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3.2 Collinear gravitons, Dirac quantization, and conformal factor

In this section, taken from [156Jump To The Next Citation Point], we discuss a number of structural issues of the 2 + 2 truncations and advocate that, as far as the investigation of the renormalization properties is concerned, the use of a proper time or Dirac quantization is the method of choice. We begin by describing what conventional graviton perturbation theory looks like in this sector.

3.2.1 Collinear gravitons

Performing a standard expansion gαβ = ηαβ + fαβ, 0 ≤ α,β ≤ 3, around the metric ηαβ of Minkowski space the quadratic part of the Einstein–Hilbert action ∫ √ --- d4x − gR (g) reads

1 ∫ [ ] S′′EH[f] = − -- d4x ∂αfβγ ∂αf βγ − 2∂ αfβγ ∂ γfβα + 2∂αf ββ ∂γf γα − ∂αfββ ∂αf γγ 4 ∫ [ ] (3.30 ) 1 4 TFαβ ( γ TF 2 TF ) 3 2 = − -- d x f 2 ∂ ∂αfγβ − ∂ fαβ − ∂α ∂βf + --f∂ f . 4 8
Indices are raised with ηαβ. In the second part of Equation (3.30View Equation) we decomposed fα β into its trace f := fαα and a trace free part fTF := fαβ − 1ηαβf αβ 4. 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 ∫ S = − 12 d4x[∂αφ∂α φ + m2 φ2] has a positive semidefinite Hamiltonian. The kinetic term for the trace f thus has the ‘wrong’ sign, which reflects the conformal factor instability. One readily checks the invariance of the action (3.30View Equation) under the gauge transformations
fα β ↦→ fαβ + ∂αξβ + ∂β ξα, (3.31 )
with ξα a being Lorentz covector field. The field equations of Equation (3.30View Equation) are
γ γ γ [ γ δ γ ] □f αβ − ∂α∂ γf β − ∂β∂γf α + ∂ α∂βf γ + ηαβ ∂ ∂ fγδ − □f γ = 0. (3.32 )

The 2-Killing vector reduction of the action (3.30View Equation) amounts to considering field configurations obeying ℒK1f αβ = 0 = ℒK2g αβ (which is equivalent to gα β = ηαβ + fαβ having two Killing vectors). In adapted coordinates where Ka = ∂∕ ∂ya, a = 1,2, the metric components depend only on the non-Killing coordinates (x0,x1). Moreover f αβ can be assumed to be block-diagonal

( f 0 ) ( η 0 ) fαβ = μν , ηαβ = μν , (3.33 ) 0 fab 0 δab
where μ,ν ∈ {0,1 }, a,b ∈ {2,3 }, and ημν has eigenvalues (− ,+ ). The component fields f22, f33, f23 retain their interpretation as parameterizing the norms and inner product of the Killing vectors, i.e. K1 ⋅ K1 = 1 + f22, K2 ⋅ K2 = 1 + f22, K1 ⋅ K2 = f23. Since (K ⋅ K )(K ⋅ K ) − (K ⋅ K )2 = 1 + f + f + O (f2) 1 1 2 2 1 2 22 33 is nonlinear in the perturbations, a (re-)parameterization in terms of a radius field ρ as in the nonlinear theory is less useful. Nevertheless the combination f22 + f33 will play a special role later on.

Entering with the ansatz (3.33View Equation) into Equation (3.30View Equation) gives the reduced action

∫ ′′ 1-- 2 [ μ a ν ν μ ab μ a b ] S [f] = − 4λ d x 2∂ f a(∂ fνμ − ∂μf ν) + ∂ f ∂ μfab − ∂ f a∂μf b ∫ [ ] (3.34 ) = − 1-- d2x f T∂ ∂ (k − f) − fTab∂2fT − 1k∂2f − 1k ∂2k + 3f ∂2f . 4λ μν μ ν ab 4 8 8
Here the Greek indices μ, ν,... are raised with μν η, and the latin a,b,... with ab δ. As in Equation (3.10View Equation) we also included Newton’s constant per unit volume of the internal space as a prefactor. The second line of Equation (3.34View Equation) is the counterpart of the second line of Equation (3.30View Equation) and we wrote
TF T 1- TF T 1- μ a fμν = fμν + 2 ημνk, fab = fab − 2δabk, k := f μ − f ;a, (3.35 )
where μν T ab T η fμν = 0 = δ fab are the individually tracefree parts of the blocks in Equation (3.33View Equation). Their traces μ f μ and a f a have been replaced by k and α μ a f = f α = f μ + f a in Equation (3.34View Equation). The wrong sign of the trace component f of course remains.

We add some remarks. As one might expect, the action (3.34View Equation) can also be obtained by “first reducing” and then “linearizing”. Indeed, from Equation (3.50View Equation) below one has

√ ---- (2) || μ ν ν 2 ρ − γR (γ)| ≃ ∂ ρ − ∂ fνμ + ∂ μf ν] + O (ρf ), (3.36 ) γμν=ημν+fμν
modulo total derivatives, and with the identifications
1- 2 ρ = 1 + 2 (f22 + f33) + O (f ), Δ = 1 + 1-(f22 − f33) + O (f2), (3.37 ) 2 ψ = f + O(f 2). 23
one recovers (3.34View Equation) from Equation (3.10View Equation). Second, the action (3.34View Equation) is invariant under the gauge transformations
fμν ↦→ fμν + ∂μξν + ∂νξμ, (3.38 )
which corresponds to Lorentz vector fields ξα of the form ξα = (ξμ(x0,x1),0,0). In addition there are the remnants of the SL (2,ℝ )∕SO (2) isometries which now generate the isometries group ISO (2) of ℝ2: two translations f22 ↦→ f22 + a1, f33 ↦→ f33 − a1 and f23 ↦→ f23 + a2, and one rotation
f22 ↦→ f22 + 2αf23, f33 ↦→ f33 − 2αf23, f23 ↦→ f23 − α(f22 − f33), (3.39 )
with α constant and modulo 2 O (α ) 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 (3.32View Equation) coincides with the field equations obtained by varying Equation (3.34View Equation). The latter are

∂2f23 = 0, 2 2 2 ν μ ν ∂ f22 = ∂ f33 = ∂ f ν − ∂ ∂ fμν, (3.40 ) [∂μ∂ν − ημν∂2](f22 + f33) = 0,
with ∂2 = ημν∂μ∂ν. These are equivalent to ∂2f23 = ∂2f22 = ∂2f33 = 0, (∂μ∂ ν − η μν∂2 )fμν = 0, and
f + f = xμv + v , (3.41 ) 22 33 μ 0
for integration constants vμ and v0. A nonzero vμ (which is not co-rotated) violates Poincaré invariance and is undesirable for most purposes. Equation (3.37View Equation) shows that on shell only two of the three free fields f , f 22 33, f 23 are independent. In fact, in a Hamiltonian formulation of Equation (3.34View Equation) the condition f22 + f33 = const can be understood as a constraint associated with the gauge invariance (3.36View Equation). Equivalently it can be viewed as a remnant of the constraints (3.15View Equation). Indeed, to linear order ℋ0 = − 2∂21ρ∕λ, ℋ1 = − 2∂1πσ, with πσ = ∂0ρ, which combined with Equation (3.37View Equation) gives f22 + f33 = const, 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 (3.30View Equation) is the transversal-traceless gauge,

[ ] ∂α fαβ − 1-ηαβfγγ = 0, 2 f = 0, (3.42 ) α0 γ f γ = 0.
The first condition is the “harmonic gauge condition”, the others are “transversal-traceless” conditions. Using the latter in the former gives ∂f = 0 iij, with i,j ∈ {1, 2,3}. Thus Equation (3.42View Equation) removes 3 + 4 + 1 degrees of freedom of the 10 components of fαβ leaving the familar 2 graviton degrees of freedom. In the 2-Killing subsector the gauge conditions read
μ ρ ab ∂ [2fμν − ημνf ρ + ημνδ fab] = 0, f00 = f01 = 0, (3.43 ) f11 + f22 + f33 = 0.
Using again the second set in the harmonic gauge condition gives f22 + f33 = const. Thus Equation (3.43View Equation) removes 1 + 3 of the 6 degrees of freedom in fμν, fab, leaving again the familiar two ‘graviton’ degrees of freedom, f22, f23, say. The only difference to the physical degrees of freedom of the full linearized theory (3.30View Equation) is that all wave vectors are aligned, that is, the gravitons are collinear. To see this recall that the general solution of Equation (3.32View Equation) subject to Equation (3.42View Equation) is a superposition of plane waves ikγxγ fαβ(x) = Aαβ (k )e + c.c., with γ kγk = 0 and A αβ(k) constrained by Equation (3.42View Equation). Both the action (3.30View Equation) and the field equations (3.31View Equation) are Poincaré invariant, so any one wave vector in a superposition can be fixed to have the form k = (k0,k1, 0,0), with k μkμ = 0. Identifying (x0,x1) with the non-Killing coordinates and 2 3 (x ,x ) with the Killing coordinates 2 3 (y ,y ), 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 k = (k0,k1, 0,0). 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 A (k) αβ 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 [2091137371].

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

∫ -1- 2 μ μ ρ ab 2α d xQ Q μ, Qν := ∂ [2fμν − ημνf ρ + ημν δ fab], (3.44 )
implementing the harmonic gauge condition is added to the action. This renders the kinetic term in Equation (3.34View Equation) 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.

3.2.2 Dirac versus covariant quantization

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 (3.10View Equation) and decompose γμν into a conformal factor eσ and a two-parametric remainder ˆγμν to be adjusted later, γ μν = e σˆγμν. Using R(2)(eσγˆ) = e−σ[R(2)(ˆγ ) − ˆγ μν ˆ∇μ ˆ∇νσ ] we can rewrite Equation (3.10View Equation) as a gravity theory for the two-parametric γˆμν

∫ 1 2 ∘ -- μν μν −2 S = 2λ- d x γˆ[2 ρR(ˆγ ) + ˆγ ∂ μρ∂ν(2σ + lnρ ) − ˆγ (∂μ Δ∂ νΔ + ∂μψ ∂νψ )ρΔ ]. (3.45 )
There are two instructive choices for ˆγμν,
( ) − n2 + s2 s ˆγμν = s 1 , (3.46 ) T ˆγμν = ¯γμρe− ¯σ (ef )ρν. (3.47 )
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 n and shift s are classically nonpropagating degrees of freedom. In a Dirac quantization one can simply fix the temporal gauge (n = 1, s = 0) and use the nondegenerate gauge-fixed action in Equation (3.45View Equation) 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 (3.47View Equation) is adapted to a covariant quantization. Here ¯γμνe¯σ is a generic (off-shell) background metric with again the conformal mode ¯σ split off. The fluctuation field f T μν is trace-free with respect to the background μν T ¯γ fμν = 0. Then Equation (3.45View Equation) describes a unimodular gravity theory (T detef = 1) and the original metric is parameterized by fTμν and fσ := σ − ¯σ as γμν = ¯γμρ(efT)ρνefσ. In a covariant formulation the degrees of freedom in f T would be promoted to propagating ones by adding a gauge fixing term to the action (3.45View Equation). 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 (3.45View Equation) 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 (3.6View Equation) 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

( ) ( ) γμν 0 ρ Δ2 + ψ2 ψ gαβ = 0 M , Mab = -- ψ 1 (3.48 ) ab Δ
with γ μν, for example, in the above lapse-shift parameterization into the Einstein–Hilbert Lagrangian. This gives Equation (3.10View Equation). For the 2D metric a convenient parameterization of the fluctuations around a generic background is Equation (3.47View Equation) which for γ μν amounts to
T γμν = ¯γμρ(ef )ρν efσ = ¯γμν + fμTν + fσ¯γμν + ..., (3.49 )
where μν T ¯γ fμν = 0 and indices are raised and with the (inverse) background metric ¯γμν. Since ¯γμν is diffeomorphic to e¯σ one has σ = ¯σ + fσ.

The counterpart of Equation (3.49View Equation) for the lower 2 × 2 block in Equation (3.48View Equation) is

fT c fρ T Mab = M¯ac (e )b e = M¯ab + fab + f ρM ¯ab + ..., (3.50 )
where M¯abfT = 0 ab. Writing √ ------ ρ¯= detM¯ one has ρ = ¯ρefρ. The trace-free part f T ab defines an element of the Lie algebra of SL (2,ℝ), though in a nonstandard basis. Writing
−1 − 1 Δ = ¯Δ + ¯ρ f1, ψ = ¯ψ + ¯ρ f2, ρ = ¯ρ(1 + fρ), (3.51 )
one finds M = ¯M + fT + f ¯M + O (f2) ab ab ab ρ ab, with
1 ( ¯ 2 ¯2 ¯ ) 1 ( ¯ ) faTb = f1--- Δ −¯ψ − ψ + f2-- 2ψ 1 . (3.52 ) Δ¯2 − ψ − 1 Δ¯ 1 0
The decomposition (3.49View Equation, 3.50View Equation) can be viewed as the counterpart of the York decomposion for the metrics (3.6View Equation). Note that conventional linear background fluctuation split would only keep the leading terms in Equations (3.49View Equation, 3.50View Equation) and base the expansion on this split.

Writing LEH (g) for Einstein–Hilbert Lagrangian with the blockdiagonal metric (3.48View Equation) one arrives at an expansion of the form

∑ T T ¯ LEH (g ) = LEH (¯g) + Ln(fμν,fσ,f ab,fρ; ¯γμν,Mab ), (3.53 ) n≥1
where Ln is of order n 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 Ln, n ≥ 3, get reshuffled. We won’t need their explicit form; the point relevant here is the invariance of L2 under the genuine gauge transformations in f μν and the background counterpart of the ISO (2) rotations (3.39View Equation). 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, ∇¯μ(fμν − 12¯γ μν¯γρσfρσ) = 0. 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 ¯ Γ B[⟨γ μν⟩,⟨Mab ⟩;¯γμν,Mab ], 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 Γ B[⟨γμν⟩,⟨Mab⟩;¯γ μν,M¯ab ] 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 (3.49View Equation) ignores the special status of the lapse and shift degrees of freedom in γ μν; all components are expanded. We know, however, that there must be two infinite series built from the components of ¯γμν and fμν that enter the left-hand-side of Equation (3.53View Equation) anyhow linearly. Concerning the lower 2 × 2 block in Equation (3.48View Equation) both the linear and the York-type decomposition will only keep the ISO (2) symmetry more or less manifest. The nonlinear realization of the SL(2,ℝ ) 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 γμν enter the left-hand-side of Equation (3.53View Equation) 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 [154Jump To The Next Citation Point155Jump To The Next Citation Point] 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 (3.14View Equation) ‘gone’ one only needs to perform a background-fluctuation split only for the remaining propagating fields Δ, ψ, ρ, σ.

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 (3.16View Equation) 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 (3.15View Equation) as renormalized composite operators. This will be done in Equations (3.100View Equation) ff.

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

1 1 12 22 1 13 Δ = ¯Δ + ξ + --¯ (ξ − ξ ) − ¯ρξ ξ + ..., 2 Δ ψ = ¯ψ + ξ2 + -1ξ1ξ2 − -1-ξ2ξ3 + ..., Δ¯ 2ρ¯ 3 (3.54 ) ρ = ¯ρ + ξ + ..., σ = ¯σ + ξ4 + -1--(ξ12 + ξ22) +--a--ξ32 + ... 4b¯Δ 4bρ¯2
Here φ¯= (¯Δ, ¯ψ, ¯ρ,¯σ ) is the reference point in field space and (ξ1,ξ2,ξ3,ξ4) is the tangent vector at φ¯ to the geodesic connecting ¯ φ and φ = (Δ, ψ, ρ,σ); the dots indicate terms of cubic and higher order in the ξi. The geodesic in question is defined with respect to the pseudo-Riemannian metric in field space 𝔥ij (see Equation (3.60View Equation) below). This metric in field space possesses a number of Killing vectors Y (not to be confused with the Killing vectors of the spacetime geometries considered) and two conformal Killing vectors related to Equation (3.27View Equation). 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.24View Equation, B.39View Equation) 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:

  1. 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.
  2. Through the use of the background effective action formalism the expectation of the quantum metric ⟨g ⟩ αβ and the background ¯g αβ are related by the condition
    δΓ B[⟨gα β⟩;g¯αβ] ----δ⟨g--⟩-----= 0. (3.55) αβ
    One does not expand around a solution of the classical field equations.
  3. 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 (3.26View Equation) and the conformal currents (3.27View Equation). 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.

3.2.3 Conformal factor

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 2 + 2 truncation. As emphazised by Mazur and Mottola [142] in linearized Euclidean quantum Einstein Gravity (based on the Euclidean version of the action (3.30View Equation)) there is really no conformal factor instability. The f ∂2f kinetic term in the second part of Equation (3.30View Equation) 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 2 + 2 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 f ∂2f term.

From the York-type decomposition (3.49View Equation, 3.50View Equation) one sees that 1 μ a 1 fσ + fρ = 2(f μ + f a) =: 2f plays the role of the (gauge-variant) conformal factor. The wrong sign kinetic term is indeed still present in the second part of Equation (3.34View Equation) and f also appears through a dilaton type coupling in the f T∂μ ∂ν(k − f) μν term. In 2D however fT μν has no propagating degrees of freedom and the term could be taken care of promoting T fμν 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 fρ = (f + k )∕4 and fσ = (f − k)∕4, viz. 1 1 2 3 2 2 4∂k ∂f + 8(∂k) − 8(∂f ) = − 4 ∂fσ ∂f ρ − 2(∂fρ). This occurs here on the linearized level but comparing with Equation (3.16View Equation) 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 − ∂ρ ∂(σ + 1ln ρ) 2. By a local redefinition of σ one can eliminate the term quadratic in ρ and in dimensional regularization used later no Jacobian arises. One is left with a − ∂ρ ∂σ term which upon diagonalization gives rise to one field whose kinetic term has the wrong sign. However ρ is a dilaton type field which multiplies all of the self-interacting positive energy scalars in the first term of Equation (3.16View Equation), and the dynamics of this mode turns out to be very special (see Section 3.3). Heuristically this can be seen by viewing the σ field in the Lorentzian functional integral simply as a Lagrange multiplier for a 2 δ(∂ ρ) 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 (3.16View Equation) 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 2 ∂ ρ field, viewed as a renormalized operator, can be verified. Since the system is renormalizable only with infinitely many couplings, the functional dependence on ρ in the renormalized Lagrangian and in the ∂2ρ 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 2 + 2 truncations. The gravity part of the action (3.10View Equation) or (3.45View Equation) arises from evaluating the Einstein–Hilbert action SEH on the class of metrics (3.6View Equation). The constant 1∕λ in Equation (3.10View Equation, 3.45View Equation) can be identified with 2 d y∕gN, i.e. with Newton’s constant per unit volume of the orbits. As such λ is an inessential parameter and its running is defined only relative to a reference operator. For the 2 + 2 truncations it turns out that the way how the action (3.10View Equation) depends on ρ has to be modified in a nontrivial and scale dependent way by a function h( ⋅) (see Equation (3.56View Equation) 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 h(⋅) remains an inessential parameter. It is thus convenient not to renormalize this overall scale and to treat λ in Equation (3.56View Equation) 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 Λ ρeσ type addition to Equation (3.56View Equation) below. In the quantum theory one is again forced to replace ρ with an scale dependent function f(ρ) in order to achieve strict cutoff independence [156Jump To The Next Citation Point]. The cosmological constant proper can be identified with the overall scale of the function f( ⋅). The function f is subject to a non-autonomous flow equation, triggered by h, but if its initial value is set to zero it remains zero in the course of the flow [156Jump To The Next Citation Point]. To simplify the exposition we thus set f ≡ 0 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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