Performing a standard expansion , , around the metric of Minkowski space the quadratic part of the Einstein–Hilbert action reads

Indices are raised with . In the second part of Equation (3.30) we decomposed into its trace and a trace free part . A critical discussion of the reconstruction of the nonlinear theory from the linear one can be found in [164]; a proper time formulation of the linearized quantum theory has been given in [141]. In the present conventions a scalar action of the form has a positive semidefinite Hamiltonian. The kinetic term for the trace thus has the ‘wrong’ sign, which reflects the conformal factor instability. One readily checks the invariance of the action (3.30) under the gauge transformations with a being Lorentz covector field. The field equations of Equation (3.30) areThe 2-Killing vector reduction of the action (3.30) amounts to considering field configurations obeying (which is equivalent to having two Killing vectors). In adapted coordinates where , , the metric components depend only on the non-Killing coordinates . Moreover can be assumed to be block-diagonal

where , , and has eigenvalues . The component fields , , retain their interpretation as parameterizing the norms and inner product of the Killing vectors, i.e. , , . Since is nonlinear in the perturbations, a (re-)parameterization in terms of a radius field as in the nonlinear theory is less useful. Nevertheless the combination will play a special role later on.Entering with the ansatz (3.33) into Equation (3.30) gives the reduced action

Here the Greek indices are raised with , and the latin with . As in Equation (3.10) we also included Newton’s constant per unit volume of the internal space as a prefactor. The second line of Equation (3.34) is the counterpart of the second line of Equation (3.30) and we wrote where are the individually tracefree parts of the blocks in Equation (3.33). Their traces and have been replaced by and in Equation (3.34). The wrong sign of the trace component of course remains.We add some remarks. As one might expect, the action (3.34) can also be obtained by “first reducing” and then “linearizing”. Indeed, from Equation (3.50) below one has

modulo total derivatives, and with the identifications one recovers (3.34) from Equation (3.10). Second, the action (3.34) is invariant under the gauge transformations which corresponds to Lorentz vector fields of the form . In addition there are the remnants of the isometries which now generate the isometries group of : two translations , and , and one rotation with constant and modulo terms.Further the operations “varying the action” and “reduction” are commuting, as expected from the principle of symmetric criticality. Thus, the reduction of the field equations (3.32) coincides with the field equations obtained by varying Equation (3.34). The latter are

with . These are equivalent to , , and for integration constants and . A nonzero (which is not co-rotated) violates Poincaré invariance and is undesirable for most purposes. Equation (3.37) shows that on shell only two of the three free fields , are independent. In fact, in a Hamiltonian formulation of Equation (3.34) the condition can be understood as a constraint associated with the gauge invariance (3.36). Equivalently it can be viewed as a remnant of the constraints (3.15). Indeed, to linear order , , with , which combined with Equation (3.37) gives , once again.For the special case of the Einstein–Rosen waves an on-shell formulation suffices and a related study linking the ‘graviton modes’ to the ‘Einstein–Rosen modes’ can be found in [23].

The linearized theory is well suited to discuss the physics content of the 2-Killing vector reduction. To this end one fixes the gauge and displays the independent on-shell degrees of freedom. The most widely used gauge in the linearized theory (3.30) is the transversal-traceless gauge,

The first condition is the “harmonic gauge condition”, the others are “transversal-traceless” conditions. Using the latter in the former gives , with . Thus Equation (3.42) removes degrees of freedom of the 10 components of leaving the familar 2 graviton degrees of freedom. In the 2-Killing subsector the gauge conditions read Using again the second set in the harmonic gauge condition gives . Thus Equation (3.43) removes of the degrees of freedom in , , leaving again the familiar two ‘graviton’ degrees of freedom, , , say. The only difference to the physical degrees of freedom of the full linearized theory (3.30) is that all wave vectors are aligned, that is, the gravitons are collinear. To see this recall that the general solution of Equation (3.32) subject to Equation (3.42) is a superposition of plane waves , with and constrained by Equation (3.42). Both the action (3.30) and the field equations (3.31) are Poincaré invariant, so any one wave vector in a superposition can be fixed to have the form , with . Identifying with the non-Killing coordinates and with the Killing coordinates , the waves in the 2-Killing subsector are of identically the same form; the only difference is that now all of the wave vectors have the form . In other words, all waves move in the same or in the opposite direction; they are collinear. This collinearity of the wave vectors must of course not be confused with the alignment of the polarization vectors; the polarization tensor here always carries two independent polarizations. Nontrivial scattering is possible despite the collinearity of the waves. It should be interesting to compute this S-matrix and to contrast it with the one in the Eikonal sector [209, 113, 73, 71].In summary we conclude that the 2-Killing vector subsector comprises the gravitational self-interaction of collinear gravitons, in the same sense as the full Einstein–Hilbert action describes the self-interaction of non-collinear gravitons.

The formulation of the perturbative functional integral in this subsector would now proceed in exact parallel to the non-collinear case: A gauge fixing term

implementing the harmonic gauge condition is added to the action. This renders the kinetic term in Equation (3.34) nondegenerate. The extra propagating degrees of freedom are then ‘canceled out’ by the Faddeev–Popov determinant. In principle a systematic collinear graviton loop expansion could be set up in this way, much in parallel to the generic non-collinear case.

For the systems at hand we now want to argue that this is not the method of choice. To this end we return to the covariant action (3.10) and decompose into a conformal factor and a two-parametric remainder to be adjusted later, . Using we can rewrite Equation (3.10) as a gravity theory for the two-parametric

There are two instructive choices for , The first choice is the densitized lapse-shift parameterization already used before (see Section 3.1.4). This is adapted to a proper time or Dirac quantization: The lapse and shift are classically nonpropagating degrees of freedom. In a Dirac quantization one can simply fix the temporal gauge (, ) and use the nondegenerate gauge-fixed action in Equation (3.45) to define the quantum theory. The Hamiltonian and diffeomorphism constraint arising from the gauge fixing then have to be defined as composite operators.The choice (3.47) is adapted to a covariant quantization. Here is a generic (off-shell) background metric with again the conformal mode split off. The fluctuation field is trace-free with respect to the background . Then Equation (3.45) describes a unimodular gravity theory () and the original metric is parameterized by and as . In a covariant formulation the degrees of freedom in would be promoted to propagating ones by adding a gauge fixing term to the action (3.45). The associated Faddeev–Popov determinant is designed to cancel out their effect again. In the case at hand this is clearly roundabout as the gauge-frozen Lagrangian (3.45) is already nondegenerate.

This setting can be promoted to a generalization of the one presented in Section 3.1.2 to generic backgrounds. In the terminology of Section B.2 one then gets a non-geodesic background-fluctuation split, which treats the nonpropagating lapse and shift degrees of freedom on an equal footing with the others. In order to contrast it with the geodesic background-fluctuation split for the propagating modes used later on, we spell out here the first few steps of such a procedure.

We wish to expand the Einstein–Hilbert action for the class of metrics (3.6) around a generic background. Technically it is simpler to “first reduce” and then “expand”. This is legitimate since all operations involved are algebraic. Recall that the reduced Lagrangian is defined by inserting the ansatz

with , for example, in the above lapse-shift parameterization into the Einstein–Hilbert Lagrangian. This gives Equation (3.10). For the 2D metric a convenient parameterization of the fluctuations around a generic background is Equation (3.47) which for amounts to where and indices are raised and with the (inverse) background metric . Since is diffeomorphic to one has .The counterpart of Equation (3.49) for the lower block in Equation (3.48) is

where . Writing one has . The trace-free part defines an element of the Lie algebra of , though in a nonstandard basis. Writing one finds , with The decomposition (3.49, 3.50) can be viewed as the counterpart of the York decomposion for the metrics (3.6). Note that conventional linear background fluctuation split would only keep the leading terms in Equations (3.49, 3.50) and base the expansion on this split.Writing for Einstein–Hilbert Lagrangian with the blockdiagonal metric (3.48) one arrives at an expansion of the form

where is of order in the fluctuations. This holds both when a York-type decomposition is adopted and when a linear split is used, just the higher order terms , get reshuffled. We won’t need their explicit form; the point relevant here is the invariance of under the genuine gauge transformations in and the background counterpart of the rotations (3.39). The genuine gauge invariance has to be gauge-fixed in order to get nondegenerate kinetic terms. The standard gauge fixing is the background covariant harmonic one, . The extra propagating degrees of freedom are canceled by the Faddeev–Popov determinant, and the construction of the perturbative functional integral proceeds as usual. One can take as the central object in the quantum theory the background effective action , which as far as the symmetries are concerned of the type characteristic for a linear background-fluctuation split. Note that this feature would not change if a nonperturbative construction of was aimed at, for example via a functional renormalization group equation.In all cases one sees that the procedure outlined has two drawbacks. First, the split (3.49) ignores the special status of the lapse and shift degrees of freedom in ; all components are expanded. We know, however, that there must be two infinite series built from the components of and that enter the left-hand-side of Equation (3.53) anyhow linearly. Concerning the lower block in Equation (3.48) both the linear and the York-type decomposition will only keep the symmetry more or less manifest. The nonlinear realization of the symmetry then has to be restored through Ward identites, iteratively in a perturbative formulation or otherwise in a nonperturbative one.

In the following we shall adopt the following remedies. The fact that the lapse and shift degrees of freedom in enter the left-hand-side of Equation (3.53) linearly of course just means that they are the Lagrange multipliers of the constraints in a Hamiltonian formulation. The linearity can thus be exploited either by a gauge fixing with respect to these variables before expanding, giving rise to a proper time formulation, or by directly adopting a Dirac quantization prodecure. By and large both should be eqaivalent; in [154, 155] a direct Dirac quantization was used, and we shall describe the results in the next two Sections 3.3 and 3.4. With the lapse and shift in Equation (3.14) ‘gone’ one only needs to perform a background-fluctuation split only for the remaining propagating fields , , , .

To cope with the second of the before-mentioned drawbacks we equip – following deWitt and Vilkovisky – this space of propagating fields with a pseudo-Riemannian metric and perform a normal-coordinate expansion around a (‘background’) point with respect to it. This leads to the formalism summarized in Section B.2.2. The pseudo-Riemannian metric on the space of propagating fields can be read off from Equation (3.16) and converts the gauge-frozen but nondegenerate Lagrangian into that of a (pseudo-)Riemannian nonlinear-sigma model. The renormalization theory of these systems is well understood and we summarize the aspects needed here in Section B.3. In exchange for the gauge-freezing one then has to define quantum counterparts of the constraints (3.15) as renormalized composite operators. This will be done in Equations (3.100) ff.

For the sake of comparison with Equations (3.49, 3.51) we display here the first two terms of the resulting geodesic background-fluctuation split:

Here is the reference point in field space and is the tangent vector at to the geodesic connecting and ; the dots indicate terms of cubic and higher order in the . The geodesic in question is defined with respect to the pseudo-Riemannian metric in field space (see Equation (3.60) below). This metric in field space possesses a number of Killing vectors (not to be confused with the Killing vectors of the spacetime geometries considered) and two conformal Killing vectors related to Equation (3.27). A major advantage of the geodesic background fluctuation split is that the associated generalized Ward identities are built into the formalism. Indeed the diffeomorphism Ward identities (B.24, B.39) become Ward identites proper for the isometries of the target space and “conformal” generalizations thereof for the conformal isometries.In summary, we find the following differences to standard perturbation theory:

- Lapse and shift viewed as infinite series in the fluctuation field are not expanded. Only the metric degrees of freedom other than lapse and shift are expanded.
- Through the use of the background effective action formalism the expectation of the quantum metric and the background are related by the condition One does not expand around a solution of the classical field equations.
- Through the use of a geodesic background-fluctuation split on the space of propagating fields the resulting background effective action is in principle invariant under arbitary local reparameterizations of the propagating fields. Among those the ones associated with isometries or conformal isometries on field space are of special interest and give rise to Ward identities associated with the Noether currents (3.26) and the conformal currents (3.27). The latter are built into the formalism, and do not have to be imposed order by order.

The first point entails that the sigma-model perturbation theory we are going to use is partially non-perturbative from the viewpoint of a standard graviton loop expansion.

Before turning to the quantum theory of these warped product sigma-models we briefy discuss the status of the conformal factor instability in a covariant formulation of the truncation. As emphazised by Mazur and Mottola [142] in linearized Euclidean quantum Einstein Gravity (based on the Euclidean version of the action (3.30)) there is really no conformal factor instability. The kinetic term in the second part of Equation (3.30) with the wrong sign receives an extra contribution from the measure which after switching to gauge invariant variables renders both the Gaussian functional integral over the conformal factor and that for the physical degrees of freedom well-defined. They also gave a structural argument why this should be so even on a nonlinear level: As one can see from a canonical formulation the conformal factor in Einstein gravity is really a constrained degree of freedom and should not have a canonically conjugate momentum.

In the truncation we shall use a Lorentzian functional integral defined through the sigma-model perturbation theory outlined above. So a conformal factor instability proper associated with a Euclidean functional integral anyhow does not arise. Nevertheless it is instructive to trace the fate of the incriminated term.

From the York-type decomposition (3.49, 3.50) one sees that plays the role of the (gauge-variant) conformal factor. The wrong sign kinetic term is indeed still present in the second part of Equation (3.34) and also appears through a dilaton type coupling in the term. In 2D however has no propagating degrees of freedom and the term could be taken care of promoting to a dynamical degree of freedom via gauge fixing and then cancelling the effect by a Faddeev–Popov determinant. As already argued before it is better to avoid this and look at the remaining propagating degrees of freedom directly. They simplify when reexpressed in terms of and , viz. . This occurs here on the linearized level but comparing with Equation (3.16) one sees that the same structure is present in the full gauge-frozen action. We thus consider from now on directly the corresponding terms proportional to . By a local redefinition of one can eliminate the term quadratic in and in dimensional regularization used later no Jacobian arises. One is left with a term which upon diagonalization gives rise to one field whose kinetic term has the wrong sign. However is a dilaton type field which multiplies all of the self-interacting positive energy scalars in the first term of Equation (3.16), and the dynamics of this mode turns out to be very special (see Section 3.3). Heuristically this can be seen by viewing the field in the Lorentzian functional integral simply as a Lagrange multiplier for a insertion. The remaining Lorentzian functional integral would allow for a conventional Wick rotation with a manifestly bounded Euclidean action. We expect that roughly along these lines a non-perturbative definition of the functional integral for Equation (3.16) could be given, which would clearly be one without any conformal factor instability. Within the perturbative construction used in Section 3.3 the special status of the field, viewed as a renormalized operator, can be verified. Since the system is renormalizable only with infinitely many couplings, the functional dependence on in the renormalized Lagrangian and in the field has to be ‘deformed’ in a systematic way; however this does not affect the principle aspect that no instability occurs.

Finally, let us briefly comment on the role of Newton’s constant and of the cosmological constant in the truncations. The gravity part of the action (3.10) or (3.45) arises from evaluating the Einstein–Hilbert action on the class of metrics (3.6). The constant in Equation (3.10, 3.45) can be identified with , i.e. with Newton’s constant per unit volume of the orbits. As such is an inessential parameter and its running is defined only relative to a reference operator. For the truncations it turns out that the way how the action (3.10) depends on has to be modified in a nontrivial and scale dependent way by a function (see Equation (3.56) below) in order to achieve strict cut-off independence. This modification amounts to the inclusion of infinitely many essential couplings, only the overall scale of remains an inessential parameter. It is thus convenient not to renormalize this overall scale and to treat in Equation (3.56) as a loop counting parameter.

A similar remark applies to the cosmological constant. Adding a cosmological constant term to the Ricci scalar term results in a type addition to Equation (3.56) below. In the quantum theory one is again forced to replace with an scale dependent function in order to achieve strict cutoff independence [156]. The cosmological constant proper can be identified with the overall scale of the function . The function is subject to a non-autonomous flow equation, triggered by , but if its initial value is set to zero it remains zero in the course of the flow [156]. To simplify the exposition we thus set from the beginning and omit the cosmological constant term in the following. It is however a nontrivial statement that this can be be done in a way compatible with the renormalization flow.

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