Symmetries and renormalization of (2 + epsilon) dimensional Unimodular Dilaton Gravity

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Universit`

a di Pisa

Dipartimento di Fisica “Enrico Fermi”

Tesi di Laurea Magistrale in Fisica

Symmetries and renormalization of (2 + )

dimensional Unimodular Dilaton Gravity

Candidato: Francesco Del Porro

Relatore: Dr. Omar Zanusso

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Chapter 1 Introduction

General Relativity (GR) is a very successful theory in describing the gravitational phe-nomena from the classical point of view. The usual Lagrangian formulation of the theory is through the Einstein-Hilbert (EH) action, which in d dimensions is:

SEH = −ZN

Z

[ddx√−g] R(g) . (1.1) Here ZN = G−1 = (16πGN)−1 is inversely proportional to the Newton’s coupling

con-stant GN, g is the determinant of the metric tensor and R is the scalar curvature.

Indeed, it is well-known that the framework in which the theory is embedded is the curved spacetime, which is considered a Lorentzian manifold (thus g < 0) with some nonzero curvature [1]. If we take into account matter, that is to say: if we consider an additional part of the action Smatter, so as our total action is Stot = SEH + Smatter,

we can derive the equation of motions (a.k.a. Einstein equations), which show that the spacetime is curved by the matter distribution:

Rµν−

1

2Rgµν = 8πGN

c4 Tµν, (1.2)

where Tµνis the energy-momentum tensor associated to some matter distribution (Tµν =

0 in the vacuum).

Classically, the huge success of the theory is highlighted by a plethora of theoretical and experimental results. We can recall the gravitational lensing, the gravitational redshift, the gravitational waves and many others [2,3, 4].

Problems arises if we try to study the quantum mechanical aspects of the theory. Even if GR has evidently the form of a field theory from its inception, traditional at-tempts to quantize it as a quantum field theory lead either to failure or to an incomplete theory. We can, in fact, build effective theories up to some energy scales, but not yet a fundamental theory which describes the physics at all energy scales.

The incompatibility between the renormalization technique and the perturbative expansion in powers of G is the root cause of this problem. It is well-known that a the-ory, in order to be pertubatively renormalizable must not have “negative dimensional” coupling constants in its bare action. Working in natural units c = ~ = 1, indeed, the action S is dimensionless, thus the Lagrangian must be of dimension [L] = d. In 4 dimensions, it implies that the coupling G has the dimension of the inverse of an energy squared [G] = −2. More generally, we can determine the dimension of G for any d. Since [g] = 0:

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Chapter 1

[R] = [∂2] + [g] = 2 =⇒ [G] = 2 − d . (1.3) Then [G] < 0 for any d > 2. Therefore, the traditional approach might not be the right way to proceed.

There are currently several attempts to modify the EH Lagrangian, in order to make gravity renormalizable. For example, one can consider the “higher derivative models”, which are built with the insertion of higher-dimensionality terms in the Lagrangian like R2_{, R}

µνRµν, RR ad so on. A higher derivative model can be described by an action

SHD which in d = 4 can be of the form

SHD = − Z [d4x√−g] Λ + ξR + αRµνRµν+ βR2 . (1.4)

A model such as SHD is perturbatively renormalizable [5] and even asymptotically

free [6]. The coupling constants have non-negative dimensions [Λ] = 4, [ξ] = 2, [α] = [β] = 01. These theories are attractive from the point of view of renormalization, but they lead to some problem with unitarity. The propagator D in the momentum space has the form D−1(p) ∼ a(p2₎2 _{+ bp}2_{. If we set b = −m}2 _{(it has the dimension of a}

squared energy) and a = 1 for simplicity, we obtain D(p) ∼ 1 p2_(p2_{− m}2₎ = 1 m2 1 p2 − 1 p2_{− m}2 . (1.5)

The presence of two poles of different signs does not satisfy the optical theorem. This problem has been faced, for example, in [9,10] by Anselmi and Piva.

We point out that the particular feature in facing gravity, instead of the other three interactions, is that we are not currently able to observe its microscopic behaviour from the experimental point of view [11]. Even if its macroscopic effects (which correspond to the “large distances” or “low momenta” limit) are clearly visible, in order to get the gravitational “short distances” effects we expect that we would have to reach energies comparable to the Planck mass Mp = G

−1/2

N ∼ 1019GeV, which is the characteristic

energy scale of the interaction. So, building a new Lagrangian (if there is one) which correctly describes also the microscopic behaviour could not be so straightforward.

From these considerations, a point of view can be developed, which interprets EH gravity as an effective theory. The prototype example that can be used to explain the concept of effective theory is “chiral dynamics”: even if we know the pion as a composite quark-antiquark bound state, there is a model in which it represents the fundamental actor. This model’s Lagrangian, which is described considering the scalar field π(x) = {πa(x)} as a three-dimensional representation of the global symmetry

SU (2) ⊗ SU (2) and it is described by the Lagrangian Lef f = −

1 2F2

π

Dµπ · Dµπ − f (Dµπ · Dµπ)2+ . . . , (1.6)

where Fπ ' 190M eV and Dµπ = ∂µπ/(1 + π2) is the covariant derivative. f = O(1)

is a numerical value to be determined by an experiment at some scale energy. If we

1_{Actually, the perturbative series is constructed in powers of the inverse of the couplings α and β.}

Instead, ξ, if needed, can be understood as a relevant parameter, comparable to the square mass of a scalar field, thus avoiding the problem of the negative dimensionality altogether. The theory can be constructed consistently without the need of scales [7,8]

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Chapter 1

manage to compute a ππ scattering amplitude A(p2) at some energy p2. For suffi-ciently small energy, the leading order is given by the first term in the Lagrangian, thus A(p2_{) ∝ p}2_/F2

π. When we take into account the next-to-leading order we take the 1

loop corrections from the first term and the leading order from the second term, thus: A(p2) = ap 2 F2 π + p 4 F4 π b lnp 2 µ2 + cf (µ) + O(p6/F_π6) . (1.7)

Here µ represent the scale at which we renormalize f . The logarithmic term is the one loop contribution of the leading-order term. We can see that, going further with energies, potentially infinitely many terms appear. Then, a theory which contains only the first contribution is an effective theory up to some energy p2 ≤ F2

π. Going to higher

energies means taking into account the internal structure of the pion.

If we consider EH theory as an effective, leading order theory, we can build the same kind of model, for example, interpreting the graviton-like fluctuations hµν of the

metric, gµν = ηµν+ hµν, as an effective degree of freedom and expanding the scattering

processes in powers of GNp2 = p2/Mp2. This type of solutions are explored, for example,

in [12] and in the introduction of [13].

In 1979, Steven Weinberg proposed an alternative idea on how to approach the problem [13]. In order to give an answer to the question about the UV behaviour of gravity (so with the goal of describing a fundamental, and not effective theory), he proposed a new way which, in modern terms, is formalized through the so-called “renormalization group” (RG) and goes under the name of Asymptotic Safety.

Asymptotic Safety (AS) is way to define a theory at all energy scale. Being asymp-totically safe means, for a theory, that its ultraviolet limit is finite. In other words: if we imagine a theory, such as gravity, that describes interactions through a coupling constant, such as G, at some renormalization group scale energy k (so if k varies, G is running, G = G(k)), an asymptotically safe theory has a finite limit k → ∞ for G: G∗ = limk→∞G(k). G∗ takes the name of “fixed point”. If this fixed point exists, we

can control the UV behaviour of the theory, even if it is not perturbatively renormaliz-able. AS is, in a precise sense, a generalization of asymptotic freedom, which replaces the Gaussian ultraviolet behavior with a nontrivial one. Under specific conditions, which will be explored later, an asymptotically safe theory can be as predictive as an asymptotically free one.

Verifying the asymptotic safeness of a theory may not be so easy. In 4 dimensions, some non-perturbative methods have been developed, in order to check if Gravity could be asymptotically safe or not [14]. However, even if these methods have the huge ad-vantage to work directly on 4 dimensional gravity, they involve explicit cutoff functions, hence scheme dependence, so the universality of the results may not be so trustworthy. Weinberg’s conjecture, however, exploits another way to reach this goal. First, let us point out that, beside the use of the aforementioned non-perturbative methods, there is the possibility to perform reliable computations by setting our theory slightly above 2 dimensions. In d = 2 dimensions SEH is perturbatively renormalizable. By extension,

in d = 2+, where 0 < 1, the theory can be constructed perturbatively in powers of . Thus, we can try to check its asymptotic safeness perturbatively in . The analytic continuation of the theory in the dimensionality, allows us to use a very convenient framework in which to develop our computation: the dimensional regularization. This provides universality, because all the divergences appear as 1/ poles and there is no scheme dependence, not in the form of a cutoff nor in the form of any other parameter.

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Chapter 1

On the other side, it does not have the advantage to work directly in 4 dimensions, that is the limit we are interested in.

Weinberg suggests an argument, that he calls “dimensional continuation” in which is observed the fact that the fixed point continuously depends on the dimensionality (we’ll see it). So it seems plausible, even if not so-well mathematically justified, to consider the non-perturbative limit → 2, once we have proven the asymptotic safeness of Gravity. We will see that this limit does not cross poles or other divergences and it seems not to present any problem when we perform it.

It is important to underline once more that this recipe of proving Gravity AS is not definitive, but it is a parallel path to explore in addition to all the other attempts, which go in the same direction. Nevertheless, exploring this idea will be the main goal of this thesis, which can be conceptually divided in two parts.

The first one, principally contained in Chap.s 2 and 3, is the presentation of the model of gravity to which we want to apply Weinberg’s conjecture.

It is a matter of fact that EH theory is not the unique formulation of gravity. It will turn out that, playing with symmetries, we can extract an equivalent theory of gravitation that will fit our computation better than Einstein and Hilbert’s. Let us briefly anticipate how.

It is well-known that the action (1.1) is invariant under change of coordinates. The group that defines the symmetry under diffeomorphisms can be denoted by Dif f . We will define, through Chap. 2, a locally equivalent model, named “Unimodular Dilaton Gravity” (UDG), where this invariance is implemented by a different group Dif f∗, which acts on the coordinates in such a way to leave the determinant of the metric unchanged (those kind of transformation are called “unimodular transformations”).

An important property that we will show is that the two symmetries Dif f and Dif f∗ are linked by an isomorphism: in this sense they are nothing but a reformulation of the same invariance.

So, why do we choose UDG instead of GR? It turns out that UDG is a version of Gravity which reproduces exactly its 2 dimensional limit. In fact, two dimensional Gravity is well understood and studied [15], also for its application to string theory [16]. The trivial limit d → 2 of the EH action, however, does not correspond to the usual formulation. As a matter of fact the two dimensional integral

Z

[d2x√−g]R (1.8)

is purely topological. The 2 dimensional limit of Gravity, however, is dynamical and is related to a model known as Liouville theory.

UDG reproduces continuously this limit in 2 dimensions. This makes the theory a sort of “bridge” between Gravity in 2 dimensions and Gravity above 2 dimensions (and hopefully Gravity in 4 dimensions) and it is only a result of consideration on symmetries.

In the second part of this thesis we try to face UDG in (2 + ) dimensions, with the attempt of proving AS. In Chap. 3 we find the 1 loop expansion in . There we will use the background field method technique, which is a well-known framework used in QFT in order to perform perturbative expansion. Then, since we are dealing with curved spacetime, we will make use of the so-called “heat-kernel method”, that is a way to covariantly dimensional-regularize the theory without computing separately all the Feynman diagrams. We will not develop all the theory about it, but we will provide,

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Chapter 1

where possible, simple example (in this case the example is made with a scalar field theory in flat spacetime) to explain how it works, before applying it.

The last part is dedicated to AS. In Chap. 4, Asymptotic Safety is explained through a modern framework: the functional renormalization group (FRG). This because of two reasons:

• FRG is an example of a quantitative, non-perturbative method to verify AS in Gravity [17]

• It provides a modern and conceptually well-defined method with which we can present the idea of Asymptotic Safety.

Finally, in Chap. 5, we will discuss Weinberg’s conjecture applied to UDG: both in its pure gravity version and with a cosmological constant. We will prove the existence of a fixed point in pure gravity UDG and we will show that its existence is not disturbed by the presence of a cosmological constant. Then we will conclude with some considerations on the predictive power of this theory, through the analysis of the UV critical surface.

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Chapter 2 Unimodular Dilaton Gravity

In this Chapter our model is presented. The goal is to start with the usual formulation of Gravity and end up with another equivalent model, that will be our framework for this thesis.

The starting point is the EH action in d dimensions: SEH = −

1 G

Z

[ddx√−g]R , (2.1)

where G = Z_N−1 is the coupling constant, proportional to GN, g = det(gµν) is the

determinant of the metric (which is negative in the Lorentzian case) and R is the scalar curvature. To clarify the notation, let us write explicitly our convention for the involved quantities: R = δ_µρgνσR_νρσµ , Rµ_νρσ = ∂ρΓµνσ− ∂σΓµνρ+ Γ µ ρλΓ λ νσ − Γ µ σλΓ λ νρ, Γρ_µν = 1 2g ρλ_[∂ µgλν+ ∂νgλµ− ∂λgµν] . (2.2)

It will turn to be convenient for us to switch from the Lorentzian framework to the Euclidean one. In a curved spacetime, it is not so obvious how the Wick rotation has to be performed. Although an idea of how to do it is given in the Appendix A, we assume that a Wick rotation exists and the Euclidean version of Einstein Gravity is described by SEH = − 1 G Z [ddx√g]R . (2.3)

So far we wrote nothing but the GR action, which, by construction, is a theory invariant under diffeomorphisms. Note that in the Euclidean case the determinant of the metric is positive, thus the minus sign in front of it disappears after the Wick rotation.

2.1 Dilaton Gravity

The first step is the insertion, “by hand”, of another degree of freedom in the form of a conformal factor. So, let us consider a conformal reparamatrization on the metric tensor:

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2.1. Dilaton Gravity Chapter 2

where Ω(x) is a scalar field, that we choose to call Dilaton. The metric ˜gµν is a

fiducial metric tensor, which role will be clear later on. The inverse of gµν must be

reparametrized with the factor Ω−2 in such a way to respect the relation gµρgρν = δµν.

2.1.1 DG action

With the insertion of the Dilaton, we modify the action that must be written in terms of ˜gµν. First of all, we look at the linear connection:

Γρ_µν = ˜Γρ_µν + Ω−1(δρ_µ∂νΩ + δνρ∂µΩ − ˜gρλg˜µν∂λΩ) = ˜Γρµν+ C ρ

µν, (2.5)

where ˜Γ has the same expression of Γ with the replacing g → ˜g. It is well-known that the connection is not a tensor. To clarify the nature of the quantity C_µνρ , let us consider the rule of transformation of Γ under change of coordinates x → y(x):

Γρ_µν(y) = ∂y ρ ∂xλ ∂xα ∂yµ ∂xβ ∂yνΓ λ αβ(x) + ∂yρ ∂xγ ∂2_xγ ∂yµ_∂yν . (2.6)

In the equation above, let us focus on the two contribution in RHS. The first one is exactly the way which we expect a (2,1) tensor transforms, the second one is a term that depends only on the form of the function y(x) we have chosen and not on the metric tensor. So, if we consider the difference (Γ − ˜Γ), that is the quantity we named C, we note that this transforms like a tensor. Then, Cµ

νρ is a (2,1) tensor, thus we can

define its covariant derivative.

We can now write down the transformation of the Riemann tensor:

Rµ_νρσ = ∂ρΓµνσ− ∂σΓµνρ+ Γ µ ρλΓ λ νσ − Γ µ σλΓ λ νρ = ∂ρ(˜Γ + C)µνσ− ∂σ(˜Γ + C)µνρ+ (˜Γ + C) µ ρλ(˜Γ + C) λ νσ− (˜Γ + C) µ σλ(˜Γ + C) λ νρ = ˜R_νρσµ + ˜∇ρCνσµ − ˜∇σCνρµ + C µ ρλC λ νσ− C µ σλC λ νρ = ˜R_νρσµ + [ ˜∇ρCνσµ + C µ ρλC λ νσ− (ρ ←→ σ)] (2.7)

For all the computations that lead to the following results (expressions (2.8)), we refer to the Appendix B. Here are reported only the final expressions, such as the transformation rules of the Ricci tensor and the scalar curvature:

Rνσ = ˜Rνσ − Ω−1[(d − 2)δσαδ β ν + ˜g αβ_g_˜ νσ] ˜∇α∇˜βΩ+ + Ω−2[(2d − 4)δα_σδ_νβ− (d − 3)˜gαβg˜νσ]( ˜∇αΩ)( ˜∇βΩ) , R = Ω−2R − Ω˜ −3(2d − 2)˜gαβ∇˜α∇˜βΩ − Ω−4(d − 4)(d − 1)˜gαβ( ˜∇αΩ)( ˜∇βΩ) . (2.8)

So a new action SD[˜gµν, Ω] ≡ SEH[Ω2g˜µν] is built, by replacing (2.8) in Eq. (2.3).

This theory, that is the first step of our model, is called Dilaton Gravity (DG), due to the new degree of freedom, the Dilaton, introduced through the conformal transformation (2.4).

However, it is customary to rewrite the dilaton field in a proper way in order to simplify the Lagrangian. If we introduce a “new dilaton” ϕ(x), defined by the relation Ω(x) = ϕ(x)d−22 , the new DG action S

D[˜gµν, ϕ] (which is nothing but a rewriting of the

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2.1. Dilaton Gravity Chapter 2 SEH[gµν] = SD[˜gµν, ϕ] = − 1 G Z [ddxp˜g] ϕ2R − 4˜ d − 1 d − 2g˜ µν_{ϕ ˜}_∇ µ∇˜νϕ . (2.9) A first look on the action points out two particular features:

• There is a pole in dimensions d = 2. This is due to the fact that, introducing ϕ instead of Ω, we perform a change of variables that is ill-defined in exactly 2 dimensions (see Sec. 2.2.3 for more details). However, our purpose is to analyze the theory in d > 2.

• The sign in front of the term ϕ ˜∇2_{ϕ is the opposite of the usual kinetic term’s one:}

this phenomenon is called Conformal mode instability. If the DG action is viewed as a gravitational action, however, the sign is right. Let us note that between 1 and 2 dimensions the instability does not arise and the kinetic term’s sign is right as well.

2.1.2 DG symmetries

DG action is a reformulation of GR, which, although it is a different theory, maintains the usual invariance under diffeomorphisms: the Lagrangian LDG has the form of a

massless scalar field coupled to gravity (apart from the kinetic term’s sign, which does not matter for this argument), that is well-known to be invariant under change of coor-dinates. We remind that, given an infinitesimal diffeomorphism ξµ_{, the GR symmetry}

is implied by the infinitesimal transformation δξgµν:

δξgµν = Lξgµν = gρν∇µξρ+ gρµ∇νξρ. (2.10)

DG’s symmetry is realized by the group Diff, also generated by the Lie derivatives. The infinitesimal transformations which imply this invariance are the following, where ξµ _{is again an infinitesimal diffeomorphism:}

δξg˜µν = Lξ˜gµν = ˜gρν∇˜µξρ+ ˜gρµ∇˜νξρ, δξϕ = Lξϕ = ξµ∂µϕ . (2.11)

Then, if we look closely at the relation (2.4), it’s easy to note that the invariance is larger than Diff. Let us see why: if we consider a transformation both on the metric ˜gµν

and the Dilaton ϕ, such that the transformed quantities g_µν0 and ϕ0 satisfy the following relation: ˜ gµνϕ 4 d−2 = g0 µνϕ 0 4 d−2 , (2.12)

then it is clear that the action is, by construction, invariant under this transformation SD[˜gµν, ϕ] = SD[gµν0 , ϕ

0_{]. The realization of this invariance is expressed by the group}

which we can call Weyl that acts on ˜gµν and on ϕ as:

˜

gµν(x) → gµν0 (x) = ω(x) 2_g_˜

µν(x) , ϕ(x) → ϕ0(x) = ω(x)1−d/2ϕ(x) , (2.13)

where the function ω(x) is a scalar function, which is the parameter that realizes W eyl. So, the introduction of the Dilaton changed the symmetries of the theory which now are expressed by the product Dif f n W eyl, which is the total DG symmetry.

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2.2. Unimodular theory Chapter 2

2.2 Unimodular theory

To define our model we want to exploit the enlarged symmetry by reducing the DG theory to its unimodular version. This theory, which is going to be our target in applying the concepts introduced in Chap. 1, takes the name of Unimodular Dilaton Gravity (UDG).

The recipe consists in breaking the total symmetry Dif f n W eyl into a special subgroup, imposing that the determinant ˜g stays fixed under the action of this new subgroup. In other words we restrict to a smaller symmetry, the “special” subgroup Dif f∗ through a pattern:

Dif f n W eyl −→ Dif f∗ (2.14) by the requirement δ∗g = 0, where the meaning of δ˜ ∗ is the variation under the action of Dif f∗. Let us see how to develop this idea.

The determinant ˜g can be rewritten in the exponential form ˜g = det(˜gµν) =

exp{tr[log(˜gµν)]}, thus its variation can be expressed generically as

δ˜g = δetr[log(˜gµν)] _{= etr}[log(˜gµν)]_δtr[log(˜_g

µν)]

= ˜gtr[δ log(˜gµν)] = ˜gtr[˜gµαδ˜gαν] = ˜g˜gµνδ˜gµν.

(2.15)

So, the request δ˜g = 0 is totally equivalent to say that the variation must be traceless δtr(˜gµν) = 0.

The idea through which realize the pattern (2.14) is that every variation caused by the action of Dif f must be compensated by the action of W eyl, in order to left the determinant ˜g unchanged. Under Dif f the original metric tensor gµν transforms as

δξgµν = gµλ∇νξλ+ gνλ∇µξλ. (2.16)

Then, rewriting Eq. (2.16) by adding and subtracting the trace part, we obtain: δξgµν = gµλ∇νξλ+ gνλ∇µξλ− 2 dgµν∇λξ λ | {z } traceless part +2 dgµν∇λξ λ | {z } trace part . (2.17)

The variation of g can be written in terms of ˜g and ϕ: δgµν = δ ˜ gµνϕ 4 d−2 = ϕd−24 δ˜g µν+ 4 d − 2g˜µνϕ 4 d−2−1δϕ . (2.18) If we want that the trace of the variation to be zero, we have to match correctly the traceless part with the term proportional to δ˜gµν and the other with the

Weyl-compensated term that contains δϕ, so we obtain the following relations: ( ϕd−24 δ˜g_µν = g_µλ∇_νξλ+ g_νλ∇_µξλ− 2 dgµν∇λξ λ 4 d−2g˜µνϕ 4 d−2−1δϕ = 2 dgµν∇λξ λ , (2.19)

which we can rewrite in terms of ˜gµν:

(

δ˜gµν = ˜gµλ∇νξλ+ ˜gνλ∇µξλ −2_d˜gµν∇λξλ

δϕ = d−2_2d ∇λξλϕ

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The covariant derivative ∇ in terms of ˜∇ and C has the form ∇µξν = ˜∇µξν+ Cµρν ξρ.

If we replace ∇ with this expression, the C-terms cancel out each other (it is an easy computation), thus we get the final expression for the variation of ˜g under the special subgroup Dif f∗, that is

(

δ∗_ξg˜µν = ˜gµλ∇˜νξλ+ ˜gνλ∇˜µξλ −_d2˜gµν∇˜λξλ = Lξg˜µν − 2_dg˜µν∇˜λξλ

δ∗_ξϕ = d−2_2d ∇˜λξλϕ + ξµ∂µϕ = Lξϕ +d−2_2d ∇˜λξλϕ

. (2.21)

The action of the group Dif f∗ is the same as the original Dif f , plus a contribution given by the degree of freedom offered by Weyl rescalings. By construction, we have tr(δ∗˜gµν) = 0: the Dilaton, to make it possible, has to change its variation in a proper

way.

2.2.1 Dif f ∼ Dif f

∗

We developed a new formulation of GR, the UDG, with different symmetries. It is a matter of fact that the new group Dif f∗ is nothing but a reformulation of the standard group Dif f of the diffeomorphisms. In this section we want to show that there exists an isomorphism between the two.

Given the infinitesimal transformation under Dif f (Eq. (2.11)), it’s easy to show (from Lie derivative’s properties) that, given two vector fields ξµ _{and ζ}µ_{, the following}

relation stands

[δξ, δζ] = [Lξ, Lζ] = L[ξ,ζ]= δ[ξ,ζ]. (2.22)

In order to prove the statement Dif f ∼ Dif f∗, it is sufficient to show that, given again ξµ _{and ζ}µ_{, which generate two one-parameter transformation dragged by the Lie}

derivative, the relation (2.23) is valid:

[δ_ξ∗, δ∗_ζ] = δ∗_[ξ,ζ]. (2.23) Dilaton Applying [δ∗_ξ, δ_ζ∗] to ϕ(x) we obtain: [δ_ξ∗, δ_ζ∗]ϕ = δ_ξ∗ Lζϕ + d − 2 2d ( ˜∇ · ζ)ϕ − (ξ ↔ ζ) = Lξ Lζϕ + d − 2 2d ( ˜∇ · ζ)ϕ +d − 2 2d ( ˜∇ · ξ) Lζϕ + d − 2 2d ( ˜∇ · ζ)ϕ − (ξ ↔ ζ) = [Lξ, Lζ]ϕ + d − 2 2d ϕ[Lξ( ˜∇ · ζ) − Lζ( ˜∇ · ξ)] = L[ξ,ζ]ϕ + d − 2 2d ϕ[Lξ( ˜∇ · ζ) − Lζ( ˜∇ · ξ)] . (2.24) If we take take the last term in the previous equation and we manage to show that stands the relation Lξ( ˜∇ · ζ) − Lζ( ˜∇ · ξ) = ˜∇λ[ξ, ζ]λ, we are done. This relation is

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proved in the following equation, using the fact that the quantity ˜∇ · V (Vµ _{is a vector}

field) is a scalar, thus we know how the Lie derivative acts on it:

Lξ( ˜∇ · ζ) − Lζ( ˜∇ · ξ) = ξµ∇˜µ∇˜νζν − ζµ∇˜µ∇˜νξν = ξµ∇˜ν∇˜µζν − ζµ∇˜ν∇˜µξν+ ξµ[ ˜∇µ, ˜∇ν]ζν− ζµ[ ˜∇µ, ˜∇ν]ξν = ˜∇ν(ξµ∇˜µζν − ζµ∇˜µξν) − ξµR˜αµζα+ ζµR˜αµξα = ˜∇ν(ξµ∂µζν − ζµ∂µξν + ξµΓ˜νµαζ α_{− ζ}µ_Γ_˜ν µαξ α₎ = ˜∇ν[ξ, ζ]ν. (2.25) Finally, if we put Eq. (2.25) into Eq. (2.24) we obtain what we want to prove:

[δ∗_ξ, δ_ζ∗]ϕ = L[ξ,ζ]ϕ + d − 2 2d ϕ[Lξ( ˜∇ · ζ) − Lζ( ˜∇ · ξ)] = L[ξ,ζ]ϕ + d − 2 2d ϕ( ˜∇ · [ξ, ζ]) = δ∗_[ξ,ζ]ϕ . (2.26) ˜ gµν field

In exactly the same way, we can prove the relation for ˜gµν. As a matter of fact, from

Eq. (2.21) we obtain, with almost the same computation done in Eq. (2.25), a similar expression for gµν as for the Dilaton. We do not repeat the calculation, since it depends

only on the Lie derivative’s properties and not on the spin of the field to which is applied: [δ_ξ∗, δ_ζ∗]˜gµν = L[ξ,ζ]˜gµν− 2 dg˜µν[Lξ( ˜∇ · ζ) − Lζ( ˜∇ · ξ)] = δ ∗ [ξ,ζ]g˜µν. (2.27)

The two equations (2.26) and (2.27) prove directly that exists a isomorphism be-tween the two algebras. So, even if EG and UDG are not the same theory, we will see later on that they are both a theory of gravity with two different, but equivalent, implementation of the invariance under changes of coordinates.

2.2.2 Unimodular and Einstein Gravity

We started from Einstein gravity (EG) in order to define our UDG theory. In doing this, we introduced the conformal reparametrization, as expressed in Eq. (2.4). The path that links EG to UDG crosses DG theory: the original gauge symmetry (Dif f ) is enlarged, then it is broken to build the unimodular theory.

We can see the process by another point of view: as a matter of fact, if we start from DG, both EG and UDG are particular realization of the broken symmetry. We can thus obtain both theories by following one of two different patterns.

Dif f ←− Dif f n W eyl −→ Dif f∗ (2.28) If we want to realize EG from DG it is sufficient to fix the value of the Dilaton ϕ = 1. The realization of UDG consists in fixing the value of ˜g.

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In some sense, then, DG is at a “higher level”, compared to the other two. Obviously, EG and UDG are not the only two theories that could be built using DG as their linking theory. In [18] other possible models are presented and is highlighted how the presence of a linking theory makes those models locally equivalent. One can go further, and break again the gauge symmetry, in order to obtain other theories. We will not go deeper in this discussion, it is sufficient to mention the fact that our model is one of the various possibilities of gravity models.

2.2.3 Why UDG?

In Eq. (2.9) we underlined how the DG action (that we remind being nothing but the Einstein action, rewritten with the introduction of the Dilaton) has a pole in d = 2. If we take it as we have written, it is not usable to make the physics in 2 dimensions. Neither the (2.4) is well-defined in terms of ϕ when we pose our theory in exactly d = 2. We can thus try to make a redefinition of the fields1, in order to drop out the pole in the DG Lagrangian. Suppose to rescale the ϕ field as:

ϕ(x) = √1

αψ(x) , α = 8 d − 1

d − 2. (2.29)

This makes the Lagrangian LDG take the form

LDG = 1 αψ 2_˜ R + 1 2( ˜∇µψ)( ˜∇ µ ψ) , (2.30)

that is properly the conformal parametrization used in [19] and [20]. Then we can rewrite the two fields ϕ and ψ in the following way:

ϕ = ϕ0+ χ , ψ = ψ0+ ρ , (2.31)

where ψ0 and ϕ0 are constant fields. If we choose ϕ0 to be ϕ0 = 1 we get equivalently

ψ0 =

√

α. Note that the background field condition ϕ0 = 1 will be used in Chap. 5 in

order to renormalize the theory at one loop level. Then, from (2.29), we have √

αϕ = √α(1 + χ) = ψ0+ ρ =⇒ χ =

ρ √

α. (2.32)

With this parametrization, the Lagrangian becomes: LDG= 1 + √ρ α 2 ˜ R + 1 2( ˜∇µρ)( ˜∇ µ_{ρ) ,} _(2.33)

which is exactly the parametrization used in [20] to describe the same model. Let us note that, in doing the previous field redefinitions we could “move” the pole from the kinetic term to the gravitational one. Note that in this case, the limit d → 2 is allowed and in exactly 2 dimensions gravity decouples from the scalar ρ.

If we choose a different parametrization for ψ, such as ψ = ψ0 + 4σ/

√

α, we have αχ = 4σ. So, if we look to the conformal factor Ω:

Ω = ϕd−22 = (1 + χ) 2 d−2 = 1 + 4σ α _d−22 , (2.34)

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then, if we substitute d = 2 + and we explicit α, we get

Ω = 1 + 4σ 8 2 = 1 + σ₂ 2 , (2.35)

and taking the limit d → 2, that is → 0, the conformal factor Ω tends to:

Ω = 1 + σ₂ 2 −−→ →0 e σ _. _(2.36)

So, the conformal factor Ω is parametrized exponentially. In this case, the 2 dimen-sional limit is well defined. The field σ is the field that is often called Liouville mode, or sometimes simply Dilaton (that is therefore linked to our Dilaton), because, if we start from the 2 dimensional EH action and introduce the field σ trough a conformal reparametrizarion gµν = e2σˆgµν, we get the so-called Liouville action:

SL=

Z

[d2xpˆg]{σ ˆ∇2_{σ + σ ˆ}_{R} .} _(2.37)

SLis exactly the 2 dimensional limit of gravity theory (see [21]). Actually, the Liouville

action does not come directly from the parametrization gµν = e2σˆgµν, but is linked to

it. For more details see Appendix C.

Even if we do not want to go deeper in this topic, we just want to underline that UDG theory is built in such a way to make the limit d → 2 continuous.

We saw that, from DG, if we impose the determinant ˜g of the metric to stay fixed under the gauge group, we obtain a different reformulation of the diffeomoprphism invariance Dif f . The new symmetry Dif f∗, we will see in the next section, engages us to use a particular parametrization for the metric when we want to expand it around its background. As a matter of fact, in order to approach the background field method, we will not use the linear split, which is common in treating EG (such as in the classical case of gravitational waves):

gµν = ¯gµν + hµν. (2.38)

Indeed the parametrization used, due to the fact that the transformation has to be unimodular (see later) is of the type

˜

gµν = ¯gµλ(eh)λν, (2.39)

where h is traceless. So, starting from gµν, we can equivalently rewrite this

parametriza-tion as

gµν = ¯gµλ(eh)λνe

2σ_. _(2.40)

With the exponential parametrization we intuitively divide the traceless terms of quan-tum perturbation, from the scalar one. The fact that these two modes are separate is good as soon as we go in 2 dimensions. As a matter of fact, it is well-known that the spin 2 field hµν does not propagate anymore when we cross down the 2 dimensions

(so, if d ≤ 2). This reparametrization of the metric, that separates the two degrees of freedom, gives the continuous passage across the 2 dimensions. This fact has the same meaning of the observation that we have done after (2.33): when we go to the 2D case, gravity decouples from the conformal mode, which remains the only degree of freedom that propagates.

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Even the “ill-definiteness” of the action written as (2.9) bring us to the same di-rections: there, indeed, when we perform d → 2, we get only the contribution of the Dilaton ϕ in the propagator, since the approach to the two dimensions makes its con-tribution arbitrarily larger compared to the other one of hµν.

Conclusion: the theory (2.9) has a kinetic pole in d = 2. This pole, after a reparametriza-tion, realizes itself only in decoupling gravity with the scalar field, but the theory is well-defined. This makes it, in some sense, a fake pole. In another parametrization of the theory, we showed that the 2 dimensional limit is leads us to the Liouville the-ory. So, UDG should be study, instead of studying EG in its standard formulation, because is built in such a way to allow the passage through the 2 dimensions without ill-definiteness.

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Chapter 3 2 + quantum gravity: one loop

expansion

In this chapter we begin to analyze the theory perturbatively above the 2 dimensions. In Chap. 1, we anticipated our approach, which is the one to study the theory slightly above d = 2. Therefore d = 2 + is set and we present here the one loop results, that are contained also in [20], in which the computation is taken at the two-loop level.

Technically speaking, the computations are made in the background field method framework, which allow us to take into account the quantum effects of the theory and the heat kernel method, which is a covariant technique (so it is very convenient in curved spacetime), with which we can dimensional-regularize the theory. Both of them are explained in dedicated sections.

3.1 Background field method

Our aim is to find the one loop expansion of Eq. (2.9). In this section, we are going to present the framework that we will use. To present the main theoretical aspects, in the first paragraph the method is presented by using a simple scalar theory with some scalar field φ(x). The results we will get, will be the same for UDG, which we will develop in the second paragraph.

So far, we have set ~ = c = 1, working in natural units. Here the dependence of the quantities by ~ will be written explicitly. This will turn out to be useful because it is well known that in QFT, power expansion in terms of ~ is equivalent to the expansion in loop orders. The main reference for this section is any QFT standard book, such as [22].

3.1.1 Background field method, an example: scalar theory

Let us consider a scalar theory with an action S[φ] in a flat spacetime. The generating functional Z[J ], with an external source J has the following expression, in Euclidean spacetime. Z[J ] = Z Dφ e−1~S[φ]+ 1 ~R Jφ. (3.1)

The generating functional is a useful quantity: every expectation values of an observable O(φ) can be computed by differentiate Z[J]

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3.1. Background field method Chapter 3 hOi = 1 Z[0] Z Dφ O(φ)e−1~S[φ] = 1 Z[0] O δ δJ Z[J ] J =0 . (3.2)

Note that the normalization factor Z[0]−1 makes every expectation value independent by every J - and φ-independent quantity in front of Z.

Then, there are two quantities which we must define. The first one is the generating functional of connected diagrams W [J ] = ~ ln(Z[J]). The second one is the effective action (EA), which is the Legendre transformed of W [J ] from the function J (x) to another function ϕ(x). In the next equations, where we define the EA, Γ[φ], we make use of the notation J · ϕ = R ddxJ (x)ϕ(x) to indicate the “product” in the functional space. Γ[ϕ] = −W [J ] + J · ϕ , δΓ[ϕ] δϕ = J | {z } Stationary condition . (3.3)

Let us note that the stationary condition in the previous equation defines the field ϕ 1

as a functional of J , ϕ = ϕ[J ].

Since we are looking for the n-loops terms, we have to compute the term proportional to ~n _{in the ~ expansion of the EA. So, it is convenient to write Γ[ϕ] as:}

Γ[ϕ] = Γ0[ϕ] + ~Γ1[ϕ] + · · · = ∞

X

k=0

Γk[ϕ]~k. (3.5)

Here, Γn is properly the n-loops term of the EA. On the other side, we can rewrite

Eq. (3.3) in this way:

Z[J ] = e1~W [J ] = e 1

~(−Γ[ϕ]+J ·ϕ)= Z

Dφe−1~(S[φ]−J ·φ). (3.6) Rearranging the terms we obtain the so-called integro-differential formula for the EA:

e−Γ[ϕ]~ = Z Dφ exp −1 ~ S[φ] − J · (φ − ϕ) = Z Dφ exp −1 ~ S[φ] − δΓ δϕ · (φ − ϕ) . (3.7)

Note that ϕ is fixed by the choice of J and the integration involves only the field φ. So, we can expand the field φ, around the fixed value ϕ, imposing φ = ϕ +√_{~χ, and} then we can match, order by order, the coefficients in order to compute the Γn. This

procedure is called background field method : ϕ(x) that is a fixed field, can be seen as a sort of background and the field χ(x) represents the quantum perturbations around ϕ. Expanding in powers of ~, here means expanding in powers of the perturbation χ.

The RHS of Eq. (3.7) can be expanded as in Eq. (3.8).

1_{Comment: what is the meaning of the background field ϕ? We used it as a Legendre-conjugated}

variable of J in Eq. (3.3). Physically, ϕ is the expectation value of the field φ. As a matter of fact, the expectation value ¡φ¿ can be written, by definition, through Eq. (3.2).

hφi = 1 Z[0] δZ δJ[0] = δW δJ [0] = ϕ (3.4) in the (3.4) we used the inverse Legendre transformation of (3.3).

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3.1. Background field method Chapter 3 S[φ] − δΓ[ϕ] δϕ · (φ − ϕ) = S[ϕ − √ ~χ] − δΓ[ϕ] δϕ · √ ~χ = = S[ϕ] +√_~ δS[ϕ] δϕ − δΓ0[ϕ] δϕ · χ + ~1 2χ · δ2S[ϕ] δφδφ · χ + O(~ 3/2 ) . (3.8)

Matching Eq. (3.5) with Eq. (3.8), order by order in ~ we see that the 0th order corresponds to Γ0[ϕ] = S[ϕ]: at the lowest order in ~, the effective action is nothing

but the classical action. Then, the terms proportional to √_{~ disappears. To evaluate} the term proportional to ~, we need to perform a Gaussian integral , which we can compute easily because of the property of the path integral (see [22], Chap. 9).

Z Dχ exp −1 2 Z ddxddy δ 2_S[ϕ] δφ(x)δφ(y)χ(x)χ(y) = N det δ2_S[ϕ] δφ(x)δφ(y) −1/2 . (3.9) The normalization constant N is fixed by the requirement on Z[0] (usually one imposes Z[0] = 1). Now, if we take Eq.s (3.9) and (3.8), put them into Eq. (3.7) and take the logarithm, we obtain Γ1L[ϕ] = S[ϕ] +~ 2ln det δ2_S[ϕ] δφ(x)δφ(y) + C = S[ϕ] + ~ 2Tr ln δ2_S[ϕ] δφ(x)δφ(y) + C , (3.10)

where C = ln(N ) is a constant, useless for our purpose, and we employed the useful property ln(det()) = tr(ln()).

The main, well-known, result, that is important to underline, is that the one loop expansion of the EA depends only on the quadratic expansion of the action. Obviously, this paragraph treats only a simple scalar field with only degree of freedom φ(x). In UDG, we have two fields to take into account: the metric tensor ˜gµν and the Dilaton

ϕ. In natural units, the expression (3.10) can be rewritten as follows (dropping C): Γ1L[ϕ] = S[ϕ] + 1 2Tr ln δ2_S[ϕ] δφ(x)δφ(y) . (3.11)

For a more compact notation, from now on the Hessian δ2_{S/δφ(x)δφ(y) will be}

indi-cated with the notation S(2)_{. We anticipate here that, in order not to make confusion,}

the notation S(2) will indicate the quadratic part of the action.

3.1.2 UDG: quadratic expansion

In the previous section we reach the result expressed in Eq. (3.11). We want to use this technique on the action (2.9). First of all, we have to split our fields in their background field and quantum fluctuation. The Dilaton is split linearly in its constant background

¯

ϕ and its quantum fluctuation. However, the metric is split in an exponential form, with hµ_µ= 0:

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3.1. Background field method Chapter 3 ϕ(x) = ¯ϕ + χ(x) , ˜ gµν(x) = ¯gµλ(x)(eh(x))λν = ¯gµν(x) + hµν(x) + 1 2hµλ(x)h λ ν(x) + ... . (3.12)

The idea of this parameterization is given by the symmetries of the theory. In UDG we want ˜g to be constant, so the background metric ¯gµν, must be related to the that

one by a unimodular transformation: ˜g = ¯g. This transformation can conveniently be parametrized exponentially, requiring tracelessness of hµν. This fact was anticipated in

Sec. 2.2.3.

The inverse has to satisfy the usual relation ˜gµσ(x)˜gσν(x) = δµν, so we can evaluate

also the expansion for ˜gµν_(x):

˜ gµν(x) = ¯gµν(x) − hµν(x) + 1 2h µλ_(x)hν λ(x) + ... := (e −h )µ_λ(x)¯gλν(x) . (3.13) Note that here the geometry of the spacetime is defined through the background metric tensor ¯gµν_{. So it is the one that can raise and lower indices (i.e. h}µν _{is defined}

as ¯gµαg¯νβhαβ). To make contact with background field method, in our case Eq. (3.10)

takes the form of Eq. (3.14).

Γ1L[¯gµν, ¯ϕ] = S[¯gµν, ¯ϕ] + ~

2Tr{ln[S

(2)_{]} ,} _(3.14)

where S(2) _{is the Hessian of the action. Thus, we have to look for the quadratic}

expan-sion of the action in terms of the quantum fluctuations χ, hµν, that we saw corresponds

to the linear order in ~ power expansion. Quadratic expansion of ˜R

In order to compute S(2) we have to expand ˜R up to the quadratic order in hµν. In

doing this, the computation is pretty heavy, so here we present only the main steps of the calculation, for the missing details we refer to the Appendix B. The starting point is the definition of the Riemann tensor as the commutator of two covariant derivatives. Given a vector Vµ_{, we have, by definition,}

[ ˜∇µ, ˜∇ν]Vα= ˜Rβµνα V

β_, _(3.15)

where ˜∇µVα = ∂µVα+ ˜ΓαµβV

β_{. So, we need to expand the linear connection ˜}_{Γ = ˜}_Γ(˜_g)

in powers of h. Let us separate the various order contribution in ˜Γ, as in the following expression

˜

Γµ_αβ = ¯Γµ_αβ + ∆µ_αβ + Σµ_αβ + ... . (3.16) Here, ∆ and Σ represent respectively the first and the second order contribution in powers of h and ¯Γ is the linear connection computed with the background metric tensor ¯

g. Thus, we can write down the covariant derivative up to the second order: ˜ ∇νVρ= ∂νVρ+ (¯Γ + ∆ + Σ)ρ_νλVλ = δ_σρ∇¯ν + (∆ + Σ)ρ_νλ Vλ. (3.17)

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3.1. Background field method Chapter 3

Once that we know how the covariant derivative acts (up to its quadratic expansion) on a generic vector Vµ_{, we can easily commute and we get the expression for the}

Riemann tensor. ˜ Rρ_σµν = ¯Rρ_σµν+ ¯∇µ(∆ + Σ)ρνσ − ¯∇ν(∆ + Σ)ρµσ+ ∆ ρ µλ∆ λ νσ− ∆ ρ νλ∆ λ µσ. (3.18) From ˜Rρ

σµν we evaluate ˜R = δρµg˜σνR˜ρσµν. Then we can write the explicit expression for

˜ R, as in the following: ˜ R = ¯R − ¯Rµνhµν + ¯∇µ∇¯νhµν+ + 1 2 ¯ Rµνρσhµρhνσ − 1 4( ¯∇µh νλ )( ¯∇µ hνλ) + 1 2( ¯∇µh µλ )( ¯∇νhνλ) . (3.19)

Taking (3.19) and inserting into (2.9), we can get the Hessian: the quadratic part in the (h, χ) fields. The expression for S(2) is:

S(2) = − 1 G Z [ddx√g]¯ −4d − 1 d − 2g¯µνχ ¯∇µ ¯ ∇νχ + χ2R + 2χ ¯¯ ϕ( ¯∇µ∇¯ν− ¯Rµν)hµν+ + ¯ϕ2 1 2 ¯ Rµνρσhµρhνσ+ 1 2( ¯∇µh µλ_{)( ¯}_∇ νhνλ) − 1 4( ¯∇µh νλ_{)( ¯}_∇µ_h νλ) . (3.20)

3.1.3 UDG: gauge fixing

The theory we developed so far has a gauge symmetry that we presented in Chapter 2: the group Dif f∗. If we want to make our computation, we had better to provide our theory with a gauge fixing procedure. To gauge-fix the theory we are going to follow the Fadeev-Popov (FP) method, which we briefly resume here.

Fadeev-Popov method

Our principal reference for more details is [22] or another introductory book of QFT. However, it is not difficult to give an idea of how we are going to gauge-fix UDG.

After the background expansion, the functional integral Z can be written as an integral on the fluctuations χ and h, since the background fields are fixed.

Z = Z

DhDχe−1~S. (3.21)

In our theory, the gauge freedom is given by the group Dif f∗, which infinitesimal transformation (here we omit the symbol∗) δϕ = δχ and δ˜gµν = δhµν are generated by

an infinitesimal diffeomorphism ξν _{as written in Eq. (2.21). Since the theory is invariant}

under these transformations, we can use this freedom to fix a proper combination of the fields χ and hµν to a specific value.

This combination, that is nothing but a function Fµ = Fµ(χ, hµν), is called gauge

condition, can be posed equal to a generic field-independent ωµ, in order to cancel the invariance. This procedure has a physical meaning, that depends on the group we are dealing with: for example, if we had to work with Einstein Gravity, the Dif f -gauge fixing, would have the physical meaning of choosing a particular coordinates

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system. Dif f∗ is a different realization of the diffeomorphism invariance, so this is not so immediate to catch the physical correspondence of our gauge-fixing condition.

First of all, we want to set the gauge condition

Fµ(χ, hµν) = ωµ. (3.22)

The FP method consists in inserting into the path integral (3.21) the following identity, that set the condition (3.22):

1 = Z DF δ(Fµ− ωµ) = Z Dξ det δFµ δξν δ(Fµ− ωµ) . (3.23)

In Eq. (3.23) we changed variables, from Fµ to ξµ. The determinant is simply the

Jacobian of that transformation (for more details, see [22]) and the quantity δFµ is

ξ-dependent, as well as the variation of χ and h.

Now, we use a trick: in (3.23) we can insert another identity, given by the Gaussian functional integrals, in order to integrate out the Dirac delta:

1 = N (λ) Z Dω exp − Z [ddx√g]¯ ωµω µ 2λ , (3.24)

where λ is a free parameter, which is called “gauge fixing parameter”. The constant N depends on it.

Once we take Eq. (3.24) and put it into (3.23), we can integrate in ω and drop the δ(Fµ− ωµ). Then, we can rewrite the Jacobian using two Grassmann variables, which

we can call ¯cµ _{and c}µ_{. The fields ¯}_cµ _{and c}µ _{are named ghosts and we can write}

det δFµ δξν = Z D(¯cc) exp − Z [ddx√g]¯¯cµδFµ δξν c ν . (3.25)

Collecting all the results and putting them into (3.23), we get, up to a redefinition of the gauge condition and the ghosts by some power of ~:

Z = N Z DhDχDξD(¯cc)e−1~Sexp − 1 G~ − Z [ddx√¯g]FµF µ 2 + G¯c µδFµ δξνc ν . (3.26) Note that we can drop out the normalization constant N , that does not depend on the fields, so, at the functional integral level, it does not matter once we want to compute the observables (see Eq. (3.2)). So, our theory is defined by:

Z = Z DhDχDξD(¯cc)e−1~Sexp − 1 G~ − Z [ddx√g]¯ FµF µ 2 | {z } Lgf + G¯cµδFµ δξν c ν | {z } Lgh . (3.27)

Gauge fixed UDG

Now, it is time to choose our gauge fixing condition. The most convenient one is the Feynman gauge, where Fµ is chosen properly in order to eliminate the off-diagonal

derivative-terms in the fields (χ, hµν). A convenient expression for Fµ is

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which is a close relative to the Feynman-de Donder gauge chosen to combine with the Hessian and properly simplify the quadratic action.

The gauge-fixing Lagrangian Lgf of Eq. (3.27) has the following form, once we

neglected boundary terms:

FµFµ= ( ¯ϕ ¯∇νhνµ− 2 ¯∇µχ)( ¯ϕ ¯∇νhνµ− 2 ¯∇µχ) =

= ¯ϕ2( ¯∇νhνµ)( ¯∇ρhρµ) − 4χ ¯∇ν∇¯νχ + 4 ¯ϕχ ¯∇ν∇¯µhµν.

(3.29) We notice that the term 4 ¯ϕχ ¯∇ν∇¯µhµν is exactly the opposite of a term in the quadratic

action S(2). Then, we have the quadratic gauge-fixing part of the action, that we can

call S₍₂₎gf: S₍₂₎gf = 1 2G Z [ddx√¯g] ¯ ϕ2( ¯∇νhνµ)( ¯∇ρhρµ) − 4χ ¯∇ν∇¯νχ + 4 ¯ϕχ ¯∇ν∇¯µhµν . (3.30) Ghost action

The last part of the gauge fixing procedure is the ghost action, which we derived in Eq. (3.25). First of all, we need to compute the first variation of Fµ w.r.t. ξµ. Varying

the gauge fixing condition, we obtain:

δFµ = ¯ϕ ¯∇ν(δhνµ) − 2 ¯∇µ(δχ) . (3.31)

We remind that the variation is made under the group Dif f∗. Then, it is easy to compute the one for the Dilaton

δχ = δϕ = ξµ∂µχ +

d − 2

2d ( ¯ϕ + χ) ¯∇µξ

µ_. _(3.32)

In order to compute the variation δh, let us consider the following property of the exponential of a certain operator M2_:

δ(eM) = eM(δM + 1 2[M, δM ] + 1 3![M, [M, δM ]] + ...) = eM ∞ X n=1 1 n![M, [M, [...[M_| _{z _} =n−1 , δM ]...]]] . (3.33)

So, we can get the expression for the infinitesimal variation of the operator M . Let us note that the Jacobian δFµ/δξν must not depend on the fields χ and hµν to compute

the Hessian. This is due to the fact that we want to compute the quadratic part of the Lagrangian in terms of all the fields we’re integrating on (χ, hµν, ¯cµ, cµ). The ghost part

of the action is, by construction, a quadratic form in the ghost fields (see (3.25)), then the variation must be taken at the 0th order in χ and hµν.

The zeroth order in M of (3.33) is given by δM = e−Mδ(eM_{) → δ}(0)_{M = δ}(0)_(eM_).

Farther, we know already the variation of the exponential: δ(eh)µ_ν = ¯gµλδ˜gλν = ¯gµλ(˜gλσ∇˜νξσ+ ˜gνσ∇˜λξσ −

2 dg˜λν

˜

∇σξσ) . (3.34)

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3.2. Heat Kernel method Chapter 3

Taken to the 0th order we have ˜∇ → ¯∇ and ˜gµν → ¯gµν. Thus, we obtain the variation

to insert in Eq. (3.31): δ(0)χ = d − 2 2d ϕ ¯¯∇µξ µ_, _(3.35) δ(0)(eh)µ_ν = δ(0)hµ_ν = ¯gµλ(¯gλσ∇¯νξσ+ ¯gνσ∇¯λξσ − 2 dg¯λν ¯ ∇σξσ) = ¯∇νξµ+ ¯∇µξν − δµν 2 d ¯ ∇σξσ. (3.36)

So, putting the two computed variations into (3.31), we get the following expression:

δFµ= ¯ϕ ¯∇ν ¯ ∇µξν + ¯∇νξµ− δνµ 2 d ¯ ∇σξσ − 2 ¯∇µ d − 2 2d ϕ ¯¯∇σξ σ = ¯ϕ( ¯∇σ∇¯µ− ¯∇µ∇¯σ+ ¯gσµ∇¯2)ξσ = ¯ϕ( ¯Rµσ+ ¯gµσ∇¯2)ξσ. (3.37)

Now we can take the variation w.r.t. ξµ _{and we can define a convenient Laplace-like}

operator for a more compact notation − GδFµ

δξν = −G ¯ϕ( ¯Rµν+ ¯gµν∇¯ 2

) := ∆gh_µν. (3.38) So, the ghost contribution to the quadratic action S₍₂₎gh take the familiar expression

S₍₂₎gh = −1 G

Z

[ddx√¯g]¯cµ∆gh_µνcν. (3.39) Now we can put all the contribution together and we can write down the complete, gauge-fixed, quadratic action. In (3.40), S(2) (and from now on) is the complete one,

and the quadratic part of UDG action is indicate here as SU DG (2) : S(2) =S(2)U DG+ S gf (2)+ S gh (2) = − 1 G Z [ddx√g]¯ 2 − 4d − 1 d − 2 χ ¯∇µ∇¯µχ + χ2R+¯ − 2χ ¯ϕ ¯Rµνhµν + 1 4ϕ¯ 2_h νρ∇¯µ∇¯µhνρ+ 1 2ϕ¯ 2_R_¯ µνρσhµρhνσ + ¯cµ∆ghµνc ν . (3.40)

3.2 Heat Kernel method

In this section a new method of computation is presented. In order to make the explicit calculation of the formula (3.14), since we are in curved spacetime, the heat kernel method is very useful: this method has the advantage to be covariant, since is built to work on a generic manifold and not only in the flat spacetime.

In the first part we are presenting some theoretical aspects. Beside them, we are providing a simple application (in the case of a scalar field in flat spacetime) to make clear on how the technique works. Since the heat kernel method is nothing but a tool for us, we leave the prove and some formal aspects to the references.

For more details and formal aspects see [23] and [24], for an introduction see the Appendix of [25] and for the uses in the case of gravity see [20] and [26].

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3.2.1 Heat Kernel: an introduction

Let us consider a generic situation: suppose to have a differential Laplace-like operator, which is local, and acts on a manifold M. We will call this operator Dx. Let D take

the general form

Dx = −I∇µ∇µ+ E(x) . (3.41)

Here, I and E indicates two operators on the internal space. Mathematically speaking, to define these operators we have to provide M with a vector bundle V . In a given point z ∈ M, V defines a vector space V (z). The form of V depends on what type of fields we are dealing with. The “internal space” which we are talking about is thus the vector space associated to our fields. For example, it can be a representation of some gauge group or of the spacetime symmetry group. That is to say: if we have to deal with spinors, V (z) is labelled with spinorial indices and I(z), E(z) operate as functions F(z) : V (z) → V (z) as the identity and an endomorphism (i.e. a matrix with spinorial indices), respectively. The covariant derivative ∇µ acts as a “Riemannian derivative”,

which contains the partial derivative ∂µand the Levi-Civita connection associated with

the metric Γ(g) plus a “bundle connection” ω, often called “spin connection”, which depends on the type of fields we are treating. The notation Dx indicates the point

x ∈ M w.r.t. which we perform the derivatives.

Named GD the Green function of the operator Dx, we can try to find it. The heat

kernel method solves exactly this problem. We remind that the Green function GD is

defined as:

DxGD(x, y) = δd(x, y) , (3.42)

where δd(x, y) is the generalization of the δ-function in the flat space. Given a scalar function f , the δ-function performs as in the flat case, with the integration over the invariant volume dv =√gdd_x:

Z _√

gddx δd(x, y)f (x) = f (y) , (3.43) The problem can be viewed in a particular way: let us consider a function H(τ, x, y; D) (where τ ≥ 0), named heat kernel, which satisfies the heat equation written below, with a well-defined boundary condition for τ → 0+_{, that is a variable τ called proper time}

(∂τ + Dx)H(τ, x, y; D) = 0 lim

τ →0+H(τ, x, y; D) = δ

d_{(x, y) .} _(3.44)

The solution for Eq. (3.44) can be written as:

H(τ, x, y; D) = hx| e−τ D|yi . (3.45) If we suppose that, when τ → ∞, the heat kernel vanishes sufficiently fast, we can compute the green function of D, that is:

GD(x, y) =

Z ∞

0

dτ H(τ, x, y; D) (3.46)

Under these assumptions, the proof of (3.46) is easy and it is due to the fact that H(τ, x, y; D) satisfies the heat equation (3.44):

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3.2. Heat Kernel method Chapter 3 DxGD(x, y) = Z ∞ 0 dτ DxH(τ, x, y; D) = = − Z ∞ 0 dτ ∂τH(τ, x, y; D) = lim τ →0+H(τ, x, y; D) = δ d_{(x, y) .} (3.47)

A property of H(τ, x, y; D) that will be useful in the later is: Z

ddy H(τ, x, y; D)H(h, y, z; D) = H(τ + h, x, z; D) , (3.48) that it is easy to prove using the expression (3.45) for H

Z

ddy H(τ, x, y; D)H(h, y, z; D) = Z

ddy hx| e−τ D|yi hy| e−hD|zi = hx| e−τ De−hD|zi

= hx| e−(τ +h)D|zi = H(τ + h, x, z; D) .

(3.49)

3.2.2 Heat Kernel method, an example: scalar field theory in

flat spacetime

To make contact with the actual computation, let us make a simple example of this technique.

Free theory

We consider a massive scalar field theory. Let us start from the free theory, in the Euclidean spacetime: L0 = 1 2∂µφ∂ µ_{φ +} m2 2 φ 2_. _(3.50)

The theory is defined in a flat spacetime, so our manifold is M = Rn_{. Here, the}

operator D assumes the expression given in Eq. (3.51). In flat spacetime gµν = δµν and

g = 1. Thus, the covariant derivative ∇µ become the partial derivative ∂µ, since the

connections vanish,

D = D0 = −∂µ∂µ+ m2. (3.51)

The free case is one of the very few cases in which the heat kernel H(τ, x, y; D0) can

be exactly computed. By substitution in (3.44), one can see that what follows is the correct expression for the heat kernel:

H(τ, x, y; D0) = 1 (4πτ )n/2e −|x−y|2 4τ −τ m 2 . (3.52)

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Scalar field theory with a potential: Seeley-de Witt expansion

Let us now introduce a potential V in our free theory. The new Lagrangian L = L0+ V (φ) provides us with a new differential operator D 6= D0.

It is not possible to solve the heat equation for a generic potential V . However, it is possible to solve the problem perturbatively. As done in [24], we can make an ansatz: we can use the free heat kernel (Eq. (3.52)) in order to make a power expansion in the proper time τ , for τ ∼ 0+_.

Our assumption is to write the heat kernel H(τ, x, y; D) in a form like the following one: H(τ, x, y; D) = H(τ, x, y; D0) ∞ X i=0 τiai(x, y; D) . (3.53)

This series expansion is called Seeley-de Witt series and the coefficients ai(x, y; D) are

called Seeley-de Witt coefficients. This expression is particularly useful if we are dealing with loop expansion. Often, especially at one loop level, we have to compute a quantity like in the following one, that, we’ll see, is nothing but Eq. (3.11)

WD =

1

2ln[det(D)] . (3.54)

It turns out that WD can be computed in terms of heat kernel. Let us see how.

First of all, we have to consider the following statement3_{, that is valid for all positive}

numbers λ > 0: ln λ = − Z ∞ 0 dt t e −tλ , . (3.55)

Making the assumption that D is positive definite (that is true in many cases), we can diagonalize it. So, we can rewrite W , exploiting the fact that the determinant is invariant by similarity, as:

ln[det(D)] = Tr[ln(D)] = −Tr Z ∞ 0 dτ τ e −τ D = −tr Z ∞ 0 dτ τ Z ddx hx| e−τ D|xi | {z } H(τ,x,x;D) . (3.56)

Here, the symbol “tr” indicates the trace over the internal indices of D.

The quantity H(τ, x, x; D) is called the coincidence limit of the heat kernel and it is often indicated with the notation H(τ, x, x; D) = H(τ ; D).

The fact that the heat kernel function appears in (3.56) only as in its coincidence limit (where x → y) makes the computation straightforward. Indeed, if we look at Eq. 3.52, the term exp{|x − y|/τ } vanishes in this limit. This, we will see soon, makes the heat kernel dumped in the small τ limit, that will be a good feature for our theory.

Now, let us assume D to have the form

Dx = −∂x2+ m2+ F (x) = D0x+ F (x) , (3.57)

3_{That statement’s proof consists in differentiating both members of the equation w.r.t. λ, then is a}

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where F is a function4.

We want to compute the theory at 1 loop level for a generic F , that is equivalent to say that we have to find the coincidence limit of the heat kernel expansion. By definition, H(τ, x, y; D) must solve

(Dx+ ∂τ)H(τ, x, y; D) = 0 , H(0, x, y; D) = δd(x, y) . (3.58)

Since we know the expression of the free heat kernel, we try to use the (3.53). Taking its derivative w.r.t. the proper time we obtain

∂τH(t, x, y; D) = [∂τH(τ, x, y; D0)] ∞ X k=0 τkak(x, y; D)+ H(τ, x, y; D0) ∞ X k=0 (k + 1)τkak+1(x, y; D) . (3.59)

Then, since we know by (3.58) that ∂τH(τ, x, y; D) = −DxH(τ, x, y; D), we can compute

also the application of Dx:

− DxH(τ, x, y; D) = −[D0xH(τ, x, y; D0)] ∞ X k=0 τkak(x, y; D)+ − F (x)H(τ, x, y; D0) ∞ X k=0 τkak(x, y; D) − H(τ, x, y; D0) ∞ X k=0 τk[D0xak(x, y; D)] + H(τ, x, y; D0)(x − y)j ∞ X k=0 τk∂x,jak+1(x, y; D) , (3.60)

where the symbol ∂x,j indicates the jth component of the gradient w.r.t. x: ∂x,j = (∇x)j.

So, if we note that in (3.60) we have (D0x+ ∂τ)H(τ, x, y; D0) = 0 by definition, we can

write down the heat equation for H(τ, x, y; D). Thus, we can obtain a recurrence relation for the Seeley-de Witt coefficients, so that:

0 =F (x)ak(x, y; D) + D0xak(x, y; D)+

+ (k + 1)ak+1(x, y; D) − (x − y)j∂x,jak+1(x, y; D) .

(3.61)

The initial condition of the heat equation implies a0(x, y; D) = 1, since H(0, x, y; D0) =

δd_{(x, y).}

In order to compute a1, one can consider the case k = 0 in Eq. (3.61), ad try to solve

for a1. If we are interested in the coincidence limit, we can make a power expansion

of the coefficients in powers of the displacement ξj _{= x}j _{− y}j_{, because that limit is}

equivalent to say ξ → 0. Therefore, we are looking for a solution like:

ak(x, y; D) = ak(x; D) + ξibk,i(x; D) + ξiξjck,ij(x; D) + O(ξ3) . (3.62)

Then, putting (3.62) into (3.61) at k = 0 (remember that ∂x,iξj = δij), we obtain a1.

4

Note that here the endomorphism F is a function F : Rn _{→ R}n_{, because the vector spaces V (x)}

coincides (is isomorphic) in this case with Rn

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0 = F (x) + m2+ a1(x, y; D) − ξj∂x,ja1(x, y; D) =

= F (x) + m2+ a1(x; D) + ξib1,i(x; D) − ξj∂x,ja1(x; D) − ξib1,i(x; D) + O(ξ2) .

(3.63) So, taking the coincidence limit of (3.63) we get the expression for the coincidence limit for a1, since ξ → 0:

a1(x; D) := a1(x, x; D) = −F (x) − m2. (3.64)

The same operation can be made for the second coefficient and the result is 2a2(x; D) =

[F (x) + m2_]2_{. That is a valid method for every k and it is not very difficult to go on.}

An explicit computation: φ4 _theory

In this paragraph we take an explicit example of a non free theory: the φ4 _{theory. The}

Lagrangian takes the form

L = 1 2∂µφ∂ µ_{φ +} m2 2 φ 2₊ λ 4!φ 4_. _(3.65)

We can make a perturbation expansion with the background field method, taking φ(x) = ϕ + χ(x), where ϕ is the expectation value of φ(x) in the vacuum, which is constant here. As in Eq. (3.10), the 1 loop contribution to the EA comes from the term which, using the notation of Eq. (3.54), we can call without confusion ~WS(2). So, in this case D = S(2), which has the expression

Γ1L[ϕ] = S[ϕ] + ~

2Tr ln(D) , D = S(2) = −∂µ∂

µ_{+ m}2₊1

2λϕ

2_. _(3.66)

The renormalization of the theory consists in looking for the counterterms to add to the Lagrangian (practically the divergent part of the Feynman diagrams). We want to show explicitly, at least in this simple case, that the results we will find using the heat kernel method coincide exactly with the canonical approach in which the loop integrals of the diagrams are regularized through dimensional regularization.

Dimensional regularization

The most familiar way to renormalize the φ4 _{theory (at least at 1 loop level) is to study}

the 2- and the 4-points functions, compute their contribution to the EA and look for the counterterms, once we have found the divergent part of their associated diagrams. The two diagrams we have to deal with are reported in Fig. 3.1.

We have to perform the loop integrals in dimensional regularization. We will regu-larize the theory in dimension n = 4 − perturbatively in .

2 points function For the 2 points functions we must analyze the left diagram in Fig. 3.1. The counterterms come from the 1/ poles in the loop integrals (the renormalization of the theory then depends also on the renormalization scheme). The integral we have to compute is the one that follows, where the written part is the only the divergent one:

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Figure 3.1: 1 loop diagrams for φ4 theory. On the left the 1 loop contribution to the 2 points function and on the right the 1 loop contribution to the 4 point functions

1 2λ Z dn_p (2π)n 1 p2_{+ m}2 = 1 2λ mn−2 (4π)n/2Γ 1 − n 2 = −m 2_vµ (4π)2 1 + · · · . (3.67) Here v stands for the dimensionless coupling constant v = µ−λ, p is the loop momentum and Γ is the Euler function. This diagrams is the correction at 1 loop level to the propagator.

4 points function In the case of the 4 pints function we have to deal with the right diagram in Fig. 3.1. This diagram give us the correction to the vertex and it goes to renormalize the coupling λ. The loop integral is written and computed in Eq. (3.68). The factor in front of the integral takes into account the symmetric consideration (for example, we have the same divergent part if we make permutations of the external legs). k is the external, conserved, momentum.

−3 2λ 2 Z dn_p (2π)n 1 (p2_{+ m}2_{)[(p + k)}2 _{+ m}2_] = = −3 2 v2_µ (4π)2Γ 2 Z 1 0 dx ln 4πµ2 k2_{x(1 − x) + m}2 /2 = −µ _v2 (4π)2 3 + ... . (3.68)

All the divergences we compute in this theory are ultraviolet divergences. As a matter of fact, the presence of a finite mass m2 _{> 0, does not allows the two integrals (3.67)}

and (3.68) diverge for p → 0. So m2_{prevents IR divergences. Therefore, in the previous}

computations, we have found the actual counterterms to add to the action. Heat kernel computation

In order to get the same results using heat kernel, we can use the general theory de-veloped above. To use the notation of Eq. (3.57), we have the special case where 2F (x) = λϕ2_{. Let us rewrite our results in a compact way:}

Tr ln(D) = − Z ∞ 0 dτ τ Z dnxH(τ, x, x; D) = = − Z dnx Z ∞ 0 dτ τ 1 (4πτ )n/2e −τ m2 ∞ X k=0 τkak(x; D) . (3.69)

Symmetries and renormalization of (2 + epsilon) dimensional Unimodular Dilaton Gravity

Universit`

a di Pisa

Dipartimento di Fisica “Enrico Fermi”

Tesi di Laurea Magistrale in Fisica

Symmetries and renormalization of (2 + )

dimensional Unimodular Dilaton Gravity

Candidato: Francesco Del Porro

Relatore: Dr. Omar Zanusso

Contents

Chapter 1

Introduction

Chapter 2

Unimodular Dilaton Gravity

2.1

Dilaton Gravity

2.1.1

DG action

2.1.2

DG symmetries

2.2

Unimodular theory

2.2.1

Dif f ∼ Dif f

2.2.2

Unimodular and Einstein Gravity

2.2.3

Why UDG?

Chapter 3

2 +  quantum gravity: one loop

expansion

3.1

Background field method

3.1.1

Background field method, an example: scalar theory

3.1.2

UDG: quadratic expansion

3.1.3

UDG: gauge fixing

3.2

Heat Kernel method

3.2.1

Heat Kernel: an introduction

3.2.2

Heat Kernel method, an example: scalar field theory in

flat spacetime

Symmetries and renormalization of (2 + )

2 + quantum gravity: one loop