltonian density given by [25] where h := det h ab , π := π a a , and (3) R denotes the scalar curvature of h ab . The integral (21) can be written where A, B and C are where is the usual ADM Lagrangian with the usual extrinsic curvature. It remains to find the determinant of A.

The determinant of A
By the symmetry of h ab , there exists an orthogonal matrix O a e such that O a e O b f h ef = λ a δ ab for some λ a , so that Now because rows (ab) = (12), (13) and (23) where λ 1 λ 2 λ 3 = det h ab has been used. Substituting this into (24) gives

Space-time covariant variables
It remains to change from the variables (h ab , N, N a ) to the space-time covariant variable g µν , we have Dg µν = Dg ab Dg 00 Dg 0a = |det J| Dh ab DN DN a where J := ∂(g ab , g 00 , g 0a ) ∂(h cd , N, N c ) .
Using the relations it is easy to check that det J = −2hN so that Dg µν = 2hN Dh ab DN DN a .

Comparison with the result of the last section
Our goal in this section is to compare the ADM measure in (26) to the final continuum path integral (20) derived in section 3 Since these two expressions have different variables, again a change of variables is necessary. As discussed in [5], one can perform a change of variables X IJ µν → (e I µ , C), 2 from X IJ µν to the tetrad e I µ together with the simplicity constraints. The resulting change in measure is given by [5] DX IJ µν =V −6 De I µ DC so that Next, we change from tetrad variables to metric variables. This can be done by fixing an arbitrary reference tetrade I µ [g] for each metric g µν , and defining a local gauge transformation Λ I J via At each space-time point, Λ I J is an element of either the 4-dimensional Euclidean group or the Lorentz group, depending on s. The change of variables (e I µ ) → (g µν , Λ I J ) yields the change of measure [26] Dg µν DΛ I J = √ gDe I µ where DΛ I J is the measure defined in [26]. Because the integral DΛ I J is independent of the metric, it can be dropped from the path integral, leaving Using the relations V s = h which is the same as (26).

Conclusions
Spin-foams are a path integral approach to quantum gravity based on the Plebanski-Holst formulation of general relativity. The basic variables of the Plebanski-Holst formulation are a Lorentz connection and the Plebanski two-form, the pull-backs of which to any Cauchy surface are conjugate to each other. The Plebanski two-form by itself completely determines the space-time geometry. In spin-foams, one sums over histories of spins and intertwiners which label eigenstates of the Plebanski two-form. Because of this, the spin-foam sum may be understood as a discretization of a Plebanski-Holst path integral in which the connection degrees of freedom have been integrated out -that is, it is a discretization of what we have called a purely geometric Plebanski-Holst path integral. In order to ensure that a path integral quantization be equivalent to canonical quantization, it is important that the correct canonical path integral measure be used. The path integral measure for Plebanski-Holst, with both connection and Plebanski two-form variables present, was calculated in the earlier work [5]. In the present work, we have calculated the pure geometric form of this path integral, whose discretization will yield the necessary measure factor for spin-foams. We have calculated the measure for this path integral in two independent ways (1.) by integrating out the connection from the path integral derived in [5], and (2.) by ensuring consistency with the canonical ADM path integral. Both methods lead to the same final measure factor, providing a check on the detailed powers of the space-time and spatial volume elements present. The next step is to discretize this measure on a spin-foam cell complex, expressing it directly in terms of spins and intertwiners. This will involve non-trivial choices which will in part be fixed by considerations of gauge-invariance. This will be discussed in a later, complementary paper.

A Integration of a path integral with quadratic action
Let (·, ·) be a symmetric, non-degenerate, but not necessarily positive definite, inner product on some real vector space V . Letâ be an invertible operator on V symmetric with respect to (·, ·), b an element of V , and c a real number. Consider the action and the path integral where Dv is any translation-invariant measure on V (unique up to rescaling). If one fixes a basis on V and uses components with respect to this basis as coordinates on V , Dv will just be a real number times the Lebesgue measure in the chosen coordinates. From the complex analytic continuation of the usual formula for the Gaussian integral, one obtains where the constant C depends exclusively on the relative scaling of the inner product (·, ·) and the measure Dv. Variation of the action gives δS| v=v0 = (δv,âv) + (v,âδv) + (δv, b) = (δv, 2âv + b).
Setting this equal to zero for all δv yields the following expression for the Substituting this into (27) directly gives S[v 0 ] = − i 4 (b,â −1 b) + c, so that (29) can be written with v 0 the unique extremum of S[v] and C depending exclusively on the relative scaling of (·, ·) and Dv.

B Background independence of the path integral
The path integral was originally discovered as a way to write transition amplitudes in quantum mechanics. In the case of a (time-)reparametrization invariant theory where the Hamiltonian is a linear combination of the constraints (such as GR), the path integral more precisely provides the projector onto physical states together with the physical inner product [1,19]. As noted in section 2, this physical inner product, together with the rest of the existing canonical loop quantum gravity framework, is enough to calculate all physically relevant quantities. Let Σ denote the spatial manifold of the canonical theory, and let M denote a region of space-time with past and future boundary Σ 1 , Σ 2 , each diffeomorphic to Σ. Select diffeomorphisms ι 1 : Σ → Σ 1 ⊂ M and ι 2 : Σ → Σ 2 ⊂ M. Select a set of observables {O i } which are functions of (the pull-back to Σ of) X IJ µν only, and whose quantum analogues  [20], generalized spin networks [21], or Livine-Speziale coherent states [22]. From (6), the path integral (20) then determines the physical inner product in LQG via where η is the 'rigging map' or 'group averaging map' from kinematical (unconstrained) states to physical states (states annihilated by the constraints) 3 , and S[X] := S BF [X, ω[X]] is the purely geometric action descending from BF theory. The path integral measure in (31) (as well as the path integral measures thus far derived in the literature for all other formulations of gravity [23,29,30], including those in equations (8) and (25)) depends on (1.) a choice of foliation Ξ of M (because of the presence of 3-dimensional volume factors) and (2.) a choice of coordinate system Φ compatible with Ξ (because the 4-and 3-volume factors are densities). That is, the measure depends on background structures. However, what matters physically is the physical inner product in (31), or more precisely, the physical inner product modulo constant rescalings. One can then ask: does the physical inner product (modulo rescalings) determined by the above equation retain this dependence on background? In this section, we show that the physical inner product is in fact background independent modulo rescalings, thus respecting an important guiding principle from general relativity.
To begin the argument, we first note that any function can always be made diffeomorphism covariant by making all background structure an additional explicit argument. Thus, if we express the transition amplitude as a function of an initial state Ψ and as a function of the choice of foliation Ξ and compatible coordinate system Φ, then it is by construction diffeomorphism covariant. As Z is a scalar-valued function, that means it is diffeomorphism invariant : for all diffeomorphisms α of M. Now, suppose (Ξ, Φ), (Ξ,Φ) are two possible choices of foliation and coordinate system. Because the foliation arises from Feynman's procedure of skeletonization in time [31], the initial and final slices Σ 1 and Σ 2 will always be leaves of Ξ andΞ. Because of this, there always exists a 4-diffeomorphism α such that α · Ξ =Ξ, and such that α is the identity on the initial and final hypersurfaces Σ 1 and Σ 2 . Because of the latter property, . Next, note that, under a change of coordinate system, the Lesbesgue measure, 3-volume and 4volume densities, and Dirac delta function in (31) change only by Jacobian factors which are constant on the space of histories. Because of this, the left hand side of the above equation is equal to where the constant is independent of Ψ {(Oi,νi)} and Ψ ′ , η(·) phys ] Φ,Ξ denote the physical inner product, as determined by the path integral using Ξ, Φ, modulo constant rescalings. (33) then tells us that [ η(·), η(·) phys ]Φ ,Ξ = [ η(·), η(·) phys ] Φ,Ξ .
C Equivalence of gauge-fixed and non-gauge-fixed path integrals In this appendix, we address the equivalence of the path integrals (3) and (4). The argument used here is based on the proof for the Yang-Mills case given by Faddeev and Popov [13]. A version of this argument is also given in [16] and [15]. However, here we keep the argument more general and give more details.

C.1 The argument
Consider a system with first class constraints C i , generating a gauge group G on shell, and an action of the form S[ζ, λ] = S o [ζ] + λ i C i , where ζ is short hand for a set of canonically conjugate variables (ϕ, π), and λ are Lagrange multipliers. We start with (3) where Dζ := DϕDπ, and where, for this appendix, we assume O is gauge invariant. Faddeev and Popov in their original paper [13] almost start with this same path integral, the only difference being that here we use phase space variables, which is the more fundamental starting point from the canonical perspective [12,14]. The Faddeev-Popov strategy [13] is to factor out the divergences in the path integral due to the integration over the gauge group. We here adapt their argument to the general path integral (34) as follows. First choose gauge-fixing functions ξ j = ξ j (ζ) which are regular, that is, have non-vanishing gradient, at ξ j ≡ 0. The ξ j then form a good set of coordinates on each gauge-orbit in a neighborhood of ξ j ≡ 0. Furthermore, given any coordinates α i → g(α) ∈ G on the gauge group G, and any phase space point ζ = (φ, π), one can define another set of coordinates on the gauge orbit containing ζ via α i → g(α) · ζ. One then has Dαδ(ξ(g(α) · ζ)) det ∂ξ j (g(α) · ζ) ∂α i = Dξδ(ξ) = 1 which can be inserted into the path integral (34) to obtain We next perform the change of variables ζ → ζ ′ := g(α) · ζ. As ζ → g(α) · ζ is a canonical transformation, and dζ = dϕdπ is the Liouville measure, we have dζ = dζ ′ . Furthermore, . In order to facilitate calculation, we now make a specific choice for g(α): For each α i , we define g(α) : Γ → Γ to be the Hamiltonian flow generated by α i C i , evaluated at unit parameter time. Equivalently, g(α) may be defined by the equationS If we let X α·C denote the derivative operator X α·C f := {α · C, f }, then the equations above implies the explicit expression The constraint factor δ(C(ζ)) can be rewritten δ(g(−α) * C(ζ ′ )), where * denotes pull-back. Because the flow generated by first class constraints is always tangent to the constraint surface, g(−α) * C i will again be a linear combination of the constraints, so that for some matrix µ(α) i j of functions on phase space, whence Turning now to the last remaining factor, we have where (δ i ) j := δ j i . Here the definition of the partial derivative has been used in the first line, equation (39) from the next subsection has been used in the second line, equation (36) has been used in the third line, and the invariance of the Poisson bracket under the gauge transformation g(sα) has been used in the last line. Define M (α) i j := 1 0 dsµ(sα) i j . Taking the determinant of (38) then yields Using this and equation (37) in equation (35) gives us At this point, all dependence on α is restricted to the inner integral in parentheses, which can be thought of as a "gauge orbit volume," with det M(α) det µ(α) ζ ′ acting as a "volume element". If we can show that this gauge orbit volume is independent of ζ ′ , then we can drop it as an overall constant, thereby proving the equivalence of the non-gauge-fixed (34, 3) and gauge-fixed (4) path integrals.
In the case of gravity, or any other theory with non-compact gauge orbits, this gauge orbit volume is infinite, so it is not clear what it means to be constant on phase space. In a moment, we will show that, in the case when the algebra of constraints under consideration has structure constants, then at least the gauge orbit volume element is constant on phase space. Assuming that the ranges of the coordinates α i on each gauge orbit are then also constant, this implies that the fully integrated gauge orbit volume itself is also constant, as required for equivalence.
The proof of the constancy of the gauge orbit volume element, det M(α) det µ(α) , when there are structure constants, is straight forward. We have µ(sα) i j C j := g(−sα) * C i = ∞ n=0 s n n! {α · C, {α · C, . . . {α · C, C i } . . . }, } where in each term there are n nested Poisson brackets, with the n = 0 term being just C i . If one has structure constants, each Poisson bracket introduces multiplication by a matrix which is constant on phase space. The product of these matrices, summed over n, is then equal to the matrix µ(sα) i j on the left hand side, which is therefore also constant on phase space, leading to M (α) i j , and hence also det M(α) det µ(α) being constant on phase space. To handle the case of structure functions, which is in the case relevant for gravity, one must look not only at the 'gauge orbit volume element', but also at the fully integrated 'gauge orbit volume'. As this volume is infinite, as already mentioned, it is not so clear what it means for it to be constant on the phase space. A better understanding of how to handle this infinite volume through an appropriate regulator, or experimentation with toy models with structure functions in which the gauge volume is finite, could shed light on how to extend the last step of this proof of equivalence to systems of interest with structure functions.

C.2 A result for non-Abelian gauge groups
In this subsection we prove equation (39) below, which has been key in the above subsection. We begin by proving a general result for linear operators, which we then apply to the relevant case at hand. In this expression, the sum over n telescopes, so that all terms in the sum cancel except the second term in the n = m case. One is thus left with I = where, in the fourth line, lemma 5 was used in the case where V is the space of functions on Γ, A = X α·C , and B = X β·C .