A non-Perturbative and Background-Independent Formulation of Quadratic Gravity

A non-perturbative and background-independent quantum formulation of quadratic gravity is provided. A canonical quantization procedure introduced in previous works, named after Dirac and Pauli, is here applied to quadratic gravity to obtain, as required by consistency, a well-defined Euclidean path integral. The theory is unitary: all probabilities are non negative and they sum up to one. We obtain path-integral expressions for the transition amplitudes, Green's functions and generic matrix elements of time-ordered products of the metric. As a byproduct, similar results are also obtained for a scalar-field four-derivative interacting model. In this way, among other things, previous perturbative and background-dependent calculations are justified. The (quantum) quadratic gravity effective action, whose field equations determine the vacuum expectation value of the metric in the presence of a generic energy-momentum tensor, is constructed. The classical limit of the effective action turns out to be equivalent to the starting classical action of quadratic gravity, whose runaway rates were previously shown to be slow enough to be compatible with observations. Finally, the constructed non-perturbative and background-independent quantum quadratic gravity is applied to quantum cosmology to obtain a path-integral expression for the wave function of the universe, which satisfies a sort of Wheeler-DeWitt equation. This application allows us to understand at the quantum level why our universe is nearly homogeneous and isotropic.


Introduction
Einstein's general relativity (GR) is an extremely successful classical theory of gravity that has been tested by several observations: recent tests include the discovery of gravitational waves, whose production is consistent with coalescing black holes [1], and the images of the black holes in the center of our galaxy and the M87 one produced by the Event Horizon Telescope [2][3][4][5][6].Furthermore, GR, appropriately extended by matter fields, is at the basis of our current understanding of cosmology, including the early universe.
However, an extension of GR is necessary because of quantum mechanics: GR is known to be nonrenormalizable by perturbative methods [7,8] and to be within the regime of validity of perturbation theory at length scales much below the Planck one.Thus, while it is possible that GR might make sense as a quantum theory at Planckian and trans-Planckian scales through some (yet unknown) non-perturbative approach, one must still extend it to make contact with large-distance physics.
Another shortcoming of quantum GR is the lack of a consistent Euclidean path integral: the Euclidean action of GR is unbounded from below, an issue that goes under the name of the "conformal-factor problem".This is a serious issue as all known quantum field theories of physical relevance are consistently defined only as analytic continuations of the corresponding Euclidean theories.
Both the non-renormalizability and the conformal-factor problem of GR could be solved by adding quadratic-in-curvature terms.Indeed, the resulting theory, which is often called quadratic gravity 1 , is renormalizable [11][12][13][14] and a simple application of the prescription [15] to determine the Euclidean action indicates that is also free from the conformal-factor problem in a sizable region of its parameter space [16].
A non-perturbative quantization leading to a well-defined Euclidean path integral has been found in Ref. [17] in the context of a simple, yet non-trivial, toy model for quadratic gravity: an interacting version of the Pais-Uhlenbeck model [18].This quantization is based on the quantum treatment of canonical variables first discussed by Pauli [19], who elaborated on a previous work by Dirac [20]; for this reason it was called the Dirac-Pauli (DP) quantization in [17].The model discussed in [17], however, features a finite number of degrees of freedom, so a question that has remained unanswered so far is whether the DP quantization can be extended to field theory, which features instead an infinite number of degrees of freedom, and ultimately to quadratic gravity.In Refs.[21][22][23] a real-time path integral was constructed for quadratic gravity using an ordinary (non-DP) quantization.As we will show here, this ordinary quantization does not allow us to construct a consistent Euclidean path integral.
As discussed in [9,17], the DP quantization implies the presence of an indefinite inner product.So one must ask whether consistent probabilities (non-negative numbers that sum up to one) can be defined in such a situation; in other words, one must ask whether this quantization respects unitarity.The answer is yes, as shown in Refs.[24][25][26].Indeed, by applying the standard frequency definition of probability as in real physics experiments, it turns out that in the Born rule one should not use the indefinite inner product but instead positive-definite inner products, which can be constructed by inserting appropriate linear operators inside the indefinite inner product.We will refer to these operators as norm operators.
The existence of a different inner product in relation with the unitarity of the theory has been discussed in a number of papers in the context of quantum mechanics and field theory with an antilinear symmetry of the Hamiltonian (see e.g.Refs.[27,28] for applications to the Pais-Uhlenbeck oscillator and Ref. [29] for a review).
Norm operators have been used so far to provide a unitary quantization of quadratic gravity through a perturbative and background dependent approach: one starts from a fixed classical background (e.g. the Minkowski or the Friedmann-Lemaitre-Robertson-Walker (FLRW) metrics) and then considers a semiclassical expansion where the perturbations are quantized with these rules [25,26,30].A perturbative and background-dependent approach is present in several roads to quantum gravity, including string theory.Needless to say, such an approach must ultimately be replaced by a non-perturbative and background-independent formulation.Indeed, one of the most important Einstein's messages on gravity is that the laws of physics should be formulated in a background-independent way.The purpose of this paper is to find a non-perturbative formulation of this type in quadratic gravity.
Note that in the perturbative background-dependent approach used so far, one first starts from the classical theory.As will be discussed in detail in this work, the classical Hamiltonian of quadratic gravity is unbounded from below (and above).In this sense quadratic gravity belongs to the class of higher-derivative models studied by Ostrogradsky in Ref. [31].Nevertheless, in [32,33] (see also [34,35] for related works) it has been found that the runaway rates in classical quadratic gravity can be slow enough 2 to ensure agreement with observations.This occurs when the typical length scales are much above a threshold that is low enough to describe the whole cosmology, i.e. the resulting theory can be called metastable rather than unstable, see Ref. [32].The possible instability that takes place when such bound is violated not only is compatible with cosmology, but would also explain why we live in a homogeneous and isotropic universe [32]: this is because this instability can be effectively activated only in sufficiently inhomogeneous and anisotropic patches of the universe.
The non-perturbative and background independent formulation that we seek in this paper can, among other things, provide us with a fully quantum and non-perturbative justification of these classical cosmological findings.Furthermore, such a formulation can also be used to make contact with other (apparently independent) perturbative and background-dependent approaches to formulate quadratic gravity as a unitarity and (meta)stable renormalizable theory: these approaches include the Lee-Wick [38][39][40][41] and the fakeon [42] programs, where all observable degrees of freedom are quantized with standard rules.
Even in ordinary quantum theories, the real-time path integral is only a formal object, whose consistency at the rigorous level is questionable.Therefore, we seek here a non-perturbative and background-independent formulation of quadratic gravity with a consistent Euclidean path integral.The real-time (Lorentzian) physics should then be defined, like in all known physically relevant quantum field theories, as an analytic continuation of the Euclidean theory.A formulation of this type can then be used to give a rigorous basis for quantum cosmology, where, as proposed by Hartle and Hawking, one describes everything through a quantum mechanical "wave function of the universe" [43].Indeed, the Euclidean path integral can be used to obtain eigenfunctions of the Hamiltonian that solve the Shroedinger equation.In the context of quantum cosmology this equation is known as the Wheeler-DeWitt equation [44,45].A consistent quantum cosmology in quadratic gravity would be interesting also because this theory contains Starobinsky's model of inflation [46], which is currently in perfect agreement with the cosmic-microwave-background observations [47][48][49].
Yet another application of a non-perturbative and background-independent formulation of quadratic gravity would be in the context of asymptotic safety [50], where one attempts to render gravity UV complete through a non-perturbative UV fixed point.Fixed points of this type have been found for quadratic gravity in Refs.[51,52].
The paper is organized as follows.In Sec. 2 we introduce the action of quadratic gravity we consider in this work and the corresponding 3 + 1 formalism, where space and time are separated.Afterwards, in Sec. 3 Ostrogradsky's canonical method is non-perturbatively applied to quadratic gravity to define the canonical variables and the Hamiltonian.The Dirac-Pauli quantization is applied to quadratic gravity in Sec. 4 to find a consistent Euclidean path integral.At the beginning of this section a review of the DP quantization and consequent norm operators is provided to render the paper self-contained.The unitarity of the theory is discussed there in the most general terms.In Sec. 4 the Euclidean path integrals for the transition amplitudes and Green's functions are studied first in a scalar-field four-derivative interacting model and then in quadratic gravity.In the same section the general covariance of the theory is also discussed.Sec. 5 is devoted to the construction of the (quantum) effective action in quadratic gravity, which is then used to study the classical limit of the theory in Sec. 6. Sec.7 is an application of the previous results to quantum cosmology.Finally, in Sec. 8 a detailed summary of the paper and conclusions are offered.The paper also includes two appendices, which give further details.
Regarding notation: since we will consider the classical limit we will leave ℏ explicit in this paper.However, we will use units where the speed of light equals 1.

The action and the 3 + 1 formalism
The action of quadratic gravity we consider is Here α, β, γ and λ are real coefficients, g is the determinant of the metric, R is the Ricci scalar and W 2 ≡ W µνρσ W µνρσ is the "square" of the Weyl tensor W µνρσ .So, in terms of the Riemann tensor R µνρσ and Ricci tensor R µν or equivalently The quantity G is the topological Gauss-Bonnet term.For simplicity we do not add to S possible boundary terms that are also quadratic in the curvature.One of our goals is to provide a Hamiltonian formulation of the theory.We thus adopt a 3 + 1 formalism [44,53,54].Such a formalism has been applied to quadratic gravity in [21][22][23]55].Unlike in these works, we shall start here with a gauge condition on the metric components to avoid dealing with complicated constraints.In this respect, our approach resembles the one used in the context of ordinary Yang-Mills theory (see e.g.[56]).Our gauge choice is the simple Gauss coordinate system, namely normal coordinates (zero shift) and geodesic slicing (unit lapse function): where latin indices are the spacial ones and t ≡ x 0 is the time coordinate.General covariance will be made manifest in Sec.4.2.The non-vanishing components of the Levi-Civita connection in Gauss coordinates are then where a dot represents a derivative with respect to t.
The action S can be expressed in terms of g ij , K ij and Kij through the following formulae, which can be derived with the help of those reported in Appendix A: (2.9) Here R 3 and D i are, respectively, the three-dimensional Riemann tensor, Ricci tensor, Ricci scalar and covariant derivative built with the three-dimensional metric g ij , the inverse threedimensional metric is represented as usual by g ij (we lower and raise the latin indices with g ij and g ij , respectively) and K ≡ K i i .The time-reversal invariance of L is manifest in Eqs.(2.7), (2.8), (2.9): the time-reversal invariant quantity Kij also appears linearly, while K ij ≡ − ġij /2, which changes sign under time reversal, only appears quadratically.The time-reversal invariant metric g ij even enters non polynomially.

Canonical quadratic gravity
The action S is a functional of g ij , ġij and gij or, equivalently, of g ij , K ij and Kij , where K ij has been defined in Eq. (2.6).We can thus use Ostrogradsky's canonical method [31] (see e.g.Ref. [9] for an introduction and references to previous works).
In this formalism one can treat g ij and K ij as independent canonical coordinates and then defines the momenta π ij and P ij conjugate to g ij and K ij , respectively, as follows: 3 Since L is independent of Kij , By using Eqs.(2.1), (2.7), (2.8) and (2.9) one obtains 3 The derivatives in (3.2) are taken as if the variables with i and j exchanged were independent.For example, The symmetry in the exchange of i and j is then reflected in the symmetry π ij = π ji and P ij = P ji . where It is easy to check that, whenever α ̸ = 0 and β ̸ = 0, is the inverse of G ijlm in the sense that so we can express Kij in terms of P ij , g ij and K ij .Inserting this expression in L one finds a functional of g ij , K ij and P ij with no dependence on π ij .Also in Ostrogradsky's canonical method, one makes use of the standard definition of the classical Hamiltonian density: As usual, the spatial integral of H (namely, the Hamiltonian) is conserved, i.e. time independent.Unlike L , the Hamiltonian density does depend on π ij , as well as g ij , K ij and P ij , but the only dependence on π ij is given by the term As mentioned in the introduction, the classical Hamiltonian is, therefore, unbounded from below, but the runaway rates in classical quadratic gravity can be slow enough to ensure compatibility with observations in our observable patch, while other highly inhomogeneous and anisotropic patches are not compatible with observers thanks to Ostrogradsky's instabilities (leading to an explanation of the nearly homogeneity and isotropy of our universe).In Sec. 7 we will discuss the quantum counterpart of these classical findings.

Euclidean path integral and Dirac-Pauli variables
As usual the dynamics of the quantum theory will be encoded in a time-evolution operator U (t) ≡ exp(−iHt/ℏ), where H is the Hamiltonian operator.As usual in a field theory, we seek here an equivalent formulation in terms of a path integral.The Euclidean path integral, followed by the analytic continuation to real time, provides us with a rigorous non-perturbative definition of our physical theories.It is thus important to construct a consistent Euclidean path integral for quadratic gravity too.In the Euclidean theory the real time t is substituted by −iτ , where τ , called the imaginary time, is treated as a real variable.The real-time (Lorentzian) theory is then defined by the analytic continuation τ → it of the resulting theory.
A first problem one faces to achieve this goal is the presence of the term (3.9) in H: if one performs an ordinary quantization where the canonical coordinates g ij and K ij (and their conjugate momenta π ij and P ij ) have a real spectrum the Euclidean path integral over π ij diverges: this is because in the Euclidean path integral the Hamiltonian appears without an overall i in the exponent of the path integrand.
A solution to this problem is the existence of an alternative quantization for the canonical variables, where these operators have instead purely immaginary eigenvalues.We will refer to this quantization as the Dirac-Pauli (DP) one [9,17,26], as it was first discussed by Pauli [19] elaborating on a previous work by Dirac [20].The existence of this alternative quantization is due to the fact that an observable A in quantum mechanics can be more generally defined as a physical quantity for which there is a complete set of states with a well defined value of A [25,26].As a result of this more general definition, A can always be described by a normal operator with respect to an appropriately defined positive-definite inner product [25,26] with eigenvectors given by the mentioned complete set of states.This extends the usual definition of observables as A can have complex eigenvalues.This is not so strange as we can construct normal non-Hermitian observables even in text-book quantum mechanics: for example, the position of a particle in a plane {x, y} can always be described by a normal operator x + iŷ with complex eigenvalues.With this definition of observable (which is adopted here) there are two possibilities to quantize a canonical variable, either the ordinary one or the DP one.Now, if we adopt the ordinary quantization for π ij and the DP one for K ij the term in (3.9) will produce a harmless purely imaginary term in the Euclidean path integral and the previously mentioned problem is solved.
To proceed it is now necessary to review some of the properties of this generalized quantization procedure.First a DP canonical variable x satisfies by definition x|x⟩ = ix|x⟩ for a complete set of states |x⟩ and real x.The operator x can be considered as a Hermitian operator with respect to an indefinite inner product for which ⟨x ′ |x⟩ = δ(x + x ′ ) rather than δ(x − x ′ ) [9].For generic states |ψ⟩ and |χ⟩ such inner product is denoted ⟨χ|ψ⟩ and the Hermitian conjugate with respect to this inner product of a generic linear operator O is denoted O † .Calling η the linear operator such that η|x⟩ = | − x⟩, it is then clear that ⟨χ|η|ψ⟩ defines a positive-definite inner product.The completeness relation then reads where in the second equality we used η 2 = 1.Also it is easy to show η † = η.If we call O † η the Hermitian conjugate of a generic linear operator O with respect to the positive-definite inner product, ⟨χ|η|ψ⟩, one finds O † η = ηO † η.More generally, for any observable A it is always possible to define a linear operator P A (satisfying P † A = P A ) such that for the complete set of eigenstates |a⟩ of A with eigenvalue α a one has ⟨a ′ |P A |a⟩ = δ a ′ a and thus ⟨χ|ψ⟩ A ≡ ⟨χ|P A |ψ⟩ is a positive-definite inner product [25,26] (when A is a DP canonical variable one has P A = η).For this reason we refer to P A as a norm operator.Adopting the standard frequency definition of probability as in real physics experiments, one finds that it is this inner product that must be used in the Born rule to define the probability p a that a measurement of A has the outcome α a [24][25][26]: Since this satisfies for any A the conditions [25,26] p a ≥ 0, where the label a ranges over all possible values, the theory is unitary.Another property that is necessary to recall is that the conjugate momentum p of a DP canonical coordinate x (satisfying the canonical commutator [x, p] = iℏ) is itself a DP variable, namely p|p⟩ = ip|p⟩ for a complete set of states |p⟩ and p real and [9,17] Therefore, the quantization that was mentioned before (the ordinary one for π ij and the DP one for K ij ) tells us that the pairs of conjugated variables {g ij , π ij } and {K ij , P ij } are respectively ordinary variables and DP ones.This agrees with the principle proposed in [17] in the simpler Pais-Uhlenbeck model [18]: coordinates that are invariant under time reversal (like g ij ) should be quantized in the standard way, while coordinates that are odd under time reversal (like ) should be DP quantized 4 .One might worry that this quantization will have the disastrous implication that the theory predicts an imaginary value for the time derivative of the metric; this does not happen, as will be confirmed in Secs. 5 and 6, because the expectation values of all types of canonical coordinates (both ordinary and DP ones) are real [26].It remains to be shown that this quantization leads to a well-defined Euclidean path integral as anticipated.This will be done in the following subsection.

Euclidean path integral for transition amplitudes and T-products
In this section it is convenient to treat a more general framework than quadratic gravity.Let us consider an arbitrary number of ordinary canonical variables q 1 , ..., q n and Dirac-Pauli variables q1 , ..., qm .A state with definite canonical coordinates is denoted here [9] |q⟩ = |q 1 , ..., q n , q1 , ..., qm ⟩ (4.5) and the corresponding state with definite conjugate momenta is We would like to evaluate the matrix element of the Euclidean time evolution operator, exp(−(τ f − τ i )H/ℏ) with the imaginary initial and final times, τ i and τ f respectively, between arbitrary states of definite canonical coordinates |q i ⟩ and |q f ⟩: Indeed, the knowledge of this object for arbitrary q i and q f is equivalent to knowing the imaginarytime dynamics of the system.The real-time dynamics can then be defined as an analytic continuation from imaginary to real time.In the expression above we have inserted the η operator so that the matrix element is taken with the positive-definite inner product ⟨χ|ψ⟩ q ≡ ⟨χ|η|ψ⟩.The matrix element without the η insertion provides an object with the same amount of information because η by definition switches the sign of all DP variables and leaves the ordinary variables unchanged, i.e. η|q⟩ ≡ |q η ⟩ ≡ |q 1 , ..., q n , −q 1 , ..., −q m ⟩. (4.8) So the desired matrix element can be written as follows: having defined |q, τ ⟩ ≡ exp(τ H/ℏ)|q⟩ and ⟨q, τ | ≡ ⟨q| exp(−τ H/ℏ).After an analytic continuation to real time, these matrix elements allows us to compute, among other things, the probability density ρ(q f ) that the system is found in the state |q f ⟩ after a time t f − t i starting from a generic state |ψ 0 ⟩ = dq ψ 0 (q)|q⟩ by using Eq.(4.2) with A = q: where in the second step we have changed the name of the integration variable, q → q η .From Eq. (4.10) it is clear that the probability density ρ(q f ) satisfies the unitarity conditions ρ(q f ) ≥ 0, (for any q f ), as a consequence of the general unitarity expressed by (4.3).We actually consider an even more general object than (4.9),where a generic time-ordered product of ordinary canonical coordinates of the form Tq i 1 (τ 1 )q i 2 (τ 2 )... is inserted: where q(τ ) ≡ exp(Hτ /ℏ)q exp(−Hτ /ℏ).This generalization will allow us in Sec.4.3 to construct the Euclidean path integral for Green's functions of the metric g ij , which, as we have seen, will be quantized as an ordinary canonical coordinate.By following steps similar to those in the absence of such insertion performed in Refs.[9,17], we obtain the path-integral representation: ) where H(q, p) ≡ ⟨p|H|q⟩ ⟨p|q⟩ (4.14) and the functional integration measure is given by The products above are extended to all values of τ ∈ [τ i , τ f ) for momenta p and for all values of τ ∈ (τ i , τ f ) for coordinates q, as the values of q at both time boundaries are fixed (we put a prime in the second product to recall this different range).Like in any quantum theory, this measure should be thought of as the zero lattice-spacing limit of an integration measure over a finite number of variables obtained by discretizing time (and all spatial dimensions in the case of field theory).As usual we can write where ⟨q f η , τ f |q i , τ i ⟩ J is the generating functional It is important to note that here the expression q(τ i ) = q i , q(τ f ) = q f appearing on the functional integral symbol represents boundary conditions for both the ordinary and DP coordinates, both at initial time τ i and final time τ f Another important aspect is that in the presence of DP variables H(q, p) does not generically coincide with the classical Hamiltonian (the space integral of the classical H given in (3.8) in the case of quadratic gravity) because the eigenvalues of the DP variables are imaginary.So, in the limit ℏ → 0, where all operators commute, H(q, p) is the analytic continuation of the classical Hamiltonian to imaginary values of the DP variables: The DP coordinates are rotated anticlockwise in the complex plane while the DP conjugate momenta are rotated clockwise because in the definition of H we are adopting, Eq. (4.14), the coordinate eigenstates appear on the right, while the momentum eigenstates are on the left.The path integral in (4.13), or equivalently that in (4.17), is not guaranteed to converge for any theory and for any choice of quantization.As we have anticipated at the beginning of Sec. 4, if we use the ordinary quantization for both {g ij , π ij } and {K ij , P ij } in quadratic gravity, H will have a term −2π ij K ij , where now π ij and K ij are real integration variables in the path integral.Since this is the only term where π ij appears, one would obtain an exponential divergence of the path integral for this quantization.In general one can only accept quantizations (and parameter values) leading to an H(q, p) that guarantees the convergence of the Euclidean path integral.Now, to specify the quantization completely we still have to uniquely define the operator H = d 3 x H, without ambiguities.The analysis of Sec. 3 was purely classical and, as well known, classical physics cannot uniquely determine quantum physics.We now define the operator H as the function of the canonical variable operators obtained by first putting all canonical momenta c-numbers on the left of all canonical coordinate c-numbers in the classical Hamiltonian and then substituting these c-numbers with the corresponding operators.In this way, H in (4.14) can be easily computed and coincides with the classical Hamiltonian in the absence of DP variables.In the presence of DP variables, H(q, p) is, now for any ℏ, the analytic continuation of the classical Hamiltonian to imaginary values of the DP variables: Note that the quantum Hamiltonian H of quadratic gravity is time-reversal invariant, T HT −1 = H.The quantum theory has inherited this symmetry property from the time-reversal invariance of the classical theory discussed in Sec. 2.

Application to a four-derivative scalar-field theory
To understand how this formalism works in a simple context, we first consider a scalar-field theory with Lagrangian density where □ ≡ ∂ µ ∂ µ , c 4 is a real coefficient governing the size of the four-derivative term and V is some potential density.In the 3 + 1 formalism we have where (∇φ Like we did for quadratic gravity, we introduce another canonical coordinate given by the time derivative of the dynamical field φ 2 ≡ φ.Following Ostrogradsky's canonical method the momenta conjugate to φ 1 ≡ φ and φ 2 ≡ φ are, respectively, The last equation can be used to express φ in terms of the canonical variables, The Hamiltonian density with the ordering specified above (all canonical momenta to the left of all canonical coordinates5 ) then reads where now {φ 1 , π 1 } and {φ 2 , π 2 } are c-numbers because are integration variables in the path integral.In The previously dangerous π 1 φ 2 has now become a harmless imaginary term iπ 1 φ 2 .The integration over π 1 and φ 2 in (4.17) then sets φ 2 = φ ′ and multiplies the integration measure by x ′ E ∆τ −1 , where ∆τ is the Euclidean time element (which becomes dτ in the zero lattice-spacing limit) and the product is over all Euclidean spacetime points x E with imaginary time τ ∈ [τ i , τ f ).We put a prime on x E to distinguish this product with that for canonical coordinates: as we have discussed below (4.15), for canonical coordinates, like φ, the product over the imaginary time ranges over τ ∈ (τ i , τ f ) while for conjugate momenta over τ ∈ [τ i , τ f ).The integration over π 2 then reads where ∆V 3 is the spatial volume element (which becomes d 3 x in the zero lattice-spacing limit).This factor of ∆V 3 appears in the measure because we have defined π 1 and π 2 through the derivatives of the Lagrangian density L rather than the Lagrangian.The integral in (4.25) converges only for c 4 > 0, which we assume, and, using a standard stationary point method, gives So we arrive at the Lagrangian version of the Euclidean path integral where 6C = S E is the Euclidean action and µ runs now over the Euclidean time and the spatial dimensions.In (4.27) the boundary conditions can be more explicitly written as We include the factor C in the definition of the integration measure in (4.27), which then reads One might wonder why we obtained precisely the Euclidean action at the end.While φ 2 ≡ φ corresponds to iφ ′ in Euclidean time and the eigenvalues of φ 2 , beeing quantized à la DP, are indeed purely imaginary, π 2 , which is also quantized à la DP, corresponds to a real quantity in Euclidean time, see Eq. (4.22).The reason why we obtained the Euclidean action is the following.After solving the Gaussian functional integral over π 2 we obtain the exponential in (4.17) computed in the stationary point of its argument with respect to variations of π 2 .This stationary point satisfies the equation where now the canonical variables are c-numbers (integration variables in the path integral).As we have seen, φ 2 is set equal to φ ′ by the integration over π 1 and φ 2 .So the equation above becomes The finiteness of (4.27) requires c 4 > 0, which has indeed been assumed in the derivation, and that the potential density V be bounded from below.Also, V should diverge fast enough to +∞ as φ → ∞.It is important to note that these conditions of convergence are on the bare parameter c 4 and the possible other bare parameters appearing in V .As in any interacting quantum field theory, the renormalized parameters are generically different quantities, not necessarily subject to the same conditions.Now let us consider the change of variable φ(τ ) ≡ φ(τ ) in the path integral in (4.27),where τ ≡ τ f + τ i − τ and the dependence on the spatial coordinate is untouched and thus understood.This corresponds to a time reversal.Since φ ′ (τ ) = − φ′ (τ ), where a prime denotes the derivative with respect to the displayed argument, and S E is invariant with respect to time reversal, i.e. S E (φ) = S E ( φ), one obtains where the new current J is defined by ) which means H † = H, or equivalently H † η = ηHη, because the path integral in (4.27) is real.In words, the quantization we have performed corresponds to having H Hermitian with respect to the indefinite inner product, ⟨χ|ψ⟩, defined at the beginning of Sec. 4. Nevertheless, the theory is unitary (all probabilities are non negative and sum up to one) as described around Eq. (4.2).As we will see, the conclusion H † = H holds in quadratic gravity too and in Sec.7 we will discuss the physical consequences.

The quadratic gravity case
The quadratic gravity case parallels the scalar-field theory one we have just discussed, but with some important differences that we now highlight.
As already mentioned, the quantization is specified by treating {g ij , π ij } as ordinary variables, {K ij , P ij } as DP variables and positioning all canonical momenta on the left of all canonical coordinates in the quantum Hamiltonian.The c-number H in (4.14) is then, for any ℏ, the analytic continuation of the classical Hamiltonian to imaginary values of the DP variables: Correspondingly, H is the same analytic continuation of the classical Hamiltonian density.
The integration over π ij and K ij in (4.17) then sets K ij = − 1 2 g ′ ij and multiplies the integration measure by x ′ E ∆τ −6 .The extra power of 6 compared to the scalar-field case is due to the fact that π ij (just as g ij , K ij and P ij ) has a total of 6 independent components at each spacetime point.
The integral over P ij is, like the integral over π 2 in the scalar-field example, a Gaussian integral that can be performed with a standard stationary point method.The stationary point satisfies now the equation which, because K ij has been set equal to − 1 2 g ′ ij , reads Similarly to the scalar-field case, the latter equation is the Euclidean version of the Lorentzian equation, Eq. (3.3), that allows us to express P ij in terms of g ij , ġij and gij in classical physics but with P ij substituted by −iP ij , compatibly with the DP quantization.Note the importance of our choice of quantization in obtaining this result.Therefore, the integral over P ij sets the exponent in the path integral (4.17) equal to where S E is the Euclidean action, and adds the extra factor due to the remaining Gaussian integral over the fluctuation η ij of P ij around the stationary point (recall that P ij is a DP variable).In (4.37) J ij is the external "current" corresponding to g ij in the generating functional (4.17).Moreover, in (4.38) the metric determinant g, the square of the Weyl tensor W 2 ≡ W µνρσ W µνρσ and the Ricci scalar R are now computed with the Euclidean metric, which for Gauss coordinates reads What remains to be done is to compute the factor C ′ in (4.39).This is done in Appendix B.
For quadratic gravity one then finally obtains Here the boundary conditions can be more explicitly written as where g lm provides initial and final conditions for the metric and its time derivative; they are functions of the three-dimensional spatial coordinates.By multiplying the result for C ′ in (B.11) by the factor x ′ E ∆τ −6 previously produced by the integration over π ij , one finds Like for the scalar field, we think this factor as part of a full integration measure, By using the methods of Appendix B (see the discussion around Eqs. (B.7), (B.8) and (B.9)), one finds that this Jacobian "matrix" produces in the path integral a Jacobian determinant (det J ) 4 , where J is the matrix with elements J k i .On the other hand, g 2 in the denominator of (4.44) produces the same factor under the same transformations, so the full measure is invariant 7 .The presence of an invariant final measure over metrics is a consequence of the fact that we started from the product of the measures associated with canonical coordinates and conjugate momenta, which is invariant.Clearly, the Euclidean action in (4.38) is also invariant.On the other hand, The invariance of the probabilities dq f ρ(q f ), with ρ(q f ) given in (4.10), tells how the states should transform.
The convergence of the Euclidean path integral in (4.41) requires the following conditions in quadratic gravity which can be derived as follows.First, consider metrics with vanishing R.These metrics can have a non-vanishing W 2 .The integration domain of the path integral include metric values for which √ g W 2 can be made arbitrarily large keeping R = 0, so this implies α > 0. Next, consider conformally-flat metrics for which W 2 = 0.In this case R, together with g, can be made arbitrarily large for some field configurations in the domain of path integration so β > 0. Finally, the last condition in (4.47) can be obtained by minimizing the Euclidean Lagrangian with respect to R and requiring S E to be always positive.Indeed, if such minimum were negative it would be possible to find field configurations with arbitrarily negative S E .The conditions in (4.47) are also sufficient conditions for the convergence of the path integration if, of course, a spacetime discretization is performed.This is opposed to the case of Einstein gravity, where there are no values of the parameters for which the Euclidean path integral could converge as the Euclidean action is always unbounded from below (the conformal-factor problem).When the continuum limit is taken by removing the cutoff, α, β, γ and λ generically diverge as a function of the cutoff to ensure the convergence of the physical quantities.In this sense they are bare rather than renormalized quantities, just like c 4 in the scalar-field example.For this reason it is not possible to infer only from (4.47) the same conditions for the renormalized quantities.For example, one could not claim only from (4.47) that the theory predicts a positive cosmological constant.Note that the discussion on the Hermiticity of H that we made for the scalar-field theory (around Eq. (4.33)) can be made here as well, with the same conclusion, H † = H.

Transition to generic coordinate systems
So far we have discussed quantum quadratic gravity and constructed the corresponding path integral in the Gauss coordinate system, where the time components of the metric are set to g 00 = −1 and g 0i = 0. Correspondingly, in Euclidean signature, g i4 = 0 and g 44 = 1, where we label with 4 the imaginary time.Let us now show how to describe quantum quadratic gravity in a generic coordinate system.
This can be done with the help of the Euclidean path integral.To this purpose let us insert as an extra factor in the integrand of (4.41), to obtain where now S E is computed in the full four-dimensional Euclidean metric.Also, we have taken advantage of the presence of δ(f ) to substitute J ij g ij with J µν g µν where J µν is an arbitrary external "current".Note that, as specified before, the product over the imaginary time inside the product over x E ranges over τ ∈ (τ i , τ f ).Therefore, the boundary values of f µ at τ i and τ f are arbitrary.As a result, the boundary conditions in (4.43) can be arbitrarily extended by adding any boundary conditions for f µ , namely for g i4 and g 44 .This reflects the freedom of choosing an arbitrary coordinate system (a gauge freedom).We can express (4.51) in a more familiar form by considering the change of coordinates, ξ µ ≡ x ′µ − x µ , that connects the Gauss coordinate system to another coordinate system with non-vanishing values of f µ .These f s are functionals of the ξs so we can write where Dg ≡ Cδgδξ, δξ ≡ and we had to insert the Jacobian det ∂f ∂ξ for the change of integration variables from the f s to the ξs.This precisely reproduces the Faddeev-Popov (FP) determinant, as we will discuss in Sec.4.3 in dealing with the path integral for Green's functions.
As we have seen around Eq. (4.46), Cδg is invariant under three-dimensional general coordinate transformations.The extra factor δξ inside Dg is invariant under generic change of coordinates that vary the values of g i4 and g 44 : this is because one can go to a further coordinate system by shifting ξ µ → ξ µ + ξ ′µ , where ξ ′µ are other spacetime functions, and shifts have unit Jacobians.Therefore, the measure Dg is invariant under four-dimensional general coordinate transformations.
As well-known (see e.g.Textbook [56]), matrix elements of the form where O(x 1 ), O(x 2 ), ... represent gauge invariant operators, are independent of the f s.Then, these f s clearly play the role of a gauge-fixing function and their choice corresponds to the choice of the coordinate system.

Euclidean path integral for Green's functions
In ordinary theories, the path integral for the matrix elements in (4.13) can be used, among other things, to construct the path integral for Green's functions.This is also true in the presence of DP variables, at least when an ordinary two-derivative theory is recovered in the large-distance limit.
The purpose of this section is to show the latter statement.Before discussing quadratic gravity it is convenient to consider a more general framework as was done at the beginning of Sec.4.1.
A generic Green's function reads where |0⟩ is the eigenstate of H with the minimal real eigenvalue E 0 , thus H|0⟩ = E 0 |0⟩ (the vacuum).One can express a Green's function in terms of the matrix elements in (4.13) by inserting the completeness relations as follows: where, for generic q and τ , we have defined and we have used H = H † , which we have shown in Sec.4.1.The quantity ψ 0 (q η ) defined above is the ground-state wave function.By applying Eq. (4.2) with A = q and |ψ⟩ = |0⟩, one finds that ψ 0 allows us to compute the probability density ρ 0 (q) of the configuration with canonical coordinates q (in the ground state): (4.59) Again we can easily check that the unitarity conditions, ρ 0 (q) ≥ 0, dq ρ 0 (q) = 1, hold.
As we have seen, the matrix elements of the T-product in the integrand in (4.57) can be obtained through functional derivatives with respect to J of the generating functional in (4.17).We then first focus on that functional.Let us choose the current J to be vanishing for τ outside the interval [τ i , τ f ].Taking two imaginary times τ 1 and τ 2 such that τ To evaluate ⟨q iη , τ i |q 1 , τ 1 ⟩ and ⟨q 2η , τ 2 |q f , τ f ⟩ in the expression above we first discretize the canonical coordinates, q, in a way that their range of variability becomes finite.At this point one would like to compute the matrix F (M ) with elements for any q i , q f and τ , where M is the matrix with elements such that F (M ) = exp(−τ M/ℏ).We are unable to show with present techniques (see [29,58]) that M (and thus H) is diagonalizable so we make use of a general decomposition, known sometimes as the Riesz-Dunford formula: where the λ n are the eigenvalues of M , the r n are their algebraic multiplicities, in our case ) and with Γ n being a closed curve in the complex plane containing only the eigenvalue λ n .Note that, knowing F (M ), for example through path-integral methods, we can extract the eigenvalue λ 0 with the smallest real part as follows: for arbitrary values of q f and q i .When the above-mentioned discretization of the qs is removed the lowest eigenvalue λ 0 tends to E 0 .Note that E 0 is real whenever ⟨q f η |e −τ H/ℏ |q i ⟩ is real and positive, which is the case both in the scalar-field example with Lagrangian (4.20) and in quadratic gravity, as we have seen in Sec.4.1.Moreover, in quadratic gravity it is possible to have E 0 non negative by choosing the bare cosmological constant λ large enough, compatibly with the convergence conditions in (4.47).We will choose λ in this way in Sec.7 to ensure that the vacuum energy is positive in agreement with observations.Analogously, in the scalar-field example with Lagrangian in (4.20), it is possible to have E 0 non negative by choosing V (0) large enough.
In both quadratic gravity and the scalar-field example the low-energy (large-distance) limit is an ordinary two-derivative theory.This can be understood by noting that the eigenvalues of M with the smaller real part are those for which the effect of the W 2 term (or the higher derivative term proportional to c 4 in the scalar-field case) is minimized: this occurs when the corresponding state is mostly a combination of nearly four-dimensional homogeneous eigenstates of the canonical coordinates 8 .In the case of quadratic gravity this ordinary theory is GR plus the scalaron field (the effective scalar associated with the R 2 term).Therefore, in the large-distance limit the spectrum of M is an ordinary one.
As a result, r 0 = 1 and in the τ 1 → −∞ and τ 2 → +∞ limits, respectively, By inserting these results in (4.60) one obtains in the τ 1 → −∞ and τ 2 → +∞ limits e E 0 (τ 2 −τ 1 )/ℏ ψ * 0 (q 1 )ψ 0 (q 2η ) On the other hand, for J = 0, but in the same limit ⟨q 2η , τ 2 |q 1 , τ 1 ⟩ = e −E 0 (τ 2 −τ 1 )/ℏ ψ 0 (q 2η )ψ * 0 (q 1 ).(4.68) Therefore, by comparing with (4.57) one obtains that the generating functional of Green's functions, Z(J) can be expressed as for arbitrary values of the boundary conditions q 1 and q 2 .In the case of quadratic gravity, using (4.52), where the prefactor indicates that one has to divide by the same quantity with J µν = 0 and we have understood the boundary conditions in the infinite past and future as they are arbitrary.Moreover, we have removed the absolute value around det ∂f ∂ξ .Indeed, det ∂f ∂ξ is a Jacobian for a change of coordinate and consistency implies det ∂f ∂ξ ̸ = 0.So the continuity of det ∂f ∂ξ requires that its sign remains constant over the full integration domain.The overall sign can be dropped in (4.70) because it cancels with the denominator, "J → 0".Similarly, we have replaced ∂f ∂ξ with δf δξ because standard and functional derivatives differ by a factor of the four-dimensional volume element, which again cancels with the denominator, "J → 0".We recognise det δf δξ as the FP determinant in the generating functional of Green's functions.
Note that there appear to be no obstacle in going from (4.70) to the generating functional of the Lorentzian Green's functions.One should replace the Euclidean "current" J µν (τ ) with a Lorentzian current with the substitutions J kl (τ ) → J kl (it)/ℏ, J k4 (τ ) → iJ k0 (it)/ℏ and J 44 (τ ) → −J 00 (it)/ℏ, where we have understood the dependence on the spatial coordinates.At the same time we should perform a change of variables in path integral of the form g kl (τ ) → g kl (it), g k4 (τ ) → −ig k0 (it) and g 44 (τ ) → −g 00 (it).As already noted, the boundary conditions in (4.70) are arbitrary so one can choose some that are compatible with a real Lorentzian metric with the chosen signature: e.g.g ′ ij = 0, g i4 = 0, g 44 = 1 both at initial and final times.So these substitutions lead to where Z(J) is the generating functional of the Lorentzian Green's functions.It should, however, be kept in mind that, as always, the Lorentzian path integral is a formal object, precisely defined only as an analytic continuation of the Euclidean one.At this point one would be interested in showing that previous perturbative calculations performed expanding around the Minkowski metric η µν are recovered when applicable.Since the FP determinant multiplies δ(f ), one only needs to compute it for vanishing ξ µ .So its ξ dependence can be dropped inside the integral in (4.71).Moreover, S is also ξ-independent as invariant under general coordinate transformations.So, when one considers the path integral for the vacuum expectation value (VEV) of T-products of gauge invariant operators, the only integrand factor that depends on ξ µ is δ(f ).For the simple f chosen in [13] (and other quantum calculations around the Minkowski background, e.g.[39]) the only role of this δ function is to eliminate the zero modes of the differential operator appearing in the linearized equations for h µν ≡ g µν − η µν .Furthermore, note that the local factors of g in the integration measure (see (4.44)) can be dropped in doing perturbative calculations around the Minkowski background based on dimensional regularization, as explained in [13].So, modulo gauge artefacts that do not affect physics, the path integral used in previous perturbative calculations around η µν is reproduced.

The quadratic gravity effective action
The "current" J µν in (4.71) can also be interpreted as an external physical source for the system, corresponding to a sort of energy-momentum tensor.Such a tensor can be produced by a set of matter fields coupled to quadratic gravity.Indeed, in this physical situation one should add to the starting action of quadratic gravity in (2.1) a matter term.Repeating the same steps that we have performed in the absence of matter fields one obtains a generating functional of the form (4.71), where, however, the Green's functions are to be obtained by performing functional derivatives with respect to J µν and then setting J µν to the physical energy-momentum tensor in question.
In this section we extend the concept of (quantum) effective action, which is typically derived for scalar fields in textbooks, to the case of the non-perturbative and background-independent formulation of quadratic gravity we have given 9 .
With the described physical situation in mind (5.1) can be interpreted as the expectation value of g µν in the presence of the physical energy-momentum tensor, which we still call J µν here for simplicity.Note that ⟨g µν ⟩ J is real because it is the expectation value of a canonical variable [26].Defining the new functional W (J) through Z(J) ≡ exp(iW (J)/ℏ) one easily finds δW δJ µν = ⟨g µν ⟩ J . ( The quadratic gravity effective action Γ can then be defined as a funcional of ⟨g µν ⟩ J by where J has to be thought of as a functional of ⟨g µν ⟩ J .From this definition it follows that Γ satisfies δΓ δ⟨g µν ⟩ J + J µν = 0. (5.4) In words, Γ is an action functional of ⟨g µν ⟩ J , whose field equations allow to determine ⟨g µν ⟩ J .This is the reason why Γ is called the quadratic gravity effective action.Therefore, this effective action, used at tree level, captures the full quantum effects.In Sec. 6 a semi-classical expansion of Γ will be performed.

The classical limit
One of the advantages of the path integral formula is the fact that it allows us to take the classical limit, ℏ → 0, easily.Taking this limit in (4.27) or (4.51) (for the four-derivative scalar-field theory and quadratic gravity, respectively) and setting the external currents to zero one obtains where S ren E is the classical Euclidean action with renormalized coefficients and computed in the solution of the imaginary-time classical equations of motion satisfying the boundary conditions q(τ i ) = q i and q(τ f ) = q f .These are generic initial and final conditions for g ij and g ′ ij (or φ and φ ′ in the scalar-field example).In other words all possible imaginary-time classical motions (for arbitrary boundary conditions) are described by the imaginary-time classical theory associated with S ren E .This action can be used to compute instantons, see e.g.[59].Now, focusing on quadratic gravity and going back to real time one obtains that the classical theory is described by our starting action in (2.1), but with renormalized coefficients10 α r , β r , γ r and λ r , instead of the bare ones, α, β, γ and λ, respectively (we call this action S ren ).Then we see that, although the extra canonical coordinates (K ij = − ġij /2, or φ 2 = φ) have a purely imaginary spectrum the corresponding classical observable quantities remain real.
This result can also be independently obtained by performing a semi-classical expansion of the (quantum) effective action.Since this expansion is well known for scalar fields we directly focus on quadratic gravity and make use of the effective action defined in Sec. 5.
In the classical limit, ℏ → 0, the path integral in (4.71) is dominated by a metric satisfying Let us call such a metric g J µν to highlight its dependence on J µν .It then makes sense to perform a change of variables g µν = g J µν +h µν in (4.71) and expand the integrand in powers of the fluctuation field h µν .This semi-classical perturbative approach allows us, among other things, to reconstruct previous perturbative quantizations of quadratic gravity performed starting from a given classical background, see e.g.[30,64,65].Doing so one obtains where the symbol ...| g J means that we are evaluating the corresponding quantity ... at g µν = g J µν and D J h is the measure Dg expressed in terms of h µν .The only property of this measure that we will use is its independence 11 of ℏ.The first term in this expansion for ℏ small gives where S| ren g J is the starting action (2.1) with renormalized coefficients and computed in the corresponding g J µν , namely the solution of The equation above can be equivalently written as So, in the classical limit one obtains g J µν = ⟨g µν ⟩ J and, using (5.3), and thus Again one finds that the classical limit is described by the classical action in (2.1) with renormalized coefficients.This action has been used to find several classical solutions, e.g.describing astrophysical objects of interest such as black holes [66][67][68][69] or horizonless ultracompact objects12 [72][73][74][75].The result obtained in this section supports these findings.
One might wonder how a quantum theory with a bounded spectrum of the Hamiltonian operator could lead in the classical limit to a theory which, as we have seen, has a classical Hamiltonian that is unbounded from below (although in the case of classical quadratic gravity the runaway rates can be slow enough to ensure compatibility with observations).
Note that in ordinary theories the non-negativity of the spectrum of the Hamiltonian operator implies that the classical Hamiltonian H c is also non negative.This can be seen from the imaginary-time Hamiltonian path integral for zero external current and its real-time counterpart dt(p q − H(q, p)) (6.10) having defined |q, t⟩ ≡ exp(iHt/ℏ)|q⟩ and ⟨q, t| ≡ ⟨q| exp(−iHt/ℏ) for real time t.In ordinary theories H is real in the classical limit because it is a real function of ordinary canonical variables with a real spectrum.Using in (6.10) the stationary phase approximation, which becomes exact in the ℏ → 0 limit, then allows us to conclude that H coincides with H c .Let us now go to the imaginary-time path integral in (6.9) and suppose that H(q, p) were negative for some q and p, say q 0 and p 0 .Choosing q i = q 0 and sending τ f → +∞ and τ i → −∞ one would obtain a divergent integral over dp(τ i ) because the path integral would feature an exponent with a positive real part of the form +∞ −∞ dτ (− H(q 0 , p 0 )) in some finite integration domain of dp(τ i ).This cannot happen if the spectrum of the Hamiltonian operator is non negative because in this case, inserting a complete set, |n⟩, of eigenstates of H with eigenvalues E n one finds which converges as τ f → +∞ and In theories with DP variables this conclusions cannot be reached when H is not real, like in quadratic gravity or the scalar-field example we have discussed (see e.g.Eq. (4.24)).The reason is that in that case the real-time Hamiltonian path integral would no longer imply that H is H c because the path integrand would feature an exponent with both an imaginary and a real part and the stationary phase approximation cannot be used to conclude H = H c .However, when the ground state energy E 0 is not negative one can still say that the real part of H is never negative, which is the real universal implication of a non-negative spectrum of H.
In theories with DP variables with a consistent Euclidean path integral, featuring a positive Euclidean action S E , the classical Hamiltonian H c should be determined by taking the classical limit to find (6.1) and then analytically continue to real time.
Let us conclude this section by noting that in a given physical situation one has to check whether the observed value of ℏ is small enough for the classical limit to be a good approximation.It is indeed possible that quantum physics overcomes classical physics in some situations of interest [76].

Quantum cosmology
In this section we apply the theory we have previously constructed to quantum aspects in cosmology.
First, let us observe that the perturbative tree-level calculations performed in quadratic gravity expanding around cosmological FLRW metrics [30,64,65] are reproduced by our non-perturbative construction.This is because in those calculations it was only assumed that canonical commutators hold and that the spectrum of the Hamiltonian is bounded from below (both these conditions are satisfied here).It is also interesting to note that the fakeon prescription [42,77], which eliminates the states with negative norm (with respect to the indefinite inner product), has led so far to the same observable predictions when applicable [78,79].Moreover, note that we can apply the Green's functions we have constructed to study non-time-ordered VEV of cosmological perturbations (which are relevant to obtain power spectra): one can consider the time-ordered product of two cosmological perturbations computed at times t + ϵ and t and then let ϵ → 0 + (and then generalize this technique to the time-ordered product of more cosmological perturbations in a straightforward way).
In quantum cosmology, as pointed out by Hartle and Hawking [43], one describes the universe through a quantum-mechanical wave function (actually a wave functional).The Hartle-Hawking construction was, however, based on Einstein's gravity, whose Euclidean path integral suffers from the conformal-factor problem.As observed in Sec.4.1, the Euclidean path integral in the quantization of quadratic gravity we have specified does not suffer from this problem.Here we can, therefore, implement the Hartle-Hawking idea in a more rigorous way.
For example, the ground-state wave functional ψ 0 (q η ), which we have previously introduced in Sec.4.3, is given here by where modulo a q f -independent proportionality factor, N 0 = ψ * 0 (q i ), that does not affect physics.This means that the ground-state wave function of the universe can be computed regardless of the past boundary condition.The result obtained here is reminiscent of the no-boundary wave function of the universe [43,80] (see also [81,82] for recent reviews), which, however, was proposed within Einstein gravity that is affected by the conformal-factor problem, unlike quadratic gravity.The smallest eigenvalue E 0 corresponding to ψ 0 , namely the vacuum energy, that appears in (7.1) can be computed using This limit is generically divergent and one should rather consider the vacuum energy density ρ V , that is E 0 divided by the three-dimensional volume element.It is easy to check, at least in the classical limit when (6.1) holds, that ρ V vanishes when λ r = 0, as expected.One can reiterate this procedure to reconstruct all the large-distance spectrum.For example, a first excited state has eigenvalue The wave functionals satisfy a sort of Wheeler-DeWitt equation [44,45].For example, for the ground state this equation in our notation simply reads Indeed, derivatives appear on the right-hand-side when we note that if {q, p} is an ordinary pair of canonical variables ⟨q|p = −i d dq ⟨q|, while if {q, p} is a DP pair of canonical variables ⟨q|p = d dq ⟨q|, as explained in [9,17].Since we are dealing with a field theory here (namely quadratic gravity), these derivatives are actually functional derivatives.
As discussed in Sec.4.3, in the large-distance limit, the smaller eigenvalues of H are those for which the effect of the W 2 term is minimized.The W 2 term vanishes for conformally-flat metrics.
This means that such eigenvalues are those whose corresponding wave functionals ⟨q η |ψ⟩ are peaked around values of the canonical coordinates, q, of the form (later in this section we will provide a more general explicit discussion about this correspondence).Note that a wave functional ⟨q η |ψ⟩ that is peaked around some metric configurations also corresponds to a probability density ρ(q) that is peaked around the same configurations: using Eq.(4.2) with A = q ρ which, as always, satisfies the unitarity conditions in (4.11).In the case of quadratic gravity a wave function ⟨q η |ψ⟩ is a functional of the metric (and its time derivative) at a given time.Therefore, the field configurations in (7.6) depend only on the three-dimensional spatial coordinate ⃗ x.
Identifying K ij with − ġij /2 in (7.6) gives a Lorentzian conformally-flat metric, which is what we want to make contact with real-time cosmology.In cosmology conformal flatness con only be broken by inhomogeneities and anisotropies as the FLRW metric is conformally flat.However, we can construct specific conformally-flat metrics that are not homogeneous and isotropic.Regarding the latter point, note that, in the large-distance limit, the smallest eigenvalues of H also corresponds to nearly constant values of the Ricci scalar R: indeed, the Euclidean action in (4.38) acquires its minimum for metrics that are conformally flat and have a spacetime-independent Ricci scalar given by R = − γ 2β .
Large inhomogeneities and anisotropies in cosmology necessarily corresponds to large values of (the real part of) the eigenvalues of H.These results are related to those of Refs.[83,84], where the finite-action principle was proposed to explain homogeneity and isotropy of the universe within quadratic gravity in the classical limit: in our fully quantum construction the configuration that leads to the smallest action corresponds to peaks in the wave functional of the ground state and are homogeneous and isotropic, while inhomogeneities and anisotropies lead to a larger action and are associated with excited states.
The link between the size of an eigenvalue E of H and the configurations that minimize the action and maximize the (absolute value) of the corresponding wave functional ⟨q η |ψ⟩ can more generally seen as follows.Note Multiplying by ⟨ψ|q f η ⟩ and integrating over q f on both sides leads to The integral dq f |⟨q f η |ψ⟩| 2 can be set to 1 with a conventional 13 choice of the normalization of |ψ⟩, which leads to Therefore, recalling that in quadratic gravity ⟨q f η , τ f |q i , τ i ⟩ is given by (4.41) (or for a generic gauge by (4.52)) with J µν = 0, the wave functionals with the smaller (real part of) E are those somewhat peaked around configurations for which the Euclidean action is small.It is interesting to note that in the small α limit14 the theory reduces to Starobinsky's model of inflation.Therefore, our construction can also be applied to study non-perturbative aspects of this successful inflationary model.However, note that in that limit the W 2 term becomes inactive and, therefore, the smallest eigenvalues of H no longer necessarily correspond to wave functionals peaked around conformally-flat metrics.
One important implication of the formulation of quadratic gravity presented here is the fact that the homogeneity and isotropy of the initial conditions for inflation are explained rather than postulated.Let us divide the space at initial time t i in causally disconnected patches.The state vector at time t i of such a system can be described by a direct product of the form .12) where the factor |ψ K ⟩ describes the Kth patch.Let us focus on a given patch at a time as its time evolution |ψ K (t i )⟩ → |ψ K (t)⟩ is independent of the others, at least for small enough t−t i .Suppose to start from a highly inhomogeneous and/or anisotropic patch at time t i .This corresponds to the short-distance limit as we have seen.Then we can be in two distinct situations.Either (i) there is a dynamical mechanism that makes the field configuration homogeneous and isotropic at large time or (ii) such a mechanism is not present.In Case (i) the homogeneity and isotropy of the initial condition for inflation are clearly explained.In Case (ii) we can either be in a situation (iia) where the initial state becomes nearly classical as time passes or in a complementary situation (iib) where the initial state remains quantum.In Situation (iia) the homogeneity and isotropy of the initial condition for inflation is explained by the classical mechanism described in Ref. [32], based on anthropic selection, which has been reviewed in the introduction.In Situation (iib) life remains impossible because classical entities, such as liquid water, are essential for life as we know it; therefore, a similar anthropic exclusion of the considered patch takes place.It is only in the nearly homogeneous and isotropic patches that inflation can occur and observations can eventually be made.If such a patch has undergone exponential expansion an internal point (far from its border) remains causally disconnected from the other patches even at large t − t i .This mechanism is an explicit quantum realization of Linde's chaotic inflation [91].The situation is illustrated in Fig. 1: the nearly homogeneous and isotropic patches are compatible with life and can inflate.
It is important to note that even if there were non standard features of the spectrum of H (i.e.H not diagonalizable with only real eigenvalues) there would be no way to observe such features.Indeed, suppose we were in such a situation and let us start from a highly inhomogeneous and anisotropic initial patch, such that the corresponding state |ψ K ⟩ could have a nonnegligible overlap with an eigenstate |ψ c ⟩ of H with a complex eigenvalue E c = E R + iE I , where E R and E I are real numbers (with E I being non negligible), and/or |ψ K ⟩ cannot be written as a linear combination of eigenstates of H. Since the time evolution is given by the operator exp(−i(t f − t i )H/ℏ), a state with E I < 0 would decay with time.However, if there were a complex eigenvalue the spectrum would also include its complex conjugate because H is time-reversal invariant 15 , T HT −1 = H .This means that E I > 0 would also be realized and we must take this possibility into account.However, in a situation where |ψ K ⟩ had a non-negligible overlap with an eigenstate |ψ c ⟩ with E I > 0, Eq. (4.10) would tell us that, as t f − t i becomes large, the density probability of patch K exponentially converges to where |ψ c ⟩ is the eigenstate with a non negligible overlap with |ψ K ⟩ and with the largest value of E I .In words, the probability density of such a patch would converge to that associated with a state |ψ c ⟩ and stay there forever.As we have seen, such a state must be highly inhomogeneous and/or anisotropic and, thus, cannot be compatible with observers.If |ψ K ⟩ could not be written as a linear combination of eigenstates of H we would obtain a similar result, except that, if there are no complex eigenvalues, the probability density would converge linearly rather than exponentially to where |ψ m ⟩ is now the eigenstate of H with the largest algebraic multiplicity, in the discrete approximation of Eq. (4.63).Again, observations in such a patch cannot be made.Furthermore, such a patch does not inflate because, as we have seen, the inhomogeneities are not diluted as time passes.

Conclusions
Here is a detailed summary of the results obtained together with conclusions.
• In the first section (after the introduction) the 3 + 1 formalism was introduced for quadratic gravity featuring both the R 2 and W 2 terms in addition to the standard Einstein-Hilbert and H is real and bounded from below; also, the corresponding smallest eigenvalue E 0 is non negative if the bare cosmological constant λ (in the case of quadratic gravity) or V (0) (in the scalar-field model) is large enough.In the same section also the Lorentzian path integral for Green's functions was obtained, which justifies previous expressions used in perturbative calculations around the flat spacetime.
• The (quantum) effective action Γ for quadratic gravity in the specified quantization is defined in Sec. 5. When used at tree level Γ captures the full quantum effects and its field equations allow us to determine the VEV of the metric in the presence of a generic (possibly vanishing) physical energy-momentum tensor.
• In the classical limit, ℏ → 0, the field equations generated by the (quantum) effective action reduce to those of the classical action of quadratic gravity, as shown in Sec. 6.In the same section it was also explained how this theory, which features a large-distance spectrum of H bounded from below, can have a classical Hamiltonian H c that is instead unbounded from below.
• Finally, in Sec.7 the constructed non-perturbative and background-independent quantum quadratic gravity was applied to cosmology.First, it was noted that the theory reduces (and thus justifies) previous tree-level calculations performed expanding around cosmological FLRW backgrounds.Next, the consistent Euclidean path integral of quadratic gravity was applied to quantum cosmology.Such path integral, indeed, allows us to find an expression for the ground-state wave function of the universe and for all the large-distance spectrum.The wave functions (which are actually functionals of the metric and its time derivative) satisfy a sort of Wheeler-DeWitt equation.In our fully quantum construction the configuration that leads to the smallest action corresponds to peaks in the ground-state wave functional that are homogeneous and isotropic, while inhomogeneities and anisotropies lead to larger actions and are associated with excited states.These results support those of Refs.[83,84], where the finite-action principle was proposed to explain homogeneity and isotropy of the universe within quadratic gravity in the classical limit.Furthermore, it was found that large inhomogeneities and anisotropies necessarily correspond to large values of (the real part of) the eigenvalues of H.When the coefficient of the W 2 vanishes one recovers Starobinsky's model of inflation, which can, therefore, be studied non-perturbatively with our construction.In the presence of the W 2 we are unable to show whether H is diagonalizable with only real eigenvalues or not.However, in this final section we found that if that were not the case there would be no way to observe such unusual features: patches where their effects are sizable are not compatible with observers.It is only in the nearly homogeneous and isotropic patches that inflation can occur and observations can eventually be made.This mechanism is an explicit quantum realization of Linde's chaotic inflation [91] and upgrades to the full quantum level the classical results of Ref. [32], which explain the near homogeneity and isotropy of our universe.The space of traceless symmetric tensors S ij t is five dimensional, while that of the trace S is one dimensional so the determinant of the "matrix" with elements Γ mn ij is which is spacetime independent.The determinant of the other matrix factor, g pm g qn , can be computed by considering, at each spacetime point, a complete set of three eigenvectors V k , with corresponding eigenvalues λ k , of the metric: where in the right-hand side there is no sum over k.Indeed, (V m k V n l + V m l V n k )/2 forms a set of six linearly-independent symmetric 3×3 matrices, which are eigenvectors of the "matrix" in question: By applying the Gaussian integration formula we then find 11)

Figure 1 :
Figure 1: Cartoon for the primordial universe.The density of wiggle represents the amount of inhomogeneity and anisotropy.

Table 1 :
Quantization for the canonical variables in the scalar theory with Lagrangian density(4.20)and in quadratic gravity.The conjugate momenta are defined in(4.22)and(3.2).
.45) Eqs.(4.38), (4.41), (4.44) and (4.45) provide us with the basic formula for the Euclidean path integral in quadratic gravity.The integration measure in (4.45) is invariant under three-dimensional general coordinate transformations.Let us see why.Under such transformations g