From Lee-Wick Models to Quantum Gravity, Fakeons and the Violation of Microcausality

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Universit`

a di Pisa

Dipartimento di Fisica “Enrico Fermi” Corso di Dottorato in Fisica

XXXI Ciclo

From Lee-Wick Models to Quantum Gravity,

Fakeons and the Violation of Microcausality

Tesi di Dottorato

Candidato: Marco Piva

Relatore: Prof. Damiano Anselmi

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Notation and conventions

• We always consider natural units ~ = c = 1, unless it is indicated. • We denote the dimension of a quantity λ, in units of mass, as [λ]. • The Riemann tensor is defined as Rµνρσ = ∂ρΓµνσ− ∂σΓµνρ+ ΓµαρΓανσ− Γ

µ ασΓανρ.

• The Ricci tensor is defined as Rµν = Rρµρν.

• The Minkowski spacetime metric tensor reads ηµν = ηµν = diag(1,−1, −1, −1).

• The integral over spacetime points of a function F of a field φ(x) is denoted by Z _√

−gF (φ) ≡ Z

d4xp−g(x)F (φ(x)) , where gµν is the spacetime metric tensor and g = detgµν.

• The functional derivative with respect to a field φ is denoted by δ/δφ. • The beta function of a parameter λ reads

βλ = µ

dλ dµ, where µ is the dynamical scale.

• In Minkowski spacetime the Fourier transform of a function f is defined as f (x) = Z d4p (2π)4f (p)e˜ −ipx , where px = ηµνpµxν. 3

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Chapter 1

Introduction

Quantum field theory has been very successful in particle physics. Indeed, the standard model accounts for three of the fundamental interactions: the strong, weak and electromagnetic ones. An important role in its formulation is played by three principles: locality, renormalizability and unitarity (plus fun-damental symmetries, such as Lorentz invariance, gauge invariance and general covariance). Locality restricts the degree of arbitrariness of the classical theory. Renormalizability further reduces such arbitrariness to a finite number of free parameters and guarantees predictivity at arbitrary energies. Finally, unitarity is necessary to have a reasonable scattering theory.

However, the gravitational interactions remained out of the picture. The formulation of a consistent theory of quantum gravity has been one of the main challenges of theoretical physics. Frameworks alternative to quantum field theory have been proposed over the course of time (such as string theory and loop quantum gravity) but failed to provide satisfactory answers to the open questions. For this reason, in this thesis we adopt a more conservative approach and take for granted that the right framework for quantum gravity is still quantum field theory.

One option to search for the right theory of quantum gravity is to relax some assumptions we are accustomed to. In this respect, an interesting subsector that is worth of attention is represented by the local, higher-derivative theories. The Hilbert-Einstein action is not renormalizable [1, 2], so if we want to keep renormalizability a fundamental principle, we must modify the theory. The first renormalizable higher-derivative model of quantum gravity was introduced by Stelle [3] in 1977, by adding new terms to the classical action, which are quadratic in the curvature tensor. However, it is easy to show that the Stelle theory is not unitary. We will understand in due time that the problem of the Stelle theory was the quantization prescription, which was the Feynman one for all the poles of the free propagator. For a long time, it seemed that unitarity and renormalizability were mutually exclusive in quantum gravity, since the Hilbert-Einstein action is unitary but nonrenormalizable, while the Stelle theory is renomalizable but nonunitary. If we consider theories with more higher derivatives, then renormalization can be futher improved and we can obtain superrenormalizability or even finiteness [4]. Quantized as usual, these theories violate unitarity, because the presence of

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6 Introduction higher-derviative terms introduces ghost degrees of freedom.

However, the formulation of higher-derivative theories turns out to be less trivial than expected. For example, when the free propagators have complex poles, the theories cannot be consistently defined in Minkowski spacetime [5], in general, because they generate nonlocal, non-Hermitian divergences, which cannot be subtracted away without destroying the nature of the theory itself.

In 1969 Lee and Wick proposed an alternative formulation of higher-derivative theories with com-plex conjugate poles [6]. They introduced a prescription for the energy integral in loop diagrams by deforming the contour integration. Such deformation is motiviated by the fact that only the imaginary parts of the amplitudes contribute in the unitarity equation. The would-be ghost poles have negative residue and their contribution to the amplitudes would break unitarity. The Lee-Wick prescription is introduced to remove the contributions to the imaginary part of the amplitudes coming from the would-be ghost poles. The Lee and Wick idea is promising because it claims to reconcile renormaliz-ability and unitarity in quantum gravity. However, several problems in this approach remained quite mysterious, so far. Some authors tried to circumvent those problems [7], but the result was not general enough to hold at higher orders in the loop expansion.

In [8] a new formulation was introduced for these models to overcome the major difficulties and prove unitarity [9, 10]. From these results and from the fundamental principles of locality, renor-malizability and unitarity, a unique theory of quantum gravity was identified by means of a novel quantization prescription [4, 11]. The theory predicts the existence of a particle of a new type, called “fake particle” or “fakeon”. In particular, in the case of gravity, at least one fakeon is predicted, with spin 2 and a mass that could be much smaller than the Planck mass. Another major prediction of the theory is the violation of microcausality, whose duration has been computed in [12]. Two events can be related in a causal way only if they are separated by a time interval that is much longer than a certain value, which depends on the mass of the fakeon. This is a peculiar property and it could be possible to detect its effects in cosmology or in black holes physics.

The new quantum theory of gravity has several properties in common with the standard model. Besides being local, strictly renormalizable and perturbatively unitary, it is not asymptotically free, since some couplings tend to zero in the UV, but others do not, pretty much like the standard model, where the QCD sector is asymptotically free but the QED one is not.

The thesis is organized as follows. In chapter 2 we give a review of the possible approaches to qua-tum gravity as a quanqua-tum field theory. In chapter 3 we introduce the Batalin-Vilkovisky formalism, which is useful to quantize gauge theories and gravity and collects the Ward-Takahashi-Slavnonv-Taylor (WTST) identities in a compact way. In chapter 4 we derive the unitarity equation. In particular, we define the so called algebraic cutting equations, which are useful to proof unitarity in several cases. In chapter 5 we reformulate the Lee-Wick models by defining them as nonanalytically Wick rotated Euclidean theories. We show that the models are intrisically equipped with the right recipe to treat the ambiguities, with no need of external ad hoc prescriptions. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are

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7 related to one another by a nonanalytic procedure. We describe these features in detail by calculating the one-loop bubble diagram and explaining how the key properties generalize to more complicated diagrams. In chapter 6 we study the perturbative unitarity of the Lee-Wick models, within the new formulation. We make explicit calculations in the cases of the bubble and triangle diagrams and address the generality of our approach. We also show a sketch of the proof of unitarity to all orders. In chapter 7 we study a class of superrenormalizable models of quantum gravity, formulated as non-analytically Wick rotated theories. We present their features and issues. In chapter 8 we introduce a new quantization prescription, which is able to turn a ghost degree of freedom into a fakeon. The new prescription leads to a unique, renormalizable, unitary theory of quantum gravity in four dimensions. We work out the one-loop renomalization of the theory and the absorptive part of the graviton self energy. The results illustrate the mechanism that makes renormalizability compatible with unitarity. In chapter 9 the action of quantum gravity is rearranged by means of auxiliary fields and standard changes of field variables. Besides the graviton degrees of freedom, the theory contains a massive scalar φ and a spin-2 fakeon χµν. These fields are collected in a graviton multiplet. We couple the theory

to matter fields of all types and compute the absorptive part of the self energy of the multiplet. The width of χµν, which is negative, shows that the theory predicts the violation of causality at energies

larger than the fakeon mass. We address this issue and compare the results with those of the Stelle theory, where χµν is a ghost instead of a fakeon. Finally, chapter 10 contains our conclusions and

outlooks.

This thesis is based on the following pubblications:

D. Anselmi and M. Piva, A new formulation of Lee-Wick quantum field theory, J. High Energy Phys. 06 (2017) 066.

D. Anselmi and M. Piva, Perturbative unitarity of Lee-Wick quantum field theory, Phys. Rev. D 96 (2017) 045009.

D. Anselmi and M. Piva, The ultraviolet behavior of quantum gravity, J. High Energy Phys. 05 (2018) 27.

D.Anselmi and M.Piva, Quantum gravity, fakeons and microcausality, J. High Energy Phys. 11 (2018) 21

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8 Introduction

Acknowledgments

I wish to thank those people that in the last three years have had an impact on my work and my life. Without their contribution, this thesis would be different.

First of all, it is a pleasure to thank prof. Damiano Anselmi for his careful guidance and patience during the development of this work, and for the discussions about physics, science and life. I am glad to share with him the way to intend physics and research. He has been an healthy example of being an honest scientist and an excellent collaborator.

I thank my dad Franco for his constant presence and support in every important decision I had to take.

My gratitude goes also

to my friend Andrea D’Amico for the hours spent talking about physics (together with a good beer) and for every time I asked him to check my results and to give me his (always honest and critique) opinion; to my friend Giovanni Rabuffo for useful discussions (even if he is betraying physics) and for the time we spent dreaming how to make our life a wonderful adventure;

to my friend Paco Giudice for our discussions about life, happiness and enjoy our passions; to Stefano Titta and Iuri Sandrin for the precious time spent with them and for EVO; to Federica Cammarata for her huge help in understanding what I have to do with my life;

to prof. Enrico Meggiolaro for the time spent together trying to be good teachers for young stu-dents;

to Matteo Archimi for the discussions about our future;

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Chapter 2

Quantum field theories of gravitational

interactions

In this chapter we review several approaches to quantum gravity in the framework of quantum field theory. We start from the quantization of general relativity and discuss its nonrenormalizability. In addition, we investigate several higher-derivative models by showing their properties and issues.

2.1 General relativity

The Hilbert-Einstein action is

SHE(g) =−

1 2κ2

Z _√

−g (R + 2ΛC) , κ2 = 8πG (2.1)

where G is the Newton’s constant, with [G] =_{−2 , Λ}C is the cosmological constant with [ΛC] = 2 and

R is the scalar curvature. This action is invariant under diffeomorphisms. A coordinate transformation xµ→ x0µ(x)

is called a diffeomorphism if both xµ_(x0_{) and x}0µ_{(x) are invertible smooth functions. The group of}

these transformations on a spacetime manifold M is denoted by Diff(M). If we couple gravity to matter through an action Sm, the equations of motion are

Rµν−

1

2gµνR− gµνΛC = κ

2_T

µν, (2.2)

where Tµν is the energy-momentum tensor of the matter sector, defined as

Tµν = √−2

−g δSm

δgµν.

For the purposes of this section we consider the case of pure gravity with vanishing cosmological constant, i.e. the Hilbert action

SH(g) =− 1 2κ2 Z _√ −gR. (2.3) 9

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10 Quantum field theories of gravitational interactions In this case the equations of motion reduce to

Rµν = 0. (2.4)

An infinitesimal diffeomorphism is given by

xµ_{→ x}0µ_{= x}µ_{+ ξ}µ_(x),

where ξµ_{(x) is an infinitesimal vector depending on the spacetime point. The metric and its inverse}

transform respectively as (0,2) and (2,0) tensors, i.e. g0_µν(x0) = ∂xµ ∂x0 α ∂xν ∂x0 β gαβ(x), g0µν(x0) = ∂x0µ ∂xα ∂x0ν ∂xβg αβ_(x).

Then it is easy to find the variation of the metric tensor under infinitesimal diffeomorphisms. To the first order in ξ, we find

g_µν0 (x0) = gµν(x)− ∂µξαgαν(x)− ∂νξαgµα(x),

while making a Taylor expansion we get

g_µν0 (x0) = g0_µν(x) + ξα_∂

αgµν(x) +O(ξ2).

We are interested in the variation of the metric in the same spacetime point δgµν ≡ gµν0 (x)− gµν(x),

so by combining the two equations we find

δgµν =−∂µξαgαν − ∂νξαgµα− ξα∂αgµν. (2.5)

We expand the metric tensor as gµν = ¯gµν + 2κhµν, where ¯gµν is a fixed background metric,

and identify the small quantum perturbation hµν as the dynamical field, called graviton field. For

semplicity we take ¯gµν to be the Minkowski flat metric ηµν. We raise and lower the indices of ∂µ, hµν

and the fields (except gµν) by means of the Minkowski metric, while the indices of gµν, the Riemann

tensor and the Ricci tensor are raised and lowered by means of gµν.

The invariance under diffeomorphisms implies that in (2.3), after the expansion, the operator quadratic in hµν has null eigenvectors. In the quantum theory the problem is solved with the same

method used in quantum gauge theories (Diff(_{M) playing the role of the gauge group), by means of} a gauge-fixing procedure.

BRST symmetry for gravity

The BRST transformation sgµν for the metric tensor is defined starting from δgµν = θsgµν obtained

by setting ξµ _{= θC}µ_{, where C}µ _{are the Faddeev-Popov ghosts and θ is an anticommuting constant.}

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2.1 General relativity 11 s2 _{= 0. Finally, a trivial gauge system made by the antighosts ¯}_Cµ _{and the Lagrange multipliers B}µ_is

added:

δBRSTgµν = θsgµν = θ(−∂µCαgαν − ∂νCαgµα− Cα∂αgµν)

δBRSTCρ = θsCρ=−θCσ∂σCρ

δBRSTC¯σ = θs ¯Cσ = θBσ

δBRSTBτ = θsBτ = 0. (2.6)

The mass dimensions are

[g] = 0, [C] = 0, [ ¯C] = 0, [B] = 1.

Now we add a gauge-fixing term to the action (2.3). In the BRST formalism this is peformed by introducing a fermionic functional Ψ, with mass dimension [Ψ] = −1, and then defining the gauge-fixing term through its BRST transformation sΨ. A convenient choice for Ψ is

Ψ = Z

¯

CµGµ(g)− κ2Bµ , (2.7)

where Gµis a local function of the metric tensor that fixes the gauge. The gauge-fixed action reads

Sgf(g, C, ¯C, B) = SH(g) + sΨ, (2.8)

which is invariant under the above BRST transformations. In fact θsSgf(g, C, ¯C, B) = θsSH(g) + θs(sΨ)

= δSH(g) + θs2Ψ = 0,

(2.9) where in the last step we have used the fact that the operator s is nilpotent and acts as a gauge transformation on the classical action.

Given a functional X of the fields, we say that X is s-closed if sX = 0, while we say that X is s-exact if X = sY , where Y is another functional. The expectation values of products of physical observables must be gauge invariant. In the BRST formalism this property is satisfied when the observables are s-closed (banning anomalies, which do not occur here). From the nilpotency of the transformations, it follows that every s-exact object is trivially s-closed. Therefore, physical observables are defined as the equivalence classes of s-closed observables that differ by s-exact observables. This definition implies that C, ¯C and B are unphysical objects, since the ghosts and antighosts are not s-closed, while the Lagrange multipliers are s-exact.

Graviton propagator

Now we work out the gauge-fixed propagator for the fluctuations hµν. Expanding the square root of

the determinant as

√

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12 Quantum field theories of gravitational interactions where h = hµνηµν, the quadratic part of the action (2.3) is

S_Hquad = ₋1 2 Z [hµνhµν− 2hµν∂µ∂ρhρν− hh + 2h∂ρ∂σhρσ] = 1 2 Z hµν_V µνρσhρσ, (2.10)

where the operator Vµνρσ is symmetrized with respect to the exchanges µ↔ ν, ρ ↔ σ and µν ↔ ρσ.

Now we have to include the contribution of the gauge-fixing term by choosing the local function Gµ.

Our choice is Gµ= ηνρ∂ρgµν− 1 2η νρ_∂ µgνρ= 2κ ∂νhνµ− 1 2∂µh , (2.11) which is the analouge of the Lorenz gauge for electrodynamics, known as de Donder gauge. For semplicity, we solve the field equation for Bµ _{and substitute the solution in}

sΨ = Z Bµ _Gµ− κ2Bµ + Sgh, (2.12) where Sgh=− Z ¯ Cµ Z _δ Gµ δgαβ sgαβ . (2.13)

The Euler-Lagrange equations for Bµ _are

Gµ− 2κ2Bµ= 0 ⇒ Bµ=

1 2κ2Gµ,

and the gauge-fixing action turns into

sΨ = 1 4κ2

Z

GµGµ+ Sgh.

The total quadratic operator in momentum space is then Vtot µνρσ(p) =−p2 Π(2)_µνρσ₋ 1 2Π (0) µνρσ+Π(1)µνρσ + 1 2Π¯ (0) µνρσ+ + 1 2 ¯ ¯ Π(0)_µνρσ (2.14) and its inverse defines the graviton propagator

hhµν(p)hρσ(−p)i₀ = i 2(p2_{+ i)} 2Π(2)_µνρσ₋Π(0)_µνρσ+ 2Π_µνρσ(1) +Π¯(0)_µνρσ₋Π¯¯(0)_µνρσ (2.15) = i(ηµρηνσ+ ηµσηνρ− ηµνηρσ) 2(p2_{+ i)} ,

where we used the Feynman prescription and the projector operatorsΠ(i)are defined in the appendix 2.4. The ghosts action reads

Sgh= Z ¯ Cµ∂ν₋1 2η µν_C_¯τ_∂ τ (gµρ∂νCρ+ gνρ∂µCρ+ Cρ∂ρgµν) (2.16)

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2.1 General relativity 13 and its quadratic part is

S_ghquad= 2κ Z

¯

CµηµνCν.

Switching to momentum space, the equation of the propagator is

− 2κp2ηµνCνC¯ρ = iδµρ, (2.17) whose solution is Cµ_C_¯ν_{= −} i 2κ 1 p2_{+ i}η µν_. _(2.18) Unitarity

Perturbative unitarity can be proved by means of the so called cutting equations. Their definition is given in chapter 4, where the problem of unitarity in a general quantum field theory is treated in detail. A preliminary check for unitarity can be done by switching to a convenient gauge fixing, where propagators become manifestly unitary. There, we count the physical degrees of freedom by looking at the poles. In the case of Yang-Mills theory this is achieved in the Coulomb gauge. In the case of gravity an analogous gauge is known as Prentki gauge and corresponds to the choice

Gµ= ηiν∂igνµ=−2κ∂ihiµ, i = 1, 2, 3 (2.19)

in (2.7). We simplify the computation of the propagator by proceeding as follows. We work in the Euclidean signature and choose a coordinate system where p1 = p2= 0. Thus, we have p2 = p24+ p23,

where p4 = ip0. Then, we impose the gauge condition ∂ihiµ = 0, which in momentum space reads

h3

µ= 0. The quadratic part in the graviton field is, in matrix notation,

S_HEquad= 1 2 Z ˜ hT V˜h. (2.20) Here, ˜h_{≡ {00, 01, 02, 11, 12, 22} (where “ij” stands of h}ij_{) and V is a 6}_{×6 matrix. The inverse matrix}

reads V−1= 1 2             p2 p4 3 0 0 − 1 p2 3 0 − 1 p2 3 0 1 p2 3 0 0 0 0 0 0 _p12 3 0 0 0 −1 p2 3 0 0 1 p2 0 −_p12 0 0 0 0 1 p2 0 −1 p2 3 0 0 − 1 p2 0 _p12             . (2.21)

There are two poles at p0 =±|p3|. Note that they are present only in the subspace S = {11, 12, 22}.

In this subspace the matrix

V−1_S = 1 2     1 p2 0 −_p12 0 _p12 0 −1 p2 0 _p12     (2.22)

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14 Quantum field theories of gravitational interactions has determinant equal to zero. This implies that there are only two independent positive residues. Indeed, the matrix of the residues can be diagonalized and gives

p2 V−1_S p2₌₀= 1 2    1 0 −1 0 1 0 −1 0 1    → 1 2    2 0 0 0 1 0 0 0 0   . (2.23) Now we check whether the ghosts have propagating degrees of freedom or not. In the gauge (2.19), the quadratic part of the action of the ghost fields is

S_ghquad =− Z ¯ Cµ(∂i∂iCµ+ ∂i∂µCi) ≡ − Z ¯ Cµ(Vgh)µνCν. (2.24) In momentum space, we rotate to have p1= p2= 0 and in the basis { ¯C0, ¯C3, ¯C1, ¯C2} and

{C0_{, C}3_{, C}1_{, C}2_{}, the matrix V} gh reads Vgh=       p2 3 p0p3 0 0 0 2p2 3 0 0 0 0 p2 3 0 0 0 0 p2 3       ⇒ V−1_gh =        1 p2 3 − p0p3 2p4 3 0 0 0 _2p12 3 0 0 0 0 1 p2 3 0 0 0 0 1 p2 3        , (2.25)

Therefore, the ghost propagators have no poles and the theory contains two massless propagating degrees of freedom, which correspond to the graviton elicities.

We stress again that this is a necessary but not sufficient condition for unitarity. A complete proof requires to show that the cutting equations are satisfied to all orders in the loop expansion. A recent proof in case of QED, Yang-Mills theory and general relativity can be found in [13].

Divergences of the theory

We recall the results about the ultraviolet behaviour of the theory. First of all, we show that general relativity is nonrenormalizable by simple power counting. In fact, the quadratic part of the action (2.3) has the correct normalization and the vertices are multiplied by powers of κ, which has negative mass dimension. This implies that we can construct infinitely many potential counterterms, with arbitrarily large dimensions, each one multiplied by a suitable power of κ. In other words, the theory is nonrenormalizable. The one-loop divergences are proportional to the following higher-derivative terms

Z _√

−gRµνRµν and

Z _√

−gR2. (2.26) Precisely, the counterterms, computed by ’t Hooft and Veltman [1] in dimensional regularization, are

∆S = µ −ε 8π2_ε Z _√ −g 1 120R 2₊ 7 20RµνR µν , (2.27)

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2.1 General relativity 15 where µ is the dynamical scale, ε = 4− D and D is the continued dimension. Since the two contribu-tions to ∆S are proportional to the vacuum field equation (2.4), they can be absorbed by means of a field redefinition. This is proved in the following way. Consider an action functional S(φi) of certain

fields φi and its equations of motion δS/δφi ≡ Si where the index i collects spacetime, Lorentz and

group indices, and summation over repeated indices includes integration over spacetime points. We can write

S(φ0) = S(φi+ Fi) = S(φi) + FiSi+O(F2)

where Fi can contain fields and their derivatives. In the case of gravity we have

g_µν0 = gµν+ aRµν + bgµνR +O(a2, b2, ab). (2.28)

With a suitable choice of the coefficients a and b, we can write

SH(g0) = SH(g)− ∆S(g) + O(a2, b2, ab), (2.29)

which shows that the one-loop divergences can indeed be removed in this way. We conlcude that pure gravity is finite at one loop in dimension 4. Actually, this is just a lucky coincidence since finiteness is spoiled by the presence of matter. If we couple gravity to matter the equations of motion are modified by the energy momentum tensor

Rµν−

1

2gµνR = κ

2_T µν.

The divergences (2.26) of the pure gravity sector are not proportional to the equations of motion any longer and cannot be completely reabsorbed by means of field redefinitions. Moreover, other divergences appear, due to loop diagrams with circulating matter fields. For instance, if we couple a scalar field ϕ to gravity through the action

S(ϕ) = 1 2

Z _√

−g∂µϕ∂νϕgµν,

we obtain new divergent terms proportional to Z _√ −g(gµν_∂ µϕ∂νϕ)2, Z _√ −gRgµν_∂ µϕ∂νϕ, Z _√ −g(∇µ_∇ µϕ)2. (2.30)

In ref. [2] Goroff and Sagnotti showed that pure gravity diverges at two loops and all the nontrivial divergent terms are proportional to

Z _√ −gRµνρσR ρσ αβR αβ µν, (2.31)

which cannot be absorbed by means of field redefinitions, neither in pure gravity, nor in gravity coupled to matter. Therefore, renormalization leads to add a term similar to (2.31) in (2.3), but this would generate other new divergent terms, which also need to be added to the action, multiplied by their own indipendent parameters, and so on. In the end, the resulting action contains infinitely many terms and parameters. Hence, the quantum version of general relativity turns out to be nonrenormalizable in 4 dimensions.

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16 Quantum field theories of gravitational interactions In higher dimension there are other divergences. For instance, in dimension 6 the counterterm (2.31) appears already at one loop [14]. Therefore, increasing the spacetime dimensions does not solve the problem. It has long been believed that supergravity could be a solution. It is known thatN = 1 supergravity is two-loop finite [15] but then it starts to be divergent. Extended supergravities,_{N = 8} for instance, are believed to be finite up to higher orders.

In conclusion, general relativity needs to be treated as an effective theory, being predictive only in a low energy approximation. A more fundamental theory should exist.

2.2 Stelle gravity

A possible way out to the nonrenormalizability problem of general relativity could be to consider a higher-derivative theory. In 1977 K.S. Stelle proposed a new model of gravity [3] where the Hilbert-Einstein action (2.1) is modified by adding two higher-derivative terms. The candidates for such additions are terms quadratic in the curvature tensors, which contain four derivatives. They modify the dominant kinetic term of the action, leading to a stabilization of the divergences and making the theory renormalizable by power counting. This is the simplest extension of the gravitational action that provides renormalizability. As shown in detail in the next section, the addition of terms with more than four derivatives can make the theory convergent above a certain order in the loop expansion, but cannot remove the one-loop divergences. Furthermore, one must choose the new terms carefully and check if they really improve the behaviour of propagators in the UV limit. For instance, the term √

−gR3 _{is not important, since it does not contain any terms quadratic in the graviton field, while}

the term √_{−gRR is relevant, but alone is not enough. Unfortunately, the price to pay to have} renormalizability is the violation of the unitarity condition. In this section we briefly review the Stelle theory, show its nonunitarity and discuss the ghost problem.

Consider the following modification of the Hilbert-Einstein action, with four-derivative terms, SHD(g) =− 1 2κ2 Z _√ −g 2ΛC+ ζR + α RµνRµν− 1 3R 2 −ξ 6R 2 , κ2 _{= 8πG.} _(2.32)

This action is the most general one with four derivatives. In fact, it is not necessary to add the functional

Z _√

−gRµνρσRµνρσ,

which is quadratic in the Riemann tensor, because the Gauss-Bonnet term Z _√

−g(RµνρσRµνρσ− 4RµνRµν+ R2)

is topological in four dimensions. In (2.32) we highlight a particular combination of the squared Ricci scalar and the squared Ricci tensor whose integral is proportional to the integral of the squared Weyl tensor, up to a total derivative. The symmetry of this action is the same as in (2.1), that is to say

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2.2 Stelle gravity 17 In this parametrization we can fix the mass dimensions in order to have a properly normalizaed dominant kinetic term. Since R2 _{and R}

µνRµν already have dimension 4 we choose

[α] = 0, [ξ] = 0, [κ] = 0, [ζ] = 2, [ΛC] = 4. (2.33)

No parameter has negative dimension. Therefore, we can construct only a finite number of countert-erms of dimension 4, which is a necessary condition to have renormalizability by power counting.

We recall the BRST transformations (2.6), which are

sgµν = −∂µCαgαν − ∂νCαgµα− Cα∂αgµν

sCρ = _−Cσ∂σCρ

s ¯Cσ = Bσ

sBτ = 0. (2.34)

The dimensions of the fields are

[gµν] = 0, [Cρ] = 0, [ ¯Cσ] = 0, [Bτ] = 1.

Adding the gauge-fixing term to the action (2.32), we obtain the gauge-fixed action

Sgf(g, C, ¯C, B) = SHD(g) + sΨ, (2.35)

where Ψ is a fermionic functional with [Ψ] =_{−1. In order to have the same behaviour in the UV for} both the ghost propagator and the graviton propagator, we choose the functional

Ψ = Z

α ¯Cµ _G

µ− λκ2Bµ , (2.36)

where = ηµν_∂

µ∂ν, Gµ = ηνρ∂ρgµν and λ is a dimensionless gauge-fixing parameter. Following the

same procedure of the previous section, we sustitute Bµ _{with the solution of its equation of motion,}

which reads

Bµ=

1

2λκ2Gµ (2.37)

and the gauge-fixed action (2.35) becomes

Sgf(g, C, ¯C, B) = SHD(g) + α 4λκ2 Z Gµ_G µ+ Sgh,

where the action Sgh of the Faddeev-Popov ghosts reads

Sgh =

Z

α ¯Cµ∂ν [gµρ∂νCρ+ gνρ∂µCρ+ Cρ∂ρgµν] . (2.38)

We show that the theory is not unitary, because some additional propagating degrees of freedom with negative residues are present. For semplicity we set ΛC = 0. Expanding the metric around a flat

background as

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18 Quantum field theories of gravitational interactions the propagator of the fluctuation hµν is

hhµν(p)hρσ(−p)i₀= i p2_{+ i} " Π(2)_µνρσ (ζ_{− αp}2₎ − Π(0)_µνρσ 2(ζ_{− ξp}2₎ − λ 2αp2 2Π(1)_µνρσ+Π¯(0)_µνρσ # . (2.39) The expression of the ghost propagator follows straightforwardly, as in (2.16), from the part of Sgh

that is quadratic in the fields C, ¯C. In momentum space we find Cµ_C_¯ν₌ i p2_{+ i} ηµν_{− p}µ_pν_/2p2 αp2 . (2.40) We identify the single poles and their residues by splitting the propagator (2.39) into partial fractions. To further simplify, we choose λ = 0, then we find

hhµν(p)hρσ(−p)iλ=0₀ = i 2ζ " 2Π(2)_µνρσ−Π(0)_µνρσ p2_{+ i} − 2Π(2)_µνρσ p2_{− ζ/α} + Π(0)_µνρσ p2_{− ζ/ξ} # . (2.41) We see that the theory has two massive poles in addition to those of general relativity. Precisely, there is one spin-2 massive pole with squared mass ζ/α and negative residue (which is a ghost) and a scalar pole with squared mass ζ/ξ. We note that if ξ > 0, then the scalar degree of freedom has both positive residue and squared mass.

We could suppress some higher-derivative terms choosing specific values for ξ or α. The choice ξ = 0 is nonsingular, since it just selects a particular combination of the coefficients in front of R2 _and

RµνRµν. However, this choice is not preserved by renormalization. We could also choose ξ = −2α,

which cancels the R2 _{terms from the action. However, neither of these limits does eliminate the}

ghost problem. The limit α → 0 is equivalent to perform a higher-derivative regularization with a cut-off Λ = α−1 and send Λ _{→ ∞ at the end. The divergences in the cut-off are always of the} from Λr_lns_{Λ = α}−r₍₋₁₎s_lns_{α, where s, r} _{≥ 0 and s + r > 0, and correspond to the divergences of}

general relativity. In the end, the nonrenormalizability problem reappears. To have power counting renormalizability, we need to keep both higher-derivative terms RµνRµν and R2, at the expense of

unitarity.

Renormalization

The theory (2.32) is perturbatively renormalizable, which means that the divergences have the same functional form as the terms of the classical action and therefore can be canceled by means of field and parameter redefinitions. This fact was proved in [3], where two methods were presented. The first one amounts to choose a specific gauge (which in our case corresponds to λ = 0) and show that it is possible to move three derivatives on the antighost by integrating by parts. Then, the superficial degree of divergence of a diagram is always negative when at least one external leg is nonphysical. The second method is more general and makes use of the WTST identities [16] to obtain an equation which determines the functional expression of the counterterms. However, in the proof a conjecture

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2.3 Inconsistencies of Minkowski higher-derivative theories 19 was assumed to hold (known as Kluberg-Stern–Zuber conjecture [17]). The conjecture was recently proved [18] in a very general setting, which also covers higher-derivative gravity. See chapter 3 and chapter 8 for more details.

Since the renormalization of the new theory of quantum gravity we propose coincides with the one of the Stelle theory, we postpone its details to chapter 8.

2.3 Inconsistencies of Minkowski higher-derivative theories

In this section we expose a recent result in higher-derivative theories, obtained by Aglietti and Anselmi in [5]. The authors show that, in general, higher-derivative theories cannot be defined directly on Minkowski spacetime but they must be defined in the Euclidean space and then Wick rotated. In fact, if a propagator has poles in the first or third quadrant of the complex energy plane, then the theory is plagued with nonlocal and non-Hermitean divergences, which cannot be absorbed away by means of field and parameter redefinitions. In [5] it is also shown that in the case of Euclidean quantum field theories of this type, nonlocal and non-Hermitean terms appear only at intermediate steps and cancel in the final result. As we will show in chapter 5, this fact forces us to define the Lee-Wick models from the Euclidean space, since those theories have complex conjugated poles in the propagator. If a theory is analytically equivalent to its Wick rotated version, then the problem of nonlocal and non-Hermitean divergences does not occur.

An example is given by a six dimensional massless scalar ϕ4_{-theory with propagator}

D(p, m) = 1 p2_{− m}2_{+ i}

M4

(p2₎2_{+ M}4. (2.42)

As shown in [5], the divergent part of the one-loop two point function has a nonlocal and non-Hermitean part, which reads

hϕ(p)ϕ(−p)inl div1-loop =−

M4 2(4π)3 h M2 (p2₎2 − i p2 i lnΛU V M2 . (2.43) In four dimensions this term is not present and the theory is safe. However, if we consider theories with derivative interactions, such as gauge theories or quantum gravity, nonlocal and non-Hermitean divergences appear again, even in four dimensions.

Higher-derivative gravity

We present the nonlocal divergences of the graviton two-point function found in [5], in the case of a relatively simple model of four-dimensional higher-derivative gravity with complex poles. The calculations are simplified as much as possible by choosing a specific action and a convenient gauge fixing.

The simplest model of higher-derivative gravity is Stelle theory. However, it is not suitable for this investigation, because its propagator does not have poles in the first or third quadrants. The simplest

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20 Quantum field theories of gravitational interactions model with the features we need is the one with action

Sgrav =− 1 2κ2 Z _√ −g R− 1 M4(∇ρRµν)(∇ ρ_Rµν_{) +} 1 2M4(∇ρR)(∇ ρ_R) , (2.44) where [κ] = _{−1, [M] = 1 and ∇}µ is the covariant derivative. The metric tensor is expanded around

flat spacetime as in the previous sections. We choose the de Donder gauge-fixing function Gµ(g) = ηνρ∂ρgµν−

1 2η

νρ_∂

µgνρ= κ(2∂νhνµ− ∂µh) (2.45)

and the fermionic functional Ψ is chosen such that, once we integrate away the field Bµ_{, the gauge-fixed}

action reads Sgf= Sgrav+ 1 4κ2 Z Gµ 1 + 2 M4 Gµ+ Sgh, (2.46) where = ηµν_∂

µ∂ν is the flat-space D’Alembertian, while the ghost action is

Sgh = Z ¯ Cµ∂ν ₋1 2η µν_C_¯τ_∂ τ 1 + 2 M4 (gµρ∂νCρ+ gνρ∂µCρ+ Cρ∂ρgµν) . (2.47)

The graviton propagator

hhµν(p) hρσ(−p)i0=

iM4

2(p2_{+ i)}

ηµρηνσ+ ηµσηνρ− ηµνηρσ

(p2₎2_{+ M}4

has the same form as the propagator (2.42), apart from the constant matrices in the numerator. Normally, the ghosts contribute to the renormalization, because they must compensate the contri-butions of the temporal and longitudinal components of the gauge fields, to give a total gauge invariant result. However, it is easy to show that in this case they can be ignored, because they cannot give nonlocal divergences at one loop. Indeed, after the redefinition ¯C0µ = 1 + 2_/M4 _¯

Cµ_{, the ghost}

action (2.47) turns into the usual one, which is Sgh = Z ¯ Cµ∂ν₋1 2η µν_C_¯τ_∂ τ (gµρ∂νCρ+ gνρ∂µCρ+ Cρ∂ρgµν) .

For this reason, the ghost contribution to the one-loop graviton two point function coincides with the usual one, which has a local divergent part.

It is sufficient to work out the three-graviton vertex, since the one-loop diagrams involving four-leg vertices are tadpoles, which can only have local divergent parts.

The one-loop nonlocal divergent part of the graviton two-point function found in [5] is hhµν(p) hρσ(−p)inl d1 = κ2_M8 240π2_(p2₎2 [(68r + i)(ηµρηνσ+ ηνρηµσ) + (373r− 4i)ηµνηρσ − 1 8p2(125ir 2_{+ 544r + 8i) (p} µpρηνσ+ pµpσηνρ+ pνpρηµσ+ pνpσηµρ) + 1 4p2(255ir 2_{− 1522r + 36i) (p} µpνηρσ+ pρpσηµν) −_2(p1₂₎₂(185r3+ 75ir2_{− 1048r + 24i)p}µpνpρpσ ln Λ 2 U V M2 , (2.48)

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2.4 Projector operators 21 where r ≡ p2_/M2_{. Coherently with the scalar result (2.43), the divergences are nonlocal and truly}

complex. It is impossible to subtract them away by means of reparametrizations and (local as well as nonlocal) field redefinitions that preserve Hermiticity. Hence, Minkowski higher-derivative theories of gravity violate the locality and Hermiticity of counterterms, when the propagators have poles in the first or third quadrants. Gauge symmetries are unable to protect those properties. The divergent term (2.48) does not appear if the theory is formulated in Euclidean space.

The gravitational action (2.44) is the simplest one that exhibits the effects uncovered in [5]. Similar effects are expected to occur in theories with actions

S_grav0 =₋ 1 2κ2 Z _√ −g R + 1 M2RµνPn(c/M 2_)Rµν₋ 1 2M2RQn(c/M 2_)R ,

where c denotes the covariant D’Alembertian and Pn, Qn are real polynomials of degree n > 0.

In conclusion, in general higher-derivative quantum field theories must be defined from the Eu-clidean space and then Wick rotated. This is a crucial point for the new formulation of the Lee-Wick models proposed in [8] and explained in chapter 5, which eventually led to the idea of a new quan-tization prescription, which is the key to the formulation of a unique, consistent theory of quantum gravity [4, 11, 12].

Appendix

2.4 Projector operators

Starting from the transverse and longitudinal projectors for vectors πµν ≡ ηµν− pµpν p2 , (2.49) ωµν ≡ pµpν p2 , (2.50)

we define the projectors for the spin 2 tensors as

Π(2)_µνρσ _≡ 1 2(πµρπνσ+ πµσπνρ)− 1 3πµνπρσ, (2.51) Π(1)_µνρσ ≡ 1 2(πµρωνσ+ πµσωνρ+ πνρωµσ+ πνσωµρ), (2.52) Π(0)_µνρσ ≡ 1 3πµνπρσ, (2.53) ¯ Π(0)_µνρσ ≡ ωµνωρσ, (2.54) ¯ ¯ Π(0)_µνρσ _{≡ π}µνωρσ+ πρσωµν. (2.55)

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22 Quantum field theories of gravitational interactions Other useful relations are:

1 2(ηµρηνσ+ ηµσηνρ) = h Π(2)+Π(1)+Π(0)+Π¯(0)i µνρσ, pµpρηνσ+ pνpρηµσ+ pµpσηνρ+ pνpσηµρ= p2 h 2Π(1)+ 4Π¯(0)i µνρσ, pµpνηρσ+ pρpσηµν = p2 ¯ ¯ Π(0)+ 2Π¯(0) µνρσ , ηµνηρσ= 3Π(0)+Π¯(0)+Π¯¯(0) µνρσ , pµpνpρpσ = (p2)2Π¯ (0) µνρσ. (2.56)

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Chapter 3

Batalin-Vilkovisky formalism

The Batalin-Vilkovisky formalism [19] is a powerful tool to quantize field theories and study their renormalization. The starting point is to define an extended action by adding source terms for the composite BRST operators. Then the generating functional and the effective action are defined as usual. In this chapter we give a brief review of the formalism, proving the properties that we need for our work. In particular, we derive the structure of the renormalized action and show that the counterterms can be subtracted by means of renormalization constants for the fields, the sources and the parameters.

We collect the physical fields in a single row

φi= gµν, Aaµ, ψ, ¯ψ, ϕ ,

where ψ and ϕ contain all the fermionic and scalar fields, respectively. We assume that a classical action Sc(φ) is invariant under certain infinitesimal transformations

φi _{→ φ}i_{+ δ} Λφi, that is to say δΛSc(φ) = Z δSc δφiδΛφ i _{= 0,}

where Λ = Λ(x) denotes the local parameter of the symmetry. Then we introduce the extended row Φα_{, which includes the ghosts, antighosts and Lagrange multipliers}

Φα = φi, C, ¯C, B .

The BRST symmetry is derived, as explained in the previous chapter, by making the substitution Λ(x)→ θC(x). The BRST transformations for the row Φα _{are denoted by}

Rα_{(Φ) = sΦ}α_.

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24 Batalin-Vilkovisky formalism We introduce the ghost number

gh(φ) = gh(B) = 0, gh(C) = 1, gh( ¯C) =−1. The action, as well as the functional measure, is invariant under

Φ→ Φexp iσgh(Φ), where σ is a constant parameter.

3.1 Composite operators

The BRST transformations involve products of fields and their derivatives in the same spacetime point, namely composite operators. The renormalization of composite operators needs to be treated separately from the renormalization of the elementary fields, since the two are not related in an obvious way. Let O be a composite operator of a certain field φ and call its renormalization constant ZO.

Then the bare operator is

OB(φB) = ZOOR.

The renormalized operator can be written in terms of the renormalized field φR as

OR= ZO−1OB(Z 1/2 φ φR),

where Zφ is the wave-function renormalization constant of φ (φB = Z_φ1/2φR). The generating

func-tional of the correlation functions is defined as Z(J) = Z [dφ]exp iSc(φ) + i Z φJ , (3.1)

where J is a source for the elementary field φ and Sc the classical action. The correlation functions

which involve composite operators must be properly defined. For this purpose we add a source term to the action

S0(φ, KO) = Sc(φ) +

Z

KOO,

where KO is a source for the composite operator. In the case of curved spacetime all the sources are

considered as scalar (vector, tensor) densities and therefore they carry a hidden√_{−g factor. Finally,} the correlation functions can be obtained by differentiating the new generating functional Z(J, KO),

[obtained by replacing Sc(φ) with S0(φ, KO) in formula (3.1)] with respect to the sources. The source

term KOO has to be treated as any other vertex, where KO, being external, is a non propagating

field. We write

KB = ZKKR

where KB and KR are the bare and renormalized sources, respectively. Since bare and renormalized

actions are the same quantities, written in different variables, we have

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3.2 Antiparentheses and master equation 25 Now we introduce the sources

Kα = (Ki, KC, K_C¯, KB)

for the composite BRST operators _Rα_{(Φ), with ghost numbers}

gh(Ki) = gh(KB) =−1, (3.3)

gh(KC) =−2, gh(K_C¯) = 0. (3.4)

Next, we add the term SK =−R RαKα to the gauge-fixed action Sgf(Φ) = Sc(φ) + sΨ(Φ) and define

the extended action as

S(Φ, K) = Sgf(Φ) + SK. (3.5)

Observe that the sources and their fields have statistics opposite to each other, since the action has bosonic statistics:

εKα = εΦα+ 1 mod 2.

3.2 Antiparentheses and master equation

Given two functionals X(Φ, K) and Y (Φ, K) we define their antiparentheses (X, Y ) as the functional (X, Y )≡ Z _δ rX δΦα δlY δKα − δrX δKα δlY δΦα , (3.6)

where the subscripts l, r in δl, δr denote the left and right functional derivatives, respectively. The

antiparenteses satify the property

(Y, X) =₋₍₋₁₎(εX+1)(εY+1)_{(X, Y )} _(3.7)

and the Jacobi identity

(₋₁₎(εX+1)(εZ+1)_{(X, (Y, Z)) + cyclic permutation = 0.} _(3.8)

Using these properties we compute the functional (S, S). For a generic bosonic functional XB we can

write (XB, XB) =−2 Z δrXB δKα δlXB δΦα . (3.9) Thus we have (S, S) =₋₂ Z _δ rSgf δKα δlSgf δΦα + δrSgf δKα δlSK δΦα + δrSK δKα δlSgf δΦα + δrSK δKα δlSK δΦα . (3.10) Since Sgf does not depend on the sources Kα, the first and the second term are zero. Then, being

sΦα₌₋δrSK δKα, the rest is (S, S) = 2 Z sΦαδlSgf δΦα + sΦ αδlSK δΦα = 2sSgf+ 2sSK. (3.11)

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26 Batalin-Vilkovisky formalism From the s-invariance of Sgf (sSgf= 0) and the nilpotency of s (s2= 0), we obtain the master equation

(also known as Zinn-Justin equation)

(S, S) = 0. (3.12) We also define the operator (S,·), then its nilpotency follows from the master equation. Indeed,

(S, (S, X)) =₋1

2(X, (S, S)) = 0, (3.13) where in the last step we used the Jacobi identity (3.8). Note that the operator (S,·) coincides with s on the fields Φα

(S, Φα_{) =}₋δrS

δKα

= sΦα_.

Moreover, the s-invariance of S follows from the master equation sS = sΦα δlS δΦα =− δrS δKα δlS δΦα = 1 2(S, S) = 0. (3.14) The master equation encodes both the symmetry and the nilpotency in a unique expression.

We call closed a functional X such that (S, X) = 0 and exact a functional Y = (S, Z) where Z in another functional.

In this formalism, given a fermionic functional Ψ, the gauge-fixing term sΨ can be written as sΨ = (SK, Ψ).

Hence the extended action is

S(Φ, K) = Sc(φ) + (SK, Ψ) + SK. (3.15)

3.3 Canonical transformations

A canonical transformationC is a map

Φα_{→ Φ}α0_{(Φ, K),} _K

α → Kα0(Φ, K)

which preserves the antiparentheses, i.e. such that

(X0, Y0)0 = (X, Y ),

where X0_{, Y}0 _{are the functionals evaluated in the transformed fields and sources and (}_{·, ·)}0 _{denotes the}

antiparenteses with respect to Φα0 _{and K}0

α. A canonical transformation is generated by a fermionic

functional_{F such that}

Φα0₌ δF δK0 α , Kα= δ_F δΦα.

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3.3 Canonical transformations 27 The generating functional Z of the correlation functions and the generating functional W of the connected correlation functions are defined by the formulas

Z(J, K) = Z [dΦ]exp iS(Φ, K) + i Z ΦαJα = exp iW (J, K), (3.16) where [dΦ] = [dφ][dC][d ¯C][dB] is the functional integration measure and Jα are external sources.

The effective action Γ(Φ, K), i.e. the generating functional of the one-particle irreducible diagrams, is defined as the Legendre transform of W (Φ, K) with respect to J, where Φα_{= δ}

rW/δJα Γ(Φ, K) = W (J(Φ), K)₋ Z ΦαJα, Jα = δlΓ(Φ, K) δΦα . (3.17)

We prove now that [dΦ] is an s-invariant functional measure in dimensional regularization. Since anticommuting fields and parameters are involved, we define the superdeterminant and the supertrace of a blocks matrix M as sdetM _≡ detA det(D_{− CA}−1_B), strM ≡ trA − trD, M = A B C D ! ,

where A, D contain commuting entries while B, C contain anti-commuting entries. Under the trans-formation Φα _{→ Φ}α0_{= Φ}α_{+ θ}_Rα_{, the functional measure changes as}

[dΦ0] =J [dΦ],

where _{J is the superdeterminant of the Jacobian matrix J =} δΦ_δΦα0β_(y)(x). The parameter θ is

anticom-muting, which implies θ2 _{= 0. Then, the superdeterminant has the exact expansion}

sdet_{J = sdet} δ_βαδ(x_{− y) +}δθR α_(x) δΦβ_(y) = 1 + str δθR α_(x) δΦβ_(y) .

In the framework of the dimensional regularization, the functional derivative inside the superdeter-minant produces a D-dimensional Dirac delta function or its derivatives, depending on whether the fields are differentiated or not. In Fourier transform, we have a sum of contributions proportional to

δ(D)_(x_{− y) =} Z _dD_p (2π)De −ip(x−y)_, _(3.18) ∂µ1. . . ∂µnδ (D)_(x_{− y) = (−i)}n Z dD_p (2π)De −ip(x−y) pµ1. . . pµn. (3.19)

Since the supertrace is understood also in the spacetime indices we have to set x = y, findingR dD_p

(2π)D1

and (_−i)nR dD_p

(2π)Dpµ1. . . pµn, which are zero in dimensional regularization as long as the number of

derivatives is finite. Finally, we have

J = 1 + str δθR

α_(x)

δΦβ_(y)

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28 Batalin-Vilkovisky formalism This ensures that, working in dimensional regularization, the functional measure is s-invariant also in the case of gravity.

Since both the action (3.5) and the measure are s-invariant, we write Z(J, K) = Z [dΦ]eiS(Φ,K)+iR ΦαJα ₌ Z [dΦ0]eiS(Φ0,K)+iR Φα0Jα = Z [dΦ]eiS(Φ,K)+iR ΦαJα+iR θRαJα _{= Z(J, K)}D_eiR θRαJαE_, (3.21) where in the last step we used the definition of the expectation value of an operator _{O in the path} integral formalism, which is

hOi = R [dΦ]Oe iS+iR Φα_J α R [dΦ]eiS+iR Φα_J α . (3.22) It follows that D eiR θRαJα E = 1.

Expanding the exponential in the parameter θ, it is sufficient to impose the first order equation Z hRα_J αi = Z hRα_{i J} α= 0. (3.23)

From the definition of the effective action we find δrZ δKα = eiWδrW δKα ⇒ − δrΓ δKα = 1 Z δrZ δKα =hRα_{i .} _(3.24) Finally, we obtain 0 =− Z δrΓ δKα Jα =− Z δrΓ δKα δlΓ δΦα = 1 2(Γ, Γ), (3.25) which is the master equation for the gamma functional. With the same argument we can prove that the average of any s-invariant composite operator does not depend on the gauge fixing, in particular on the choice of the fermionic functional Ψ. Let_{O be a composite operator. From the s-invariance of} the measure and the action we can write

Z−1(J, K) Z [dΦ]OeiS+iR Φα_J α_| J =0 = Z−1(J, K) Z [dΦ](O + θsO)eiS+iR Φα_J α_| J =0, (3.26) whence hsOi0 = 0, (3.27)

where the subscript 0 denotes that the external sources J are set to zero. Now, consider s-closed composite operators Θα(x) and the correlation function

hΘα1(x1) . . . Θαn(xn)i0. (3.28)

Suppose that we vary the fermionic functional with an arbitrary deformation δΨ. Then the action varies by s(δΨ). Therefore the first order variation of (3.28) is

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3.4 Renormalization 29 This ensures that the value of (3.28) is the same for every gauge choice. If the correlation functions (3.28) satisfy certain properties in some gauge, then we can conclude that such properties hold in every gauge. For instance, in QED, using a manifestly unitary gauge such as the Coulomb gauge, we can easily check that the propagating degrees of freedom are just the two physical polarizations.

3.4 Renormalization

The renormalization can be achieved by making canonical transformationsCk and parameters

redefi-nitions Rk. We restore the ~ unit and denote any quantity renormalized up to the order ~n included

by the subscript n. Renormalization is performed by the operations

Sn(Φn, Kn, %n) = (Cn◦ Rn)Sn−1(Φn−1, Kn−1, %n−1),

S0(Φ0, K0, %0) ≡ SB(ΦB, KB, %B),

where % collects all the parameters of the theory and the subscript B indicates the bare quantities. The renormalization up to the order ~n _reads

Sn(Φn, Kn, %n) = ℘nSB(ΦB, KB, %B), (3.30)

where

℘n=Cn◦ Rn◦ . . . C1◦ R1. (3.31)

Observe that since the field redefinitions are canonical transformations, the master equation is pre-served at every order

Sn, Sn n= 0, Γn, Γn n= 0, where ·, ·

n are the antiparentheses with respect to Φn and Kn. Expanding Γn in powers of ~, the

(n + 1)-th order of the master equation for Γn reads

Γn= ∞ X m=0 ~mΓ(m)n ⇒ n+1 X k=0 Γ(n+1−k)_n , Γ(k)_n n= 0. (3.32)

We know that the functionals Γ(k)n are convergent for k≤ n. Thus, the divergent part of (3.32) gives

Γ(0)_n , Γ(n+1)_{n div}

n= 0. (3.33)

Since Γ(0)n = S0 = S we have

S, Γ(n+1)_{n div} = 0. (3.34)

In order to work out the general solution of the cohomological problem (3.34), we inductively assume that all the subdivergences are subtracted by appropriate counterterms, so Γ(n+1)_{n div} is a local functional. Furthermore, it is the integral of a linear combination of functionals with dimension D and ghost

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30 Batalin-Vilkovisky formalism number zero. Moreover, the counterterms are linear in the sources Kα, since terms quadratic in Kα

have either dimension greater than D or ghost number different from zero. Working out the closure condition for Γ(n+1)_{n div}, the general solution can be written as

Γ(n+1)_{n div} = ˜Gn+1+ S, ˜Xn+1, (3.35)

where ˜G(Φ, K) is local functional such that S, ˜Gn = 0 and ˜X(Φ, K) is another local functional of

ghost number minus one. Moreover, the solution (3.35) can be reorganized in the form

Γ(n+1)_{n div} = Gn+1(φ) + S, Xn+1, (3.36)

where G is a s-invariant local functional of the physical fields (and then also gauge invariant) and X(Φ, K) is a local functional of fields and sources. This fact was already known in the case of gauge theories [20, 21, 22]. A general proof, valid also for theories of gravity (both renormalizable and nonrenormalizable) is given in ref. [18].

Using a canonical transformation we can remove the exact term. Let I(Φ, K0) =

Z Φα_K0

α

be the generator of the identity, then the transformation generated by

Fn+1(Φ, K0) =I(Φ, K0)− Xn+1(Φ, K0) (3.37)

gives

S0(Φ0, K0) = S(Φ, K)− S, Xn+1(Φ, K), (3.38)

plus higher orders. The term G can be subtracted by a redefinition of the parameters in the clas-sical action Sc. By iterating this procedure, the renormalization to all orders is performed by the

transformation ℘R= ℘∞.

Additional properties of the solution Γ(n+1)_{n div} depend on the theory and on the choice of the gauge fixing. In chapter 8 we show such properties in the case of Stelle action (2.32), since the new quantum gravity theory we propose is based on the same classical action.

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Chapter 4

Unitarity

In this chapter we derive the unitarity equation and the necessary assumptions to state that a certain quantum field theory is unitary. A set of equations, named cutting equations [23], are obtained diagrammatically order by order. They can be worked out from the so called largest time equation [24], which is an identity that holds in quantum field theory. The additional step to prove unitarity is to ensure that, at every order, the cutting equations can be interpreted as the diagrammatic version of the unitarity equation SS†= 1, where S is the scattering matrix. Moreover, it is possible to derive a more general version of cutting equations, called algebraic cutting equations [25]. They generalize the standard ones (which we denote as diagrammatic cutting equations) and allow to simplify some steps in the proof of unitarity. In section 4.2 we introduce the basic definitions and examples for the use of algebraic cutting equations, while in section 4.3 we reconsider the standard bubble diagram and study its discontinuity. We generalize the usual derivation [26] in various directions, to prepare the extension to the Lee-Wick models.

4.1 Diagrammatic cutting equations

A quantum field theory is said to be unitary if its scattering matrix S is a unitary matrix, i.e.

SS†= 1. (4.1) By splitting the S-matrix in S = 1 + iT, the unitarity condition becomes

− i(T − T†) = TT†. (4.2) Let V denotes the space of physical states, then (4.2) can be written in terms of matrix elements Tf i=hf|T|ii as − i(Tf i− T∗if) = X n Z dΠ(n)Tf nT∗in, (4.3) 31

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32 Unitarity

Figure 4.1: Cut propagator

where |ii, |fi ∈ V , dΠ(n) _{is the Lorentz invariant measure on the phase space of n-particle final}

states and the sum is understood over all possible final states. Given a process, the associated matrix element of T is related to the amplitude _{M of such process in the following way}

Tf i=M(i → f)(2π)4δ(4)(pf − pi), (4.4)

where pf and piare the sum of final and initial four momenta, respectively. Then the untarity equation

reduces to − i[M(i → f) − M∗(f _{→ i)] =}X n Z dΠ(n)_M∗(f _{→ n)M(i → n)(2π)}4δ(4)pi− X j qj , (4.5) which, in the particular case of process that is symmetric under the exchange i_{↔ f, simplifies into}

− iDiscM = 2ImM =X n Z dΠ(n)_|M|2(2π)4δ(4)pi− X j qj , (4.6) where qj are the integrated four momenta. Equation (4.5) is also known as optical theorem and it

gives a set of identities obeyed by the scattering amplitudes. The diagrammatic version of these identities amount to a set of cutting equations [23], which involve a diagram together with its variants obtained by cutting it in various ways. To obtain a cut diagram it is necessary to introduce a new set of Feynman rules. In the case of a scalar theory1_{, a cut propagator is obtained by means of the}

substitution

i

p2_{− m}2_{+ i} → 2πθ(±p

0_)δ(p2_{− m}2_), _(4.7)

where the sign in front of p0 _{is determined by the direction of the energy flow through the cut. A}

cut propagator is graphically shown in fig. 4.1. Then, in a general diagram, every propagator and vertex which lies in the shadowed region, i.e. the right side of the cut in fig. 4.1, is substituted with its complex conjugate, while the rest remains unchanged.

Given a diagram G, the cutting equation states that G + ¯G =₋ X

cuttings C

GC, (4.8)

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4.2 Algebraic cutting equations 33 where ¯G is the complex conjugate of G, GC is a cut diagram and the sum runs over all possible

cuttings. In a quantum field theory, if the right hand side of (4.8) can be interpreted as (minus) the right-hand side of (4.5) for every diagram and to all orders, then the theory is said to be perturbatively unitary.

More generally, the cutting equations can be collected into the pseudounitarity equation

− i(T − T†) = T†HT, (4.9) where H is a diagonal matrix having eigenvalues 1, 0 and -1. Then the condition for unitarity can be rephrased as follows. If there exists a subspace V of the total Fock space W of states of the free theory, such that equation (4.9) holds with H = 1 when the external legs and the cut legs are projected onto V , then the pseudounitarity equation implies perturbative unitarity, expressed by equation (4.2).

In chapter 6 we check explicitely the cutting equations for the bubble and triangle diagrams in the case of a scalar Lee-Wick theory and in section 6.6 we give a sketch of the proof to all orders [10]. For those purposes, it is useful to introduce a set of identities which are more general than those we derived in this section, namely the algebraic cutting equations.

4.2 Algebraic cutting equations

The algebraic cutting equations [25] are particular polynomial identities associated with Feynman diagrams. We intorduce the following definitions

a) Given a diagram, two legs with a vertex in common are said to have coherent orientations if the orientation2 _{of one leg points to the vertex in common and the orientation of the other leg}

points away from the vertex in common.

b) Given a diagram, a loop is oriented if the orientations of all its legs are coherent.

c) A loop is called minimal if it is not the union of two loops that have a vertex in common. d) A polar number is a variable equipped with a polarity, denoted by + or−.

e) Given a diagram G, equip every leg with a polar number. A loop γ of G is said to be polarized if the polar numbers associated with the legs of γ are arranged so that, moving along γ, the polarization flips if and only if the leg orientation flips.

Figure 4.2: ACE propagators

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34 Unitarity e) Given a diagram G, a polarized monomial is a product of polar numbers, one for each internal

leg, where at least one loop is polarized.

Let _{σ_i+, τ_i+, σ−_i , τ_i−_{}, i = 1, . . . N, denote N sets of polar numbers. Each set is associated with a} leg of a diagram. We say that σ+

i , τi+ (resp. σ − i , τ

−

i ) are positive (negative) polar numbers and use

them to define the propagators ηi= σi++ σ − i , wi = τi++ τ − i , ui = σi++ τ − i , vi = σi−+ τ + i . (4.10)

Consider a connected Feynman diagram G and equip its I internal legs with orientations. Assign an independent energy ei to each internal leg, where i = 1, . . . , I, so that each energy flows according

to the leg orientation. Then, impose the energy conservation at each vertex. For semplicity, give zero energies to the external legs. This leaves L = I− V + 1 independent energies e1, . . . , eL, where V is

the number of vertices in G. The orientations and the energies can be arranged so that the energy flowing in each internal leg is a linear compbination of e1, . . . , eL with coefficient 0 or 1 and the flow

of each energy defines a minimal orented loop. A diagram G obtained by means of this construction is called oriented diagram. Build a variant GM of an oriented diagram G by choosing any number of

vertices and marking those vertices.

The algebraic cutting equation associated with G is X

markings M

PM =PG, (4.11)

where the sum runs over all possible markings,_PGis a linear combination of polarized monomials and

PM is the value of GM, which is defined as follows. Assign one propagator, among those in fig. 4.2,

to each internal leg of GM, where the dots denote the marked vertices. Then, PM reads

PM = (−1)m I

Y

i=1

p(i)_M, (4.12) where p(i)_M is the propagator of the ith leg and m is the number of marked vertices.

The polynomial identity (4.11) is powerful, because it collects the terms that do not contribute to the diagrammatic cutting equations inside_PG. In fact, in physical applications, the polarity refers to

the position of the poles with respect the integration path on the loop energy, e.g. above or below the real axis. If we consider a product of polar numbers with the same polarity, i.e. a polarized loop, then its integral over the loop energy is zero by the residue theorem, being all the poles located on the same side.

To give a few examples, consider the diagrams of fig. 4.3. The oriented loops of the first diagram are 213 and 354, while 2145 is a nonoriented loop. If we equip such loops with the polar monomials σ₂+σ+₁τ₃+, σ₃−τ₅−τ₄− and σ₂+σ+₁σ₄−τ₅−, respectively, we obtain polarized loops. Examples of polarized monomials are σ+₁σ₂+τ₃+σ₄+τ₅−and σ₁+σ₂+σ−₃τ₄−τ₅−. Examples of polarized loops for the second diagram of fig. 4.3 are 213 with the monomial σ+

2 σ+1τ3+, 2145 with σ+2σ1+τ4+σ5+ and 345 with σ3+σ − 4τ

− 5 .

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4.3 The standard bubble diagram revisited 35 2 5 1 4 3 2 5 1 4 3

Figure 4.3: Two-loop oriented diagrams

We stress that the algebraic cutting equations are more general than the diagrammatic ones, since the polarity assignments are the only assumptions we need to make about the polar numbers. For example, we can keep the infinitesimal width of the Feynman propagator different from zero and arbitrary. Therefore, the cut propagators (typically corresponding to ui and vi) are not necessarily

distributions of compact support. In the case of theories of the Lee-Wick type it is crucial to be able to work at 6= 0, as we show in section 6.6.

4.3 The standard bubble diagram revisited

In this section we check the cutting equation in the case of standard bubble diagram. We use the dimensional regularization and work in a generic Lorentz frame, instead of choosing, say, the external momentum p = (p0_{, p) of the form (p}0_{, 0). One reason is that this choice is only allowed for timelike}

external momenta. Moreover, in the Lee-Wick models it is crucial to keep the external space momen-tum p different from zero, to obtain well defined amplitudes everywhere on the real axis, as explained in details in chapter 5.

We also take different masses m1, m2, and independent infinitesimal widths 1, 2, which we keep

nonvanishing as long as we can. The loop integral reads

i_{M(p) =} λ 2 2 Z _dD_k (2π)D 1 k2_{− m}2 1+ i1 1 (k− p)2_{− m}2 2+ i2 ≡ λ 2 2 B, (4.13) where _{M(p) is the amplitude and λ is the coupling constant. We can equivalently write (4.13) as}

i_{M(p) =} λ 2 2 Z _dk0_dD−1_k (2π)D 2 Y j=1 1 (ej− ωj+ ij)(ej+ ωj − ij) , (4.14) where e1 = k0, e2 = k0− p0, ω1 = pk2+ m2₁ and ω2 = p(k − p)2+ m2₂. In going from (4.13) to

(4.14), we have expanded the denominators for 1, 2 small and rescaled such widths.

We perform the integral on k0 _{by using the residue theorem and closing the integration path in}

the lower half k0 _{plane. The relevant poles are located at k}0 _{= z}

1 and k0 = z2, where

From Lee-Wick Models to Quantum Gravity, Fakeons and the Violation of Microcausality

Universit`

a di Pisa

From Lee-Wick Models to Quantum Gravity,

Fakeons and the Violation of Microcausality

Tesi di Dottorato

Contents

Notation and conventions

Chapter 1

Introduction

Chapter 2

Quantum field theories of gravitational

interactions

2.1

General relativity

2.2

Stelle gravity

2.3

Inconsistencies of Minkowski higher-derivative theories

Appendix

2.4

Projector operators

Chapter 3

Batalin-Vilkovisky formalism

3.1

Composite operators

3.2

Antiparentheses and master equation

3.3

Canonical transformations

3.4

Renormalization

Chapter 4

Unitarity

4.1

Diagrammatic cutting equations

4.2

Algebraic cutting equations

4.3

The standard bubble diagram revisited