Quantum cosmology: effective theory

In homogeneous models, the dynamics is completely determined by one equation for gravitational degrees of freedom (such as the Friedmann equation) and one for matter (such as the continuity equation). The Friedmann equation of isotropic models,

$$\begin{equation} \left(\frac{\dot{a}}{a}\right)^2 +\frac{k}{a^2}= \frac{8\pi G}{3}\rho , \end{equation} \tag{ 1 }$$

depends only on first-order derivatives and is therefore a constraint, to be satisfied by initial values of second-order equations of motion. When interpreted as a constraint (the Hamiltonian constraint), it is usually written in the form of an energy-balance law:

$$\begin{equation} H:= -\frac{3}{8\pi G}(\dot{a}^2a+ka)+ E_{\rm matter}=0 \end{equation} \tag{ 2 }$$

with the matter energy E_matter = ρa³ contained in some region of unit coordinate volume. (See [4] for a detailed discussion of coordinate factors when the volume is not fixed, especially in the context of quantization.) In this form, the Hamiltonian constraint of gravity is obtained by varying the action by the lapse function $N=\sqrt{-g_{00}}$. (For more on constraints and canonical gravity, see [5].)

Given the Friedmann equation and the continuity equation

$$\begin{equation} \dot{\rho }+3\frac{\dot{a}}{a}(\rho +P)=0, \end{equation} \tag{ 3 }$$

with pressure P, one can derive a second-order equation of motion by taking a time derivative of (1) and eliminating $\dot{\rho }$: the Raychaudhuri equation

$$\begin{equation} \frac{\ddot{a}}{a} = -\frac{4\pi G}{3} (\rho +3P). \end{equation} \tag{ 4 }$$

At this stage, we have all equations expected from the components of the isotropic Einstein tensor: the time-time component providing the Friedmann equation, the (identical) diagonal components of the spatial part amounting to the Raychaudhuri equation, and all off-diagonal components vanishing identically. The equations obtained are automatically consistent with each other: by construction, the time derivative of the Friedmann equation vanishes if the Raychaudhuri equation holds. Therefore, if the constraint imposed by the Friedmann equation holds for initial values at some time, it holds at all times.

This latter property is realized for a large class of systems describing versions of isotropic (or homogeneous) cosmology, not just for the classical one resulting from Einstein's equation. As a phase-space function, H(a, p_a) in (2), with the momentum $p_a= -3(4\pi G)^{-1} a\dot{a}$ as it follows from the variation $\partial L_{\rm grav}^{\rm iso}/\partial \dot{a}$ of the Einstein–Hilbert action reduced to isotropy, plays the role of the Hamiltonian generating all equations of motion in proper time. The Raychaudhuri equation indeed follows from the Hamiltonian equations of motion $\dot{a}=\partial H/\partial p_a=\lbrace a,H\rbrace$ and $\dot{p}_a=-\partial H/\partial a=\lbrace p_a,H\rbrace$ (the first of which is identical to the definition of the momentum p_a). The matter terms ρ and P are realized by

$$\begin{equation} \rho =\frac{E_{\rm matter}}{a^3}\quad \mbox{ and }\quad P=-\frac{1}{3a^2}\frac{\partial E_{\rm matter}}{\partial a}, \end{equation} \tag{ 5 }$$

the negative change of energy by volume change. Matter equations of motion follow once E_matter in (2) is expressed in terms of canonical degrees of freedom.

The phase-space function H(a, p_a) itself evolves according to the same Hamiltonian law, $\dot{H}(a,p_a)=\lbrace H(a,p_a),H\rbrace =0$ and is automatically constant in time, no matter what form H(a, p_a) has. In homogeneous situations, the single Hamiltonian constraint that determines evolution is automatically preserved and consistent with evolution equations. We can easily modify H by any form of quantum corrections without encountering consistency problems, issuing a powerful license to cosmological model builders.

Consistency remains valid when we consider different choices of time. So far, the equations were in proper time τ. All other choices t in homogeneous models are related to τ by τ(t) = ∫^tN(t') dt', with a lapse function N that enters Hamiltonian equations as well: if df(a, p_a)/dτ = {f, H}, we have

$$\begin{equation*} \frac{\,{\rm d}\,f(a,p_a)}{\,{\rm d}\, t} = \frac{\,{\rm d}\,\tau }{\,{\rm d}\,t}\frac{\,{\rm d}\,f(a,p_a)}{\,{\rm d}\,\tau } =N\lbrace f,H\rbrace \approx \lbrace f,N H\rbrace \end{equation*}$$

for any other time with dτ/dt = N. In the last step, we are allowed to pull the lapse function N inside the Poisson bracket, keeping the equality satisfied as a 'weak' one, one that is valid provided the constraint H = 0 holds. Applying the general law to the constraint itself, we again observe consistency: dH/dt = {H, NH} ≈ 0.

The Hamiltonian constraint generates not only evolution with respect to a given time choice, but also the transition between different choices as a gauge transformation. Infinitesimally, with N close to one, we have τ = t + with = ∫(N − 1) dt, and for any function f,

$$\begin{equation} \delta _{\epsilon }f= f(\tau )-f(t)=\epsilon \frac{\,{\rm d}\,f}{\,{\rm d}\,t} =\epsilon \lbrace f,H\rbrace \approx \lbrace f,\epsilon H\rbrace . \end{equation} \tag{ 6 }$$

The Friedmann equation, amounting to the Hamiltonian constraint, is automatically invariant under changes of time, and so are its solutions. Also the evolution equations are invariant if we use the Jacobi identity for Poisson brackets:

$$\begin{eqnarray*} &&\left\lbrace \frac{\,{\rm d}\,f}{\,{\rm d}\,t},\epsilon H\right\rbrace = \lbrace \lbrace f,NH\rbrace ,\epsilon H\rbrace = \lbrace \lbrace f,\epsilon H\rbrace ,NH\rbrace + \lbrace f,\lbrace NH,\epsilon H\rbrace \rbrace \\ &&{\hphantom{\left\lbrace \frac{\,{\rm d}\,f}{\,{\rm d}\,t},\epsilon H\right\rbrace}}= \frac{\,{\rm d}\,\lbrace f,\epsilon H\rbrace }{\,{\rm d}\,t}- \lbrace f,(\,{\rm d}\,\epsilon /\,{\rm d}\,t) H\rbrace + \lbrace f,(\delta _{\epsilon }N)H\rbrace . \end{eqnarray*}$$

Rearranging, we see that the gauge-transformed f evolves in agreement with the gauge-transformed df/dt, with a correction taking into account a possible gauge transformation of N and time dependence of .

The presence of a single Hamiltonian constraint therefore ensures dynamical consistency and invariance, even if quantum modifications occur. For this reason, homogeneous minisuperspace models are a simple and popular tool to investigate possible consequences of quantum gravity and cosmology. But for the very same reason, extreme care must be exercised when such models are used for physical predictions: the trivialization of consistency conditions does not hold in a more general context, and therefore spurious results can easily be produced in their absence.

Compared with minisuperspace models, inhomogeneous cosmology presents a very different situation regarding consistency, even if inhomogeneity is small and treated perturbatively. For gravity and matter to fit together in Einstein's equation or a quantum modification thereof, a version of the contracted Bianchi identity must hold. But this identity, relating different types of dynamical equations, is easily destroyed if quantum corrections, for instance those found in minisuperspace models, are inserted blindly. When amending inhomogeneous equations by quantum corrections, one must face the problem of anomaly freedom or covariance. Certain relations between different dynamical equations and gauge generators, classically implemented by the contracted Bianchi identity, must be preserved in the presence of quantum corrections.

This well-known classical fact is often overlooked in quantum treatments, especially those making use of gauge fixing or deparameterization. Gauge generators disappear when the gauge is fixed. A simple but illegitimate way out of difficult consistency problems is therefore to fix the gauge before quantum corrections are inserted. However, one then dispenses with ways to check consistency and cannot be sure that results obtained are physically viable. Most cases in which cosmological perturbations have been computed by gauge fixing in loop quantum cosmology, for instance, have by now been shown to be incorrect; important effects such as signature change have been overlooked. See also the instructive discussion of [6] in this context. We will come back to this issue later and for now continue with a classical discussion to provide more details.

The first implication of the Bianchi identity is the presence of constraints. If we write ∇_νG^ν_μ = 0 in the form

$$\begin{equation} \partial _0G^0_{\mu }= -\partial _aG^a_{\mu }-\Gamma ^{\nu }_{\nu \kappa }G^{\kappa }_{\mu }+ \Gamma ^{\kappa }_{\nu \mu }G^{\nu }_{\kappa }, \end{equation} \tag{ 7 }$$

with spatial indices 'a', it becomes evident that the components G⁰_μ of the Einstein tensor cannot contain second-order time derivatives: on the right-hand side, all factors in the three terms are at most second order in time, and there is one explicit time derivative on the left-hand side, leaving only the option of first time derivatives in G⁰_μ. These components of the Einstein tensor (minus 8πG times the corresponding stress-energy components T⁰_μ if there is matter) are constraints on initial values, while the remaining components provide evolution equations. In contrast to minisuperspace models, we are dealing with a larger constrained system of four independent and functional constraints, the Hamiltonian constraint H = G⁰₀ − N^aG⁰_a and the diffeomorphism constraint D = NG⁰_a, which are to be imposed pointwise or for all possible multiplier functions N and N^a in

$$\begin{equation} \fl H[N]=-\int {\rm d}^3x N\big(G^0_0-N^aG^0_a\big)\quad \mbox{ and }\quad D[N^a]= -\int {\rm d}^3x N^a N G^0_a. \end{equation} \tag{ 8 }$$

We are dealing with an infinite number of constraints. (To define these integrations, one introduces a foliation of space-time into spatial surfaces t = const, also used to set up canonical variables. The time direction t^a at each point, used to define time derivatives in evolution equations, may be different from the normal direction n^a to spatial slices in space-time, a freedom parameterized as t^a = Nn^a + N^a with the lapse function N and the shift vector field N^a. The linear combinations of G⁰₀ and G⁰_a in (8), depending on the shift N^a and lapse N, take into account that constraints refer to directions normal and tangential to spatial slices, not to coordinate directions such as the zero-index of the Einstein tensor for a component along the time-evolution vector field t^a.)

As before, for consistency the constraints must always hold provided they are imposed for initial values, a feature that is guaranteed by the Bianchi identity as well. If we combine the Einstein tensor and the stress-energy tensor T_μν of matter in (7), we see that ∂₀(G⁰_μ − 8πGT⁰_μ) vanishes at any time provided the constraints G⁰_μ − 8πGT_μ⁰ themselves (and therefore their spatial derivatives) vanish at that time and the evolution equations hold. By virtue of the contracted Bianchi identity, the constraints are consistent with evolution. Finally, as a third consequence, we see that all equations, Einstein's equation and the contracted Bianchi identity as a consistency condition, are covariant and independent of coordinates used. Therefore, solutions will be covariant. All equations hold irrespective of the choice of coordinates, a well-known feature in perturbative cosmology which allows one to express all equations explicitly in terms of gauge-invariant variables [7, 8].

As in our discussion of minisuperspace models, a canonical view is useful to analyze which consistency conditions are satisfied automatically and which ones are non-trivial. There is a Hamiltonian constraint, and therefore Hamiltonian equations $\dot{f}=\lbrace f,C\rbrace$ are generated by a constraint C. However, in the inhomogeneous context, the constraint is not unique (up to a pre-factor N), nor is time evolution. We have a much larger choice of possible time variables to generate evolution. The most general version of equations of motion is obtained if we use all our constraints in a linear combination, defining H[N, N^a] := H[N] + D[N^a]. For fixed N and N^a, the Hamiltonian flow $\dot{f}=\lbrace f,H[N,N^a]\rbrace$ is then equivalent to Lie derivatives $\dot{f}=\mathcal{L}_tf$ along the time-evolution vector field t^a = Nn^a + N^a in space-time, foliated by spatial slices with unit normals n^a.

For all constraints to be preserved by the evolution equations they generate, we need {H[M, M^a], H[N, N^a]} ≈ 0 for all M, N, M^a and N^a. If this condition is satisfied, the constraints are said to form a first-class system. Unlike in homogeneous models, where H[M] always commutes weakly with H[N], the general condition is highly non-trivial and provides strong restrictions on consistent modifications of Einstein's equation. Quantum corrections can no longer be inserted at will.

The same constraints that generate evolution provide gauge transformations, classically equivalent to coordinate changes. We use the same general combination as before, H[, ^a], but interpret the multipliers and ^a differently, not related to a time-evolution vector field. Instead, the gauge transformation $\delta _{\epsilon ^{\mu }}f= \lbrace f,H[\epsilon ]+D[\epsilon ^a]\rbrace$ with the classical constraints is equivalent to a coordinate transformation or the Lie derivative $\mathcal{L}_{\xi }f$ along the space-time vector field ξ^μ with components such that = Nξ⁰ and ^a = ξ^a + N^aξ⁰ [9]. The factors of N and N^a again result because space-time coordinate changes and the components of ξ^μ refer to coordinate directions, while constraints refer to directions normal and tangential to spatial slices with normal n^a = N⁻¹(t^a − N^a). Also regarding gauge invariance, the condition of a first-class constraint algebra is then sufficient for consistency: in this case, all constraints are gauge invariant, δH[N, N^a] = {H[N, N^a], H[, ^a]} ≈ 0, and so are the evolution equations they generate. With this full set of gauge transformations, one can freely change the constant-time spatial surfaces used to define canonical variables and to integrate the constraints (8). We are then dealing with a covariant theory of space-time, not just with a theory on a fixed spatial foliation.

The crucial consistency condition for any classical constrained system, its quantization, or an effective theory thereof, is therefore that it be first class: all constraints H[N, N^a] must have Poisson brackets that vanish when the constraints are imposed,

$$\begin{equation} \lbrace H[M,M^a],H[N,N^a]\rbrace \approx 0. \end{equation} \tag{ 9 }$$

For gravity, or any generally covariant space-time theory, the specific form is the hypersurface-deformation algebra [10]

$$\begin{eqnarray} \lbrace D[M^a],D[N^a]\rbrace &=& D[\mathcal{L}_{N^b}M^a] \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \lbrace H[M],D[N^a]\rbrace &=& H[\mathcal{L}_{N^b}M] \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \lbrace H[M],H[N]\rbrace &=& D[q^{ab}(M\nabla _bN-N\nabla _bM)] \end{eqnarray} \tag{ 12 }$$

with the spatial metric q^ab.

For a consistent quantization, an algebra of this form must be realized with commutators for constraint operators instead of Poisson brackets, and an effective constrained system must have quantum-corrected constraints such that a first-class algebra holds with Poisson brackets. If this is realized, no gauge transformations are broken by quantization and the quantum or effective theory is called anomaly-free. If this condition is satisfied, all consistency conditions that are classically implied by the contracted Bianchi identity hold in the presence of quantum corrections, and the (quantum) theory describes space-time rather than just a family of spatial slices. This property may be achieved with exactly the same form of the algebra, or with one that shows quantum corrections not just in the constraints but also in the structure functions of the algebra, as long as two constraints still commute up to another constraint. One universal example found in loop quantum gravity, as the most prominent result regarding hypersurface deformations with quantum corrections, has (10) and (11) unchanged, but (12) modified to [11]

$$\begin{equation} \lbrace H[M],H[N]\rbrace = D[\beta q^{ab}(M\nabla _bN-N\nabla _bM)] \end{equation} \tag{ 13 }$$

with some phase-space function β. If the classical hypersurface-deformation algebra is modified, gauge transformations no longer correspond to Lie derivatives by space-time vector fields. Not just the dynamics but even the structure of space-time may be modified by quantum effects. We will discuss specific examples and results in later parts of this review.

By analyzing quantum or effective constraints and their algebra, one can draw conclusions about quantum space-time structures. For instance, once the full hypersurface-deformation algebra is known, one can specialize it to Poincaré transformations by using linear N and N^a. With N(x) = Δt + vx, for instance, we have a combination of a time translation by Δt and a boost by v. With $\beta \not=1$ in (13), the usual Poincaré relations are modified. Although this may look like a version of deformed special relativity [12–14], there is no direct relation: in deformed special relativity, one has non-linear realizations of the Poincaré algebra, with structure constants depending on the algebra generators. In (13), we have corrections of structure functions depending on phase-space degrees of freedom, not directly on the algebra generators H[N] and D[N^a]. A deformed version of special relativity would require a relation between phase-space variables, such as extrinsic curvature, and some space-time generators, such as energy. Relations of this form do exist in some regimes, for instance in asymptotically flat ones using the ADM energy, but not in general.

In general terms, the problem of canonical quantum cosmology can be formulated as follows: find quantum corrections in H[N, N^a], perhaps motivated by some full theory of quantum gravity, such that these constraints remain first class. This statement presents a well-defined mathematical problem of classifying deformed algebras. For every consistent version that may exist, we can compute and analyze equations of motion by standard means, as explicitly written out above in the general classical case. At the present stage, several examples of consistent deformations of constraints remaining first class are known, mainly from loop quantum gravity, but there is no general classification of these infinite-dimensional algebras.

As already mentioned, there are attempts to shortcut through the difficult calculation of these algebraic structures by fixing the gauge before quantum corrections or other modifications are put in. If the gauge is fixed, one would no longer consider H[, ^a] as gauge generators, but only use H[N, N^a] as constraints and to generate evolution. Some consistency conditions still need to be satisfied because the constraints are to be preserved by evolution, but this can usually be achieved more easily than in the non-gauge fixed case in which more fields are present. However, even if formally consistent versions of preserved constraints can be found in this way, such as those in [15], they are not guaranteed to be consistent because only a subclass of the constraint algebra can be tested when some modes are eliminated beforehand. Even if these are classical gauge modes, some consistency conditions and physical effects are overlooked. Moreover, the procedure is intrinsically inconsistent because one would first fix the gauge as it is determined by the classical constraints, and then proceed to modify the constraints that generate the gauge.

A consistent scheme could be obtained only when the modified gauge structure is taken into account from the very beginning, but for that one would have to know a consistent version of non-gauge fixed constraints, not just of the gauge-fixed ones. Finally, having fixed the gauge, there is no way of calculating general gauge-invariant observables. Also here, one could only refer to the classical invariant variables, whose form however must be modified when quantum corrections are put into the constraints. Note that quantum space-time structures and modified constraints in most cases imply departures from classical manifold pictures, as we will see explicitly in the examples provided later. One can no longer refer to the usual form of coordinate transformations to compute gauge-invariant variables without using the constraints explicitly. In modified space-times, the constraints are the only means to compute gauge flows and invariants, but this can be done only if the gauge has not been fixed.

Algebraic conditions can be simplified even more when gauge fixing is combined with deparameterization. With the latter procedure, one chooses a phase-space degree of freedom, for instance from matter, to rewrite constraint equations as relational evolution equations with respect to this variable. Not just gauge transformations but even constraints then disappear from the system, and no strong consistency conditions remain. A popular example is the coupling of a free, massless scalar field ϕ, whose homogeneous mode $\bar{\phi }$ in an expansion $\phi =\bar{\phi }+\delta \phi$ around some background can be treated as a global time function. The canonical scalar Hamiltonian

$$\begin{eqnarray} &&H_{\,{\rm scalar}}= \frac{1}{2}\int \,{\rm d}\,^3x \left(\frac{p_{\phi }^2}{\sqrt{\det q}}+ \sqrt{\det q} q^{ab}(\partial _a\phi )(\partial _b\phi )\right) \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} &&\hphantom{H_{\,{\rm scalar}}}= \frac{1}{2} \left(\frac{\bar{p}_{\phi }^2}{\sqrt{\det \bar{q}}}+ \int \,{\rm d}\,^3x \delta (\cdots )\right) \end{eqnarray} \tag{ 15 }$$

with the momentum $p_{\phi }=\bar{p}_{\phi }+\delta p_{\phi }$ then has a purely kinetic background term and is completely independent of $\bar{\phi }$. The momentum $\bar{p}_{\phi }$ is therefore a constant of motion and never becomes zero unless it vanishes identically. The background scalar $\bar{\phi }$ has no turning points and is monotonic, serving as a global internal time along classical trajectories.

The Hamiltonian generating evolution with respect to this variable is obtained by solving the Hamiltonian constraint H_gravity + H_scalar = 0 (with fixed N as part of the gauge choice) for $\bar{p}_{\phi }=H_{\bar{\phi }}(q_{ab},p^{ab},\delta \phi ,\delta p_{\phi })$: we have $\dot{\bar{\phi }}=\lbrace \bar{\phi },H_{\bar{\phi }}\rbrace =1$ and $\dot{\bar{p}_{\phi }}=\lbrace \bar{p}_{\phi },H_{\bar{\phi }}\rbrace =0$, consistent with evolution with respect to $\bar{\phi }$ where the dot stands for ${\rm d}/{\rm d}\bar{\phi }$, as well as Hamiltonian equations for the remaining variables, such as $\dot{q}_{ab}=\lbrace q_{ab},H_{\bar{\phi }}\rbrace$ and $\dot{\delta }\phi =\lbrace \delta \phi ,H_{\bar{\phi }}\rbrace$. There is just one Hamiltonian generating all evolution equations, instead of a set of infinitely many constraints. Almost as in minisuperspace models one can then implement in $H_{\bar{\phi }}$ any quantum corrections one may desire. (In the context of cosmology, this approach has been suggested for instance in [16] in an analysis of Gowdy models and possible quantizations.)

Deparameterization is a powerful mathematical tool to derive properties of physical Hilbert spaces, for which no general method exists in the absence of deparameterizability. (See also section 4.1.5.) In quantum cosmology, this method has been applied in [17], and used recently to derive several aspects of self-adjointness and unitarity of evolution [18–20]. But it cannot serve as a valid procedure to evade the anomaly problem and do physical evaluations. To start with, most realistic systems are not globally deparameterizable, with one variable having no turning points at all. Moreover, also this procedure, on its own, cannot weed out physical inconsistencies in spite of its formal consistency. One has the same drawbacks as in the gauge-fixed approach, and on top of that one has distinguished (or even introduced) one degree of freedom as time. For physical consistency, one should then show that results for observables do not depend on one's choice of time after quantization, but no systematic procedures exist to this end, constituting part of the problem of time in quantum gravity [21–23]. (See [24] for a discussion in the context of deparameterization or reduced phase-space quantization.) And even if one thinks that the distinguished time $\bar{\phi }$ should be sufficient for all physical purposes and does not worry about relating results in different times, the fact that there is no analog of the Bianchi identity to constrain modifications should arouse suspicion.

Gauge fixing and deparameterization are not necessarily bad, but they provide reliable results only when they are used after a consistent version of quantum-corrected constraints has been found. When this is the case, it is clear that all equations are consistent and gauge covariant, and instead of computing complete gauge-invariant variables for the corrected constraints, one may well pick a gauge or choose an internal time and work out physical implications. There is also a possibility of providing consistent results with gauge fixing or deparameterization before quantization, but then one would have to show that all possible gauge fixings or deparameterizations would lead to the same physical observables. This requirement can be achieved in cases of simple constraints, such as the Gauss constraint of gravity in triad or connection variables as described at the end of section 5.1, but presents a much more involved problem for the complicated Hamiltonian constraint. Problems and physical inconsistencies arise always when the gauge or time is chosen according to the classical system, and modifications of the constraints and thereby gauge transformations are inserted later.

Effective equations, in general terms, describe quantum evolution by a smaller, more manageable number of degrees of freedom compared with the full quantum theory considered. When we go from classical physics to quantum physics, every degree of freedom we have at first is replaced by infinitely many parameters, for instance values the whole wave function takes at all points in configuration space. It is difficult to deal with values of wave functions in physical terms. Another parameterization, more convenient for effective equations as it turns out, is the set of expectation values of basic operators, $\langle \hat{a}\rangle$ and $\langle \hat{p}_a\rangle$ in Wheeler–DeWitt quantum cosmology, together with fluctuations and higher moments

$$\begin{equation} \Delta (a^bp_a^c):= \langle (\hat{a}-\langle \hat{a}\rangle )^b (\hat{p}_a-\langle \hat{p}_a\rangle )^c\rangle _{\rm Weyl} \end{equation} \tag{ 16 }$$

with operators in totally symmetric, or Weyl, ordering. For generic states, all these moments with integer b and c such that b + c ⩾ 2, are independent of one another and of the expectation values, and therefore provide infinitely many degrees of freedom. (For b + c = 2, for instance, we have the two fluctuations Δ(a²) = (Δa)² and Δ(p²_a) = (Δp_a)², and the covariance $\Delta (ap_a)= \frac{1}{2}\langle \hat{a}\hat{p}_a+ \hat{p}_a\hat{a}\rangle - \langle \hat{a}\rangle \langle \hat{p}_a\rangle$.)

Any set of effective equations provides a description between the classical limit and the full quantum theory of the system considered, making use of finitely many degrees of freedom for each classical one. There may be additional degrees of freedom compared with the classical theory, but not infinitely many more. By the new degrees of freedom included and their coupling to expectation values—the variables with a direct classical analog—quantum corrections result. This is our general definition of effective theories, which we distinguish from coarse-grained theories. The latter may remove some quantum degrees of freedom by integrating them out, but typically leave whole towers of moments intact and therefore present infinitely many variables for some classical degrees of freedom.

The behavior of the moments computed in simple unsqueezed Gaussian states with fluctuation parameter σ [25],

$$\begin{equation} \Delta \big(a^bp_a^c\big)= 2^{-(b+c)} \hbar ^c\sigma ^{b-c}\frac{b!\,c!}{(b/2)! (c/2)!} \quad\, {\rm if}\, b \,{\rm and}\, c\, {\rm are}\,{\rm even} \end{equation} \tag{ 17 }$$

and Δ(a^bp^c_a) = 0 otherwise, suggest a natural organization of effective theories of different degrees. A Gaussian state saturates the uncertainty relation

$$\begin{equation} (\Delta a)^2(\Delta p_a)^2- \Delta (ap_a)^2\ge \frac{\hbar ^2}{4} \end{equation} \tag{ 18 }$$

and, for position and momentum fluctuations of the same order, has σ = Δa = O(ℏ^1/2). The product ℏ^cσ^{b − c} ∼ O(ℏ^{(b + c)/2}) then shows that moments of order n = b + c are of the order n/2 in ℏ. We use this observation to generalize the notion of semiclassical states from simple Gaussians to a much larger class: if the moments of a state behave as Δ(a^bp^c_a) ∼ O(ℏ^{(b + c)/2}) (said to obey a ℏierarchy), the state is called semiclassical. Moments of higher orders then affect only terms of high orders in ℏ and can be neglected in an approximation. Ignoring all moments of order higher than some n, only finitely many quantum degrees of freedom are left, and we obtain an effective theory. With n + 1 moments of order n, an effective theory with moments up to order n has $\sum _{k=2}^{n}(k+1)=\frac{1}{2}n^2+\frac{3}{2}n-2$ state parameters in addition to the two basic expectation values.

Effective solutions, describing the evolution of a quantum state, depend not only on classical variables but also on the quantum state used, specified for instance by initial values of its moments. In quantum cosmology, one first promotes the constraint (2) expressed in canonical variables a and p_a as

$$\begin{equation} H(a,p_a)= -\frac{2\pi G}{3} \frac{p_a^2}{a}+E_{\,{\rm matter}} \end{equation} \tag{ 19 }$$

to an operator $\hat{H}$, choosing some ordering of $\hat{a}$ and $\hat{p}_a$. Dirac quantization then implements the classical constraint equation H(a, p_a) = 0 by the condition $\hat{H}|\psi \rangle =0$ on physical states. In particular, the expectation value $\langle \hat{H}\rangle$ must vanish in all physical states. If we express this equation as a functional equation on the space of expectation values and moments parameterizing states, we obtain an expression such as

$$\begin{equation} \langle \hat{H}\rangle = -\frac{2\pi G}{3} \left(\frac{\langle \hat{p}_a\rangle ^2}{\langle \hat{a}\rangle }+ \frac{(\Delta p_a)^2}{\langle \hat{a}\rangle }+\cdots \right)+ \langle \hat{E}_{\,{\rm matter}}\rangle \end{equation} \tag{ 20 }$$

where we have used $\langle \hat{p}_a^2\rangle = \langle \hat{p}_a\rangle ^2+ (\Delta p_a)^2$ and the dots indicate additional terms that contain the covariance of a and p_a as well as higher moments, and depend on the specific ordering chosen. If we take into account only expectation values in $\langle \hat{H}\rangle =0$, the classical Hamiltonian and the classical constraint surface are obtained. With fluctuations and higher moments, quantum corrections result that couple moments to expectation values and change the classical constraint surface. A systematic derivation and analysis of such moment terms, starting with expectation values of Hamiltonians and constraints, is the key ingredient of effective theories.

The moments couple to expectation values, thereby providing quantum corrections to the classical motion, and their values therefore enter effective equations. A complete effective theory cannot leave the moments in equations as unknowns and instead provides evolution equations or other conditions for them as well. But some freedom always remains, for instance in the initial values chosen for evolution equations of moments. If effective theories are formulated for expectation values without including additional degrees of freedom such as moments to describe the evolution of classes of states, the specific states must either be restricted to provide unique effective equations, or be parameterized for a more general set. Often, such a state dependence enters effective theories only implicitly, for instance in the unique-looking low-energy effective action [26] free of any state parameters. This effective action describes low-energy effects of states near the vacuum of the interacting theory. In this way, the class of states is specified, certainly a rather small set compared to all possible states.

In quantum cosmology, we are often interested in high-energy, Planckian phenomena and cannot restrict attention to the low-energy effective action. Even in low-energy regimes, which would be all we need to make contact with potential observations, it is not clear what low-energy state should be used. Quantum cosmology in its non-perturbative form, or quantum gravity in general, does not have a vacuum or other distinguished low-energy state. Effective theories of quantum cosmology must therefore be more general than the low-energy effective action, leaving more freedom for states parameterized in some suitable way.

Even if we restrict attention to semiclassical regimes, the class of states to be considered may be large: simple Gaussians provide at most a two-parameter family within a large set of semiclassical states when they are fully squeezed. For uncorrelated Gaussians, wave functions

$$\begin{equation} \psi _{\sigma }(a)= N\exp \left(-\frac{(a-\langle \hat{a}\rangle )^2}{4\sigma ^2}\right) \exp (-{\rm i}a\langle \hat{p}_a\rangle /\hbar ) \end{equation} \tag{ 21 }$$

with a normalization constant N, we have, besides the two expectation values $\langle \hat{a}\rangle$ and $\langle \hat{p}_a\rangle$ on which the state is peaked, only one quantum parameter, the variance σ. The most general Gaussian state is of the form ψ_z(a) = exp ( − z₁a² + z₂a + z₃) with three complex numbers z_i such that Rez₁ > 0 for normalizability. Out of these six real parameters, the two contained in z₃ do not matter for moments because the real part is fixed by normalization and the imaginary part contributes only a phase factor. For the remaining parameters, writing z₁ = α₁ + iβ₁ and z₂ = α₂ + iβ₂ with real α_i and β_i, we compute expectation values

$$\begin{equation} \langle \hat{a}\rangle = \frac{\alpha _2}{2\alpha _1},\quad \quad \langle \hat{p}_a\rangle = \hbar \frac{\alpha _1\beta _2-\alpha _2\beta _1}{\alpha _1} \end{equation} \tag{ 22 }$$

and second-order moments

$$\begin{equation} \fl (\Delta a)^2 = \frac{1}{4\alpha _1},\quad \quad (\Delta p_a)^2 = \hbar ^2\alpha _1+\hbar ^2\frac{\beta _1^2}{\alpha _1},\quad \quad \Delta (ap_a) = -\hbar \frac{\beta _1}{2\alpha _1}. \end{equation} \tag{ 23 }$$

One easily confirms that the uncertainty relation (18) is always saturated.

Deviations from saturation are much more difficult to parameterize, but easily occur for evolved semiclassical states even if they start out as a Gaussian. And even if one stays close to the saturation condition and does not vary second-order moments much beyond the values they can take for Gaussians, a Gaussian determines all higher moments in terms of the real and imaginary parts of z₁. Our general semiclassicality condition Δ(a^bp^c_a) ∼ O(ℏ^{(b + c)/2}) provides an infinite-parameter family instead of the special 2-parameter one realized for Gaussians. Unless one can motivate Gaussians by other means, for instance proximity to the Gaussian harmonic-oscillator ground state or the vacuum of a free field theory, using them exclusively may easily be too restrictive. (Given the large parameter space of states, one may be tempted to refer to probabilistic arguments to pick 'likely' states, following ideas that go back to an analysis of inflation in quantum cosmology [27, 28]. However, such probability considerations, though popular, are difficult, if not impossible, to make sense of in quantum cosmology [29].)

Not just the absence of a vacuum state but several other special properties of quantum cosmology are important when we consider possible states.

• Quantum cosmology considers long-term evolution. Even if we may be able to choose a specific form of semiclassical states at large volume and small curvature, it may change much when states are evolved back to high densities to infer possible implications at the big bang.
• As already mentioned, even the form of semiclassical states is unclear. It is customary to explore semiclassical features using simple and nicely peaked Gaussian states. In quantum mechanics, such states provide interesting information, and they are realized in exactly this form as coherent states or the ground state of the harmonic oscillator. In quantum field theory, Gaussian states are then close to the perturbative vacuum even for interacting theories. The dynamics of quantum cosmology, however, is not near that of the harmonic oscillator (except for some special models), and the unquestioned use of Gaussians is more difficult to justify. But going beyond Gaussians is complicated in terms of wave functions, whose parameters then become much less controlled.
• It is not clear how precisely quantum cosmology can be derived from some full theory of quantum gravity, but the number of degrees of freedom is certainly reduced either by exact symmetries or by perturbing around some background. In such situations, when degrees of freedom are eliminated, pure states easily become mixed. For general evolution equations able to model a full state, we should therefore allow for the possibility of mixed states, again going beyond pure Gaussians or other specific wave functions. Moments provide a parameterization of mixed states as well since they are based only on the notion of expectation values.
• The question of covariance affects also the choice of classes of states. A state chosen in a minisuperspace model must have a chance of being the reduction of a full state that does not break covariance. This question may be difficult to analyze at the level of wave functions, but it also shows that a sufficiently large freedom in the choice of states must be included in considerations that aim to provide a consistent formulation of quantum cosmology.

Since state properties are important for effective actions and quantum back-reaction, we should be as general as possible with the choice of states we consider. With wave functions or density matrices for mixed states, such a generality is difficult to achieve, but it is possible with parameterizations such as the one by moments. Moments provide a general form of semiclassical states, as already introduced. In effective theories, they are subject to their own evolution equations which show how they may change as high-density regimes are approached. They go well beyond the two-parameter family of squeezed Gaussians, and describe pure and mixed states alike.

Covariance at the level of effective equations with quantum back-reaction can be addressed by combining the moment parameterization with the methods of the previous section, deriving consistent constrained systems amended by quantum corrections that include the moments. For the last question, we must be able to fit moments into a phase-space structure, so as to be able to compute Poisson brackets such as {H[M, M^a], H[N, N^a]} with constraints that may include moment terms, such as an inhomogeneous version of (20).

This construction is indeed possible: together with expectation values, the moments form a quantum phase space with Poisson brackets defined by the commutator [30], first for expectation values of arbitrary operators:

$$\begin{equation} \lbrace \langle \hat{A}\rangle ,\langle \hat{B}\rangle \rbrace = \frac{\langle [\hat{A},\hat{B}]\rangle }{\,{\rm i}\,\hbar }. \end{equation} \tag{ 24 }$$

This bracket satisfies the Jacobi identity and is linear. If we extend it to polynomials of expectation values by imposing the Leibniz rule, all laws for a Poisson bracket are satisfied, and we can apply the definition to moments. For instance, we obtain $\lbrace \langle \hat{a}\rangle ,\langle \hat{p}_a\rangle \rbrace =1$ and $\lbrace \langle \hat{a}\rangle ,\Delta (a^bp_a^c)\rbrace =0= \lbrace \langle \hat{p}_a\rangle ,\Delta (a^bp_a^c)\rbrace$ for all b and c. The moments, as defined here, are symplectically orthogonal to expectation values, a convenient feature for calculations. Poisson brackets between different moments have been computed explicitly but are lengthy; see [30] and the correction of a typo in [25]. For low orders, it is usually more convenient to compute Poisson brackets directly from the definition (24). For instance, we calculate

$$\begin{eqnarray} &&\fl \lbrace (\Delta q)^2,(\Delta p)^2\rbrace = \lbrace \langle \hat{q}^2\rangle -\langle \hat{q}\rangle ^2, \langle \hat{p}^2\rangle -\langle \hat{p}\rangle ^2\rbrace \nonumber\\ &&\fl {\hphantom{\lbrace (\Delta q)^2,(\Delta p)^2\rbrace }}= \frac{\langle [\hat{q}^2,\hat{p}^2]\rangle }{\,{\rm i}\,\hbar } - 2\langle \hat{p}\rangle \frac{\langle [\hat{q}^2,\hat{p}]\rangle }{\,{\rm i}\,\hbar } - 2\langle \hat{q}\rangle \frac{\langle [\hat{q},\hat{p}^2]\rangle }{\,{\rm i}\,\hbar } + 4\langle \hat{q}\rangle \langle \hat{p}\rangle \frac{\langle [\hat{q},\hat{p}]\rangle }{\,{\rm i}\,\hbar }\nonumber \\ &&\fl {\hphantom{\lbrace (\Delta q)^2,(\Delta p)^2\rbrace }}= 2\langle \hat{q}\hat{p}+\hat{p}\hat{q}\rangle - 4\langle \hat{q}\rangle \langle \hat{p}\rangle = 4\Delta (qp) \end{eqnarray} \tag{ 25 }$$

using the Leibniz identity and (24).

If we know how a state enters effective constraints via its moments, a question which we will address in due course, we can compute Poisson brackets of effective constraints and see whether they provide a consistent deformation of the classical constraint algebra. If a consistent deformation is realized, the system can be analyzed further by standard canonical means to arrive at observables and dynamical equations. Note, however, that the Poisson tensor for moments truncated to some order is in general not invertible, for instance on the three second-order moments which form an odd-dimensional Poisson manifold: with (25) and similar calculations for the other second-order moments, we obtain

$$\begin{eqnarray} \lbrace (\Delta q)^2,(\Delta p)^2\rbrace &=& 4\Delta (qp), \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} \lbrace (\Delta q)^2,\Delta (qp)\rbrace &=& 2(\Delta q)^2, \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} \lbrace (\Delta p)^2,\Delta (qp)\rbrace &=&-2(\Delta p)^2, \end{eqnarray} \tag{ 28 }$$

the three-dimensional Poisson manifold of second-degree polynomials. Symplectic geometry therefore cannot be used for effective theories, while Poisson geometry is available from the definition (24). All relevant properties of constrained systems, such as the distinction between first and second class or properties of gauge transformations, can be formulated at the level of Poisson geometry [31].

In summary of this section, we note that there are several specific issues in the derivation of effective theories for quantum cosmology, compared to other fields in which such methods are in use. The central theme is covariance, an issue which can be addressed only if one goes beyond the traditional quantum-cosmological realm of minisuperspace models. The notion of covariance and the question whether it is realized consistently may be affected by new quantum space-time structures introduced by the specific form of quantum geometry in one's approach to quantum gravity.

The second, less obvious source of potential modifications of covariance is the form of dynamical quantum states used. Effective actions or equations depend on the classes of quantum states whose evolution they approximate, which should be specific solutions of some underlying quantum theory of gravity. Even if a theory such as quantum gravity is covariant, the selection of specific solutions may always break this symmetry. Covariance may then be realized in a deformed way, or only partially within one effective theory. For instance, even if the states used are peaked on covariant classical fields, giving rise to covariant effective terms depending on expectation values, their fluctuations or higher moments may not be fully covariant. Changing the gauge in quantum gravity may then require the transition to a different effective theory, in which different state parameters have been used (reminiscent of the application of background-field methods in standard quantum field theory). On the other hand, if the class of states is sufficiently large, all gauge-related parameters could be encompassed within one effective theory. These considerations highlight the importance of the selection of states in the derivation of effective theories.

Both quantum geometry and quantum dynamics must be part of one consistent quantum theory of gravity; in a complete treatment, their effects therefore cannot be separated from each other. However, they are derived by different means, making use of different expansions of the expectation-value function $\langle \hat{H}\rangle$ of the Hamiltonian or Hamiltonian constraint in the class of states used. In what follows, as in most derivations in the literature, we will split the treatment into one of quantum-geometry corrections first, as they are easier to see, followed by a discussion of quantum-dynamics corrections. When their expressions are known, we can compare implications of different effects and see if some are more relevant than others in specific regimes.

In the canonical setting, quantum-geometry corrections are specific to loop quantum gravity which has given rise to many results regarding background-independent quantization [32–34]. The effects in $\langle \hat{H}\rangle$ are not unique, but characteristic enough to show implications for quantum space-time structure. After an overview of these terms in the next section, we will discuss dynamical quantum back-reaction, obtained from a further expansion of $\langle \hat{H}\rangle$ by the moments parameterizing states. We will see the relation of canonical formulations to the low-energy effective action used in particle physics and the role of higher time derivatives in effective equations. Finally, we will put together our results to find properties of general effective actions taking into account all possible effects of canonical quantum gravity, and thereby shed light on quantum space-time structure.

A constraint operator $\hat{H}[N]$ restricts states by $\hat{H}[N]|\psi \rangle _{\rm phys}=0$ and generates a gauge flow $|\psi \rangle _{\epsilon }= \exp (-{\rm i}\hat{H}[\epsilon ]/\hbar )|\psi \rangle$. Physical states, on which $\exp (-{\rm i}\hat{H}[\epsilon ]/\hbar )$ acts as the identity, are gauge-invariant, and the gauge flow need not be considered separately if first-class constraint operators are solved. But exact solutions are complicated to find, and when quantum corrections are computed by systematic approximation schemes, the situation is quite different. Expectation values are constrained by $\langle \hat{H}[N]\rangle$, a weaker condition than $\hat{H}[N]|\psi \rangle =0$. We have additional constraints $\langle (\hat{f}-\langle \hat{f}\rangle )\hat{H}[N]\rangle =0$, an enlarged set of constraints which all vanish in physical states, for arbitrary $\hat{f}$. These constraints in general are all independent, constraining not just expectation values but also moments. In an effective theory, as spelled out in detail later, one solves this infinite set of quantum phase-space constraints order by order in the moments. To any given order, the sharp condition $\hat{H}[N]|\psi \rangle _{\rm phys}=0$ for physical states is not fully implemented, and on the corresponding solution space there is a non-trivial gauge flow by quantum constraints. Moreover, the effective treatment at this stage is more general because one does not assume a (kinematical) Hilbert-space structure when solving the constraints for moments; therefore the standard argument of a trivial gauge flow does not apply. At the effective level, there are non-trivial gauge transformations by quantum constraints, bringing solution procedures closer to classical ones: one solves phase-space constraints and factors out their gauge.

For an effective theory, the key ingredient is therefore the expectation value $\langle \hat{H}[N]\rangle$ in a sufficiently large class of states, or the general expression parameterized by the moments of states. As in (20), moments then appear in quantum corrections that change the constraint surface. With a modified constraint surface, the gauge flows must receive quantum corrections as well for them to be tangential to the constraint surface mapping solutions to constraints into other solutions, as required for a first-class system. Indeed, the expectation value of the constraint, an expression that includes quantum corrections, determines the gauge flow on the quantum phase space with Poisson brackets (24) by

$$\begin{eqnarray} &&\fl \delta _{\epsilon }\langle \hat{O}\rangle (\epsilon )= \epsilon \frac{\delta }{\delta \epsilon } \langle \psi |\exp (\,{\rm i}\,\hat{H}[\epsilon ]/\hbar ) \hat{O} \exp (-\,{\rm i}\,\hat{H}[\epsilon ]/\hbar )|\psi \rangle = \frac{\langle [\hat{O},\hat{H}[\epsilon ]]\rangle }{\,{\rm i}\,\hbar }= \lbrace \langle \hat{O}\rangle ,\langle \hat{H}[\epsilon ]\rangle \rbrace\nonumber\\ \end{eqnarray} \tag{ 29 }$$

for expectation values $\langle \hat{O}\rangle (\epsilon )= {}_{\epsilon }\langle \psi |\hat{O}|\psi \rangle _{\epsilon }$, and therefore for any quantum phase-space function such as the moments by using the Leibniz rule. If the constraint surface changes by corrections in $\hat{H}$, so do the gauge transformations. For the quantum corrected constraint surface and the gauge flow we are therefore required to compute $\langle \hat{H}[\epsilon ]\rangle$ in general kinematical states.

If quantum constraint operators are represented in an anomaly-free way, such that any commutator $[\hat{C}_1,\hat{C}_2]$ is an operator of the form $f(\hat{q},\hat{p})\hat{C}_3$ with another constraint operator $\hat{C}_3$ and structure functions f(q, p), the quantum constraints $\langle \hat{C}_i\rangle$ form a consistent first-class system: $\lbrace \langle \hat{C}_1\rangle ,\langle \hat{C}_2\rangle \rbrace = \langle [\hat{C}_1,\hat{C}_2]\rangle /{\rm i}\hbar = \langle f(\hat{q},\hat{p})\hat{C}_3\rangle$. Since all expressions such as $\langle f(\hat{q},\hat{p})\hat{C}_3\rangle$ vanish when computed in physical states, they provide effective constraints (in general independent of $\langle \hat{C}_3\rangle$ as phase-space functions). If all these constraints are imposed, the effective constrained system is first class and has a consistent gauge flow generated by all these constraints [37–39].

Such systems of infinitely many constraints for each local classical degree of freedom can be difficult to analyze. However, just as the moments obey a ℏierarchy in semiclassical regimes, the effective constraints can be truncated to finite sets to any given order in ℏ or in the moments. The leading corrections can be found in direct expectation values $\langle \hat{C}_i\rangle$, restricting expectation values with quantum corrections that depend on the moments. For instance, (20), when imposed as a constraint, shows how the classical constraint surface and the gauge flow of $\langle \hat{a}\rangle$ and $\langle \hat{p}_a\rangle$ are changed by fluctuations (Δp_a)². These corrections depend on the value of (Δp_a)², which is constrained and subject to gauge flows by higher-order constraints such as $\langle \hat{p}_a\hat{H}\rangle$. When all constraints have been solved and gauge flows factored out to a certain order, the condition $\hat{H}|\psi \rangle =0$ has been implemented for states used in expectation values and moments. One has then computed observables in physical states, sidestepping the complicated problem of computing an integral form of the physical inner product. This is one example for the use of effective methods to tame complicated technical and conceptual issues in quantum gravity. Even the problem of time can be solved, at least at the semiclassical level: different choices of time are related by mere gauge transformations in the quantum phase space [40–42]. We will come back to these conceptual question in section 4.1.5, and for now note the lesson that expectation values of constraints supply the key ingredient for effective gauge theories.

We should then apply effective techniques to the Hamiltonian constraint of gravity, computing $\langle \hat{H}[N]\rangle$. Expressions for the Hamiltonian constraint are rather complicated even classically, given by

$$\begin{equation} H_{\,{\rm grav}}\,(q_{ab},\pi ^{cd})= -\frac{16\pi G}{\sqrt{\det q}} \bigg(\pi _{ab}\pi ^{ab}- \frac{1}{2}(\pi ^a_a)^2\bigg)+ \frac{\sqrt{\det q}}{16\pi G} {}^{(3)}R \end{equation} \tag{ 30 }$$

in ADM variables, the spatial metric q_ab and its momentum

$$\begin{equation*} \pi ^{cd}= \frac{\sqrt{\det q}}{16\pi G} (K^{cd}-K^a_aq^{cd}) \end{equation*}$$

in terms of extrinsic curvature (with the spatial Ricci scalar ⁽³⁾R).

Loop quantum gravity uses a different set of variables, the densitized triad E^a_i instead of the spatial metric $q^{ab}=E^a_iE^b_i/\det (q_{cd})$ and the Ashtekar–Barbero connection Aⁱ_a = Γ_aⁱ + γKⁱ_a [43, 44], defined by combining the spin connection Γⁱ_a compatible with the densitized triad and extrinsic curvature $K_a^i=K_{ab}E^{bi}/\sqrt{|\det (E^c_i)|}$. The basic Poisson brackets are

$$\begin{equation} \lbrace A_a^i(x),E^b_j(y)\rbrace = 8\pi \gamma G \delta _a^b \delta _j^i \delta (x,y) . \end{equation} \tag{ 31 }$$

The Barbero–Immirzi parameter γ > 0 [44, 45] does not play a role classically but appears in quantum spectra and corrections. In these variables, the Hamiltonian constraint is

$$\begin{equation} \fl H_{\rm grav}\big(A_a^i,E^b_j\big) = -\frac{E^a_iE^b_j\epsilon ^{ij}{}_k}{16\pi G \sqrt{|\det (E^c_l)|}} \big(F_{ab}^k+(1+\gamma ^{-2}) \epsilon ^k{}_{mn} \big(A_a^m-\Gamma _a^m\big)\big(A_b^n-\Gamma _b^n\big)\big) \end{equation} \tag{ 32 }$$

with the curvature F^k_ab of the Ashtekar–Barbero connection.

Both expressions are rather complicated to quantize, owing for instance to the expressions of ⁽³⁾R or Γⁱ_a, to be written in terms of the metric or densitized-triad operators. Fortunately, there are several characteristic features, common to both versions: the Hamiltonian constraint is quadratic in the connection or extrinsic curvature, and it requires an inverse of the spatial metric or the densitized triad. These two features, when combined with quantum-representation properties, imply characteristic structures of quantum geometry and associated corrections to classical equations. These corrections, in turn, can be analyzed for potential implications even if their precise form cannot be determined from a complete calculation of $\langle \hat{H}\rangle$ in semiclassical states. And even if expectation values could be computed for a specific $\hat{H}$ and some states, ambiguities in the construction of $\hat{H}$ or the choice of states would be so severe that only general features and characteristic effects could be trusted.

In a Wheeler–DeWitt quantization, we have formal quantum constraint equations with π_ab in (30) replaced by functional derivatives with respect to q_ab, acting on wave functions ψ[q_ab]. The formal nature leaves precise representation properties unclear, and therefore does not show specific quantum effects; one simply takes the classical expression and performs the usual substitution of momenta by derivative operators. One does not expect strong quantum-geometry effects, simply because quantum geometry has not been completed in this setting. Loop quantum gravity, on the other hand, has celebrated its greatest success so far at this level of quantum representations, and indeed sheds considerable light on questions of quantum corrections resulting from quantum geometry.

Loop quantum gravity looks closely at representation properties of basic operators. To eliminate the need for formal functional derivatives and the associated delta functions in quantizations of (31), the classical fields Aⁱ_a and E^b_j are smeared or integrated over suitable sets in space: holonomies $h_e(A_a^i)= \mathcal{P}\exp (\int _e {\rm d}\lambda t_e^a A^i_a \tau _i)$ and fluxes F^(f)_S(E^b_j) = ∫_Sd²yn^S_aE^a_ifⁱ with the tangent vector t^a_e to a curve e and the co-normal n^S_a to a surface S (on which an su(2)-valued smearing function fⁱ is chosen). Allowing for all curves and surfaces, all information about the fields can be recovered. By the integrations, the delta function in {Aⁱ_a(x), E^b_j(y)} = 8πγGδ^b_aδⁱ_jδ(x, y) is eliminated, and a well-defined holonomy-flux algebra results:

$$\begin{equation} \lbrace h_e(A_a^i),F_S(E^b_j)\rbrace = 8\pi \gamma G h_{e\rightarrow x}\tau _if^i(x) h_{x\rightarrow e} \end{equation} \tag{ 33 }$$

if there is only one intersection point {x} = e∩S, denoting by h_{e → x} and h_{x → e} the holonomy along e up to x and starting at x, respectively. With strong uniqueness properties of possible quantum representations [46–49], spatial quantum geometry in this setting is under excellent control.

Representations of the holonomy-flux algebra then provide operators to be inserted in (32) to obtain a quantized Hamiltonian constraint. The kinematical Hilbert space is spanned by cylindrical states $\Psi (A_a^i)= \psi (h_{e_1}(A_a^i),\ldots , h_{e_n}(A_a^i))$, each of which depends on the connection via a finite number of holonomies. The full state space, has no restriction on the number of holonomies that may appear, thereby representing the continuum theory rather than some lattice model. The inner product is obtained by integrating the product of two cylindrical functions over as many copies of SU(2) as there are non-trivial dependencies on holonomies in both states, using the normalized Haar measure. By completion of the space of cylindrical functions, the kinematical Hilbert space is obtained [50]. Holonomies $\hat{h}_e$ then act as multiplication operators, changing the dependence of a cylindrical state on h_e, or creating a new dependence if the state was independent of h_e before acting. Flux operators, representing the densitized triad, become derivative operators in the connection representation used, and can be expressed in terms of invariant derivative operators on SU(2) (or angular-momentum operators). They have discrete spectra, indicating modifications to the classical spatial structure [51–53]: distance, areas or volumes computed from E^a_i can, after quantization, no longer increase continuously even if the underlying curves, surfaces, or regions are deformed by homotopies.

As a further consequence of discreteness, it turns out that holonomy operators do not continuously depend on the curves used. A cylindrical state depending only on h_e and one depending only on $h_{e\circ e^{\prime }}$ with a piece e' appended to the curve refer to two different holonomies, and are orthogonal according to the inner product just described, even if e is a one-point set. While we classically have $t_{e^{\prime }}^a A_a^i(x)\tau _i= \lim _{e^{\prime }\rightarrow \lbrace x\rbrace } (h_{e^{\prime }}(A_a^i)-1)/|e^{\prime }|$, where $t_{e^{\prime }}^a$ is the tangent vector of e' at x and |e'| its coordinate length, the sequence $\hat{h}_{e^{\prime }}|\psi \rangle$ of states after quantization does not converge for any |ψ〉 if e' is changed. The edge dependence disappears when one implements the diffeomorphism constraint, which has not been assumed in the previous constructions. States $\hat{h}_{e^{\prime }}|\psi \rangle$ for different e' are no longer orthogonal, but they all give rise to the same state when diffeomorphism are factored out. The limit can then be taken, but always equals zero. Again, there is no way of deriving a connection operator.

Unlike with classical expressions, it is not possible to take a derivative of h_e(Aⁱ_a), for instance by the endpoint of e, to obtain an expression for a quantized Aⁱ_a. Loop quantum gravity does not offer connection operators; all connection dependence in the Hamiltonian constraint must be expressed in terms of holonomies. No quadratic function as it appears in the constraint can exactly agree with a linear combination of exponentials, and modifications arise, motivated by background-independent quantum geometry.

When the Hamiltonian constraint is quantized in loop quantum gravity, holonomies are used instead of connection components [35], providing a 'regularization' necessary to render the constraint expressible by basic operators. However, the limit in which the 'regulator'—the specific curves used for holonomies—is removed does not exist; we are not dealing with a proper regularization. (In [36], the limit is argued to exist and be trivial if spatially diffeomorphism-invariant states are used. But with this assumption there is no handle on the full off-shell algebra and its anomaly problem.) Instead, one usually interprets the difference of a holonomy-modified constraint or its effective versions with the classical expression as a series of higher-order corrections, amending the classical Hamiltonian by higher powers of the connection, or intrinsic and extrinsic curvature. In this viewpoint, the modification is similar to expected higher-curvature corrections—except for the issue of covariance that we will have to address. (In some models of loop quantum cosmology, it is possible to represent an exponentiated version of the constraint in terms of holonomy operators without introducing modifications to the classical expression [54]. However, the procedure seems to depend sensitively on specific properties of the model used and is not available in general.)

Holonomy corrections therefore appear as higher-order terms such as

$$\begin{equation} \fl {-}2 {\,{\rm tr}}\,\big(\tau _ih_e\big(A_b^j\big)\big)-t_e^aA_a^i(x)= -\case{1}{24}(t_e^aA_a^i(x)(t_e^bA_b^j(x))(t_e^cA_{cj}(x))+ t_e^at_e^b \partial _bA_a^i(x)+\cdots \end{equation} \tag{ 34 }$$

obtained from a Taylor expansion of the exponential and a derivative expansion of the integration. These terms depend on the routing of the curve e, to be chosen suitably for a quantization of the Hamiltonian constraint. The condition of anomaly-freedom puts strong restrictions on the possible routings [55] which, however, are difficult to evaluate. Moreover, the expansion is done in an effective derivation, requiring the calculation of expectation values that lead to additional moment terms. All these calculations are difficult to perform explicitly, but the form of corrections is clear and quite characteristic: higher orders as well as higher spatial derivatives in the connection. With these types of corrections, suitably parameterized, one can look for consistent deformations of the constraint algebra to find versions for a physical evaluation of these corrections, or to derive further restrictions on the quantization choices made.

In effective equations, the use of holonomies as basic operators of the quantum theory has another consequence: the holonomy-flux algebra (33) is not canonical, with non-constant Poisson brackets. Moments of states used for effective descriptions are based on expectation values of basic operators, holonomies and fluxes in loop-quantized models. The general constructions of effective theories still go through: Poisson brackets on the quantum phase space, the quantum Hamiltonian or constraints, and so on. However, the explicit Poisson relations (24) evaluated between individual moments of the form Δ(h^bF^c) are different from the canonical case. In particular, moments no longer Poisson-commute with expectation values. This feature requires care and complicates some calculations. These problems, however, are only technical; see the examples provided later in section 4.3.2 and in [56, 57].

Fluxes are linear in the densitized triad and do not suggest the same kind of modification as holonomies. Nevertheless, there is a characteristic effect associated also with them. Flux operators have discrete spectra, containing zero as an eigenvalue. Such operators do not have densely-defined inverses, and yet we need an inverse of the densitized triad for the classical Hamiltonian constraint. To obtain such quantizations, a more indirect route is taken in loop quantum gravity, which does result in well-defined operators but introduces another kind of correction, called inverse-triad correction.

To see the form of these corrections, we show a lattice calculation of inverse-triad operators. As proposed in [36, 58], the combination of triad components required for the Hamiltonian constraint (32) is first rewritten as

$$\begin{equation} 2\pi \gamma G \epsilon ^{ijk}\epsilon _{abc} \frac{E^b_jE^c_k}{{\sqrt{|\det E|}}}=\bigg\lbrace A_a^i,\int {\sqrt{|\det E|}}\,\mathrm{d}^3x\bigg\rbrace . \end{equation} \tag{ 35 }$$

In the new form, no inverse is needed, Aⁱ_a can be expressed by holonomies, the volume operator can be used for $\int {\sqrt{|\det E|}}\,\mathrm{d}^3x$, and the Poisson bracket be turned into a commutator divided by iℏ. The resulting operators are rather contrived, especially with SU(2) holonomies and derivatives involved. But the presence of corrections and their qualitative form can be illustrated by a U(1)-calculation, for which we also assume regular cubic lattices.

On a cubic lattice, we can assign a unique plaquette to each link e, by which we then label fluxes. Our basic operators are $\hat{h}_e$, which as a multiplication operator by exp (i∫_eA_at^a_edλ) takes values in U(1), and $\hat{F}_e$ for a fixed set of edges e in a regular cubic lattice. We are therefore computing inverse-triad operators and their expectation values for a fixed subset of cylindrical states, but the lattice can be as fine as we want, allowing us to capture all continuum degrees of freedom. For this U(1)-simplification, the holonomy-flux algebra, quantizing an Abelian version of (33), reads $[\hat{h}_e,\hat{F}_e]= -8\pi \gamma \ell _{\rm P}^2\hat{h}_e$ while all operators commute if they belong to different edges. Moreover, the U(1)-valuedness of holonomies implies the reality condition $\hat{h}_e\hat{h}_e^{\dagger }=1$, which we will make use of below.

We first rewrite (35) in terms of holonomies instead of connection components, and express the volume $V=\int \sqrt{|\det E|}{\rm d}^3x$ by lattice fluxes $\sqrt{|F_1F_2F_3|}$ per vertex, with the three fluxes through plaquettes in all three directions around the vertex: $t_e^a\lbrace A_a,V\rbrace ={\rm i}h_e\lbrace h_e^{-1},\sqrt{|F_1F_2F_3|}\rbrace$ or, more symmetrically, $\frac{1}{2}{\rm i}(h_e\lbrace h_e^{-1},\sqrt{|F_1F_2F_3|}\rbrace - h_e^{-1}\lbrace h_e,\sqrt{|F_1F_2F_3|}\rbrace )$, computes the Poisson bracket at the vertex. Since h_e commutes with all but one of the F_I, we can focus on one of them, $\sqrt{|\hat{F}_e|}$. Keeping the power more general, the quantization of some inverse power of flux takes the form

$$\begin{equation} \widehat{(|F|^{r-1} {\,{\rm sgn}}\,F)_e}= \frac{\hat{h}_e^{\dagger }|\hat{F}_e|^r\hat{h}_e- \hat{h}_e|\hat{F}_e|^r\hat{h}_e^{\dagger }}{16\pi Gr\gamma \ell _{\,{\rm P}}\,^2} =: \hat{I}_e. \end{equation} \tag{ 36 }$$

For any 0 < r < 1 we quantize an inverse power of F but need not use any inverse in the commutator.

Following [59], we can now easily simplify these operators, if we observe the relations of the U(1)-holonomy-flux algebra, together with the reality condition. These relations imply

$$\begin{equation*} \hat{h}_e^{\dagger }|\hat{F}_e|^r\hat{h}_e= |\hat{F}_e+8\pi \gamma \ell _{\,{\rm P}}\,^2|^r ,\quad \hat{h}_e|\hat{F}_e|^r\hat{h}_e^{\dagger }= |\hat{F}_e-8\pi \gamma \ell _{\,{\rm P}}\,^2|^r , \end{equation*}$$

such that

$$\begin{equation} \hat{I}_e= {\frac{|\hat{F}_e+8\pi \gamma \ell _{\,{\rm P}}\,^2|^r- |\hat{F}_e-8\pi \gamma \ell _{\,{\rm P}}\,^2|^r}{16\pi Gr\gamma \ell _{\,{\rm P}}\,^2}}. \end{equation} \tag{ 37 }$$

Eigenvalues of this operator can easily be computed, with eigenstates equal to flux eigenstates [60, 61]. All eigenvalues are finite, as required for a densely-defined operator, and show how the classical divergence of |F|^{r − 1} at F = 0 is cut off. For inverse-triad corrections in effective Hamiltonians, however, we need expectation values of $\hat{I}_e$ in semiclassical states. Explicit calculations would require good knowledge of semiclassical wave functions or coherent states.

For general effective equations it is sufficient, and even more useful, to perform a moment expansion, keeping the specific state free and parameterized by moments. Staying at the expectation-value order of effective expressions, we have

$$\begin{equation} \langle \hat{I}_e\rangle = {\frac{|\langle \hat{F}_e\rangle +8\pi \gamma \ell _{\,{\rm P}}\,^2|^r- |\langle \hat{F}_e\rangle -8\pi \gamma \ell _{\,{\rm P}}\,^2|^r}{16\pi Gr\gamma \ell _{\,{\rm P}}\,^2}}+ \mbox{moment terms}. \end{equation} \tag{ 38 }$$

Already to this order we see characteristic corrections (depending on ℏ via the Planck length). Inverse-triad corrections therefore have a contribution independent of quantum back-reaction.

Interpreting $\langle \hat{F}\rangle =:L^2$ as the discrete quantum-gravity scale (the lattice spacing as measured by flux operators), we find the correction function

$$\begin{equation} \alpha _r(L):= \frac{\langle \hat{I}\rangle }{I_{\,{\rm class}}}= \frac{|L^2+8\pi \gamma \ell _{\,{\rm P}}\,^2|^r- |L^2-8\pi \gamma \ell _{\,{\rm P}}\,^2|^r}{16\pi \gamma r\ell _{\,{\rm P}}\,^2}L^{2(1-r)} \end{equation} \tag{ 39 }$$

that will appear in an effective Hamiltonian constraint. To leading order in an expansion by ℏ (or ℓ²_P/L²), the correction function equals one. But even if no moment terms are included, there are quantum corrections in the full form of α(L). Corrections are strong for L² ∼ 8πγℓ²_P or smaller, typically in the deep quantum regime, where α(L) drops to zero at L = 0. However, even for larger L, α_r(L) is not identical to one and implies interesting corrections.

In addition to corrections contained in (39) and quantum back-reaction from moment terms, the flux dependence implies corrections from a derivative expansion of the integrations involved, as already seen for holonomies. Moreover, non-Abelian holonomies do not lead to exact cancellations in the substitution of h_e{h⁻¹_e, V} for t^a_e{A_aⁱ, V} and rather imply additional higher-order corrections by powers of Aⁱ_a [62]. As noted in the context of holonomy corrections, such extra terms mix with higher-curvature corrections. The leading term in (39), on the other hand, shows a different dependence on parameters that distinguish a given cosmological regime and are more characteristic. Their effects can thus be studied in isolation.

The quantum potential is defined as a function on the infinite-dimensional quantum phase space of expectation values and moments, whose Poisson structure is given by (24). In a quantum Hamiltonian

$$\begin{equation} H_Q=\langle \hat{H}\rangle = \frac{1}{2m}(\langle \hat{p}\rangle ^2+ \Delta (p^2))+ V_Q(\langle \hat{q}\rangle , \Delta (q^n)), \end{equation} \tag{ 43 }$$

we therefore have terms generating a dynamical flow of the moments by

$$\begin{equation} \dot{\Delta }(q^bp^c)= \lbrace \Delta (q^bp^c),H_Q\rbrace . \end{equation} \tag{ 44 }$$

The coupled set of equations for expectation values and moments, (40) and (44), is equivalent to the Schrödinger flow of quantum mechanics, but its solutions do not provide wave functions but rather variables directly related to observations. It can be solved with different approximations, most importantly a semiclassical one by the order of moments, sometimes combined with an adiabatic one. In the latter case, applied to anharmonic oscillators, effective equations are equivalent to those of the low-energy effective action [30]. The validity and usefulness of the canonical effective scheme is thereby established.

In the context of quantum gravity, the feature of higher time derivatives in effective equations, a crucial ingredient of higher-curvature corrections, is of particular interest. The moments are related to such terms, although not in a direct way. Equation (40) combined with (41) already shows that a specific linear combination of the moments, with coefficients depending on expectation values, amounts to the time derivative of the momentum, or the second derivative of the position expectation value. Higher than second time derivatives of $\langle \hat{q}\rangle$ can be computed by taking further derivatives of (40) and inserting $\dot{\Delta }(q^n)$ and, for higher than third order, $\dot{\Delta }(q^bp^c)$ according to equations of motion (44) generated by the quantum Hamiltonian. Different combinations of the moments therefore provide all higher time derivatives of $\langle \hat{q}\rangle$. With the scheme just sketched, it is difficult to invert the equations to find expressions for moments in terms of higher time derivatives, or to eliminate all moments and end up with a higher-derivative equation just for $\langle \hat{q}\rangle$ instead of the moment-coupled (40), (41) and (44). But with more-refined methods, as well as an adiabatic expansion, this task can be performed. For quantum cosmology, we learn that we must study quantum back-reaction to see all terms relevant for higher-curvature corrections.

In semiclassical regimes, the moments by definition obey the ℏierarchy Δ(q^bp^c) ∼ O(ℏ^{(b + c)/2}), as can easily be verified in Gaussians; see (17) and [25]. We can therefore consider the first term $\frac{1}{2} V^{\prime \prime }(\langle \hat{q}\rangle ) (\Delta q)^2$ for n = 1 in (42) as the leading semiclassical correction, providing a quantum Hamiltonian

$$\begin{equation} H_Q=\langle \hat{H}\rangle = \frac{1}{2m}\langle \hat{p}\rangle ^2+ V(\langle \hat{q}\rangle )+\frac{1}{2m} (\Delta p)^2+ \frac{1}{2} V^{\prime \prime }(\langle \hat{q}\rangle ) (\Delta q)^2. \end{equation} \tag{ 45 }$$

(The kinetic term contributes (Δp)²/2m, potentially of the same order as $\frac{1}{2}V^{\prime \prime }(\langle \hat{q}\rangle )(\Delta q)^2$. But it does not appear in a product with expectation values and therefore does not cause quantum back-reaction.) For equations of motion of expectation values and second-order moments, relevant to this order, we use the Poisson brackets (26). Applied to our second-order quantum Hamiltonian, we find

$$\begin{eqnarray} \frac{\,{\rm d}\, \langle \hat{q}\rangle }{\,{\rm d}\,t} &=& \frac{\langle \hat{p}\rangle }{m} \end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray} \frac{\,{\rm d}\, \langle \hat{p}\rangle }{\,{\rm d}\,t} &=& -V^{\prime }(\langle \hat{q}\rangle )- \frac{1}{2}V^{\prime \prime \prime }(\langle \hat{q}\rangle ) (\Delta q)^2 \end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray} \frac{\,{\rm d}\,(\Delta q)^2}{\,{\rm d}\,t} &=& \frac{2}{m}\Delta (qp) \end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray} \frac{\,{\rm d}\,\Delta (qp)}{\,{\rm d}\,t} &=& \frac{1}{m} (\Delta p)^2- V^{\prime \prime }(\langle \hat{q}\rangle ) (\Delta q)^2 \end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray} \frac{\,{\rm d}\,(\Delta p)^2}{\,{\rm d}\,t} &=& -2V^{\prime \prime }(\langle \hat{q}\rangle )\Delta (qp). \end{eqnarray} \tag{ 50 }$$

For a given potential, one may solve these equations numerically. However, it would be more instructive to compute (Δq)² and insert it in (47) to see what quantum corrections result. So far, all equations are coupled to one another and one cannot solve independently for (Δq)² (unless V'' is constant, the case of the harmonic oscillator, a constant force or a free particle). But with an additional adiabatic approximation for the moments, decoupling can be achieved.

To zeroth order in an adiabatic approximation, we assume the moments (but not expectation values) to be time independent. We will denote the adiabatic order by an integer subscript. Equations (48) and (50) then imply Δ₀(qp) = 0 at zeroth adiabatic order, and (49) shows that $(\Delta _0 p)^2=mV^{\prime \prime }(\langle \hat{q}\rangle ) (\Delta _0 q)^2$. With the last equation, we see that the zeroth-order adiabatic approximation cannot be valid unless $\langle \hat{q}\rangle$ is constant in time as well. To avoid such a restrictive condition, we proceed to higher adiabatic orders, from order i to order i + 1 by inserting time derivatives of Δ_i(q^bp^c) on the left-hand sides of (48)–(50) to compute Δ_{i + 1}(q^bp^c) on the right-hand sides. (For a systematic implementation of the adiabatic approximation, see [30, 70].) With time derivatives known from preceding orders, the equations to solve for the moments are initially algebraic, but additional consistency conditions relating different orders sometimes imply differential equations for coefficients, as we will see in this example.

To first adiabatic order,

$$\begin{eqnarray} &&\Delta _1(qp)=\frac{1}{2}m \frac{\,{\rm d}\,(\Delta _0 q)^2}{\,{\rm d}\,t} = -\frac{1}{2V^{\prime \prime }(\langle \hat{q}\rangle )} \frac{\,{\rm d}\,(\Delta _0p)^2}{\,{\rm d}\,t} \end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray} &&\hphantom{\Delta _1(qp)}= -\frac{1}{2}m\left( \frac{V^{\prime \prime \prime }(\langle \hat{q}\rangle )}{V^{\prime \prime }(\langle \hat{q}\rangle )} \frac{{\rm d}\langle \hat{q}\rangle }{\,{\rm d}\,t} (\Delta _0q)^2+ \frac{\,{\rm d}\, (\Delta _0q)^2}{\,{\rm d}\,t}\right) \end{eqnarray} \tag{ 52 }$$

using (48), (50) and our zeroth-order condition relating (Δ₀p)² to (Δ₀q)². The two lines can both hold only if

$$\begin{equation} \frac\,{{\rm d}\,(\Delta _0q)^2}{\,{\rm d}\,t} =- \frac{1}{2} \frac{V^{\prime \prime \prime }(\langle \hat{q}\rangle )}{V^{\prime \prime }(\langle \hat{q}\rangle )} \frac{\,{\rm d}\,\langle \hat{q}\rangle }{\,{\rm d}\,t} (\Delta _0q)^2, \end{equation} \tag{ 53 }$$

solved by

$$\begin{equation} (\Delta _0q)^2= \frac{C}{\sqrt{V^{\prime \prime }(\langle \hat{q}\rangle )}} \end{equation} \tag{ 54 }$$

with a constant C. Our zeroth-order adiabatic relation between the moments then shows that $(\Delta _0p)^2= mV^{\prime \prime }(\langle \hat{q}\rangle ) (\Delta _0q)^2= mC \sqrt{V^{\prime \prime }(\langle \hat{q}\rangle )}$. Inserting these solutions in the quantum Hamiltonian (45), we obtain a correction

$$\begin{equation} \frac{1}{2m}(\Delta _0p)^2+ \frac{1}{2}V^{\prime \prime }(\langle \hat{q}\rangle ) (\Delta _0q)^2= C\sqrt{V^{\prime \prime }(\langle \hat{q}\rangle )} \end{equation} \tag{ 55 }$$

to the classical Hamiltonian.

As one goes to higher orders in the adiabatic approximation, one takes more and more time derivatives of Δ₀(q^bp^c). We can see this feature already with the low-order equations found here. So far, we have used the first adiabatic order only to restrict the zeroth-order solutions. But with the solution (54) found for (Δ₀q)², we obtain from (51) the moment

$$\begin{equation} \Delta _1(qp)= \frac{1}{2}m\frac{\,{\rm d}\,(\Delta _0q)^2}{\,{\rm d}\,t}= -\frac{1}{4} Cm \frac{V^{\prime \prime \prime }(\langle \hat{q}\rangle )}{V^{\prime \prime }(\langle \hat{q}\rangle )^{3/2}} \frac{\,{\rm d}\langle \hat{q}\rangle }{\,{\rm d}\,t}, \end{equation} \tag{ 56 }$$

depending on a first-order derivative of $\langle \hat{q}\rangle$. We do not need Δ(qp) in the quantum Hamiltonian, but this pattern continues for all moments at higher adiabatic orders, including Δ_i(qⁿ). When we go beyond second adiabatic order and insert solutions into expectation-value equations, higher-derivative effective equations will be obtained; see [70] for explicit derivations. Quantum back-reaction by moments is responsible for these higher-derivative corrections, but there is no direct correspondence between the moments as independent quantum degrees of freedom and new degrees of freedom that appear in higher-derivative equations because more initial values need to be specified. It is not the moment expansion itself which gives rise to higher derivatives, but rather the adiabatic expansion of individual moments. The order of moments corresponds to a semiclassical expansion, according to Δ(q^bp^c) ∼ O(ℏ^{(b + c)/2}) in semiclassical states, not to a derivative expansion. Any fixed order in ℏ can produce arbitrarily high orders of time derivatives if the adiabatic expansion is pushed further.

The parameter C in (54), related to second-order moments, is of the order ℏ in semiclassical states; the correction (55) is therefore the first-order semiclassical correction under the assumption of zeroth adiabatic order for the moments. We cannot choose arbitrary values for C because the uncertainty relation (18) must be obeyed, such that $C=m^{-1/2} \Delta _0q\Delta _0p\ge \frac{1}{2}\hbar /\sqrt{m}$. Requiring the uncertainty relation to be saturated determines C. In general, this condition may be too strong because we would assume saturation at all times, amounting to the existence of a dynamical coherent state which is not guaranteed for general potentials. But to zeroth adiabatic order, with the solutions found here, such an assumption is consistent: all dependence on $\langle \hat{q}\rangle$ drops out in the product of (Δ₀q)²(Δ₀p)² (and we have Δ₀(qp) = 0).

Without additional assumptions on the states solved for, or initial conditions for the moment equations (48)–(50), the constant C remains undetermined. One possibility to fix C, in the class of models of this example, is to assume that solutions are close to the harmonic-oscillator vacuum or some other specific state. If the potential $V(q)=\frac{1}{2}m\omega ^2q^2$ is harmonic, $(\Delta _0q)^2= C/\sqrt{m\omega ^2}$ is constant—in this case there are states for which the adiabatic approximation is exact—and equals the Gaussian spread σ² in a coherent state: we may write $C=\sigma ^2\sqrt{m\omega ^2}$. For the harmonic oscillator, dynamical coherent states do exist and the uncertainty relation may be satisfied at all times. In this case, $C=\frac{1}{2}\hbar /\sqrt{m}$, or $(\Delta _0q)^2= \frac{1}{2}\hbar /m\omega$, the correct relation for position fluctuations in the ground state. With $(\Delta _0p)^2=\frac{1}{2}m\hbar \omega$, the non-classical terms in the quantum Hamiltonian amount to the zero-point energy $\frac{1}{2}\hbar \omega$.

For a general potential V, we do not have the frequency parameter ω to refer to, but we can define it as the square root of 2/m times the coefficient of the quadratic term in a Taylor expansion $V(q)=V_0+ V_1q+\frac{1}{2}m\omega ^2q^2+\cdots$, assuming that the coefficient is not zero. In this way, we treat higher than second-order terms in the potential as an anharmonicity. Specifying the class of states solved for as those that are close to a harmonic-oscillator ground state, we can therefore write $(\Delta _0q)^2= \frac{1}{2}\hbar /\sqrt{mV^{\prime \prime }(\langle \hat{q}\rangle )}$. The correction $\frac{1}{2}\hbar \sqrt{V^{\prime \prime }(\langle \hat{q}\rangle )/m}$ in the effective Hamiltonian (55) then agrees with that found for the low-energy effective action [26], a relation that holds to higher adiabatic orders as well [30].

The canonical picture of quantum back-reaction provides an interpretation of moment-coupling terms as an analog of loop diagrams in quantum field theory, with moments taking the place of n-point functions. A formulation of the canonical effective scheme for quantum field theory is not fully worked out yet, but its implications for quantum gravity and cosmology can nevertheless be seen. Already in minisuperspace models there are characteristic effects which show cosmological implications of quantum corrections.

The WKB approximation is often seen as implementing a semiclassical regime, in the sense that leading terms in powers of ℏ are considered in the quantum evolution equation for states, expanded as ψ(q) = exp (iℏ⁻¹∑^∞_{n = 0}ℏⁿS_n(q)) with an asymptotic series. With this ansatz in the Schrödinger equation, one can solve order by order in ℏ to find expressions for the S_n: in quantum mechanics,

$$\begin{equation} \frac{1}{2m} \left(\frac{\,{\rm d}\,S_0}{\,{\rm d}q}\,\right)^2+ V(q)=E ,\quad \,{\rm i}\,\frac{\,{\rm d}\,^2S_0}{\,{\rm d}\,q^2}+ 2\frac{\,{\rm d}\,S_0}{\,{\rm d}\, q} \frac{\,{\rm d}\,S_1}{\,{\rm d}\,q}=0 \end{equation} \tag{ 57 }$$

for zeroth and first order in ℏ implies $S_1=-\frac{1}{4}{\rm i} \log (2m(E-V(q))+{\rm const}$, while S₀ satisfies the classical Hamilton–Jacobi equation.

Solutions obtained by the WKB approximation do not directly provide observables such as expectation values, for which additional integrations would be necessary. Such integrations are usually complicated to perform not just analytically but also numerically, given the strongly oscillating nature of WKB solutions in semiclassical regimes. Moreover, WKB solutions do not show how quantum corrections can be included in classical equations as the dominant quantum effects. In particular, although quantum back-reaction is implicitly contained in solutions to the WKB equations, it does not appear in the form of effective potentials or quantum forces useful for intuitive explanations of quantum effects. In the WKB approximation, S₀ satisfies exactly the classical Hamilton–Jacobi equation, without any quantum corrections. Corrections to the dynamics arise by higher orders of S_n in the wave function, but they do not appear in a form added to the Hamilton–Jacobi (or another classical) equation.

While the WKB approximation, as an expansion in ℏ, does have a semiclassical flavor, it can more generally be viewed as a formal expansion to produce solutions for wave functions. The WKB equations are obtained by solving the Schrödinger equation exactly at every order of ℏ: an equation ∑^∞_{n = 0}E_nℏⁿ = 0 is interpreted as implying E_n = 0 for all n. From a semiclassical perspective, on the other hand, one would interpret an equation ∑^∞_{n = 0}E_nℏⁿ = 0 as providing a tower of quantum corrections ∑^∞_{n = 1}E_nℏⁿ to the classical expression E₀, and then be interested in solutions to the equations ∑^N_{n = 0}E_nℏⁿ = 0 cut off at finite orders of ℏ. Additional consistency conditions are needed to determine the E_n showing up in quantum corrections. Usually, the E_n for n > 0 depend on state parameters such as fluctuations, while E₀ depends only on expectation values and equals the classical expression. A dynamical equation ∑^N_{n = 0}E_nℏⁿ = 0 then encodes the quantum back-reaction of state parameters on the expectation values, implying deviations from classical behavior, as derived systematically by effective equations.

In principle, one could derive such quantum corrections from WKB solutions by computing expectation values of the ℏ-expanded wave functions. But the WKB approximation does not automatically arrange the terms in its equation by semiclassical relevance. While canonical effective equations have a direct correspondence to the low-energy effective action, as already seen, the WKB approximation does not produce all terms [71]. Another question, important in the context of quantum gravity and quantum cosmology, is the treatment of quantum constraints (or the physical Hilbert space), which remains open in the context of WKB solutions. (For instance, one may solve $\hat{H}|\psi \rangle =0$ with WKB techniques, but for approximate solutions, the gauge flow $\exp (-{\rm i}\hat{H}[\epsilon ]/\hbar )|\psi \rangle _{\rm WKB}$ does not automatically vanish.) Canonical effective techniques, on the other hand, apply to constrained systems as well and even help to solve some long-standing conceptual problems of quantum gravity related to constraints and gauge.

As already indicated in section 3.1.1, a quantum constrained system with constraint operators $\hat{C}$ produces quantum constraints $C_Q:=\langle \hat{C}\rangle$, defined just like a quantum Hamiltonian (43), but also independent quantum phase-space functions $C_f=\langle (f(\hat{q},\hat{p})-\langle f(\hat{q},\hat{p})\rangle )\hat{C}\rangle$ (in this ordering) constrained to vanish in physical states [37, 38]. In semiclassical expansions, calculating order by order in the moments, polynomial f(q, p) are sufficient. To fixed order in the moments, only finitely many constraints are then present. Their number is larger than the number of classical constraints because they remove not only expectation values of constrained degrees of freedom but also the corresponding moments.

With the ordering of constraint operators to the right of f(q, p) chosen in effective constraints, they are automatically first class if the constraint operators are first class. There are then constraint equations to be solved, and gauge flows to be factored out. The gauge flow is computed using the Poisson brackets (24), affecting also the moments. Standard techniques of constrained systems can then be used, except that moments truncated to a fixed order usually define a non-symplectic Poisson manifold. This feature requires some care and may have several consequences, for instance that the number of independent gauge flows does not equal the number of first-class constraints. Nevertheless, the usual classification of constraints and gauge flows applies [31].

To see the treatment of effective constraints we consider a Hamiltonian constraint operator $\hat{C}=\hat{p}_{\phi }^2- \hat{p}^2+W(\hat{\phi })$ for a free, massless relativistic particle (q, p) coupled to a second degree of freedom (ϕ, p_ϕ) with an arbitrary ϕ-dependent potential W(ϕ). Depending on the form of W(ϕ), p_ϕ may become zero along trajectories generated by the Hamiltonian constraint, in which case ϕ does not serve as global internal time. On the other hand, with a q-independent Hamiltonian constraint, we could deparameterize by q, obtaining evolution by the classical Hamiltonian $p= \pm \sqrt{ p_{\phi }^2+W(\phi )}$. We have equations of motion ${\rm d}\phi /{\rm d}q= \pm p_{\phi }/\sqrt{ p_{\phi }^2+W(\phi )}$ and ${\rm d}p_{\phi }/{\rm d}q= \mp \frac{1}{2}W^{\prime }(\phi )/\sqrt{ p_{\phi }^2+W(\phi )}$. The momentum p_ϕ evolves, and could indeed become zero.

The constraint operator gives rise to the effective constraints [40, 41]

$$\begin{eqnarray} \fl C_Q&=&\langle \hat{p}_{\phi }\rangle ^2-\langle \hat{p}\rangle ^2+ (\Delta p_{\phi })^2-(\Delta p)^2+W(\langle \hat{\phi }\rangle )+ {\textstyle \frac{1}{2}}W^{\prime \prime }(\langle \hat{\phi }\rangle )(\Delta \phi )^2 \end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray} \fl C_{\phi }&=& 2\langle \hat{p}_{\phi }\rangle \Delta (\phi p_{\phi })+ {\,{\rm i}}\, \hbar \langle \hat{p}_{\phi }\rangle -2p\Delta (\phi p)+W^{\prime }(\langle \hat{\phi }\rangle ) (\Delta \phi )^2 \end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray} \fl C_{p_{\phi }}&=& 2\langle \hat{p}_{\phi }\rangle (\Delta p_{\phi })^2- 2\langle \hat{p}\rangle \Delta (p_{\phi }p)+ W^{\prime }(\langle \hat{\phi }\rangle )(\Delta (\phi p_{\phi })- {\textstyle \frac{1}{2}}{\,{\rm i}}\,\hbar ) \end{eqnarray} \tag{ 60 }$$

expanded to second order in the moments, together with additional constraints C_q, C_p, C_qp and so on, which we will not make use of. These constraints can be solved to find the quantum-corrected constraint surface, and their gauge flows can be computed to find moments of observables in physical states. Once the non-symplectic nature of the Poisson manifold of second-order moments is taken into account, these calculations are not very different from standard procedures.

The effective constraints shown here illustrate another important feature: the complexity of constraints and their solutions. It comes about because effective constraints, to be first class, are defined in a non-symmetric ordering, while moments are by definition Weyl ordered. Reorderings required to express effective constraints as functions of the moments then introduce imaginary contributions by the commutator $[\hat{q},\hat{p}]={\rm i}\hbar$. For C_ϕ and $C_{p_{\phi }}$ to vanish, some moments must be complex. While moments before the imposition of constraints, belonging to a kinematical Hilbert space, should be real as the expectation values of Weyl-ordered operators, after solving the constraints one moves to the physical Hilbert space, in general not related to a subspace of the kinematical one. After solving the constraints, the original kinematical moments may therefore take complex values, as long as physical observables of the quantum constrained system are subject to reality conditions.

For further consequences, we study the problem of time in this system, using the variable ϕ as internal time even though it does not deparameterize the system globally. At the full quantum level, local internal times, free of turning points only for finite ranges of evolution, cannot easily be made sense of: if internal time exists only for a finite range, evolution cannot be unitary even in this range. (See for instance the discussion in [72–74]. If states are evolved by local internal times past their turning points, evolution freezes: expectation values are stuck at constant values [75, 76].) This consequence is the reason why the problem of time is much more severe at the quantum level, compared to the classical one. At the effective level, as we will see, the problem of time can be overcome, allowing consistent derivations of observables without using artificial deparameterizations [40–42].

If ϕ is used as (local) internal time, it is not represented as an operator on the resulting physical Hilbert space, whatever it may be. No generally manageable techniques are known to derive physical Hilbert spaces and evolution in non-deparameterizable systems (for some possibilities of Hilbert-space derivations, see e.g. [75–78]). At the effective level, it is sufficient to distinguish ϕ as non-operator time by requiring that its moments in effective constraints vanish,

$$\begin{equation} (\Delta \phi )^2=\Delta (\phi q)=\Delta (\phi p)=0 \end{equation} \tag{ 61 }$$

while its expectation value 〈ϕ〉 (denoted without the hat to indicate that $\hat{\phi }$ no longer acts as an operator) will become the time parameter. In fact, the conditions (61) implement a good gauge fixing of the second-order constraints C_ϕ, $C_{p_{\phi }}$, C_q and C_p after quantization. (With constraints on a non-symplectic Poisson manifold, only three gauge-fixing conditions are required for four constraints. The remaining second-order moment involving ϕ, Δ(ϕp_ϕ), is fixed by the constraints, as we will see shortly.) In the terminology of [40], these conditions implement the Zeitgeist during which 〈ϕ〉 as local internal time is current. Imposing (61) initiates the transition to physical moments—moments computed for states in the physical Hilbert space on which ϕ does not act as an operator.

Solving the effective constraints in the given Zeitgeist, we have $\Delta (\phi p_{\phi })=-\frac{1}{2}{\rm i}\hbar$ from C_ϕ = 0, which then implies

$$\begin{equation*} (\Delta p_{\phi })^2= \frac{\langle \hat{p}\rangle ^2}{\langle \hat{p}_{\phi }\rangle ^2} (\Delta p)^2+ \frac{1}{2}{\,{\rm i}}\,\frac{W^{\prime }(\langle \phi \rangle )\hbar }{\langle \hat{p}_{\phi }\rangle } \end{equation*}$$

from $C_{p_{\phi }}=0$. Inserted in (58), this implies the reduced constraint

$$\begin{equation} C=\langle \hat{p}_{\phi }\rangle ^2-\langle \hat{p}\rangle ^2+ \frac{\langle \hat{p}\rangle ^2- \langle \hat{p}_{\phi }\rangle ^2}{\langle \hat{p}_{\phi }\rangle ^2}(\Delta p)^2+ \frac{1}{2}{\,{\rm i}}\,\frac{W^{\prime }(\langle \phi \rangle )\hbar }{\langle \hat{p}_{\phi }\rangle } + W(\langle \phi \rangle ) \end{equation} \tag{ 62 }$$

amounting to the quantum constraint $C_Q=\langle \hat{C}\rangle$ on the space on which C_ϕ and $C_{p_{\phi }}$ are solved in the given Zeitgeist. Solving C = 0 for $\langle \hat{p}_{\phi }\rangle$, we obtain the time-dependent Hamiltonian for 〈ϕ〉-evolution, including quantum back-reaction. However, it still contains complex terms.

In (62), all terms except the last two are expected to be real-valued because $\langle \hat{p}\rangle$ and Δp are physical observables, and $\langle \hat{p}_{\phi }\rangle$ can be interpreted physically as the local energy value. The constraint can then be satisfied, only if we allow for an imaginary part of 〈ϕ〉, calculated from

$$\begin{equation} \frac{1}{2}{\,{\rm i}}\,\frac{W^{\prime }(\langle \phi \rangle )\hbar }{\langle \hat{p}_{\phi }\rangle }+ W(\langle \phi \rangle )=0. \end{equation} \tag{ 63 }$$

For semiclassical states, to which this approximation of effective constraints refers, we can Taylor expand the potential

$$\begin{equation*} \fl W(\langle \phi \rangle )=W({\rm Re}\langle \phi \rangle +{\rm i} {\rm Im}\langle \phi \rangle )= W({\rm Re}\langle \phi \rangle )+{\rm i} {\rm Im}\langle \phi \rangle W^{\prime }({\rm Re}\langle \phi \rangle )+ O(({\rm Im}\langle \phi \rangle )^2) \end{equation*}$$

by the imaginary term, expected to be at least of the order ℏ because it vanishes classically. To this order, the imaginary contribution ImC = 0 to C in (62) implies that

$$\begin{equation} {\,{\rm Im}}\,\langle \phi \rangle = -\frac{\hbar }{2\langle \hat{p}_{\phi }\rangle }. \end{equation} \tag{ 64 }$$

The remaining terms,

$$\begin{equation} {\,{\rm Re}}\,C= \langle \hat{p}_{\phi }\rangle ^2-\langle \hat{p}\rangle ^2+ \frac{\langle \hat{p}\rangle ^2- \langle \hat{p}_{\phi }\rangle ^2}{\langle \hat{p}_{\phi }\rangle ^2}(\Delta p)^2 + W({\,{\rm Re}}\,\langle \phi \rangle )=0 \end{equation} \tag{ 65 }$$

provide the physical Re〈ϕ〉-Hamiltonian upon solving the constraint equation for $\langle \hat{p}_{\phi }\rangle$. At this stage, the Hamiltonian and its solutions, corresponding to evolving observables with respect to ϕ, are all real: physical reality conditions are imposed and we have solutions corresponding to states in the physical Hilbert space.

Although imaginary parts may be unexpected, a detailed analysis of this and other models shows that they are fully consistent [41]. In models in which one can compute a physical Hilbert space, results equivalent with those shown here are obtained. Within the effective treatment of constraints, if we transform to a different internal time such as q, which is done by a gauge transformation in the effective constrained system so that a new Zeitgeist—the gauge-fixing (61)—is realized, the imaginary parts are automatically transferred from 〈ϕ〉 to 〈q〉, in such a way that observables remain real. By successive gauge transformations, one can evolve through turning points of local internal times, without freezing the evolution of physical observables; see in particular the cosmological example analyzed in [42]. The imaginary part of time can be seen as a remnant of non-unitarity problems of evolution in local-time quantum systems, but unlike in Hilbert-space treatments, it does not pose any problems at the effective level.

Gauge transformations in effective constrained systems show that physical results are independent of the choice of (local) internal time. One may deparameterize the effective system in different ways to solve the resulting equations, without affecting observables. This conclusion, one example for effective solutions to the traditional problems of canonical quantum gravity, indicates that deparameterization can be used consistently. However, in complicated systems subject to ambiguities such as factor-ordering choices, each deparameterization must be formulated in a specific way so that they all can result from one non-deparameterized system, effective or not. At the effective level, all quantum constraints and Zeitgeists must be computed and implemented with the same operator $\hat{C}$ for different local time choices to produce mutually consistent results. In many constructions of physical Hilbert spaces, however, one quantizes a system with a specific deparameterization in mind, choosing factor orderings and using possible simplifications. In such a case, there is no guarantee that results can agree with those obtained from other parameterizations, and the independence of physical results of the choice of time is put at risk.

Effective deparameterized equations are generated by the quantum Hamiltonian $\langle p_{\phi }(\hat{a},\hat{p}_a)\rangle$,

$$\begin{equation} \frac{\,{\rm d}\,\langle \hat{O}\rangle }{\,{\rm d}\,\phi }= \frac{\langle [\hat{O},p_{\phi }(\hat{a},\hat{p}_a)]\rangle }{\,{\rm i}\,\hbar }= \lbrace \langle \hat{O}\rangle ,\langle p_{\phi }(\hat{a},\hat{p}_a)\rangle \rbrace \end{equation} \tag{ 75 }$$

using the Poisson brackets (24). The absolute value in p_ϕ(a, p_a) makes a completely general expansion in moments complicated, but for |p_ϕ| not close to zero, there is a simple effective Hamiltonian. If we can ensure positivity of $\widehat{ap_a}$ in evolved states, the absolute value can be dropped. This is possible in particular for an initial state supported solely on the positive part of the spectrum of $\widehat{ap_a}$ (an operator for which we will assume Weyl ordering). Since ap_a is preserved by the motion it generates, also after quantization, the evolved state will remain supported on the positive part of the spectrum of $\widehat{ap_a}$. Unless |p_ϕ| is close to zero, it is easy to find initial states supported only on the positive part of the spectrum of $\widehat{ap_a}$ and with specified initial expectation values for $\hat{a}$ and $\hat{p}_a$: projecting out negative contributions will not change the basic expectation values much. We are then allowed to write (75) as

$$\begin{eqnarray} &&\frac{\,{\rm d}\,_+\langle \hat{O}\rangle _+}{\,{\rm d}\,\phi } = \pm \sqrt{\frac{4\pi G}{3}} \frac{_+\langle [\hat{O},\widehat{|ap_a|}]\rangle _+}{\,{\rm i}\,\hbar }= \pm \sqrt{\frac{4\pi G}{3}} \frac{_+\langle [\hat{O},\widehat{ap_a}]\rangle _+}{\,{\rm i}\,\hbar }\nonumber \\ &&{\hphantom{\frac{\,{\rm d}\,_+\langle \hat{O}\rangle _+}{\,{\rm d}\,\phi }}}= \pm \sqrt{\frac{4\pi G}{3}} \lbrace _+\langle \hat{O}\rangle _+, _+\langle \widehat{ap_a}\rangle _+\rbrace \end{eqnarray} \tag{ 76 }$$

using $\widehat{|ap_a|}|\psi \rangle _+= \widehat{ap_a}|\psi \rangle _+$ (and $\widehat{|ap_a|}^{\dagger }|\psi \rangle _+= \widehat{ap_a}^{\dagger }|\psi \rangle _+$) on states |ψ〉₊ with support only on the positive part of the spectrum of $\widehat{ap_a}$.

On such positively supported states, the ϕ-Hamiltonian is quadratic and can easily be expanded in moments. We have the quantum Hamiltonian

$$\begin{equation} H_Q= \pm \sqrt{\frac{4\pi G}{3}} (\langle \hat{a}\rangle \langle \hat{p}_a\rangle +\Delta (ap_a)), \end{equation} \tag{ 77 }$$

free of coupling terms of expectation values and moments: there is no quantum back-reaction. We compute and solve equations of motion for expectation values, resulting in

$$\begin{equation} \fl \langle \hat{a}\rangle (\phi )= \exp (\pm \sqrt{4\pi G/3}\: \phi ) \quad \mbox{and} \quad \langle \hat{p}_a\rangle (\phi )= \exp (\mp \sqrt{4\pi G/3}\: \phi ). \end{equation} \tag{ 78 }$$

To transform to proper time, we solve ${\rm d}\phi /{\rm d}\tau = p_{\phi }\exp (\mp \sqrt{12\pi G}\phi )$ for

$$\begin{equation} \phi (\tau )= \pm \frac{\log (\pm \sqrt{12\pi G}p_{\phi }\tau )}{\sqrt{12\pi G}} \end{equation} \tag{ 79 }$$

and obtain $\langle \hat{a}\rangle (\tau )= (\pm \sqrt{12\pi G} p_{\phi }\tau )^{1/3}$, the classical dependence on proper time with a stiff matter source. (In particular, even after Wheeler–DeWitt quantization the system remains singular: infinite density $p_{\phi }^2/(2\langle \hat{a}\rangle ^3)$ is reached at finite proper time.)

In addition to these expectation-value solutions, the state evolves such that its second-order moments change by

$$\begin{equation} \frac{{\rm d}\,(\Delta a)^2}{\,{\rm d}\,\phi }= \pm 2\sqrt{\frac{4\pi G}{3}} (\Delta a)^2 ,\qquad \frac{{\rm d}\,(\Delta p_a)^2}{\,{\rm d}\phi }= \mp 2\sqrt{\frac{4\pi G}{3}} (\Delta p_a)^2 \end{equation} \tag{ 80 }$$

and dΔ(ap_a)/dϕ = 0, with solutions such that $(\Delta a)/\langle \hat{a}\rangle$ and $(\Delta p_a)/\langle \hat{p}_a\rangle$ are constant. Semiclassicality is preserved exactly throughout evolution in this harmonic model, even at high density. Note that Δa and Δp_a change nonetheless, but have often been assumed constant when state evolution was modeled by Gaussians. Wrong quantum corrections then result, which is especially significant in the presence of quantum back-reaction when the harmonic model is generalized. Especially curvature fluctuations are important because they grow when one evolves to high density, and they do show up in effective constraints such as (20) as a simple example. This issue is another illustration of the importance of complete effective equations including the moment dynamics.

With additional ingredients such as spatial curvature, a cosmological constant, a scalar mass or self-interaction, anisotropy or inhomogeneity, the system is no longer harmonic and becomes subject to quantum back-reaction. Deviations from the classical trajectory will then occur, to be captured by effective equations. (Some of these ingredients also remove deparameterizability, but at the effective level we can still use local internal times.) With a cosmological constant, for instance, the ϕ-Hamiltonian becomes $p_{\phi }(a,p_a)= \pm \sqrt{4\pi G/3}\: a \sqrt{p_a^2- 4\Lambda a^4}$, a non-quadratic expression that entails coupling terms between expectation values and moments in a quantum Hamiltonian.

At first sight, it seems that quantum-geometry corrections in loop quantum cosmology imply quantum back-reaction by deviations from the quadratic nature if (66) is used. This expectation is correct for inverse-triad corrections, but holonomy corrections, although they change the quadratic nature by higher-order terms, still lead to a harmonic model free of quantum back-reaction [56].

To see this, we first change to connection variables $c=\gamma \dot{a}= -(4\pi \gamma G/3) p_a/a$ and p = a², with {c, p} = 8πγG/3. The ϕ-Hamiltonian is still quadratic in these variables, proportional to |cp|, but the holonomy modification leads us to replace c by sin (ℓc)/ℓ with some ℓ that may depend on p. After this, the Hamiltonian p_ϕ(c, p) is no longer quadratic in c and p. However, if we introduce a new variable J := 3pexp (iℓc)/8πγG, we have a linear ϕ-Hamiltonian

$$\begin{equation} p_{\phi }= \pm 2\sqrt{\frac{4\pi G}{3}} \frac{|\,{\rm Im}\,J|}{\ell } \end{equation} \tag{ 81 }$$

if ℓ is constant. More generally, we can assume a power law for the p-dependence of ℓ(p) = ℓ₀p^x, and define new basic variables

$$\begin{equation} V:= \frac{3p^{1-x}}{8\pi \gamma G(1-x)} ,\quad J:=V\exp (\,{\rm i}\,\ell _0 p^xc) \end{equation} \tag{ 82 }$$

so that the ϕ-Hamiltonian remains linear (and is just multiplied with 1 − x compared to (81)). Moreover, and importantly, we have a (non-canonical) closed algebra of basic variables,

$$\begin{equation} \lbrace V,J\rbrace = -{\rm i}\ell _0J,\quad \lbrace V,\bar{J}\rbrace ={\,{\rm i}}\,\ell _0\bar{J},\quad \lbrace J,\bar{J}\rbrace = 2{\,{\rm i}}\,\ell _0V, \end{equation} \tag{ 83 }$$

with the Hamiltonian a linear combination of the generators.

Given these properties, upon quantization the Ehrenfest equations still provide closed equations for expectation values, without coupling to moments and quantum back-reaction. The Hamiltonian operator

$$\begin{equation} \hat{p}_{\phi }= \pm \sqrt{\frac{4\pi G}{3}} (1-x) \left|\frac{\hat{J}-\hat{J}^{\dagger }}{\,{\rm i}\,\ell _0}\right| \end{equation} \tag{ 84 }$$

is linear in $\hat{J}$ and its adjoint, and the quantum Hamiltonian is linear in $\langle \hat{J}\rangle$ and its complex conjugate. The only additional condition to impose is a reality condition because we have used partially complex variables. If we initially keep J and $\bar{J}$ as independent variables, valid solutions must satisfy $J\bar{J}=V^2$.

We quantize by turning V and J into operators, choosing an ordering of J with the exponential to the right. The classical Poisson algebra is then replaced by the closed commutator algebra

$$\begin{equation} \fl [\hat{V},\hat{J}]= \ell _0\hbar \hat{J},\quad [\hat{V},\hat{J}^{\dagger }]= -\ell _0\hbar \hat{J}^{\dagger },\quad [\hat{J},\hat{J}^{\dagger }]= -\ell _0\hbar (2\hat{V}+\ell _0\hbar ), \end{equation} \tag{ 85 }$$

where the last ℏ comes from reordering exponentials. (Note that this non-canonical algebra implies (V, J)-moments not commuting with expectation values on the quantum phase space.) The reality condition, with the same ordering, reads $\hat{J}\hat{J}^{\dagger }-\hat{V}^2=0$, which implies conditions on moments upon taking an expectation value, possibly preceded by multiplication with basic operators. For second-order moments, we have

$$\begin{equation} |\langle \hat{J}\rangle |^2-(\langle \hat{V}\rangle +\ell _0\hbar /2)^2= (\Delta V)^2- \Delta (J\bar{J})+\case{1}{4}\ell _0^2\hbar ^2. \end{equation} \tag{ 86 }$$

(For conditions on higher moments, see [82].) The reality condition is a Casimir of the commutator algebra of type ${\rm sl}(2,{\mathbb R})$ and therefore commutes with the quantum Hamiltonian proportional to ${\rm i}(\hat{J}-\hat{J}^{\dagger })$. If reality holds for initial expectation values and moments, it holds at all times. This statement is true not only for the harmonic model but for any Hamiltonian because the Casimir commutes with all $\hat{V}$, $\hat{J}$ and $\hat{J}^{\dagger }$ individually, and therefore with any function of these variables.

Solutions for expectation values obtained from the linear quantum Hamiltonian are

$$\begin{eqnarray} \langle \hat{V}\rangle (\phi )&=& A\exp (C\phi )+B\exp (-C\phi )\quad \mbox{ and} \end{eqnarray} \tag{ 87 }$$

$$\begin{eqnarray} \langle \hat{J}\rangle (\phi )&=& A\exp (C\phi )-B\exp (-C\phi )+ \frac{\,{\rm i}\,\ell _0}{C} p_{\phi }, \end{eqnarray} \tag{ 88 }$$

with two integration constants A and B as well as the constant $C=\pm 2\sqrt{4\pi G/3}(1-x)$. The reality condition then requires that

$$\begin{equation} |\langle \hat{J}\rangle |^2-\langle \hat{V}\rangle ^2= -4AB+ \frac{\ell _0^2p_{\phi }^2}{C^2} \end{equation} \tag{ 89 }$$

is of the order $\langle \hat{V}\rangle \ell _0\hbar$ (the size of semiclassical fluctuations (ΔV)² and $\Delta (J\bar{J})$ in (86)), much smaller than p²_ϕ for a universe which has a large amount of matter and is semiclassical at least once. (Recall that it is sufficient to impose the reality condition at just one time, for instance when semiclassicality is realized.) The product AB must therefore be positive, close to ℓ²₀p_ϕ²/C², and the function $\langle \hat{V}\rangle (\phi )\propto \cosh (C\phi -C\phi _0)$ never becomes zero. Holonomy modifications in the harmonic model replace the classical singularity by a bounce.

As already seen in the beginning of this section, the bounce property of holonomy-modified equations is very sensitive to quantum back-reaction (or other possible corrections). With quantum back-reaction, the equations become more complicated but can still be analyzed numerically as long as moments do not become large. The approach to high densities can therefore be studied, but the Planck regime remains poorly controlled. In general, it is not known whether loop models always exhibit a bounce.

As in the case of holonomy-modified constraints, one can put effective equations into the form of quantum Friedmann equations. In the case of harmonic loop quantum cosmology, we eliminate $\langle \hat{J}\rangle$ in favor of $\dot{\langle \hat{V}\rangle }$ by using the effective equations and, importantly, the reality condition.

We have ${\rm d}\langle \hat{V}\rangle /{\rm d}\phi =\lbrace \langle \hat{V}\rangle , \langle \hat{p}_{\phi }\rangle \rbrace \propto {\rm Re} \langle \hat{J}\rangle$ using (84) and (85), which we turn into a proper-time derivative, as needed for a Friedmann-type equation, using dϕ/dτ = p_ϕ/a(ϕ)³ where a(ϕ) is taken as some power of $\langle \hat{V}\rangle (\phi )$, depending on x according to (82). (More precisely, we write the Friedmann equation in terms of V by $\dot{a}/a= \dot{V}/(2V(1-x))$ and then use $\langle \hat{V}\rangle (\phi )$. There is no direct relation between $\langle \hat{a}\rangle$ and $\langle \hat{V}\rangle$ because these are expectation values of different powers of the basic $\hat{V}$ unless x = 1/2.) The real part of $\langle \hat{J}\rangle$, which is proportional to ${\rm d}\langle \hat{V}\rangle /{\rm d}\phi$, is related to the imaginary part by the reality condition (86). The imaginary part, finally, is proportional to the deparameterized quantum Hamiltonian p_ϕ. When all these relations are used and p_ϕ is expressed via the energy density of the free massless scalar assumed here, we have

$$\begin{equation} {\rm Re}\,\langle \hat{J}\rangle = \pm \langle \hat{V}\rangle \sqrt{1-(8\pi G/3)\gamma ^2(\ell a)^2\rho _Q} \end{equation} \tag{ 90 }$$

with

$$\begin{equation} \rho _Q=\rho _{\,{\rm free}}\,+ (3/8\pi G)(\gamma \ell a)^{-2} (\Delta (J\bar{J})-(\Delta V)^2)/\langle \hat{V}\rangle ^2 \end{equation} \tag{ 91 }$$

the free energy density corrected by moment terms. If (90) is squared and suitable factors are inserted to express all terms by a, an equation for $(\dot{a}/a)^2$ follows:

$$\begin{equation} \left(\frac{\dot{a}}{a}\right)^2= \frac{8\pi G}{3}\rho _{\,{\rm free}}\, \left(1-\frac{8\pi G}{3}\gamma ^2(\ell a)^2\rho _Q\right). \end{equation} \tag{ 92 }$$

Except for the fluctuation terms in ρ_Q, this is equation (67).

There is another, more important difference: the derivation of (92) holds only in the harmonic model, in which moments enter just by the reality condition, not by quantum back-reaction. The quantum Friedmann equation is valid in this form only if the sole matter source is a free, massless scalar, and there is no spatial curvature, a cosmological constant, or deviations from isotropy. When any one of these conditions is violated, quantum back-reaction results and there are additional corrections, not just in ρ_Q but also in the general form of (92): as already anticipated by considering holonomy expansions in section 4.2, a whole series of corrections in a density expansion appears. The quantum Friedmann equation reads

$$\begin{eqnarray} &&\fl \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\left(\rho \left(1- \frac{8\pi G}{3}\gamma ^2(\ell a)^2 \rho _Q\right) \pm \frac{1}{2}\sqrt{1-\frac{8\pi G}{3}\gamma ^2(\ell a)^2\rho _Q} \:\eta W+ \frac{a^6W^2}{2p_{\phi }^2}\eta ^2 \right) \nonumber\\ \end{eqnarray} \tag{ 93 }$$

where W(ϕ) is a possible scalar potential, and we have a general quantum parameter η = ∑_kη_{k + 1}(a⁶W/p²_ϕ)^k with coefficients η_k that depend on the moments, especially correlation parameters [83, 84]. The expansion by a⁶W/p²_ϕ can be interpreted as one by (ρ − P)/(ρ + P) with pressure P. (For a free, massless scalar, ρ = P.) A more-specific evaluation requires the detailed computation of quantum back-reaction to analyze how the moments evolve and what values they take especially at high density. Techniques for numerical studies have been provided in [82].

While it is difficult to find general information about the values of moments, it is clear that they contribute, among other effects, the canonical analog of higher-time derivatives as they appear in higher-curvature corrections. Moment terms and quantum back-reaction should therefore be large in high-density regimes, near the big bang. Reliable results in loop quantum cosmology can only state that the singularity is avoided by a bounce when matter is kinetic dominated, in which case W ≪ ρ_kin and the η-dependent terms in (93) can be ignored unless η is extremely large. This conclusion coincides with numerical investigations [85–87] of the underlying difference equation for wave functions. However, such numerical studies suffer from the choices required for wave functions, for instance by an initial state. With such methods, it is difficult to capture general-enough effects, which in quantum cosmology with its lack of distinguished states make robust conclusions difficult. As shown by the generality of the quantum Friedmann equations displayed here, effective techniques allow one to draw conclusions and confirm the regime-dependent validity of some effects even when no specific states are chosen.

There are additional and more-surprising properties of the high-density regime that invalidate a traditional bounce interpretation even in the harmonic model. First, staying in the isotropic context, there is cosmic forgetfulness [88, 89]: when crossing the bounce in ϕ-evolution, some moments change in ways so sensitive to initial values that the pre-bounce state cannot be recovered precisely from what may be known post bounce. Using solutions of moment equations [89] or considerations of semiclassical wave functions [90], one can derive an inequality

$$\begin{equation} \left|1-\frac{(\Delta V)_{\phi \rightarrow \infty }}{(\Delta V)_{\phi \rightarrow -\infty }}\right|\le \frac{\Delta p_{\phi }/\langle \hat{p}_{\phi }\rangle }{(\Delta V/\langle \hat{V}\rangle )_{\phi \rightarrow \infty }} \end{equation} \tag{ 94 }$$

bounding the ratio of volume fluctuations at early and late times. For a state with fluctuations symmetric around the high-density regime near the minimum of $\langle \hat{V}\rangle (\phi )$, the left-hand side would be near zero. The estimate can therefore be used to shed light on the question of how much a quantum state may change while evolving through high density, and how much possible changes can be controlled.

With matter fluctuations $\Delta p_{\phi }/\langle \hat{p}_{\phi }\rangle$ usually much larger than geometry fluctuations $(\Delta V/\langle \hat{V}\rangle )_{\phi \rightarrow \infty }$ at large volume, where quantum field theory on curved space-time should be a good approximation, the right-hand side of the inequality is much larger than one, and (ΔV)_{ϕ → −∞} can differ significantly from (ΔV)_{ϕ → ∞}. The inequality can be saturated by highly squeezed dynamical coherent states [89], showing that control on pre-bounce fluctuations cannot be improved unless states are restricted further, more strongly than by semiclassicality.

Several classes of specific states, especially ones with weak correlations of the canonical variables—the volume and the Hubble parameter—show more-symmetric behavior of volume fluctuations. Most wave functions that can be constructed explicitly, for instance using ${\rm sl}(2,{\mathbb R})$-coherent states based on the algebra (85) as used in [91], are only weakly correlated and do not show all possible asymmetries. Again, the effective viewpoint using moments instead of wave functions provides larger generality. And even though wave functions are not provided in explicit terms, one can show that wave functions for the moment solutions even at saturation of (94) do indeed exist: examples for such states are dynamical coherent states [57] saturating the uncertainty relation, whose existence can be shown by general methods well-known from quantum mechanics.

The second feature preventing a bounce interpretation brings us back to quantum space-time structure. For a reliable cosmological model, we must embed a holonomy-modified isotropic version within a consistent deformation of the constraint algebra. Only then can we be sure that the model describes consistent evolution of quantum space-time. No complete extension of holonomy corrections to inhomogeneity is known, but as we will see in the next section, existing versions of holonomy modifications at high density imply drastic modifications with signature change, turning space-time into a quantum version of four-dimensional Euclidean space. This happens right where the bounce would be, but without time and evolution in Euclidean space, a bounce interpretation is not valid even though the model remains non-singular.

To spell out the general procedure, let us assume that we have a Hamiltonian or Hamiltonian constraint H(q, p) depending on canonical fields q(x) and p(x) in space. We introduce δH/δp(x) ≕ v(x) as a new independent variable in place of p, and then expand equations by this newly defined v. If H is a Hamiltonian generating evolution in some fixed time parameter, we have $v(x)=\dot{q}(x)$ by Hamilton's equations. If H is a Hamiltonian constraint in the absence of an absolute time, v = (δN)⁻¹{q, H[δN]} is the derivative of q in a direction normal to spatial slices. For gravitational variables with q the metric or triad, v would therefore be related to extrinsic curvature. This change of variables amounts to what is needed for a Legendre transformation from (q, p) with Hamiltonian H to (q, v) with Lagrangian L = pv − H, whose form will result as a solution of the v-expanded equations.

Note that we cannot always assume the Hamiltonian to be local and free of derivatives of p, which would imply that partial derivatives could be used to compute v. Holonomy corrections in loop quantum gravity, for instance, introduce higher spatial derivatives of the momentum conjugate to the densitized triad. In such situations, there is no local relation between the Hamiltonian and the Lagrangian, and explicitly performing a Legendre transformation is complicated.

Instead of computing the transformation, we intend to calculate the Lagrangian directly from the constraint algebra, assuming from now on the relations of the (deformed) hypersurface-deformation algebra. Using the definition of v and

$$\begin{equation} \left.\frac{\delta H}{\delta q(x^{\prime })}\right|_{p(x)} = -\left.\frac{\delta L}{\delta q(x^{\prime })}\right|_{v(x)} \end{equation} \tag{ 105 }$$

for the unsmeared Hamiltonian constraint and the Lagrangian density, we write the Poisson bracket (13) of two Hamiltonian constraints as

$$\begin{eqnarray} &&\fl \lbrace H[N],H[M]\rbrace = -\int {\rm d}^3x\int {\rm d}^3y \frac{\delta L(y)}{\delta q(x)} v(x) N(y)M(x)- (N\leftrightarrow M) \end{eqnarray} \tag{ 106 }$$

$$\begin{eqnarray} &&\hphantom{\fl \lbrace H[N],H[M]\rbrace }= \int {\,{\rm d}}\,^3x \beta D^a(x) (N\nabla _aM-M\nabla _aN) \end{eqnarray} \tag{ 107 }$$

with the local diffeomorphism constraint D^a. Taking functional derivatives by N and M, we arrive at the functional equation

$$\begin{equation} \frac{\delta L(x)}{\delta q(x^{\prime })}v(x^{\prime }) + \beta (x)D^a(x)\nabla _a\delta (x,x^{\prime })-(x\leftrightarrow x^{\prime })= 0 \end{equation} \tag{ 108 }$$

for L(x), which can be solved once an expression for the diffeomorphism constraint D^a is inserted, depending on whether q refers to gravity or some matter field. In all cases, D^a is linear in the momenta. A linear equation for L is thus obtained [128]. If (10) and (11) are unmodified, standard expressions for D^a can be used. (See section 3.2 for a discussion of possible further modifications.)

To illustrate the regaining procedure, we look at a scalar matter field ϕ without derivative couplings, whose Hamiltonian obeys the (deformed) hypersurface-deformation algebra on its own, without adding the gravitational piece. The Lagrangian density must be of the form $\mathcal{L}= \sqrt{\det g} L(\phi ,v,\psi )$ where v = (δN)⁻¹{ϕ, H[δN]}, as before, is the normal scalar velocity and ψ = q^ab∇_aϕ∇_bϕ is the only remaining scalar that can be formed from ϕ and its derivatives, to a total derivative order of at most two. Higher derivatives may easily result in interacting matter or quantum-gravity theories, with higher time derivatives from quantum back-reaction and higher spatial ones, additionally, from possible discretizations. (See e.g. [129–132] for information about discretized space-time theories.) For now, however, we look for modifications implied by corrections that leave the classical derivative order unchanged.

With the canonical variables of a scalar field and its diffeomorphism constraint D^a = p_φ∇^aϕ, equation (108) assumes the form

$$\begin{equation} \frac{\delta L(x)}{\delta \phi (x^{\prime })} v(x^{\prime })+ \beta \frac{\partial L(x)}{\partial v(x)} (\nabla ^a\phi (x)) \nabla _a\delta (x,x^{\prime }) - (x\leftrightarrow x^{\prime })=0. \end{equation} \tag{ 109 }$$

As in [128], we write

$$\begin{equation*} \frac{\delta L(x)}{\delta \phi (x^{\prime })}= \frac{\partial L(x)}{\partial \phi (x)} \frac{\delta \phi (x)}{\delta \phi (x^{\prime })}+ 2\frac{\partial L(x)}{\partial \psi (x)} (\nabla ^a\phi (x)) \nabla _a\delta (x,x^{\prime }) . \end{equation*}$$

It follows that

$$\begin{equation*} A^a:= (\nabla ^a\phi ) \left(\beta \frac{\partial L}{\partial v}+ 2v\frac{\partial L}{\partial \psi }\right) \end{equation*}$$

satisfies the equation A^a(x)∇_aδ(x, x') − (x↔x') = 0, shown in [128] to imply A^a = 0. Thus,

$$\begin{equation*} \beta \frac{\partial L}{\partial v}+ 2v\frac{\partial L}{\partial \psi }=0 \end{equation*}$$

and L must be of the form L(ϕ, ψ − v²/β). With non-trivial deformation, $\beta \not=1$, the scalar field therefore obeys a modified dispersion relation. The kinetic term of the Lagrangian does not depend on ψ − v² = g^μν(∇_μϕ)(∇_νϕ) in space-time terms, but has its time derivatives in ψ − v²/β rescaled by the correction function β. The resulting modified dispersion relation is in agreement with the wave equation (104).

At the canonical level, for comparison, we begin with a matter Hamiltonian density of the form

$$\begin{equation} H= \nu \frac{p_{\phi }^2}{2\sqrt{\det q}}+ \frac{1}{2}\sigma \sqrt{\det q}\psi + \sqrt{\det q} \: W(\phi ) \end{equation} \tag{ 110 }$$

with general inverse-triad correction functions ν and σ, and some potential W(ϕ). The corresponding Lagrangian density, with $v=\nu p_{\phi }/\sqrt{\det q}$, takes the form

$$\begin{equation} \fl L= \sqrt{\det q}\left(\frac{v^2}{2\nu }-\frac{\sigma \psi }{2}- W(\phi )\right)= -\sqrt{\det q}\:\frac{\sigma }{2}\left(\psi - \frac{v^2}{\beta }\right)- \sqrt{\det q} W(\phi ). \end{equation} \tag{ 111 }$$

This function has the same kinetic dependence as derived above, provided that β = νσ, exactly the requirement for an anomaly-free constraint algebra in the presence of inverse-triad corrections, where β = α² [11]. Purely canonically, this consistency condition can be seen to ensure causality in the sense that gravitational waves on quantum space-time travel at the speed of light, modified by a factor of $\sqrt{|\beta |}$ compared to the classical speed [133]. No super-luminal propagation happens when anomaly-freedom is taken into account, even if propagation speeds may be larger than the classical speed of light for α > 1.

For gravity, one example has been worked out in quite some detail with these methods: inverse-triad corrections of loop quantum gravity. These corrections have a characteristic component independent of higher derivatives, and therefore can be analyzed already at the level of second-order equations. The resulting second-order effective Lagrangian, regained from a modified constraint algebra with a correction function β independent of curvature components, is

$$\begin{equation} L_{\beta }=\frac{1}{16\pi G} \sqrt{\det q} \left(\frac{\,{\rm sgn}\, \beta }{\sqrt{|\beta |}} \frac{v_{ab}v^{ab}- v^a_a v^b_b}{4} + \sqrt{|\beta |} \,{}^{(3)}\!R-2\lambda \right) \end{equation} \tag{ 112 }$$

with 'velocities' v_ab, defined again as normal derivatives [102]. For classical gravity, v_ab = 2K_ab would be proportional to extrinsic curvature. Compared with the classical action obtained for β = 1, the notion of covariance has changed: space and time derivatives are corrected by different coefficients $\sqrt{|\beta |}$ of ⁽³⁾R and |β|^−1/2sgn(β) of $\frac{1}{4}(v_{ab}v^{ab}- v^a_av^b_b)$, respectively. This result is consistent with the fact that the underlying constraint algebra is modified, taking the form (13). For this reason, as already mentioned in the context of line elements, the manifold structure required to integrate L_β to an effective action is not clear.

Nevertheless, the effective Lagrangian shows several characteristic effects. First, inverse-triad corrections, having implications even without higher-derivative or higher-order terms in v_ab, can easily be separated from holonomy effects and higher-curvature corrections. They are especially significant for small fluxes, where β becomes small. With these constructions, it is possible to use inverse-triad corrections even when they imply strong modifications, regimes in which one would otherwise have to include full holonomy and higher-curvature corrections as well. These latter types of corrections are indeed present, but affect only higher orders in the v-expansion. Inverse-triad effects up to quadratic order in v are reliable even if the other corrections are not known precisely.

Time derivatives are then dominant compared to the spatial Ricci scalar when β approaches zero at small fluxes, indicating that near-singular geometries are controlled by homogeneous dynamics, strengthening the classical BKL scenario [134] by a no-singularity scenario in loop quantum gravity [135]. While holonomy effects cannot easily be seen at this level since they would manifest themselves only at higher orders in v_ab and mix with higher-derivative terms, they have one drastic implication at high density if they are dominant. Then, the holonomy correction function β becomes negative, a feature taken into account by the sign factors in (112). When the sign changes, the signature does too: at the Lagrangian level, one can interpret the effect by turning t into it in (112) with v_ab interpreted as first-order time derivatives. (The constraint algebra again provides a rigorous interpretation of this signature change; see section 5.2.) At Planckian densities, holonomy effects are so strong that they turn space-time into a quantum version of four-dimensional Euclidean space, lacking time and evolution. Temporal interpretations of high-density holonomy implications, such as bounces, are incorrect. Only large higher-derivative terms could prevent β from turning negative, but then other holonomy effects such as bounces would go away too.