Wavefunction of the universe: Reparametrization invariance and field redefinitions of the minisuperspace path integral

We consider the Hartle-Hawking wavefunction of the universe defined as a Euclidean path integral that satisfies the"no-boundary proposal."We focus on the simplest minisuperspace model that comprises a single scale factor degree of freedom and a positive cosmological constant. The model can be seen as a non-linear $\sigma$-model with a line-segment base. We reduce the path integral over the lapse function to an integral over the proper length of the base and use diffeomorphism-invariant measures for the ghosts and the scale factor. As a result, the gauge-fixed path integral is independent of the gauge. However, we point out that all field redefinitions of the scale factor degree of freedom yield different choices of gauge-invariant path-integral measures. For each prescription, we compute the wavefunction at the semi-classical level and find a different result. We resolve in each case the ambiguity in the form of the Wheeler-DeWitt equation at this level of approximation. By imposing that the Hamiltonians associated with these possibly distinct quantum theories are Hermitian, we determine the inner products of the corresponding Hilbert spaces and find that they lead to a universal norm, at least semi-classically. Quantum predictions are thus independent of the prescription at this level of approximation. Finally, all wavefunctions of the Hilbert spaces of the minisuperspace model we consider turn out to be non-normalizable, including the no-boundary states.


Introduction
According to the inflationary paradigm, a tiny Planckian region of space underwent a period of rapid accelerated expansion, and grew large enough to encompass the entire observable universe. Quantum effects are crucial to understand the state of the universe at the initial stages of this inflationary era. Indeed, it is widely believed that quantum fluctuations seed the primordial density perturbations, which lead eventually to the large scale structure of the universe today. Hence, it would be desirable to obtain the wavefunction of the universe, and to uncover a statistical interpretation favouring initial conditions amenable for inflation.
A step toward this direction was initiated by Hartle and Hawking [1] in the context of Einstein's theory of gravity for closed universes, in the presence of a positive cosmological constant Λ > 0. In particular, they proposed a definition for the "ground-state wavefunction," which Vilenkin has interpreted as the probability amplitude for creating from "nothing" a three-dimensional universe with metric h ij [2][3][4][5]. In practice, this wavefunction is computed via a Euclidean path integral according to the "no-boundary proposal." This path integral involves a sum over compact four-geometries that end on a particular spatial slice with induced metric h ij . Notice that the denomination of "ground state" is somehow misleading, since in quantum gravity all states associated with a closed universe (including the matter content) are degenerate, with vanishing energy. Moreover, as we will see, such wavefunctions are not necessarily normalizable.
In this work, we address and clarify four issues related to this path integral approach to quantum gravity: (i) To begin with, since general relativity is invariant under diffeomorphisms, special attention must be paid to the gauge fixing of this symmetry. This problem can be analyzed in the simpler framework of minisuperspace models, where the universe is assumed to be homogeneous, with a finite number of degrees of freedom depending only on time. In the literature, this gauge fixing of time-reparametrizations has not always been implemented appropriately for wavefunctions of the universe defined as path integrals, since the results depend on the chosen gauges [6].
In the present work, we consider the simplest such minisuperspace model, corresponding to a homogenous and isotropic universe with a single dynamical degree of freedom, namely the scale factor a. The model can be interpreted as a non-linear σ-model, where Euclidean time parametrizes a base manifold which is a line segment. Thus the scale factor is a coordinate in a one-dimensional target space. After gauge fixing of the Euclidean-time reparametrizations, the path integral over the lapse function reduces to an integral over the moduli space of the base manifold, which is parametrized by the proper length of the line segment. Moreover, we use diffeomorphism-invariant path-integral measures for the Faddeev-Popov ghosts and the scale factor. The outcome is a gauge-fixed path integral consistently independent of the choice of gauge.
(ii) Field redefinitions of the scale factor, a = A(q), leave the classical action invariant. However, at the quantum level, the path-integral measures Da and Dq are not equivalent, as they are related to each other by a Jacobian. Since the fields a and q correspond to different coordinate systems of the σ-model target space, there is no preferred choice among these measures. As a result, infinitely many acceptable definitions of wavefunctions exist, which are associated with all possible field redefinitions of the scale factor degree of freedom.
For each choice of path-integral measure Dq, we compute the ground-state wavefunction using the steepest-descent method. This amounts to deriving all instanton solutionsq and the corresponding values of the modulus¯ of the base segment, and to computing the quantum fluctuations and modulus fluctuations around these solutions at quadratic order. This is achieved by evaluating functional determinants by applying and refining the methods presented in Ref. [7].
(iii) Given a choice of measure Dq, the set of wavefunctions describing the allowed states of the universe are the solutions of a Wheeler-DeWitt equation [8], which is the quantum analogue of the vanishing of the classical Hamiltonian on shell. However, in the process of canonical quantization of the Hamiltonian, the question of the ordering of the canonically conjugate variables q and π q gives rise to an ambiguity in the exact form of the Wheeler-DeWitt equation. 1 Thanks to our derivation of the ground-state wavefunctions via the steepest-descent method, we are able to compare the results with the solutions of the Wheeler-DeWitt equation found by applying the WKB approximation. This allows us to resolve the ambiguity of the equation at the semi-classical level for every choice of measure Dq in the path integrals.
(iv) A question related to points (ii) and (iii) arises then. Do the different choices of measures Dq in the prescription of the wavefunctions, as well as the distinct associated Wheeler-DeWitt equations, lead to different quantum gravity theories with the same classical limits? We show that the answer to this question is negative, at least at the semi-classical level.
In fact, by imposing that the Hamiltonians of these possibly distinct quantum theories are Hermitian, we find that the inner products of the corresponding Hilbert spaces take different forms in each case. However, the point is that the norms of the wavefunctions in the WKB approximation turn out to be universal, i.e. independent of the prescription. A non-trivial crosscheck of our results is provided by the particular choice of measure Dq associated with a field q with a quadratic kinetic term. In that case we find that the Wheeler-DeWitt equation reduces to the time-independent Schrödinger equation for vanishing energy, and that the inner product of the Hilbert space takes the standard form encountered in quantum mechanics.
Moreover, in the minisuperspace model that comprises a single scale factor degree of freedom, it turns out that all wavefunctions of the two-dimensional Hilbert space are nonnormalizable, including that of the "ground-state." To construct normalizable states, extra degrees of freedom must be added, rendering the dimension of the Hilbert space infinite.
In some cases, it could then be possible to construct square-integrable wavefunctions by superimposing infinitely many solutions of the Wheeler-DeWitt equation [8].
In Sect. 2, we review the definition of the ground-state wavefunction in the minisuperspace model we consider. The implementation of the gauge fixing of the Euclidean-time diffeomorphisms introduces a Faddeev-Popov Jacobian. The latter is expressed in Sect. 3 as a path integral over anticommuting ghosts, which is evaluated by using gauge-invariant measures. The outcome of the computation is that the Faddeev-Popov determinant is a trivial (and irrelevant) constant. The following two sections are devoted to the evaluation of the gauged-fixed path integral by applying the steepest-descent method. This is done in Sect. 4 by using the measure Da of the scale factor, while in Sect. 5 the result is generalized to any choice of gauge-invariant measure Dq associated with a field q obtained by redefining the scale factor degree of freedom. We also clarify the extent to which our calculations differ with previous works such as Refs. [6,[14][15][16], leading to different results. In Sect. 6, we lift at the semi-classical level the ambiguities arising in the Wheeler-DeWitt equations corresponding to the wavefunctions with arbitrary prescriptions Dq for the path-integral measure. The equivalence at this level of approximation of all Hilbert spaces is demonstrated in Sect. 7.
Our conclusions and perspectives can be found in Sect. 8. Various technical complements are reported in the Appendix.

Ground state wavefunction
In this section, we present the formal definition of the ground-state wavefunction of a closed, homogeneous and isotropic universe. The wavefunction is expressed in terms of a Euclidean path integral over the lapse function and the scale factor. Our goal is to carry out carefully the gauge fixing of the Euclidean-time reparametrization group. In the following sections we present the computations of the Faddeev-Popov determinant and the path integral over the scale factor.

Minisuperspace of dimension one
We are interested in Einstein's theory for spatially-closed universes in the presence of a positive cosmological constant Λ > 0, formulated on Lorentzian four-manifolds M with space-like boundaries ∂M. The appropriate action reads where R is the scalar curvature associated with the metric g µν (x) on M, while K is the trace of the extrinsic curvature on ∂M whose metric is denoted h ij (x b ).
In this work, we consider the minisuperspace version where the degrees of freedom are reduced to a single scale factor depending only on time, a(x 0 ). This amounts to restricting the manifolds M to be homogeneous and isotropic. The invariant infinitesimal length squared is given by where N (x 0 ) ≡ g 00 (x 0 ) is the lapse function and dΩ 3 is the volume element of the unit 3-sphere. Moreover, ∂M is composed of 3-spheres at initial and final times x 0 i and x 0 f , and the boundary action becomes where v 3 = 2π 2 is the volume of the unit 3-sphere. This term cancels a similar boundary term generated upon integrating by parts the second derivative of the scale factor arising from the Ricci scalar. As a result, the full action is reduced to Notice the negative sign of the kinetic energy compared to that of a conventional matter scalar field. In this form, the action is expressed in terms of a Lagrangian involving only the scale factor and its first derivative (along with N ), which is suitable for canonical quantization, as well as for deriving classical equations of motion, keeping the scale factor fixed on the boundaries.

Ground-state wavefunction as a Euclidean path integral
The quantum state of the universe can be described by a wavefunction that depends on h ij , defined as a path integral. Hartle and Hawking have proposed to sum over all manifolds M with a space-like boundary, on which the induced metric is h ij [1]. Specifying other conditions on the class of paths summed over amounts to characterizing the state. For the ground state, the Hartle-Hawking prescription is formulated in terms of the Euclidean action, while the four-manifolds should have no other boundary than that of metric h ij . This is the "no-boundary proposal" for the ground-state wavefunction. It follows from this definition, that the wavefunction can be interpreted as the amplitude for creating the three-geometry from the empty set, i.e. from "nothing" [2][3][4][5]. 2 In minisuperspace, the boundary geometry is fully characterized by the scale factor a 0 on the boundary 3-sphere. As a result, a possible definition of the ground-state wavefunction is given by (2.5) where we keep explicit the reduced Planck constant . The following comments are in order: • Two possible prescriptions for the continuation to imaginary time have been advocated in the literature, given by Hartle and Hawking [1] take s = −1, while Vilenkin [2][3][4][5]13] and Linde [9] argue for s = +1.
In various minisuperspace models, the predictions associated with the two choices can be drastically different. For instance, when the path integral is approximated by the steepestdecent method, with only one instanton solution taken into account, the wavefunction for s = −1 leads to the conclusion that the cosmological constant is likely to be null [10]. On the other hand, when s = +1 conditions amenable for inflation are favored [9,20]. In the following, we consider both options, so that the Euclidean action S E = −iS becomes where we have defined Moreover, the paths a(x 0 E ) must obey a(x 0 Ef ) ≡ a f = a 0 and the "no-boundary condition" a(x 0 Ei ) ≡ a i = 0. Since both choices of s yield actions S E [g 00 , a] not bounded from below, special attention must be paid for defining convergent path integrals. We will come back to this issue in Sect. 4.3.
• The action S E describes a non-linear σ-model where Euclidean time parametrizes a base manifold which is a segment of metric g 00 , while the scale factor is a coordinate in a one-dimensional target space of metric (2.9) In other words, the system can be viewed as describing a worldline a(x 0 E ) in a dimension-one target minisuperspace, by analogy with the worldline trajectory of a particle in spacetime, or a worldsheet embedded in spacetime in string theory.
• The action being invariant under Euclidean-time reparametrizations, all diffeomorphismequivalent metrics g 00 yield an overcounting of physically equivalent configurations. As a result, the path-integral measure Dg 00 must be divided by the volume of this group, which is denoted by Diff[g 00 ]. 3 • For field configurations a(x 0 ) and a(ξ(x 0 )), where ξ ∈ Diff[g 00 ], to be truly equivalent, the path-integral measure Da must be invariant under this symmetry group.
• However, diffeomorphism-invariant measures are not unique. The definition of the wavefunction in Eq. (2.5) uses a particular choice of coordinate in the target space, namely the scale factor. As a result, Ψ(a 0 ) is not equivalent to the wavefunction defined with the measure Dq, where q = Q(a) is a field redefinition for an arbitrary function Q. This is despite the fact that such a transformation leaves the action invariant, as it corresponds to a change of coordinate in the target space. Since there is no preferred variable q in the target space, there is no preferred measure Dq in the definition of the wavefunction. All choices yield wavefunctions solving different Wheeler-DeWitt equations. In Sects. 5 and 6, we will determine the relations between all these avatars and see in Sect. 7 how they yield equivalent predictions, at least at the semi-classical level.

Gauge fixing of the Euclidean-time reparametrizations
In this subsection, we rewrite in a more practical way the path integral over the metrics g 00 , which is weighted by the inverse of the volume of the diffeomorphism group.
For any metric g 00 defined on the domain [x 0 Ei , x 0 Ef ], let us denote the action of a change of coordinate as follows, For an infinitesimal diffeomorphism ξ = Id + δξ, this transformation rule yields where ∇ 0 is the covariant derivative associated with g 00 . Notice that not all metrics g 00 are equivalent up to diffeomorphisms since such transformations cannot change the proper length of the line segment, As a result, the set of metrics can be divided in equivalence classes distinguished by the value of ∈ R + . In other words, a line segment admits a moduli space of real dimension one parametrized by . 4 In practice, varying the modulus of a metric g 00 amounts to rescaling it while keeping fixed its domain of definition.
Let us define reference metrics that will serve as gauge-fixed representatives of each where Diff[g 00 ] Id is the subgroup connected to the identity.
Next we replace the path integral over g 00 by two integrals: An integral over the moduli space and the other one over the orbit of each equivalence class, in order to cancel the volume 4 We review in Appendix A.3 a formal proof of the fact that there is no other modulus than the length . Since the proof uses ingredients from Sect. 3.1, the reader can wait until then before reading it. 5 In fact, an infinitesimal diffeomorphism that leaves the metric invariant must satisfy δg 00 = 0. Hence, which implies δx 0 E = C √ g 00 for some constant C. However, the definition of the metric includes its domain of definition, which must also be left invariant by the diffeomorphism. Hence, δx 0 of the diffeomorphism group. This operation yields a Jacobian that can be determined by applying the method of Faddeev and Popov, which is valid for any local symmetry group. Dξ Dg 00 where L E is the Euclidean Lagrangian density associated with the action S E . Performing the path integral over g 00 and renaming a into a ξ leads to (2.17) In the above expression, the action is diffeomorphism invariant. As stressed in the previous subsection, we also use a scalar-field measure satisfying this symmetry (see Sect. 4.3 for an explicit construction at the semi-classical level), Dξ δ g 00 −ĝ ξ . (2.19) In the second line, we use the fact that the δ-functional is diffeomorphism invariant, as will be seen explicitly in the next section. Moreover, the third line follows from the change of variable ξ = ξ −1 •ξ and the invariance of the measure, Dξ = Dξ , which is the case because the symmetry of reparametrizations is anomaly free. 6 Finally, the group Diff[ĝ ξ 00 [ ]] is by definition diffeomorphism invariant. As a result, the wavefunction simplifies to give (2.20) where the action is defined on the fixed domain of the fiducial metrics All dependency on ξ being now trivial, the path integral over the diffeomorphisms can be carried out, cancelling the volume factor. The ground-state wavefunction therefore takes the form of a gauged fixed path integral

Introducing ghost fields
Since the reversal of the orientation of the segment is the only isometry, the path integral appearing in the expression is twice the contribution of the diffeomorphisms connected to the identity, Dξ . (3.2) In that case, the δ-functional enforces only = and ξ = Id. In the vicinity of this point, the total variation of the fiducial metric is where the diffeomorphisms δx 0 E are required to vanish at the boundariesx 0 Ei andx 0 Ef of the domain ofĝ 00 [ ]. Otherwise the δ-functional implies that they do not contribute at linear order.
By writing the δ-functional as a Fourier transform, the previous expression becomes (3.5) where the measure Dβ is assumed to be diffeomorphism invariant for the δ-functional to be invariant as well. Notice that we have not made any reference to the contravariant or covariant nature of the tensors in the measures Dδx E and Dβ because the contractions in the argument of the exponential can be made in an arbitrary way. We justify at the end of the section why this does not introduce ambiguities in the measures.
At this stage, it is relevant to introduce a notation that will be used extensively in the following. Let f 0···0 and h 0···0 be two tensors with m ∈ Z contravariant indices, where m < 0 actually means that the tensors have |m| covariant indices. We define the reparametrizationinvariant quantity it is convenient to switch all integration variables into Grassmann ones, where b 00 and c 0 are ghost fields. This operation inverts the r.h.s. of Eq. (3.5) up to an irrelevant numerical factor α [21]. Hence, we obtain where the second equality follows by Berezin integration over λ.

Choice of fiducial metric and mode expansions
Even though the content of this subsection can be obtained without specifying the fiducial metric, we find it simpler to present it for a convenient choice, remembering that is independent of this choice. So let us choose a constant lapse function, where the variablex 0 E is denoted τ for the sake of simplicity, while∇ ≡ ∂ 0 . Notice that τ is proportional to the "cosmological Euclidean time t E ," which satisfies dt E = dτ .
In Eq.
The domain of definition of β 00 can be extended to R by imposing β 00 to be even on [−1, 1] and 2-periodic. Similarly, the domain of definition of δx E0 can be extended to R by imposing δx E0 to be odd on [−1, 1] and 2-periodic. As a result, they can be expanded in the Fourier basis {1/ √ 2, cos(kπτ ), sin(kπτ ), k ∈ N * } restricted to the even modes for β 00 and odd modes for δx E0 .
In that case, we can expand β 00 , δx E0 and the ghost fields in orthonormal basis, where β k , γ k and anticommuting b k and c k are arbitrary constants, while Using these conventions, we first obtain two pieces of Eq.
Next, we determine the correctly normalized path-integral measures of the ghosts. In the case of commuting tensors β 00 and δx E0 , the norms associated with the inner products in Eq. (3.10) yield In the anticommuting case of interest, we thus adopt the analogous prescriptions, Taking into account these results, Eq. (3.8) becomes, up to an ambiguous sign that depends on the precise ordering of the Grassmann integration variables, where the last equality follows by the zeta regularization formulas is actually a Jacobian, α must be such that ∆ FP [ 2 ] is a real non-negative number.
The important outcome of the above calculation is that the Jacobian is a trivial constant, i.e. it is independent of . Note that this is not something that could be inferred at the outset, as this result applies to the case of a base manifold that is a segment in a specific way. Indeed, one can check that in the case of a base manifold with the topology of a circle, the Jacobian proves to be a constant times 1/ , where is again the proper length of the circle [22]. 8 To conclude this section, let us mention that since this is important, we make even

Scale-factor path integral and modulus integral
Having implemented the gauge fixing of Euclidean-time reparametrizations, we now proceed to compute the path integral over the scale factor. We will work in the simplest gauge where the lapse function is constant, as displayed in Eq.
where ∆ FP is an irrelevant constant and the action (2.7) is Since the action is not quadratic, we will make use of the method of steepest-descent to approximate the evaluation of the wavefunction. In practice, one has to expand S E [ 2 , a] around its extrema and integrate over the fluctuations at quadratic order only. The validity of this approximation is guaranteed when → 0, which corresponds to the semiclassical limit. We would like to stress that our treatment to come differs to what is done in previous works [6,[14][15][16], as will be explained in Sect. 5.3.

Instanton solutions
In order to apply the method of steepest-descent, we extremize the action with respect to both and a(τ ). Denoting such an extremum as (¯ 2 ,ā), we first have to solve Moreover, writing the equation of motion for the scale factor reads where V a ≡ dV /da.
where E is an arbitrary integration constant, which can be related to the total energy of the universe. Using this relation, Eq. (4.3) gives 0 = sE, showing that it suffices to solve the Friedmann equation in order to extremize the action with respect to and a(τ ).
The latter equation takes the form with solution λā(τ ) = | sin(λ¯ τ + cst.)|. Imposingā(0) = 0 implies the constant to be zero, while the conditionā(1) = a 0 fixes the modulus¯ of the base segment. When can take discrete real values. However only two yield smooth instanton solutions, The line elements of the associated Euclidean spacetimes are showing that the instantons describe portions of a 4-sphere of radius 1/λ. Actually, = +1 corresponds to a cap smaller than a hemisphere (see Fig. 1a), while = −1 describes a cap bigger than a hemisphere (see Fig. 1b). The actions associated with these solutions are 9 Other instanton solutions exist, which are continuous but not differentiable, The associated four-dimensional Euclidean spacetimes look like "necklaces" composed of n beads (4-spheres).
The series terminates with a small or big cap. However, if one restricts the path integral sum to include smooth four-manifolds only, these solutions should be excluded. It is nevertheless interesting to relax this requirement, as in Ref. [23]. Then because the actions of these Euclidean solutions arē In this section, the semi-classical approximation of the wavefunction Ψ(a 0 ) will be carried out by applying the steepest-descent method to evaluate the path integral when 0 < λa 0 < 1.
The expression in the regime λa 0 > 1 will be determined to some extent in Sect. 6 by analytically continuing the solutions of the Wheeler-DeWitt equation. 10

Expansion to quadratic order
The action (4.2) can be expanded around the extremal solutions (¯ 2 ,ā ). Setting where δa(τ ) obeys the boundary conditions of Eq. (4.4), one obtains to quadratic order All linear terms vanish thanks to the equations of motion and O(δ 3 ) stand for terms at least cubic in fluctuations. In the above equation, Q is a quadratic differential operator, The linear operator Q can be seen as an endomorphism of this Hilbert space. 11 Its adjoint, denoted Q T , satisfies by definition and can be determined by integrating by parts the left hand side, Let us then introduce the "symmetric" and "antisymmetric" parts of Q , They satisfy which means that S is self-dual while A satisfies (δa, A δa)¯ = 0 for all δa. As a result, we obtain Using the explicit expression of the instanton solution in Eq. (4.9), the differential operator S can be written in the following form, In the next subsection, we will show that the operator S is invertible when 12 Assuming for the moment this fact, we can diagonalize the integrand appearing in Eq. (4.13) when λa 0 is restricted in the above range, The above expressions deserve a comment. A priori, V a (ā ) is a function which satisfies Hence, in Eq. (4.23), V a (ā ) should be thought of as its Fourier expansion in the above basis, which indeed belongs to the Hilbert space.
Applying the steepest-decent method, the wavefunction in Eq. (4.1) reads 13 where we have defined Since S −1 V a (ā ) belongs to the Hilbert space, so does δǎ. As a result, we may perform a change of variable (actually field redefinition) in the path integral from δa to δǎ, with 12 Actually, it is a regularized version of S that will satisfy this property. 13 Since the actionsS E approach each other when λa 0 approaches 1, we do not absorb one of the two instanton contributions in the corrections O( ).
identical vanishing boundary conditions at τ = 0 and τ = 1. Furthermore, taking into account that the measures are related to each other by a trivial Jacobian (4.28)

Scale-factor quadratic fluctuations
For the sake of simplicity we make the change of notation δǎ → δa in the above expression of Z (a 0 ). Our goal in the present subsection is to compute this contribution to the wavefunction.

Mode expansion
Since S is a self-dual endomorphism of the Hilbert space, it is diagonalizable in an orthonormal basis. Denoting φ k the eigenvector with eigenvalue ν k , we have 14 Expanding the scale factor fluctuation as the path-integral measure is derived from the norm (4.31) 14 We will show later on that the eigenspaces are of dimension 1, so that an index k is sufficient to label the eigenvectors unambiguously. Moreover, this label takes values in N * since 2/¯ sin(kπτ ), k ∈ N * is another (orthonormal) basis.
Hence, we obtain where the last equality follows by zeta regularization and det S = k≥1 ν k . Notice that the prescription that we apply to the path integral is the following: Each mode coefficient δa k is integrated from −∞ to +∞ when sν k > 0, and from −i∞ to +i∞ when sν k < 0. 15  where it is not [7,24].
Before we go any further, let us mention that an explicit check of the fact that Z (a 0 ) is independent of the choice of fiducial metric is provided in Appendix A.4.

Computation of the determinants
To compute the determinant of S we apply the method presented in Ref. [7]. Let us consider where ν is real while τ ∈ (0, 1) will serve as a regulator to be sent to 0 at the end of all computations. Since this is a linear differential equation of order 2, the initial conditions on ϕ ν and its derivative select a unique solution for any ν. When such a solution satisfies ϕ ν (1) = 0, we know that there is k ∈ N * such that ν = ν k . Moreover, the dimension of any eigenspace of S cannot be of dimension > 1 since otherwise there would be two or more solutions of the differential system for some ν = ν k . 16 We may order them such that 15 The case ν k = 0 is excluded since we assumed that S is invertible, as will be shown later on. 16 φ k and ϕ ν k /(ϕ ν k , ϕ ν k )¯ are thus equal up to an arbitrary sign.
We can take a more general look to the same system for two operators of the Hilbert space that are identical to S up to the terms involving no derivative. They can be written as It then turns out that [7] det . (4.36) Indeed, because both sides of this equality are meromorphic functions of ν with identical simple zeros and poles, they are proportional. Moreover, thanks to the fact that W (1) and W (2) are bounded on [τ , 1], the two sides of the equality tend to 1 when ν → ∞ in the complex plane, which shows that they are equal for all ν ∈ C. Hence, there is a constant 17 To apply this formula to det S , we have to compute ϕ 0 (1) by solving Eq. (4.33) for ν = 0.
the system becomes The general solution of the differential equation can be written as where A , B are constants determined by the initial conditions. 19 In the limit θ * → 0, they Therefore, the mode proportional to B is dominated by the mode proportional to A , so

45)
19 It is at this stage that we see the relevance of introducing a cutoff τ i.e. θ * . With θ * = 0 we would have A = B = 0 which is not allowed for an eigenvector. 20 In fact, this is true unless the solutions of the system do not generate a dimension one vectorial space. This fact happens for some choices of W (1) and ν for which S in terms of which we obtain There is therefore a choice for which all eigenvectors and eigenvalues can be found trivially. They are given by where C k are arbitrary constants. It is then straightforward to compute in this case the determinant by zeta regularization, (4.49) Next, we recompute this determinant by applying Eq. (4.38). We thus consider the (4.52) Integrating, we finď which we can use to get Identifying the above expression with Eq. (4.49), we obtain and so S + and S − are invertible, as announced before.

Quadratic fluctuations of the length
The computation of the Gaussian integral over the fluctuation δ of the modulus in the expression of the wavefunction in Eq. (4.28) requires the evaluation of K given in Eq. (4.26).
Therefore, we determine S −1 V a (ā ), which is the unique solution of the system where τ is the regulator close to 0 that was used in the derivation of det S in the previous subsection. As explained below Eq.  The general solution of the differential equation is where D , E are constants determined by the initial conditions. 21 In the limit θ * → 0, they (4.62) As a result, the function f (τ ) is The quadratic fluctuations of the length around the extremal value¯ then yield a contribu- provided the domain of integration is from −∞ to +∞ when s > 0, and from −i∞ to +i∞ when s < 0.
We are now ready to insert in the expression of the wavefunction given in Eq. (Eq. (4.11)), the quadratic fluctuations Z (a 0 ) of the scale factor (Eqs. (4.32), (4.56)) and those of the modulus (Eq. (4.65)), to obtain the final result In this expression, C s (θ * ) is a regulator-dependant coefficient which is an irrelevant multiplicative factor. Notice that even though the wavefunction defined in Eq. (2.5) seems to be real, it turns out to be complex due to the factor 1/

Field redefinitions
The classical action in Eq.

Distinct gauge-invariant field measures
To better understand the issue, let us apply a field redefinition of the scale-factor degree of freedom, where Q = A −1 is an invertible function defined for a > 0. 22 In fact, q(τ ) is a field defined on [0, 1] satisfying fixed boundary conditions at τ = 1 and τ = 0, Around the small-or big-cap instanton solution, the fluctuations satisfy and where "primes" denote derivatives.
Introducing the cutoff τ as in the previous section, δa(τ ), δq(τ ) and A (q (τ ))φ k (τ ) are all elements of the Hilbert space of functions in L 2 ([τ , 1]) that are vanishing at τ and 1. 23 Hence, we may expand where the coefficients δa k were already defined in Eq. where the equality is to be understood as a relation between column matrices δa and δq.
Using this matrix notation, the regularized path integral of the fluctuations δa reads where P −1 M S M T P is diagonal. Two possible outcomes to this calculation can be proposed.
Firstly, we can compute the Gaussian integral as in Eq. (4.32) and check that it yields the same answer, up to O( ) terms: This is a matrix proof of the fact that a "change of field" can be applied in the path integral in analogy with a "change of variable" in a conventional integral. In fact, | det M | is nothing but the Jacobian of this transformation.
However, comparing Eq. (5.7) with the two first lines of Eq. (5.6), we can also write where Z (q 0 ) is a path integral based on the field δq,
Being represented by a real symmetric matrix, it is self-adjoint.
The conclusion to the above discussion is that for every field redefinition a = A(q) in the classical action, it is legitimate to define a quantum wavefunction as in Eq. (4.1), which is characterized by its own path-integral measure Dq. In the following we compute Ψ(q 0 ) in the semi-classical limit.

Steepest-descent approximation
The quadratic expansion of the classical action in Eq. (4.13) (and Eq. (4.20)) can be expressed in terms of δq and δ , 14) The explicit form of the self-dual operator S is obtained from its definition in Eq. (5.11), which yields where all functions A and their derivatives are evaluated atq . Proceeding as we did for Ψ(a 0 ) between Eqs. (4.23) and (4.28), the steepest-descent approximation leads to Notice that we could have reached this result by simply substituting Z (a 0 ) with Z (q 0 ) (related to each other by Eq. (5.9)) in Eq. (4.28). In that case, K would have not replaced K . Consistently, we will find that K = K .

Determinants of the fluctuations of q
As in Eq. To compute them we can work exactly as we did in Sect. 4.3.2. This leads to the result det S = N ϕ 0 (1) , (5.18) where N is a constant to be determined and ϕ 0 is the solution of the system However, the change of unknown function where K is a constant, transforms the differential equation back to a form involving the much simpler operator S , Hence, choosing K = A (q (τ )), the functions ϕ 0 and ϕ 0 (encountered in the computation of det S ) satisfy the same equation and initial conditions, as can be seen in Eq. (4.33).
Therefore, they are equal, ϕ 0 = ϕ 0 , which leads to In this expression, it is important to note that which depends only on the cutoff θ * (and not on a 0 ). Hence, we obtain

Computation of the normalization constants N
The computation of N can be done by following steps similar to those presented in Sect. 4.3.3.
Let us start by deriving explicitly the infinite product of the eigenvaluesν (1) k of an operator where W (1) (τ ) is a function on [τ , 1] to be chosen suitably. These eigenvalues are such that the system admits a dimension one vectorial space of solutions. By defining a new Euclidean-time variableζ satisfying¯ dτ = A(q (τ )) A (q (τ )) dζ (5.27) and a new unknown functionˇ ψ k (ζ) by the differential equation takes a canonical form Choosing W (1) such that the term in brackets vanishes identically, the system becomes identical to the one analyzed in Eq. (4.46). As a result, we obtain that The next step is to obtain an alternative form of the determinant in terms of the constant N . The latter being "universal," it enters the formula analogous to Eq. (5.18), Applying the change of unknown function ϕ (1) 0 (τ ) = r(ζ)ˇ ϕ (1) 0 (ζ) , (5.33) this system becomes 3 4 |A (q(τ ))| A (q(τ )) .

(5.34)
Integrating and multiplying by r(ζ), one obtains which can be evaluated at τ = 1 to find The identification of the above expression with Eq. (5.30) leads to In the above expression, sign Q is a sign independent of a 0 because the function Q(a) is invertible and thus monotonic. Moreover, we also have Q (a) = 0 for all a > 0 since otherwise A (q) = 1/Q (a) would not be defined at some q = Q(0). 24 Hence, we conclude that: • The small-and big-cap instantons yield invertible operators S when 0 < a 0 < 1.
• When Q and A are increasing, we have det S + > 0 and det S − < 0. On the contrary, when Q and A are decreasing, the converse is true i.e. det S + < 0 and det S − > 0.

Quadratic fluctuations of the length
What remains to be done to obtain the expression of Ψ(q 0 ) in the semi-classical approximation is to find the value of K defined in Eq. (5.16). To this end, we determine S −1 V q (q ), 24 We may however have Q (0) = 1/A (Q(0)) ∈ {0, +∞, −∞}.
which satisfies , (5.40) the above system can be re-written in terms of the operator S , Since it turns out to be identical to the one given in Eq. (4.58), we conclude that f = f .
As a result, we obtain that which was anticipated before and shows that the result in Eq. (1 + O( )) , 0 < λa 0 < 1 , Alternatively, we may express the wavefunction explicitly in terms of q 0 as follows, (1 + O( )) . (5.45) 25 Notice that we also obtain that δǎ ≡ δa Hence, it is equivalent to present the computation in Eq. (5.6) as a change of variable from δa to δq or from δǎ to δq (see Eq. (4.28)).

Difference with previous works
Before concluding, we would like to stress that even if the analysis presented in our paper shows similarities with those developed in previous works, it is to a large extent different and leads to different results.
For instance, the starting point in Ref. [15] is Eqs. (4.1) and (4.2) (considered in Lorentzian time), which describe the amplitude for the universe to evolve from a = 0 to a = a 0 in a proper time . Following Refs. [6,14], the authors then apply the following change of time

Wheeler-DeWitt equation
The Wheeler-DeWitt equation is the differential equation whose solutions are the wavefunctions associated with the states of the entire quantum gravity Hilbert space [8]. However, its derivation leads to an ambiguity that arises when applying canonical quantization on its classical counterpart. In this section, we lift this ambiguity for each given choice of measure Dq in the definition of the wavefunctions, at least at the semi-classical level.

Ambiguity in the Wheeler-DeWitt equation
In this subsection, we derive the Wheeler-DeWitt equation satisfied by the wavefunctions corresponding to the choice of measure Dq in the path integrals. This will be done by working in Lorentzian signature.

Hamiltonian constraint
The starting point is to note that the path integral of a total functional derivative vanishes, Indeed, the right-hand side contains a multiplicative factor that is a boundary term, and which vanishes for a suitable contour of integration of N (x 0 ), Using this result, we have where the path integral is over any class of paths denoted by C.
The conjugate variables of the generalized coordinates q and N are given by 5) and the classical Hamiltonian obtained by a Legendre transformation reads which can be used in Eq. (6.4) to obtain The above result, which is valid for any class of paths C, is equivalent in the Hilbert-space formalism to saying that all matrix elements of the quantum Hamiltonian divided by the lapse function are vanishing. This quantum operator is thus annihilating all kets, In terms of wavefunctions Ψ C , the canonical quantization of the classical expression of H/N given in Eq. (6.6) is obtained by replacing Indeed, this prescription provides a representation of the commutator relation [q, π q ] = i on the Hilbert space of wavefunctions. However, because classically π 2 q AA 2 = 1 AA 2 ρ 1 ρ 2 π q ρ 1 π q ρ 2 for any functions ρ 1 (q 0 ), ρ 2 (q 0 ) , (6.11) the process of quantization yields an ambiguity in the form of the Wheeler-DeWitt equation, where A, A along with ρ, ω depend on q 0 .

Alternative formulation
In some cases, it is convenient to formulate the dependance of the wavefunctions on boundary data in terms of the scale factor a 0 ∈ [0, +∞) instead of q 0 = Q(a 0 ) ∈ [Q(0), Q(+∞)). This corresponds to applying the change of variable Notice that even though the functions Ψ AC depend on a 0 , they still correspond to path integrals based on the measure Dq. For instance, among all wavefunctions Ψ AC (a 0 ), the one denoted Ψ A (a 0 ) corresponding to the ground-state is to be distinguished from Ψ(a 0 ) given in Eq. (4.66), which corresponds actually to the particular case A(q) = q so that Da ≡ Dq.
Applying the change of variable in the Hamiltonian constraint in Eq. (6.14), one obtains the equivalent equation In fact, the above result can be obtained by deriving the Hamiltonian constraint in terms of the conjugate variables a, π a and applying the canonical quantization prescription while taking into account the ambiguity of the classical quantity a −1 π 2 a .

Resolution of the ambiguity
In Sect. 5, we have determined at the semi-classical level the ground-state wavefunction Ψ(q 0 ), which must be a particular solution of the Wheeler-DeWitt equation. Hence, we can lift the ambiguity of the equation by imposing that this is indeed the case.
To this end, we put Eq. (6.14) in the following form, and apply the WKB method [26]. This is done by writing where F 0 , F 1 are functions of q 0 , leading to The unknown function ω is absorbed in the O(1) terms and thus cannot be determined at the semi-classical level. On the contrary, F 0 is found by identifying the contributions O(1/ 2 ), while F 1 is such that the term O(1/ ) vanishes. Two cases can be considered: • When P(q 0 ) > 0 i.e. 0 < λa 0 < 1, we find two imaginary solutions for F 0 , where s denotes either +1 or −1. Consistently, there are two linearly independent modes which lead to 22) where N C are two integration constants. Comparing this result with Eq. (5.45) we can identify ρ,ρ Using the relation between ρ and ρ A , we also obtain the alternative form of the wavefunctions which is consistent with Eq. (5.43). Hence, we have ρ(q 0 ) > 0 and ρ A (a 0 ) > 0 for 0 < λa 0 < 1 (thanks to the remark below Eq. (5.38)). Moreover, the values N of the mode coefficients N C (with arbitrary normalization) that select the ground state are given by . (6.25) • For completeness, let us mention that in the second case P(q 0 ) < 0 i.e. λa 0 > 1, the two solutions for F 0 are real, which lead to (1 + O( )) , λa 0 > 1 , (6.27) where M C are integration constants. However, we do not have an explicit expression of a solution of the Wheeler-DeWitt equation derived by evaluating a path integral when λa 0 > 1. 26 Hence, we cannot determine ρ(q 0 ) when λa 0 > 1. 27 Applying the change of variable from q 0 to a 0 , we also have (1 + O( )) , λa 0 > 1 . (6.28) 26 We hope to provide such a computation at the semi-classical level in a forthcoming work. 27 However, notice that the constants N C and M C can be related, whatever the functions ρ and ω are. This can be done by solving the Wheeler-DeWitt equation in the neighbourhood of λa 0 = 1, where the solutions are expressed in terms of Airy functions. It is then possible to match the solutions and thus the integration constants in the three regions 0 < λa 0 < 1, λa 0 1 and λa 0 > 1.
It should be noted that the asymptotic form of the function ρ A (a 0 ), which governs the ambiguity of the Wheeler-DeWitt equation at the semiclassical level, can be fixed by imposing boundary conditions for the modulus of the wavefunction at a 0 = 0 and/or a 0 → ∞ [27].
A more complete theory of quantum gravity may also lead to a resolution of the ambiguities -see e.g. Ref. [28] for work towards this direction, in the context of loop quantum cosmology.
However, our approach in this work is different. We do not prescribe specific boundary conditions to select ρ A (a 0 ) and solve for the wavefunction. Rather, we construct a large family of no-boundary wavefunctions, based on the various choices Dq of gauge-invariant pathintegral measures, and then proceed to determine the Wheeler-DeWitt equation each such wavefunction satisfies. Equivalently, for each positive function ρ A (a 0 ), we can construct, via a suitable field redefinition, a path-integral wavefunction that implements the no-boundary proposal. In the next section, we will show that, at least at the semi-classical level, all these prescriptions yield identical observable predictions, despite the different asymptotic behavior the corresponding no-boundary wavefunctions exhibit. In particular, the inner product measure that renders the Hamiltonian Hermitian is such that the quantum probability density is universal, irrespectively of the choice Dq. In terms of Euclidean time t E = τ , the action S E defined in (5.12) is

Crosscheck with the Schrödinger equation
A relevant choice of target-minisuperspace coordinate q is such that the metric of the σ-model action is canonical. Indeed, imposing  Therefore, the Wheeler-DeWitt equation (6.14) reduces to Omitting the contribution of O( 2 ) to the potential, we recognize the time-independent Schrödinger equation for a stationary state of vanishing energy. This is the expected result for the following reasons.
In quantum mechanics, the Lorentzian action of a particle of unit mass moving on a line of coordinate q and subject to a force derived from the potential 3v 3 V (q) is The Lagrangian is opposite to that corresponding to the gravity system (see Eq. (2.4) in terms of the field a = A(q)), which has a negative kinetic term and a positive sign in front of the potential 3v 3 V (q). The amplitude for the particle to travel from an initial position q i to a final one q 0 in a real time is given by By discretizing time and varying the final position, one derives using standard manipulations that this amplitude satisfies the Schrödinger equation [29], The stationary states Ψ part Hence, for the gravity problem for which the total (kinetic + potential) energy is −E = 0, the exact Wheeler-DeWitt equation is Eq. (6.32), with ω = 0.

Quantum equivalence at the semi-classical level
In the previous section, we have identified for each choice of measure Dq in the path integrals the expression of the function ρ A (or ρ) appearing in the Wheeler-DeWitt equation. This expression for ρ A is valid for 0 < λa 0 < 1. However, we will show in this section that all these prescriptions yield identical observable predictions, at least at the semi-classical level. In fact, the identification of ρ A turns out to be somehow superfluous at this level of approximation, provided it is assumed to be positive. Moreover, any such function can be obtained for a suitable choice of field redefinition q = Q(a) and associated path integral measure Dq, at least when 0 < λa 0 < 1, as can be seen by integrating Eq.

Hermiticity of the quantum Hamiltonians
To define probability amplitudes, we need to specify for each choice of path integral measure Dq an inner product of the corresponding Hilbert space of wavefunctions. To this end, let us set where Ψ A1 , Ψ A2 are at this stage arbitrary complex functions. Moreover, µ A is a real positive function that we would like to interpret as a "measure" on the space of functions.
Alternatively, we may apply the change of variable shown in Eq. (6.15) and define where µ is related to µ A so that the two Hermitian forms are equivalent,

Determining µ A and µ
In order to find µ A , we are going to impose that the Hamiltonian differential operator H A /N in Eq. (6.16) is Hermitian on a suitable Hilbert space. To this end, we derive the following identity by integrating by parts, where we have defined For the Hamiltonian to be Hermitian, two conditions must be met: which can be seen to be equivalent to If κ is non-vanishing, we obtain a separated differential equation by dividing by κ , which leads upon integration to an expression of Ψ AC in terms of κ ρ A . However, this cannot be true since Ψ AC is arbitrary. Hence, κ ≡ 0 i.e. κ is a constant. Because the normalization of wavefunctions is irrelevant, we may take κ = 1 so that µ A (a 0 ) = a 0 ρ A (a 0 ) , λa 0 ≥ 0 , and µ(q 0 ) = A(q 0 ) A (q 0 ) 2 ρ(q 0 ) . (7.9) As a result, since µ A and µ are positive, the functions ρ A (a 0 ) and ρ(q 0 ) appearing in the Wheeler-DeWitt equations must also be positive for all λa 0 ≥ 0, and not only in the range 0 < λa 0 < 1 as shown in Sect. 6.2. This is necessary for the whole picture to be consistent.
• Moreover, the boundary term in Eq. (7.5) must vanish for all functions of the Hilbert space, . (7.10) In Appendix A.5, we recover the result µ A = a 0 ρ A and derive for any complex functions Ψ A 1 , Ψ A2 the following identity valid for all λa 0 ≥ 0, Applying this result to wavefunctions annihilated by the Hamiltonian, the left hand side is identically zero and we obtain As a result, the vanishing of the boundary term in Eq.

Crosscheck with quantum mechanics
We can check our conclusions, by applying them to the case where the path integral measure Dq corresponds to a field q with canonical kinetic term. As seen in Sect. 6.3, this amounts to choosing q = A −1 (a) given in Eq. (6.30), for which we know that ρ(q 0 ) = (6v 3 ) 3 4 when 0 < A(q 0 ) < 1. Indeed, using Eq. (7.9), we find that µ(q 0 ) is a constant This is the correct answer, as it reproduces the well known prescription in quantum mechanics where the wavefunction norm involves a constant measure.

Universality at the semi-classical level
The considerations of the previous subsection are valid for any choice of positive function ρ A (a 0 ), λa 0 ≥ 0, in the WDW equation and corresponding measure µ A in the Hilbert space of wavefunctions. An immediate consequence is that at the semi-classical level, the probability amplitudes √ µ A Ψ AC are universal, since Eqs. (6.24) and (6.28) yield (1 + O( )) , λa 0 > 1 . (7.14) In fact, the choices of functions ρ A and ω A in the Wheeler-DeWitt equation (6.16) can only affect corrections in to the Hermitian product defined in Eq. (7.2) beyond the semi-classical level. 28

Normalizability
Another consequence is that none of the wavefunctions of this Hilbert space is normalizable.
In these cases the "norms" are infinite due to the logarithmic divergences of the integrals at a 0 → +∞, while for M C+ = M C− = 0 the norm is of course vanishing. Hence, in the simplest minisuperspace model we are considering, there are no normalizable states in the Hilbert space, including the no-boundary "ground state." So at best we can use these wavefunctions to define relative probabilities, in terms of ratios of the probability densities evaluated at different points of minisuperspace. Interesting work on defining probabilities and constructing observables in quantum cosmology includes Refs. [13,[30][31][32][33].
In more realistic cases, the minisuperspace models comprise extra matter degrees of freedom, besides the scale factor a. DeWitt considers at least one matter field associated with a dust filling universe [8], while Hartle and Hawking include a conformally coupled scalar field [1]. One may also consider including an inflaton field with appropriate potential, for applications to inflationary cosmology. As a result, the Hilbert spaces of solutions of the corresponding Wheeler-DeWitt equations become infinite dimensional. As discussed in Ref. [8], one may choose a suitable basis labelled by quantum numbers associated with the matter Hamiltonian, and define a proper Hermitian product. Presumably, in the more involved general cases, one can construct normalizable solutions of the Wheeler-DeWitt equation, in terms of superpositions of the basis states. Moreover, there should be superpositions exhibiting classical behavior, which can be interpreted as wave-packets "moving" in minisuperspace [8]. It would be desirable to construct normalizable solutions in terms of path integrals, implementing the no-boundary proposal. Notice however that for the conformally coupled scalar field of [1], the Wheeler-DeWitt equation can be cast into a separable form, and the large a 0 asymptotic behavior of the solutions becomes independent of the matter Hamiltonian quantum number. So in this case, the wavefunction factorizes in the large a 0 limit, exhibiting similar asymptotic behavior to the one described in this work. As a result, the solutions in this conformally coupled scalar field case continue to be non-normalizable.

Comparison with the literature
We can compare our analysis with that of Ref. [8]. In this work, the minisuperspace model does not implement a cosmological term but takes into account, as outlined above, mat- It is also interesting to mention the discrepancy between our results and some conclusions reported in Ref. [1], where the discussion of the measure µ A is presented in the case of the minisuperspace model we study. They consider the Wheeler-DeWitt equation in a form where ρ A (a 0 ) = a p 0 and ω A (a 0 ) = 0. From our analysis, we conclude that the Hilbert space measure must be µ A (a 0 ) = a p+1 0 . However, the authors of Ref. [1] take µ A (a 0 ) = a p 0 , which leads to the conclusion that all solutions of the Wheeler-DeWitt equation are normalizable.
The origin of the disagreement is that instead of imposing hermiticity of the Hamiltonian H A /N , they impose hermiticity of what they call the "Wheeler-DeWitt operator," which is a 0 H A /N . In practice, this amounts to changing µ A → µ A a 0 everywhere in our derivation of Sect. 7.1, and thus to replacing Eq. (7.9) with µ A a 0 = a 0 ρ A i.e. µ A = ρ A .

Conclusion
In this work, we have considered the minisuperspace model describing a homogeneous and isotropic universe with positive cosmological constant. More specifically, we focused on the wavefunction defined as a Euclidean path integral satisfying the "no-boundary proposal." The invariance under redefinitions of the lapse function imposes a gauge fixing of the reparametrization symmetry of Euclidean time. To do so, we have applied a procedure from first-quantized string theory, where quantum amplitudes are defined as path integrals of two-dimensional field theories. In both cases, the gauged-fixed path integrals involve integrals over the moduli spaces of metrics, and the results do not depend on the gauge choice. On the contrary, the redefinitions of the scale factor lead to inequivalent choices of diffeomorphism-invariant path-integral measures. All prescriptions yield different forms of the Wheeler-DeWitt equation and thus a priori distinct quantum theories. However, the quantum Hilbert spaces of wavefunctions lead to universal predictions, at least at the semiclassical level. It would be highly interesting to determine whether this remains true beyond this level of approximation, or even exactly.
In our analysis, the ground-state wave functions are evaluated using the steepest-descent method when the scale factor on the boundary 3-sphere satisfies 0 < λa 0 < 1. We hope to extend this result in the near future to the case λa 0 > 1, for which the instanton solutions are complex. Computing the path integrals for λa 0 = 0 is also challenging since there exists a one-parameter family of instanton solutions in this case, implying that a functional determinant possesses a vanishing eigenvalue.
The techniques we have used to fix the gauge of the diffeomorphisms as well as to com-pute the quantum fluctuations of the scale factor can be implemented in richer setups. In particular, it would be very natural to consider minisuperspace models involving scalar fields with potentials possessing non-negative (local) minima, and possibly to account for the statistically natural emergence of a long enough period of inflation. Taking into account extra degrees of freedom also allows for the existence of wave-packets "moving" in minisuperspace [8]. Hence, it would be very interesting to see whether normalizable states constructed as path integrals obeying the no-boundary proposal exist. In this work, we have considered the amplitude describing the transition from "nothing", the "space" reduced to the empty set, to a 3-sphere. This serves as the definition of the "ground state" wavefunction of Hartle also leads to ξ(x 0 Ei ) = x 0 Ei . Applying Eq. (2.12) forĝ 00 [ ] and using the above results, we have finally which implies that ξ(x 0 Ef ) = x 0 Ef since g 00 is in class .

A.2 Isometries of the line segment
Theorem 2. The isometry group of a line segment with metric g 00 is Z 2 generated by orientation reversal.
Any isometry I satisfies g I 00 = g 00 . Since g 00 in class can be obtained by acting onĝ 00 [ ] with a diffeomorphism ξ, this equality can be written asĝ I The isometry group is thus The moduli correspond to the deformations of the metric that are "orthogonal" to those obtained by diffeomorphisms. Let us consider any metric g 00 in class and defined on [x 0 Ei , x 0 Ef ]. We look for an infinitesimal deformation, denoted δ g 00 , that is orthogonal to all deformations δ ξ g 00 , where ξ ∈ Diff[g 00 ].
Using Eq. (2.11) and the definition (3.6), we have to solve 0 = δ g, δ ξ g = δ g, −2∇δx E where we have used the fact that √ g 00 δx 0 E is diffeomorphism invariant and, therefore, can be expressed in the coordinate system associated with the metric in class given in Eq. (A.5).
However, we know that where the classes are indicated in brackets. Thus the only moduli deformation of a line segment is that associated with the variation of its length .
Notice that the above equation also shows that the diffeomorphisms acting on g 00 [ ] are actually acting only on g 00 [1]. Therefore, the reparametrization group of the line segment is independent of its length. As a result, where the last equality means that the group depends only on the topology of the line segment.

A.4 Independence on the fiducial metric
In this Appendix, we check explicitly that the Faddeev-Popov Jacobian and the contributions The commuting tensors and the ghosts in Sect. 3.2 transform as where the expansion modes are given by χ00(ŷ0 E ) = Moreover, denoting ξ(τ ) ≡ τ (ŷ0 E ), the scalar field δa in Sect. 4.3.1 is transformed into δa ξ , which can be expanded as The modes φ ξ k are eigenvectors of the operator S ξ in the new coordinate system, with eigenvalues ν k , since As a result, the argument of the exponential in the definition of Z (a 0 ) in Eq. This shows the gauge invariance of Z (a 0 ).

A.5 Hermiticity of the Hamiltonian
In this subsection, we retrieve the results of Sect. 7.1 using refined arguments that provide insight. The following proofs include a generalization of some treatment given in Ref. [8] in order to take into account a positive measure µ A (a 0 ) to be determined and an unknown function ρ A (a 0 ) in the definition of the operator H A /N .

Let us consider the integral
where Ψ A1 , Ψ A2 are arbitrary complex functions. Moreover, Ξ, which is also complex, is an arbitrary test function. This means that it is identically vanishing outside some closed interval [a min , a max ] ⊂ R * + . Using the notation of Eq. (7.2), we can write the first term as follows