Can quantum fluctuations differentiate between standard and unimodular gravity?

We formally prove the existence of a quantization procedure that makes the path integral of a general diffeomorphism-invariant theory of gravity, with fixed total spacetime volume, equivalent to that of its unimodular version. This is achieved by means of a partial gauge fixing of diffeomorphisms together with a careful definition of the unimodular measure. The statement holds also in the presence of matter. As an explicit example, we consider scalar-tensor theories and compute the corresponding logarithmic divergences in both settings. In spite of significant differences in the coupling of the scalar field to gravity, the results are equivalent for all couplings, including non-minimal ones.

quantum gravity, offering, in this viewpoint, some advantages [11][12][13][14][15][16][17][18][19][20][21][22][23], see, however, [24][25][26]. At the classical level, GR and UG are "almost equivalent", in the sense that (1) can be seen merely as a gauge condition imposed on GR. There is however a subtlety that prevents a full equivalence: the identity which follows from (1), is a relation between observables, and cannot be seen as a gauge condition 2 . One must conclude that GR has one degree of freedom more than UG -just one, not one per spacetime point [8]. Thus, classical UG is equivalent to a version of classical GR where the total volume of spacetime is held fixed. This can be seen in the Lagrangian formalism by adding to the action a term where Λ has to be thought of as a Lagrange multiplier enforcing that the spacetime volume is equal to V 3 . It is natural to expect that this distinction will affect the infrared properties of the theory, but not its behavior at short scales. In this paper we shall not discuss the global, large scale properties and when we say "GR" we shall implicitly mean "GR with fixed total volume".
It is then natural to ask whether this "almost equivalence" holds also for the quantum versions of the theories. The unimodularity condition (1) restricts the invariance group from diffeomorphisms (Diff ) to special (volume-preserving) diffeomorphisms (SDiff ) 4 , and one may expect that this could lead to different quantum theories. For example, if we choose a unimodular gauge for GR, this requires a Faddeev-Popov determinant, while in the quantization of UG, the unimodularity condition is present ab initio and no Faddeev-Popov determinants are necessary.
In this work, we prove in general, based on formal path integral arguments, that there exists a quantization procedure that preserves the "almost equivalence" between these theories. The proof goes through for any Diff -invariant action and in this sense extends beyond ordinary GR.
Of course, there may be other definitions of the quantum theories that break the equivalence, but 2 When the volume is infinite, one has to regulate it by "putting the system in a box" and impose (2) on the regulated system. 3 This point of view has been used by Hawking in Euclidean quantum gravity, where he interpreted the resulting partition function as the "volume canonical ensemble", see [27] 4 In the recent literature, the group SDiff is often referred to as TDiff , where T stands for "transverse". in the absence of other independent arguments in their favor, we think that the one we describe here is more natural. Our argument is in the same spirit as the one presented in [24,25] and extends the results of [23,36,46] beyond one-loop order. We should remark that both GR and UG are not renormalizable in perturbation theory and the formal path integrals should be ultraviolet (UV) regularized. Our formal proof relies on the use of the background field method, but we leave the parameterization of the metric, i.e., the way that we split the full metric in background and fluctuating parts, generic. Hence, this also extends previous results [23,36] which made explicit use of the so-called exponential split of the metric to impose the unimodularity condition [30].
Our proof of equivalence is given initially for pure gravity and one may again worry that as soon as matter degrees of freedom are introduced, the equivalence would fall apart. This is due to the different vertex structures. In GR, the determinant of the metric produces infinitely many vertices between gravitons and matter fields that are absent in UG. Hence, Feynman rules are different in the two settings and one might expect that it is very unlikely that in the computation of an observable, miraculous cancellations lead to equivalent results. Yet, there are results in the literature explicitly showing that this happens, see, e.g., [37,38,47]. In fact, we shall see that our formal proof of equivalence extends also to the case when matter is present.
As an explicit check, we shall consider gravity non-minimally coupled to a scalar field and show that the one loop UV divergences are the same for GR and UG. This disagrees with [43], who claimed that a particular dimensionless combination of couplings, called ∆, has different beta functions in the two settings. In our calculation, the beta functions turn out to be the same. What is perhaps more important, we find that the beta functions of ∆ are gauge-dependent, which may at least in part explain the discrepancy. Furthermore, the implementation of the unimodularity condition adopted in [43] is different from the one we use in this paper. We therefore think that the question whether different formulations of quantum UG can lead to different physical predictions than GR remains still open.
The paper is structured as follows: In Sect. II, we define the path integral of diffeomorphisminvariant theories and formally show that it is possible to partially fix the gauge so as to reduce it to the one of UG. In Sect. III, we perform an explicit computation in order to verify the claim of Sect. II in the presence of matter. We consider scalar-tensor theories with a non-minimal coupling.
We then proceed to the calculation of the one-loop beta functions in these theories both in the full diffeomorphism and special diffeomorphism invariant cases. In particular, we compute the running of ∆ and discuss how it depends on the choice of gauge. We also display the results of the running of ∆ in GR with linear parameterization of the metric, in a generic linear covariant gauge, and compare our findings with the available literature. We collect our conclusions and perspectives in Sect IV. Appendix A is mainly intended for users of the functional renormalization group and explains how to extract the logarithmic terms of the beta functions. Appendix B contains some long expressions that are omitted in the main text.

II. EQUIVALENCE OF PATH INTEGRALS
The starting point of our analysis is the (Euclidean) 5 path integral defined by a gravitational action S Diff (g µν ), g µν = g µν (ḡ; h) being the metric,ḡ µν a fixed background metric and h µν the fluctuating field which is integrated over. The fluctuating field h µν does not need to be small, i.e., a perturbation aroundḡ µν . Moreover, the split of the full metric g µν in background and fluctuating parts is also general, not being restricted to the standard additive (linear) split. The action is assumed to be invariant under diffeomorphisms (but it is not restricted to be the Einstein-Hilbert action), and so is the functional measure Dh µν . Formally, the path integral is expressed as The factor V Diff stands for the volume of the diffeomorphism group.
In most practical calculations within a continuum quantum-field theoretic setting, a gaugefixing term must be introduced in (4). This is typically achieved by the Faddeev-Popov procedure.
The redundancy is generated by vector fields ǫ µ which can be decomposed as with ∇ µ ǫ µ T = 0 and ∇ µ the covariant derivative defined with respect to the metric g µν . The transverse vectorfields ǫ µ T generate the group SDiff of special (volume-preserving) diffeomorphisms. Instead of introducing a single gauge-fixing condition for the entire group of diffeomorphism, we introduce two different conditions, first breaking Diff to SDiff, and then breaking SDiff. This strategy has been discussed and worked out in a different way in [24], see also [25] and [48] for a general discussion of partial gauge fixing. In the first step we choose a gauge-fixing functional F(g) and insert the standard Faddeev-Popov identity given by 5 The Euclidean signature is not essential at this stage and the results could be equally deduced in the Lorentzian case.
where ∆ F (g) denotes the Faddeev-Popov determinant. The notation g φ denotes the transformation of the metric generated by the longitudinal vectorfield ∇ µ φ: We can now plug (6) in (4) leading to Following the standard steps we now use the gauge invariance of the measure, of the Faddeev-Popov determinant and the action and redefine the integration variable, to get In [23,36] it was shown that where V SDiff denotes the volume of the SDiff group. Hence, An explicit example of this first stage of gauge fixing is the unimodular gauge defined by ω(x) being a fixed scalar density. The delta function in (11) enforces that the full dynamical metric is unimodular. The corresponding Fadeev-Popov determinant is The contribution due to ω 2 (x) in (13) can be absorbed in a normalization factor of the path integral and thereby it is harmless. Finally, by plugging (13) into (11), yields Due to the presence of the delta functional in (14), the action in the Boltzmann factor collapses to by ω(x) and when expanded in h µν , the constraint F(g) = 0 must be imposed. Eq. (14) is the path integral of UG with the unimodular measure (Dh µν ) UG defined by i.e, One particular parameterization which is well-suited for the implementation of the unimodularity condition is the exponential split Unimodularity of g µν is achieved by requiring the background to be unimodular (detḡ = ω 2 (x)) and that the flucuations h µν are traceless 6 .
In order to complete the gauge-fixing procedure, one applies again the Faddeev-Popov method for a gauge condition which fixes the SDiff invariance. This is achieved, e.g., by taking the standard linear covariant gauges in quantum gravity and applying the transverse projector to it. We refer to [23,30,31,33,36,44,45,50] for more details.
We remark that eq.(16) does not rely on the specific form of the gravitational action. Morever, if matter interactions were included (also of arbitrary form), the equivalence would still hold.
In this case, the matter action S Diff g → ω and fluctuations satisfying the constraint defined by the delta functional in (15). Thus, we expect that gravity-matter systems in a full diffeomorphism-invariant setting are equivalent, quantum-mechanically, to gravity-matter systems in the unimodular framework.

III. NON-MINIMAL COMPARISONS IN SCALAR-TENSOR THEORIES
This section is devoted to the explicit calculation of one-loop divergences in gravity-matter systems, illustrating the quantum equivalence between Diff -and SDiff -invariant theories. In particular, we focus on scalar-tensor theories including non-minimal couplings between gravity and the scalar field.
This system has been discussed recently in [43], where it was claimed that through consideration of a suitable dimensionless combination of couplings ∆, it is possible to distinguish GR from UG. This would seem to contradict our results. While for us the beta function ∆ is the same in the two theories, it turns out to be gauge dependent, thus weakening the significance of this test. We should also stress that in [43], the authors employ a different way of implementing the 6 Another efficient method is the "densitized" parameterization, see, e.g., [22,49]. If one opts for less efficient implementations, the unimodularity condition becomes difficult to implement in practical calculations. Nevertheless, for the partial gauge-fixing associated with the gauge freedom (7), there seem to be no generation of quartic ghost terms [48] due to the fact that one just introduces a ghost-antighost pair.
unimodularity condition, and we do not exclude the existence of quantization schemes that break the equivalence between GR and UG.

A. Action
The beta functions of GR coupled to a scalar have been derived previously in, e.g., [51] for the general class of actions This includes an arbitrary potential V and arbitrary non-minimal couplings parametrized by the function F . If one expands V and F in Taylor series in φ, with the additional assumption of We are especially interested in the dimensionless couplings ξ and λ, whose leading one-loop beta functions are universal, and in dimensionless ratios of the dimensionful couplings, such as G N Λ, G N m 2 , Λ m 2 , since their beta functions are also known to be less gauge-and parameterizationdependent. In [51], the beta functions were computed by the use of the functional renormalization group (FRG) equation which is based on a cutoff-like regularization. Thus, power-law divergences are also taken into account. In [43], on the other hand, the authors employed dimensional regularization which is blind to the power-law divergences. For a direct comparison, we would have to extract from the FRG the "universal" contributions, i.e., those related to logarithmic running. This is discussed in Appendix A 1. In the next section we directly extract the beta functions from the logarithmic divergences, calculated with heat kernel methods.

B. Dynamical gravitons: UG or GR in exponential parametrization
We start from GR in the exponential parametrization (17) and follow the procedure of [52]. We decompose the metric fluctuation in its irreducible spin 2, 1 and 0 components: with∇ µ h TT µν = 0,∇ µ ξ µ = 0 andḡ µν h µν = h. A redefinition of the fields σ and ξ µ is performed in order to cancel the Jacobian generated by the York decomposition (21), We take the background metricḡ µν to be a four-dimensional Euclidean maximally symmetric space. Then we choose the "unimodular physical gauge", which consists of setting to zero the spin one field ξ ′ µ and the spin-0 field h. With these choices, the gauge fixed Hessian is As a further simplification we note that defining 7 the gauge fixed Hessian becomes diagonal, The unimodular physical gauge produces Faddeev-Popov ghost determinants with the subscripts 0 and 1 denoting the spin of the fields that the corresponding operators act on.
Thus the one-loop partition function reads where This agrees with the standard result for GR with a cosmological constant, except for the appearance of the additional scalar determinant.
Consider now the same calculation in UG. The trace fluctuation h is absent from the degrees of freedom from the start and it is therefore not necessary to fix the corresponding gauge. The SDiff gauge can be fixed again by setting ξ ′ = 0. Altogether this produces the Faddeev-Popov On the other hand, as discussed in [23,36], the factorization of the volume of SDiff produces and additional determinant det(−∇ 2 ) which cancels the determinant coming from the integration over σ ′ , so that the final result is again exactly (27). Notably, such an equivalence holds irrespective of the choice of F (φ).
In a standard perturbative approach, the beta functions can be read off from the logarithmic divergences. The one-loop effective action is and its divergent parts can be obtained from with Λ standing for an ultraviolet cutoff and µ being a reference scale. The first two contributions in (31) only give terms of order R 2 and are not relevant for the beta functions of interest. For ∆ S we have Inserting (19) and (20) and expanding in powers of φ we obtain the beta functions We note that the leading terms are the same as for the pure scalar theory, discussed in Appendix A.
Here we only kept correction terms linear in G. The explicit one-loop computation reported above leads to the same results in GR and UG, since (31) is the same in both cases. Moreover, we have kept only the contributions generated by the universal Q-functionals. The remaining terms are associated to power divergences and are not universal. The conclusion of this explicit calculation agrees with our statement in Sec. II. In particular, the non-minimal scalar-gravity coupling does not change this conclusion.
As is well-known, quantum-gravity contributions to matter beta functions can be gauge dependent. The "unimodular physical gauge" can be obtained from the standard two-parameter linear covariant gauge condition for Diff-invariant theories, namelȳ with b µ being a fixed function, by taking the limits α → 0 and β → −∞. For generic α, β, the beta functions β V and β G are left unchanged, while the others become, The contribution F(α, β, ξ) is lengthy and collected in the Appendix B. In the limit β → −∞, the beta functions turn out to be α-independent. It is also worth mentioning that the first two of these beta functions are also independent of another parameter that can be introduced in the definition of the measure, namely the use of a densitized metric as a quantum field, see, e.g., [49].
As a side comment, at one-loop order, the universal gravitational correction to the quartic coupling λ at vanishing non-minimal coupling is negative, irrespective of the values of the gauge parameter β, provided that 8 α < 3, At α = 3 or β → ±∞, the contribution vanishes at one loop. In particular, in the unimodular physical gauge, the gravitational contribution vanishes at vanishing ξ. However, in such a gauge, if the non-minimal coupling is included, the contribution is always negative at leading order in G.
Hence, such a contribution can balance the non-gravitational contribution to the one-loop running 8 In an Euclidean setting, α has to be non-negative.
of λ -which is positive and leads to the well-known triviality problem. In order to circumvent the issues due to the gauge dependence, and of the non-universal power-law terms, one will have to to compute a gauge invariant physical observable possibly along the lines of [53].

C. A universal beta function?
In [43], it was argued that the dimensionless combination of couplings has a universal beta function and carries a physical meaning. By quantizing UG in the presence of non-minimally coupled scalar fields, the authors claim that the results differ in GR and UG. More precisely, taking into account the differences in notation, their result for UG is while their result for GR is Hence, ∆ would be a physical quantity able to distinguish GR and UG if the scalar field is nonminimally coupled to the gravitational field.
Using our previous calculations, we cannot distinguish UG and GR non-minimally coupled to scalars at one-loop simply because the path integrals are the same. In particular, in the unimodular physical gauge, we obtain which differs from either of the results above. These discrepancies may be ascribed to the fact that we are using a different parameterizarion of the metric and a different implementation of the unimodularity condition.
What is perhaps more important is that, even if we stick to our computation scheme, the quantity ∆ is gauge dependent. In fact, in the linear covariant gauge (38), the result is with Thus, even in the absence of ξ, the beta function of ∆ is gauge dependent and comparing results for GR and UG would be problematic. We also remark that, in the limit β → ±∞, eq.(45) reduces to (44) irrespective of α.

D. Dynamical gravitons: GR in linear parametrization
So far, the explicit one-loop computations were performed using the exponential parameterization of the metric. In this parameterization, the unimodularity condition simply amounts to removing the trace mode of the gravitational field fluctuation h µν . While field redefinitons, properly done, should not affect the result of physical quantities, there are several subtleties when changing from one parameterization to another in quantum gravity. In this section, we present the one-loop results for the scalar-gravity system with a non-minimal coupling in the so-called linear parameterization, i.e., in the linear covariant gauges (38). This system was studied, e.g., in [51], but the beta functions were computed with the functional renormalization group and contained also non-universal terms. Here, we select just the universal contributions, that are related to logarithmic divergences.
In a general gauge (α, β), the result is completely equivalent to (39) apart from the beta function of the non-minimal coupling β ξ which reads where the explicit expression for G(α, β, ξ) is reported in Appendix B. In particular, if we take α → 0 and β → ±∞, we obtain which differs from F(0, ±∞, ξ). The beta function of ∆ in a general linear covariant gauge and in linear parameterization is the same as (45). Hence, although gauge-dependent, ∆ seems to display some kind of universality as far as different choices of parameterization is concerned. This fact is not very surprising given that, the only beta function in the linear parameterizarion that differs from the exponential parameterization at one-loop is β ξ and it does not enter the definition of β ∆ .
In [54], Kamenshchick and Steinwachs (see, also, [55]) investigated the one-loop divergences of a more general theory than the one considered in this work. In particular, they have considered a scalar-gravity action S KS [g, Φ] expressed as 9 where a = 1, . . . , N and N is a positive integer. The functions V and F depend onΦ = δ ab Φ a Φ b .
The gauge condition used in [54] is with ϕ a the scalar field fluctuations and n a = Φ a Φ . Unfortunately, our gauge condition (38) is not deformable to this, and therefore we cannot directly compare our results with theirs. However, we can extract from their work the beta function of ∆ in the gauge (51). The authors employed the linear parameterization of the metric and the reduction to the a single-scalar non-minimally coupled to gravity is achieved by taking N → 1, Φ a = n a = 1 andΦ → φ. Moreover, in order to have the same scalar-tensor action we discussed in this work, one has to take G(Φ) → 1 and their quantity s has to be identified as The beta functionals of V and F are β V = 2α 1 , β F = 2α 2 , where α 1 and α 2 are given in their equations (48) and (49). From there we read off which gives This confirms once more that the first and last term in the fraction are universal, but not the other ones.

IV. CONCLUSIONS
Disregarding the single global spacetime volume degree of freedom, we have shown at a formal path integral level that the classical equivalence between general (Diff -invariant) and unimodular (SDiff -invariant) versions of gravity theories, can be maintained at the quantum level 10 .
This is true independently of the choice of the action and also in the presence of matter.
We have then given an explicit one-loop example of this, by computing the universal parts of the beta functions of scalars coupled to gravity. In spite of significant differences in the two cases, the beta functions turn out to be the same. We have then compared these results to those of [43], who also made the same comparison. Our beta function for the dimensionless combination ∆ differs from theirs both for GR and UG. The differences can probably to be ascribed at least in part to the different way they implement unimodularity. A more detailed analysis has shown that the beta function of ∆ is actually gauge-dependent, so that it is not a sufficiently good test. There are two terms in the beta function of ∆ that are the same in all gauges and are the same across all calculations we could find in the literature, whereas other terms have strong gauge dependence.
For the future, it will be important to identify a genuinely universal combination of couplings, or another observable that can act as a benchmark.
We conclude with some comments on the cosmological constant. In UG, a "cosmological term" Λ 8πG ∫ d 4 x ω in the Lagrangian is just an additive, field-independent term that does not affect the equations of motion and can be absorbed in the overall normalization of the functional integral.
Thus, it has no physical effect. GR is only (classically) equivalent to UG if we impose that the total volume of spacetime is fixed. In this restricted theory the cosmological term in (3) is a Lagrange multiplier, whose value is ultimately related to the volume through the equations of motion.
Computations of the beta functions performed in the so-called unimodular gauge [52] show that the cosmological constant decouples from the system of beta functions. This resembles simpler calculations involving the functional renormalization group, where a field-independent contribution is generated by the flow and can be cancelled by a proper normalization of the vacuum energy. This suggests that its quartic running is unphysical. This is in line with other hints coming from different directions [56][57][58][59]. This and related issues deserve to be investigated further. In the literature, the beta functions of gravity, with or without matter, have been often calculated in the Functional Renormalization Group (FRG) framework, see, e.g., [60][61][62][63] for reviews on the subject). Since the FRG is based on a momentum cutoff, the beta functions contain terms proportional to powers of the cutoff, that are not seen with other techniques. In this appendix we discuss the way in which one can recover from the FRG the standard one loop beta functions that one would see, e.g. in dimensional regularization. For a more detailed discussion of the relation between the FRG and dimensional regularization we refer to [64].
In the FRG, a cutoff function R k is introduced by hand in the quadratic part of the action, in order to suppress the contribution to the functional integral of modes with (Euclidean) momenta smaller than a cutoff scale k. This leads to a coarse-grained effective action Γ k which coincides with the full effective action at k = 0. The flowing action Γ k obeys the flow equation where Γ (2) k is the Hessian constructed from Γ k . For our present purposes it will be enough to consider the simple case of scalar fields in a background metric, with a Hessian of the form Γ (2) k = ∆ where ∆ is a Laplace-type operator: This derives from a scalar action containing a potential and a non-minimal coupling to gravity.
Then the r.h.s. of the flow equation is a function W (∆) that, for constant φ, can be evaluated as with the Q-functionals defined as In eq.(A4), n ∈ R, W (k) (z) stands for the k-th derivative of W with respect to z. If n > 0, then k = 0. Otherwise, k is a positive integer such that n + k > 0. The heat kernel coefficients are In the flow equation, we are interested in computing Q-functionals of the form where P k (z) = z + R k (z). If, m = n + 1, then one can show that is "universal", i.e. independent of the shape of R k . For certain cutoff schemes the denominator in the function W is P k + E, and are, in general, non-universal quantities. Nevertheless, one can extract universal parts of each Q-functional defined in eq.(A8) by expanding in E: and exploiting the linearity of the Q-functionals to pick up the contribution which satisfies n = m + 1.
Consider first a "type III" cutoff, see, e.g., [60,65]. The beta functional iṡ Only the last term is universal. Thuṡ The relevant terms (up to linear order in R which are not total derivatives) are From here one reads off the beta functions The same result can be obtained in a more laborious way using a "type I" cutoff. In this casė The universal terms come from all three pieces in this expression, when one expands in E: the third term in the expansion for Q 2 , the second for Q 1 and the leading term for Q 0 . In the latter term, B 4 (−∇ 2 ) is of order R 2 and does not concern us. The rest iṡ which is clearly the same as before, and therefore leads to the same beta functions. Note that the universal parts of the beta functions are those that come from the dimensionless Q-functionals and therefore are independent of k, see eq.A7.
As for the linear parameterization, the expression for G(α, β, ξ) in (48) is