Quantum scalar field theories with fractional operators
aa r X i v : . [ h e p - t h ] F e b February 5, 2021
Scalar and gravity quantum field theories withfractional operators
Gianluca Calcagni
Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain
E-mail: [email protected]
Abstract:
We study a class of perturbative scalar and gravitational quantum field the-ories where dynamics is characterized by Lorentz-invariant or Lorentz-breaking non-localoperators of fractional order and the underlying spacetime has a varying spectral dimension.These theories are either ghost free or power-counting renormalizable but they cannot beboth at the same time. However, some of them are one-loop unitary and finite, and possiblyunitary and finite at all orders. One of the theories is unitary and infrared-finite and canserve as a ghost-free model with large-scale modifications of general relativity. ontents T [ ∂ γ ] T [ ∂ + ∂ γ ] T [ ∂ γ ( ℓ ) ] T [ (cid:3) γ ] γ
33– i –.2 Theory T [ (cid:3) + (cid:3) γ ] γ T [ (cid:3) γ ( ℓ ) ] γ T [ ∂ γ ] T [ ∂ + ∂ γ ] T [ ∂ γ ( ℓ ) ] T [ (cid:3) γ ] T [ (cid:3) + (cid:3) γ ] T [ (cid:3) γ ( ℓ ) ] C.1 Derivation of (5.17) 54C.2 Derivation of (5.21) and (5.33) 55
Some time ago, it was noted that all theories of quantum gravity are characterized bydimensional flow: whenever a continuous spacetime arises, fundamentally or effectively,its dimension varies with the probed scale [1–3]. Since a varying dimensionality requiresspacetime geometry to be endowed with intrinsic time and length scales, a theory with– 1 –imensional flow is said to be multi-scale . All quantum gravities are multi-scale in certaincorners of their parameter space and many examples of dimensional flow in quantum gravityhave been studied, or re-studied, since then [4–7]. Among other features, it emerged thatnot all multi-scale spacetimes can be called multi-fractal indiscriminately, but only thosewhose Hausdorff, spectral and walk dimensions are related to one another in a certain way[4]. In parallel, it was suspected that dimensional flow could be related to the taming ofinfinities in the ultraviolet (UV). This possibility is natural in the view of the widely suc-cessful dimensional regularization scheme [8, 9], according to which the divergence of four-dimensional integrals appearing in the summation of Feynman diagrams can be singled-out,and eventually subtracted by counter-terms, by artificially letting the topological dimen-sion D to vary, then setting it to D = 4 at the end of the calculation after subtraction. Ifone could devise an exotic geometry with a physically meaningful non-integer dimension,the infinities of a quantum field theory (QFT) might be cured. This idea was investigatedboth in a model-independent fashion and in the specific framework of multi-fractional the-ories [5], field theories living in multi-scale spacetimes whose measure and kinetic operatorsare factorizable in the coordinates. At first, having quantization of gravity in mind, theexpectation was that a spacetime with reduced dimensionality in the UV would improvethe renormalizability properties of a QFT [3]. However, two counter-examples were foundwhere QFTs living in spacetimes with varying dimension were no better behaved thantheir ordinary analogue on standard Minkowski spacetime [11].Since then, the trajectory of multi-fractional theories has been oscillating betweenphenomenology and the striving for their original goal of renormalizability. To understandwhat led us to the present point, a short historical digression may be helpful. Intereston fractal spacetimes ran for some time as a subject per se [12–22]. Some proposals wereinspired by dimensional regularization [12, 16, 17], others [18–20] by early results about aquantum-gravity fractal foam [23, 24], yet others attempted to reconcile quantum mechanicsand general relativity by extending the principles of special relativity to intrinsic scales[21, 22], while a few more were simply born out of pure curiosity [13–15]. Soon, however,attention was drawn to the booming and blooming of quantum-gravity theories and thediscovery, between the early 1990s and the late 2000s, of their fractal-like properties atshort scales [25–37] (more references can be found in [4–7, 38] and in section 6.2.3). Itwas at this point that a general pattern in dimensional flow was recognized [1, 2] andresearch on field theories on spacetimes with varying dimension was revived from differentperspectives involving fractional integration measures or fractional operators: QFTs onmulti-fractal spacetimes [3, 39, 40] were followed by multi-fractional spacetimes [38, 41–44](see [5] for a review and more references) and, soon afterwards, Trinchero’s model of scalarQFT [45–47] and a scalar model related to causal sets [48–52]. These models are indebtedto the pioneering works on fractional powers of the d’Alembertian [53–63]. The circle closes These examples are the multi-fractional theories dubbed T v and T q with, respectively, weighted and q -derivatives. The first ( T v ) lives in a spacetime with varying Hausdorff dimension and constant spectraldimension [10]; this spacetime is not a multi-fractal. The second ( T q ) lives in a multi-fractal spacetime withvarying Hausdorff and spectral dimension. – 2 –ecause Bollini and Giambiagi, authors that studied the fractional d’Alembertian [53, 55–57], also proposed dimensional regularization in the first place [8], in both cases with theobjective of controlling the UV divergences appearing in QFT.Coming back to the multi-fractional paradigm, while the spacetime measure is uniquelydefined parametrically [5, 43], different choices of Lagrangian symmetries can produce threetypes of kinetic terms, either with ordinary but measure-dependent derivatives (theories T , T v and T q , in the labeling of [5]), or with fractional operators (collectively labeled T γ ), i.e.,integro-differential operators studied in the branch of mathematics known as fractionalcalculus [64–67]. Although the first multi-fractional theories that were considered did em-ploy fractional derivatives [38, 42, 68, 69], later developments focused on the proposals T with ordinary derivatives and T v and T q with measure-decorated ordinary derivatives,for the reason that they were more manageable and generated a wealth of easily testablephenomenology [5]. After near completion of this program, we have recently turned ourinterest back to the original problem, left open in [42, 69], of how to define a field theorywith scale-dependent fractional derivatives [5, 70], with the hope that it would not sufferfrom the non-renormalizability problems of the other multi-fractional candidates.In this paper, we set the foundations of the class of multi-fractional theories T γ withfractional operators and work out some of their classical and quantum properties. Wefind that there are eight inequivalent theories in this class (Fig. 1). A first sub-divisiondepends on whether the kinetic operator is made of fractional derivatives or the fractionalLaplace–Beltrami (or d’Alembertian) operator. The first option violates Lorentz invariance,while the second preserves it. Then, for each case, one starts with a kinetic term made ofjust one multiplicative operator with fixed fractional exponent γ (theories we call T [ ∂ γ ] and T [ (cid:3) γ ] ). Taken as stand-alone theories, γ must be close to 1 in order to respect allknown experimental constraints on Standard-Model and gravitational physical observables.However, T [ ∂ γ ] and T [ (cid:3) γ ] can also be taken as the basis to develop multi-fractional theories,either by adding ordinary derivative operators (theories T [ ∂ + ∂ γ ] and T [ (cid:3) + (cid:3) γ ] ) or bytaking a scale-dependent fractional exponent (theories T [ ∂ γ ( ℓ ) ] and T [ (cid:3) γ ( ℓ ) ] ).While T [ ∂ γ ] and T [ (cid:3) γ ] have constant Hausdorff and spectral dimension, the dimen-sional flow of all the other theories is the same, with a constant Hausdorff dimension anda varying spectral dimension. We will carry out a systematic study in the case of a realscalar field theory, which is a good training ground in preparation for gravity, which we willalso touch upon.For the theory T [ ∂ γ ] , we will show that the propagator in Lorentzian signature is non-analytic in momentum space, due to the presence of the absolute value | k | . To overcome thisproblem, one can take two roads: either one defines the quantum theory in Euclidean sig-nature and then analytically continues only momenta in external legs of Feynman diagrams(Efimov analytic continuation [71–73]) or, in alternative, one defines the quantum theoryin Lorentzian signature but restricts the anomalous differential structure to spatial direc-tions only, thus providing a sort of fractional extension of theories with higher-order spatialLaplacians, such as Hořava–Lifshitz gravity [74]. The multi-fractional versions T [ ∂ + ∂ γ ] and T [ ∂ γ ( ℓ ) ] will also be sketched.At the same time, we will propose an alternative (theory T [ (cid:3) γ ] ) where the propaga-– 3 – γ theories withfra tional operators T [ ∂ γ ] fra tional derivatives T [ (cid:3) γ ] fra tional d'Alembertian T [ ∂ + ∂ γ ] multi-fra tional derivatives T [ ∂ γ ( ℓ ) ] variable fra tional derivatives T [ (cid:3) + (cid:3) γ ] multi-fra tional d'Alembertian T [ (cid:3) γ ( ℓ ) ] variable fra tional d'Alembertian Figure 1 . The theories with fractional operators studied in this paper. tor is analytic, Lorentz invariance is preserved, the dynamics is simpler and soluble andthere are no ghosts for an order γ of the fractional operators. The theory is power-counting renormalizable only in the different range γ > , thus complying with the originalexpectation of an improved renormalizability but at the unforeseen price of losing unitarity.Almost the converse holds for all other multi-fractional theories T , T v and T q : they arealways unitary but their power-counting renormalizability is never improved, at least forspacetime measures without logarithmic oscillations [5]. Still, by explicit calculations weshow that the theory T [ (cid:3) γ ] is one-loop finite and possibly finite at all orders in the rangewhere it is also unitary, except for some specific values of γ . This implies that the theory T [ (cid:3) + (cid:3) γ ] cannot be unitary and renormalizable at the same time, a problem which couldbe solved by a realization of the multi-fractional Ansatz via a scale-dependent operator(theory T [ (cid:3) γ ( ℓ ) ] ). – 4 –n essence, the theory T [ (cid:3) γ ] with fractional d’Alembertian with γ > is problematicbecause its derivatives of non-integer order have the same advantage (suppression of thepropagator in the UV) and disadvantage (loss of unitarity [75]) of higher-order derivatives.When γ < , the theory is no longer power-counting renormalizable but it is at least one-loop finite. The theories T [ (cid:3) γ ≈ ] and T [ (cid:3) γ ( ℓ ) ] where the kinetic term consists of only onefractional operator with, respectively, exponent very close to 1 or with scale-dependentexponent, can be regarded as fundamental quantum theories. Otherwise, if the kineticterm is composed by a d’Alembertian plus a fractional d’Alembertian with γ < (theory T [ (cid:3) + (cid:3) γ ] ), then its properties in the UV are those of a standard QFT but, as a payback,it displays modifications dominant in the infrared (IR) without problems of stability, withpossible consequences for cosmology.All these considerations apply, mutatis mutandis , to gravity when treated as a pertur-bative QFT. We will write down the non-linear action and full equations of motion in allsix fractional cases, as well the linearized equations for the graviton. The action of thetheory T [ (cid:3) + (cid:3) γ ] is similar to the one for non-local quantum gravity with asymptoticallypolynomial operators. Both theories are non-local but T [ (cid:3) + (cid:3) γ ] (like all the theories T γ )is characterized by branch cuts, while the other only has simple poles.The plan of the paper is as follows. In section 2, the general setting is introduced.The scalar theory T [ ∂ γ ] with fractional derivatives D γ is discussed in section 3, whilethe Lorentz-invariant scalar theory T [ (cid:3) γ ] with fractional d’Alembertian ( m − (cid:3) ) γ , with orwithout mass, will be presented and analyzed in section 4. Sections 3 and 4 are independentand the reader unfamiliar with fractional calculus may skip section 3 at the first readingand concentrate on T [ (cid:3) γ ] , which is simpler and more intuitive than T [ ∂ γ ] . Classical andquantum gravity with fractional operators is discussed in section 5 and argued to inherit allthe properties of its scalar counterpart. A comparison with other results in the literatureof scalar fields and gravity with fractional operators is made in section 6. Section 7 isdevoted to conclusions and an outlook. Appendix A contains technicalities about fractionalderivatives, in appendix B we give an alternative proof of unitarity of the Lorentz-invariantscalar theory T [ (cid:3) γ ] and the equations of motion of the fractional gravitational theories arederived in appendix C. In this section, we will introduce the main structure of the theory of quantum gravity wewant to build. First, we will consider a scalar field theory and customize it to reproduce (i)an effect found in all quantum-gravity theories, namely, dimensional flow and (ii) a charac-teristic desired in a perturbative QFT approach, namely, power-counting renormalizability.In later sections, we will refine this sketch and also extend the lessons learned in the scalarcase to gravity.Consider a real scalar field φ living in D -dimensional Minkowski spacetime with metric η µν = diag( − , + , · · · , +) . An ordinary field theory with self-interaction V ( φ ) is governed– 5 –y the action S = Z d D x (cid:20) φ (cid:3) φ − V ( φ ) (cid:21) , (2.1)where d D x is the Lebesgue measure in D topological dimensions and (cid:3) = η µν ∂ µ ∂ ν isthe d’Alembertian. In general, quantum-gravity effects can modify the integro-differentialstructure of (2.1) in the UV, namely, the measure d D x and the kinetic operator (cid:3) . Theseeffects gradually disappear as the observer moves to IR lengths or low energy scales. Themagnitude and nature of the modifications strongly depend on the specific theory and theremay even be cases where geometry is not continuous at small scales and all corrections arenegligible when the continuum limit (2.1) is reached. Here we will focus our attention ontheories defined on a continuum and where the d’Alembertian is corrected by a fractionalderivative term, (cid:3) → K ≃ ℓ − γ ∗ (cid:3) + O ( ∂ γ ) , (2.2)where ℓ ∗ is a fundamental length scale and γ is a real constant. The ensuing theories canalso be regarded as models of other quantum gravities in their continuum limit but herewe will treat them as stand-alone proposals, thus demanding self-consistency in the formof unitarity and renormalizability.With the aim of introducing the theory as a modification of standard QFT or gravity,it is more natural to attach the ℓ ∗ factor to the fractional-derivative term: K ≃ (cid:3) + ℓ γ − ∗ O ( ∂ γ ) . However, for estimating UV divergences eq. (2.2) is slightly more convenient.The two choices are physically equivalent. The most general action for a real scalar with operators of fractional order in Minkowskispacetime is S = Z d D x v ( x ) (cid:20) φ K φ − V ( φ ) (cid:21) , (2.3)where v ( x ) is a non-trivial measure weight and K is the kinetic operator (2.2). The measure weight v ( x ) can be either 1 or a specific multi-scaling parametric profile dic-tated by basic dynamics-independent requirements on the Hausdorff dimension of spacetime[5, 43]. This profile is universal in the sense that it applies to any theory of quantum gravity,although each specific dynamics will fix the parameters in v in a unique way. Two of theseparameters are the UV scaling of the measure in the time direction ( α ∈ R ) and the UVscaling in the spatial directions ( α ∈ R ), so that [ d D x v ] ir = − D , [ d D x v ] uv = − α − ( D − α . (2.4)where square brackets denote the engineering (energy) dimension. Throughout the paper,energy-momentum scales normally as [ k µ ] = 1 .At the classical level, a non-trivial v ( x ) does not present any major difficulty, but atthe quantum level it leads to non-delta-like distributions for vertex contributions, which– 6 –ake the Feynman expansion challenging if not impossible [11]. In some cases with simplekinetic terms, as the multi-fractional theories T v and T q , this problem is avoided by amathematical trick (a mapping to a non-physical frame where the QFT simplifies), but nosuch a stratagem exists for more complicated K s. Therefore, in this paper v ( x ) = 1 , α = 1 = α , (2.5)and the Hausdorff dimension of spacetime is constant, d h = − [ d D x v ] = D . In the restof the section, however, for the sake of generality we will keep the scaling of the measureweight arbitrary, letting only α = α , so that d h = Dα in the UV. In the next sections, we will discuss two choices for K , one which enjoys Lorentz symmetryand another which does not. Other possibilities were discussed in [5]. In both cases,assuming that the fractional term dominates in the UV, from (2.2) one gets [ K ] ir = 2 , [ K ] uv = 2 γ , (2.6)which, combined with (2.4), yields the dimensionality of the scalar, which is given in theUV by [ φ ] uv = − [ d D x v ] − [ K ]2 = Dα − γ , (2.7)while in the IR one effectively recovers the engineering dimension of an ordinary field: [ φ eff ] ir = D − , φ eff = ℓ Dα − γ − D − ∗ φ . (2.8) In order to get the superficial degree of divergence in a cut-off scheme, one can employthe usual power counting in ordinary quantum field theory. This argument about renor-malizability has been put forward from the onset of the multi-fractional proposal [3, 39]and repeated since then time and again [5, 38]. Let us revive it here, warning the readerthat it will actually fail for the specific theories under inspection because it will grosslyoverestimate the actual degree of divergence of Feynman diagrams.For a non-trivial measure weight v ( x ) = 1 scaling as (2.4), there corresponds a multi-fractional measure in momentum space d D k w ( k ) such that [ d D k w ] = Dα in the UV.Consider a potential V ( φ ) = λ N φ N , where the coupling has energy dimension [ λ N ] = Dα − N [ φ ] (2.7) = 2 Dα − N ( Dα − γ )2 . (2.9)In a one-particle-irreducible graph with L loops, N I internal propagators, N V vertices and N E external legs, in the UV each momentum-space loop contributes [ d D k w ] = Dα powersto the momentum integral defining the amplitude of the graph, while [ ˜ G ] = − γ for thepropagator in momentum space (see below). Interaction vertices are dimensionless forthe theory (2.3) because the coupling λ N is a constant which does not contribute to the– 7 –ivergence of the integral. The divergent part of the graph in the UV scales as an energycut-off Λ UV to some power δ . Therefore, the superficial degree of divergence δ in the UVis equal, by definition, to [ loop ] L + [ propagator ] N I = DαL − γN I . Since N I > L , then δ L ( Dα − γ ) and the graph is convergent if γ > Dα/ . Also, using the topologicalrelations N N V = N + 2 N I and L = N I − N V + 1 , one ends up with δ = DαL − γN I = [ λ N ](1 − N V ) . (2.10)Since N V > , to have δ the constant λ N must be positive semi-definite, which happensif [ λ N ] > ⇐⇒ γ > Dα N DαDα − γ . (2.11)In particular, for γ = Dα/ the theory is power-counting renormalizable for any power N .We can summarize the results for multi-fractional theories:• The theory T has α = 1 and γ = 1 (multi-scale measure and plain derivatives) ispower-counting renormalizable for α D/ but it is difficult to work out QFT inthese spacetimes due to their lack of symmetries [5].• In the theory T v with weighted derivatives, α = 1 and γ = 1 (multi-scale measureand plain derivatives with weights). The power-counting argument does not show itbut, in fact, the theory does not have improved renormalizability [11].• The theory T q with q -derivative corresponds to the case γ = α with < α < (multi-scale measure and multi-scale derivatives). The superficial degree of divergence neverbecomes negative and it vanishes only in the delicate limit α → , which exists [5] butgoes beyond the scope of the present paper. Again, infinities in QFT are not tamed[11].• The class of theories T γ considered here have α = 1 and γ = 1 (plain measure andfractional operators), so that power-counting renormalizability is achieved when γ > D N DD − γ D =4 −→ γ > N − γ . (2.12)For γ = D/ ( = 2 in four dimensions) the theory is power-counting renormalizable(all couplings are positive) for any power N . As is well known, this theory withfourth-order derivatives is not unitary because it has a ghost; in the case of gravity,it corresponds to Stelle theory [75]. In this section, we outline the structure of the theories T [ ∂ γ ] , T [ ∂ + ∂ γ ] and T [ ∂ γ ( ℓ ) ] .To get a derivative operator with anomalous scaling, there are three options. One is todress first- and second-order derivatives with measure factors; the theory T v with weightedderivatives v − β ∂ x ( v β · ) and the theory T q with q -derivatives ∂ q := v − ∂ x are of this type,– 8 –ut they do not have improved renormalizability [5]. Another possibility is to considerfractional derivatives [64–67], which have numerous applications in engineering, percolationand transport theory, chaos theory, fractal geometry and complex systems [42, 64, 65, 68, 69]and, in general, in any system with dissipation [39, 42] or a nowhere-differentiable geometry[76]. The third option is to consider non-integer powers of the d’Alembertian; this will bedone in section 4.The theory T [ ∂ + ∂ γ ] is made of multi-fractional derivatives with explicit scaling, op-erators that we have to construct on demand for QFT applications [70]. We will first studythe theory T [ ∂ γ ] introducing fractional derivatives of a fixed order γ in one dimension andthen generalize them to D dimensions and to operators with a scale dependence. The basic operators in fractional calculus are derivatives of non-integer order γ . There aremany inequivalent ways to define these operators [77]. Here we select two specific choices,which are preferred among the others for various technical and physical reasons [42], mainlybecause we want the fractional derivative of a constant to be zero and we need to integrateon the whole axis. Namely, we choose the Liouville derivative ∞ ∂ γ f ( x ) := 1Γ( m − γ ) Z + ∞−∞ dx ′ Θ( x − x ′ )( x − x ′ ) γ +1 − m ∂ mx ′ f ( x ′ ) , m − γ < m , (3.1)and the Weyl derivative ∞ ¯ ∂ γ f ( x ) := 1Γ( m − γ ) Z + ∞−∞ dx ′ Θ( x ′ − x )( x ′ − x ) γ +1 − m ∂ mx ′ f ( x ′ ) , m − γ < m , (3.2)where Θ is Heaviside’s left-continuous step function: Θ( x ) = ( , x , x > . (3.3)The Liouville derivative has memory of the past (integration from −∞ to x ), while theWeyl derivative encodes a pre-knowledge of the future (integration from x to + ∞ ). Theirproperties, which can be found in [42, 64–67], are summarized in appendix A.Fractional derivatives can be combined into the mixed operator [42, 69, 78] D γ := c ∞ ∂ γ + ¯ c ∞ ¯ ∂ γ . (3.4)Its properties can be derived from the elementary properties of fractional operators collectedin appendix A. In the following, − e iπ .1. Limit to ordinary calculus (eqs. (A.2) and (A.11)): lim γ → n D γ = ( c + ¯ ce iπn ) ∂ n , n ∈ N . (3.5)2. Linearity (eqs. (A.3) and (A.12)): D γ [ c f ( x ) + c g ( x )] = c ( D γ f )( x ) + c ( D γ g )( x ) . (3.6)– 9 –. Composition rule (eqs. (A.4) and (A.13)): for all γ, β > , D γ D β = ( c + ¯ c ) D γ + β + c ¯ c ( ∞ ∂ γ ∞ ¯ ∂ β + ∞ ¯ ∂ γ ∞ ∂ β − ∞ ∂ γ + β − ∞ ¯ ∂ γ + β ) . (3.7)4. Kernel (eqs. (A.5) and (A.14)): D γ x β = ( ce iπγ + ¯ c ) Γ( γ − β )Γ( − β ) x β − γ . (3.8)Equation (3.8) vanishes for β = 0 , , , . . . , m − and is ill-defined for β = γ . Thelatter case can be treated separately. From the definitions of the operators ∞ ∂ γ and ∞ ¯ ∂ γ , for m − < γ < m the mixed fractional derivative of a polynomial f m − oforder m − is zero, D γ f m − = 0 . (3.9)The mixed derivative of a constant is always zero and the kernel of a mixed derivativewith < γ < is trivial.5. Eigenfunctions (eqs. (A.7) and (A.16)): D γ e ikx = (cid:16) c e iπ γ + ¯ c e − iπ γ (cid:17) k γ e ikx , (3.10)where we used i = e iπ/ .6. Leibniz rule (eqs. (A.8) and (A.17)): D γ ( f g ) = + ∞ X j =0 Γ(1 + γ )Γ( γ − j + 1)Γ( j + 1) ( ∂ j f )( D γ − j g ) . (3.11)A more symmetric version of this rule exists for pure fractional derivatives [66,eq. (15.12)] but its generalization to the mixed derivative D γ would require extrawork not done here.7. Integration by parts (eq. (A.9)). If ¯ c = ± c ∗ , then Z + ∞−∞ dx f D γ g = ± Z + ∞−∞ dx ( D γ ∗ f ) g . (3.12)The choice c = 12 e iθ , ¯ c = ± e − iθ , (3.13)where θ is a phase, clarifies the above formulæ and eventually fixes θ . In fact, calling D γ ± the mixed derivative with the ± choice in (3.13), equations (3.5) and (3.10) become lim γ → n D γ + = ( cos θ ∂ n n even i sin θ ∂ n n odd , (3.14) D γ + e ikx = cos h πγ k ) θ i | k | γ , (3.15)– 10 –nd lim γ → n D γ − = ( i sin θ ∂ n n even cos θ ∂ n n odd , (3.16) D γ − e ikx = i sgn( k ) sin h πγ k ) θ i | k | γ . (3.17)Equations (3.14) and (3.16) are meaningful only for even n or odd n , but not for all n atthe same time. In fact, in order to reproduce even-order or odd-order derivatives, one mustrescale the operator D γ ± either by a factor cos θ or by i sin θ . Therefore, D γ + and D γ − arethe generalization of, respectively, even and odd (or odd and even) integer derivatives. Inparticular, the mixed derivative with c = ¯ c = 12 , θ = 0 , (3.18)is the generalization of even integer derivatives of order n = 0 , , , . . . , and (3.15) is D γ + e ikx = cos (cid:16) πγ (cid:17) | k | γ e ikx , n even , (3.19)giving the correct limit γ → n . When instead c = − ¯ c = 12 , θ = 0 , (3.20)one gets the generalization of odd integer derivatives of order n = 1 , , , . . . , and (3.17)becomes D γ − e ikx = i sin (cid:16) πγ (cid:17) sgn( k ) | k | γ , n odd . (3.21)With these choices of phase and coefficients, D γ ± = 12 (cid:0) ∞ ∂ γ ± ∞ ¯ ∂ γ (cid:1) . (3.22) T [ ∂ γ ] From the discussion of section 3.1, it becomes clear that the Liouville and Weyl derivativemust coexist in the dynamics, since integrating by parts transforms one into the other.Therefore, we take a fractional kinetic term with mixed derivatives. The action (2.3) with v = 1 can be written in two ways: S + = Z d D x (cid:20) η µ φ D γ + µ φ − V ( φ ) (cid:21) , (3.23) S − = Z d D x (cid:20) η µν φ D γ − µ D γ − ν φ − V ( φ ) (cid:21) = Z d D x (cid:20) − η µν D γ − µ φ D γ − ν φ − V ( φ ) (cid:21) , (3.24)where µ = 0 , , . . . , D − , η µ = ( − , , . . . , , Einstein summation convention is used (in(3.23), K = −D γ + D γ + · · · + D γD − ) and we integrated by parts via (3.12). Note that D γ − D γ − = D γ − due to (3.7): D γ − D γ − = 12 ( D γ + − ∞ ∂ γ ∞ ¯ ∂ γ ) = D γ − . (3.25)– 11 –here we used eqs. (A.5) and (A.14) to find that ∞ ∂ γ ∞ ¯ ∂ γ = ∞ ¯ ∂ γ ∞ ∂ γ . Equation (3.25)also implies that (3.23) and (3.24) are inequivalent, D γ − D γ − = D γ + . This can also be seenfrom eqs. (3.19) and (3.21): D γ + e ikx = cos( πγ ) | k | γ e ikx =: − b γ + | k | γ e ikx , (3.26) D γ − D γ − e ikx = − sin (cid:16) πγ (cid:17) | k | γ = −
12 [1 − cos( πγ )] | k | γ =: − b γ − | k | γ e ikx , (3.27)where | k | γ := −| k | γ + D − X i =1 | k i | γ , (3.28)Guidance in the choice between (3.23) and (3.24) is given by the fact that our aim is tobuild a classical and quantum field theory encompassing both the matter and the gravitysector. Equation (3.23), however, does not allow for a generalization to a generic curved,non-diagonal metric g µν , while (3.24) does. The latter will be our pick. From now on, wewill omit the − subscript in the symbol D . Variation of the action with respect to δφ yields η µν D γµ D γν φ − V ′ ( φ ) = 0 , (3.29)where a prime denotes derivation with respect to φ . Expanding the scalar field into Fouriermodes, for the massive free case V = m γ φ / we get the dispersion relation [ b γ − | k | γ + m γ ] φ k = 0 , (3.30)where b γ − = [1 − cos( πγ )] / and m is a mass. Note that b γ − for all γ , while if wehad chosen b γ + we would have had windows in γ where b γ + < , with possible consequenceson the unitarity of the theory. In this theory, dimensional flow is trivial. Due to (2.5), the theory with multi-fractionalderivatives lives on a spacetime with constant Hausdorff dimension both in position and inmomentum space: d h = d k h = D . (3.31)This means that rulers and clocks measure the same lengths and time intervals as in anordinary setting.The spectral dimension is the dimension felt by a probe particle let diffusing in space-time and it depends on the Hausdorff dimension of momentum space and on the kineticterm [10, 69, 79]: d s = 2 d k h [ K ] . (3.32)– 12 –n the case of T [ ∂ γ ] , the spectral dimension is constant and takes the same anomalous valueat all scales: d s = Dγ . (3.33)Notice that d s is well-defined only when γ > , since there is no physical meaning of negativedimensions.In section 6.2.3, we will compare this and the other types of dimensional flow discussedin sections 3.3.3 and 3.4.3 with others in quantum gravity. T [ ∂ + ∂ γ ] We define the multi-fractional derivative with explicit scaling any linear combination ofmixed fractional derivatives of different order: D µ := X γ u γ D γµ . (3.34)The scaling is said to be explicit because the coefficients u γ are dimensionful and depend onone or more fundamental scales of spacetime geometry, such as ℓ ∗ introduced in eq. (2.2).To recover standard QFT or gravity in some regime, we assume that γ = 1 is included inthe sum.Just like for the theory with fixed γ , we take the derivatives D − and a kinetic term φ K φ in the action. Also, to make dimensional flow only one scale is sufficient [5], so that,overall, we choose as kinetic term K the combination (we omit the − subscript in D ) K = η µν D µ D ν . (3.35)Of all possible D -dimensional fractional derivatives, we choose the most symmetricone with γ µ = γ for all µ because, otherwise, we would have to include extra scales in D .Another reason is that we want to treat all spacetime coordinates on the same ground, asort of fractional covariance principle [5]. However, later we will also consider a differentrealization where γ = −∞ (no fractional derivative in the time direction): D µ = ℓ − γ ∗ ∂ µ + δ iµ D γµ , i = 1 , , . . . , D − , (3.36)corresponding to a fractional version of Hořava–Lifshitz field theories. The k term isdropped in the definition of | k | γ in eq. (3.28). Variation of the action with respect to δφ yields η µν D µ D ν φ − V ′ ( φ ) = 0 . (3.37)This equation is simpler than the most general one presented in [38] for a non-trivial mea-sure. Since plane waves are eigenfunctions both of the d’Alembertian (cid:3) = η µ ∂ µ and of the– 13 –ixed fractional derivatives, we can again expand in Fourier modes. In the free massivecase, h ℓ − γ ) ∗ k + b γ − | k | γ + m γ i φ k = 0 , (3.38)where b γ − is defined in (3.27) and k := k µ k µ = − k + P D − i =1 k i . In the theory T [ ∂ + ∂ γ ] with the measure choice (2.5), the Hausdorff dimension is constant,eq. (3.32), while the spectral dimension varies. In contrast, the Hausdorff dimension ofthe multi-fractional theory T v with weighted derivatives varies and the spectral dimensionis constant, while both the spectral and Hausdorff dimensions vary in the multi-fractionaltheory T q with q -derivatives [10].We can distinguish two types of geometry for the multi-fractional operator (3.35), plusanother one for a generalization with two scales.• When γ > , the fractional operator in (3.35) grows faster in momentum space thanwhen γ = 1 and it dominates the dynamics in the UV: d s UV ≃ Dγ , d s IR ≃ D , (3.39)so that there is a dimensional flow from an anomalous value at short scales to thetopological dimension at large scales. Here “short” and “large” mean, respectively,below and above the microscopic scale ℓ ∗ .• When γ < , the spectral dimension is anomalous in the IR and standard in the UV: d s UV ≃ D , d s IR ≃ Dγ . (3.40)For this theory, the quantum properties in the UV are the same as an ordinary QFT,so that there is no chance that renormalizability is improved. However, if ℓ ∗ is largeenough the theory can work as a model with IR modifications which, in the case ofgravity, can have interesting consequences at the cosmological level.• A third possibility is to extend the multi-fractional kinetic operator (3.35) to threeoperators with three different exponents and two scales ℓ < ℓ : D µ = ∂ µ + ℓ γ − D γ µ + ℓ γ − D γ µ , γ < < γ . (3.41)This dimensional flow has three regimes, a UV one where one can study the renor-malizability of the theory, a mesoscopic one where standard classical and quantumfield theory is recovered, and an IR or ultra-IR one which may have applications tocosmology: d s UV ≃ Dγ < D , d s meso ≃ D , d s IR ≃ Dγ > D . (3.42)This is the most interesting case among the three but we will not study it in thispaper, which is focussed on basic questions about the QFT in geometries of the firstand second kind. – 14 – .4 Theory T [ ∂ γ ( ℓ ) ] Another possibility is to define dimensional flow in terms of fractional derivatives of variableorder, which corresponds to integrate over all scales [38]. In the case where spacetimegeometry has only one fundamental scale ℓ ∗ , S = 1 ℓ ∗ Z + ∞ dℓ w ( ℓ ) Z d D x (cid:20) η µν φ D γ ( ℓ ) µ D γ ( ℓ ) ν φ − V ( φ ) (cid:21) , (3.43)where ℓ is the probed scale and w ( ℓ ) is a dimensionless weight function. Variation of the action with respect to δφ yields η µν D γ ( ℓ ) µ D γ ( ℓ ) ν φ − V ′ ( φ ) = 0 , (3.44)where omit the integration in ℓ because the integrand should vanish at any given scale.Note that also the dimensionality of φ changes with the scale, according to our conventionsfor the weight w ( ℓ ) .The corresponding dispersion relation for the massive free case is h b γ ( ℓ ) − | k | γ ( ℓ ) + m γ ( ℓ ) i φ k = 0 , (3.45)where b γ ( ℓ ) − is defined in (3.27), now with a scale dependence. Just like for the theory T [ ∂ + ∂ γ ] , we can distinguish three cases.• With the profile [68, 69] γ ( ℓ ) = 1 + ( ℓ ∗ /ℓ ) γ ℓ ∗ /ℓ ) , (3.46)for any positive γ one gets the dimensional flow (3.39). In the UV regime ℓ ≪ ℓ ∗ , γ ( ℓ ) → γ , while in the IR regime ℓ ≫ ℓ ∗ one has γ ( ℓ ) → .• With the profile γ ( ℓ ) = γ + ( ℓ ∗ /ℓ ) ℓ ∗ /ℓ ) , (3.47)for any positive γ one recovers (3.40).• With a profile γ ( ℓ ) [68, 69] γ ( ℓ ) = 1 + ( ℓ /ℓ ) γ + [ ℓ / ( ℓ − ℓ )] γ ℓ /ℓ ) + [ ℓ / ( ℓ − ℓ )] , (3.48)with three plateaux , γ and γ and two scales ℓ < ℓ , again we get the flow (3.42),with the UV regime corresponding to ℓ ≪ ℓ , the mesoscopic regime to ℓ ∼ ℓ andthe IR regime to ℓ ≫ ℓ . Here, however, we do not have to impose the condition γ < < γ . – 15 – .5 Propagator and the quantum theory The propagator of the three theories can be read from the corresponding dispersion relation.Let us take the one for T [ ∂ + ∂ γ ] , which has the most general structure. From eq. (3.38),we get the bare Green function in momentum space ˜ G ( k ) = − ℓ − γ ) ∗ k + b γ − | k | γ + m γ . (3.49)The same expression can be found rigorously from the Green’s equation K G ( x ) = δ D ( x ) ,as done in [38] for non-mixed fractional derivatives.The propagator (3.49), or its versions for T [ ∂ γ ] and T [ ∂ γ ( ℓ ) ] , is not analytic in k µ .The Osterwalder–Schrader conditions [80, 81], necessary and sufficient for a Euclidean fieldtheory to admit an analytic continuation to Lorentzian signature, include analyticity in thetime component k of the D -momentum. Thus, there is a tension between this requirementand reality of the kinetic operator spectrum that must be solved before quantizing thetheory. We can do this in two ways. The first solution to the analyticity problem is to define the theory with Efimov analyticcontinuation [71–73], used successfully in non-local quantum gravity [73]. As we mentionedin the introduction, in this prescription scattering amplitudes are analytically continuedto (or defined with) Euclidean momenta and both internal loops and external legs carrypurely imaginary energies, denoted respectively as k = − ik and p = − ip . Then, onereconstructs the amplitudes for real values of the external energies by analytic continua-tion at the end of the calculation. The latter operation corresponds to integrate along asophisticated path C in the complex plane of the energies circulating in the loop integrals.The main difference with respect to traditional Wick rotation is that the contour C is notrotated back clockwise in a rigid fashion but is deformed in a certain way to avoid the polessprinkling the complex plane. In other words, one follows the same steps as in ordinaryLorentzian QFT: internal and external momenta in Lorentzian amplitudes are promotedto complex variables and a closed contour C is chosen so that integration on the real axiscan be rewritten as an integration along another path, with the difference that, thanks toJordan’s lemma, in ordinary QFT this path is exactly the imaginary axis, while in nonlocalQFTs such as those considered here it is a deformed contour.Since internal legs are analytically continued to (or defined with) Euclidean momenta,one can write the factor | k | γ = P Di =1 ( k i ) γ and perform all calculations until the end, whenexternal momenta are continued back to Lorentzian signature. The analyticity problem can be bypassed also by considering ordinary time derivatives andfractional spatial derivatives, i.e., definition (3.36). In fact, the spatial momenta k i play therole of spectator parameters in the proof of the Osterwalder–Schrader conditions [80, 81]and analyticity of the propagator in k i is not required.– 16 –his is a sort of fractional extension of Hořava–Lifshitz gravity [35, 74, 82–84] wheretime derivatives are second-order and spatial derivatives are of higher but non-integer order[85]. Power-counting renormalizability has already been discussed and the scalar field theoryis ghost free for the same reason as for the ordinary Hořava–Lifshitz scalar: the propagatorhas a simple mass pole with positive residue [74]. However, the absence of ghosts in thegravitational theory (present in some versions with ordinary derivatives) is much less obviousand may be worth a separate investigation of its own. The theories with fractional derivatives have two other problematic features that cannot becircumvented by Efimov analytic continuation or a choice of anisotropic derivatives.• Although they are recovered in the IR, Lorentz and Poincaré invariance are broken inthe UV and there is no other obvious symmetry replacing them. The absence of a rulecan quickly lead the theory out of control in term of naturalness of the Lagrangian,proliferation of operators, phenomenology, and so on. In particular, the Feynman rulefor vertices is no longer a Dirac distribution of the sum of external momenta, whichnotably hinders the calculation of diagrams.• The Leibniz rule (3.11) may complicate otherwise elementary calculations already atthe classical level.These issues add to the several technical points noted in [42], where a pessimistic view onthe use of mixed derivatives was endorsed.
We ended the previous section by listing three problems for the theory with fractionalderivatives: non-analyticity of the propagator, lack of symmetries, and a complicated Leib-niz rule. There is an easy way to solve the first and second problem at the same time. T [ (cid:3) γ ] A derivative operator K that scales anomalously, preserves Lorentz symmetry and can bereadily generalized to a curved background is (cid:3) γ , the non-integer power of the d’Alember-tian. Allowing for a mass, we want to keep the pole structure as simple as possible andavoid complex poles. To that purpose, we incorporate the mass term inside the fractionaloperator K = ( m − (cid:3) ) γ , (4.1)so that the action (2.3) with v = 1 reads S = Z d D x (cid:20) φ ( m − (cid:3) ) γ φ − V ( φ ) (cid:21) , (4.2)– 17 –here the potential includes only non-linear interactions. Of this theory, we will discuss thepropagator, unitarity and renormalizability. Generalizing it to more fractional operatorsand fundamental scales is straightforward.The fractional massive operator (4.1) can be written in a convenient position-spaceSchwinger representation [86]. Suppose γ < n , where n ∈ N . From the definition of thegamma function, ( m − (cid:3) ) γ = ( m − (cid:3) ) n ( m − (cid:3) ) γ − n = 1Γ( n − γ ) Z + ∞ dτ τ n − − γ ( m − (cid:3) ) n e − τ ( m − (cid:3) ) . (4.3)One can use this formula to integrate by parts and derive the equation of motion δS/δφ = 0 from the action (4.2): ( m − (cid:3) ) γ φ − V ′ ( φ ) = 0 . (4.4) The dimensional flow of this theory is trivial as in T [ ∂ γ ] (section 3.2.3). The Hausdorffdimension in position and momentum space is constant, eq. (3.31), and so is the spectraldimension: d s = Dγ . (4.5)This deviation from D could violate experimental bounds on local spacetime geometryand particle-physics observables, unless γ = 1 ± ε with ε ≪ . At present, we have notheoretical argument explaining why a fundamental theory would have such a fine tuning.We can still keep this as a possibility but in this paper we will mainly use the theory T [ (cid:3) γ ] as a spearhead to understand the other, multi-fractional theories.The spectral dimension is a meaningful geometric indicator only when γ is positive, γ > , (4.6)a condition that will shrink the range on γ allowed by unitarity (section 4.1.7). Let us now come to the problem of how to represent solutions of the free-field equation K ( (cid:3) ) φ ( x ) = 0 , (4.7)with a generic function K of the d’Alembertian (cid:3) in Lorentzian signature. We will follow[61, 63], where the energy k is analytically continued to the complex plane (Re k , Im , k ) .Decompose the scalar field into momentum modes, φ ( x ) = Z Γ d D k e − ik · x ˜ φ ( k ) , Z Γ d D k := Z Γ dk Z + ∞−∞ d D − k , (4.8)which looks similar to a Fourier transform but with the difference that integration in k isnot done along the real axis but on the path Γ , which runs from −∞ to + ∞ along Im k > b – 18 –nd from + ∞ to −∞ along Im k < − b for a given number b ∈ N (see Fig. 2 below). ˜ φ ( k ) is analytic in the domain C b = { k : | Im k | > b } and ˜ φ ( k ) / ( k ) b is bounded continuous inthe domain C = b = { k : | Im k | > b } .Consider now a function K ( − k ) analytic on the quotient domain C b / C β and such that K ( − k ) / ( k ) β is bounded continuous in C = b / C = β , for some β ∈ N . Then, applying K ( (cid:3) ) to(4.8) gives a well-defined expression: K ( (cid:3) ) φ ( x ) = Z Γ d D k e − ik · x K ( − k ) ˜ φ ( k ) , (4.9)where Γ now runs from −∞ to + ∞ for Im k > b + β and from + ∞ to −∞ for Im k < − b − β .Therefore, φ is a solution of the free-field equation of motion (4.7) when eq. (4.9) vanishesand this happens whenever the function a ( k ) := K ( − k ) ˜ φ ( k ) is entire and analytic onthe (Re k , Im k ) plane, so that by Jordan’s lemma its integral on any closed path is zero,including on Γ : Z Γ dk e ik x a ( k , k ) = 0 . (4.10)Note that this implies that K − ( − k ) has the same analytic properties of ˜ φ ( k ) .Define ∆ ± ( K ) := 1 K [( k + iǫ ) − | k | ] ± K [( k − iǫ ) − | k | ] . (4.11)The path Γ can be deformed around the singularities of K − ( − k ) and can be split intotwo paths Γ − and Γ + sandwiched around the branch cuts on the real axis, if any, plus theloops Γ i circling around isolated singularities in the complex plane. In this way, the solutionreads as the sum of the discontinuity functional ∆ − ( K ) [61, 63], plus the contributions ofthe poles [48]: φ ( x ) = Z Γ d D k e − ik · x a ( k ) K ( − k )= Z + ∞−∞ d D k e − ik · x a ( k ) ∆ − ( K ) + X i Z Γ i d D k e − ik · x a ( k ) K ( − k ) , (4.12)where Γ = Γ + ∪ Γ − S i Γ i . When all the singularities of the propagator K − ( − k ) are onthe real axis Im k = 0 , the last contribution in eq. (4.12) vanishes and from the realitycondition a ( − k ) = a ∗ ( k ) we get φ ( x ) = Z + ∞−∞ d D k h a ( k ) e − ik · x − a ∗ ( k ) e ik · x i Θ( k ) ∆ − ( K − ) . (4.13)In the specific case of the fractional kinetic term (4.1), Fig. 2 shows the contour Γ andits deformation Γ + ∪ Γ − around the symmetric branch cut, where ω := p | k | + m . (4.14)Note that ∆( K ) = 0 for − ω k ω . – 19 – Γ Γ + Γ - - ω ω Re k Im k Figure 2 . Contour Γ (dashed lines) in the (Re k , Im k ) plane and its deformation Γ + ∪ Γ − (solidthick curves) around the branch cuts k − ω and k > ω (gray thick lines). Defining the real-variable distributions [87, section 3.2] x λ + := Θ( x ) x λ , x λ − := Θ( − x ) | x | λ , (4.15)where Θ is defined in (3.3) such that Θ(0) = 0 , one can calculate the power of a complexvariable when approaching the real axis from above and below [87, section 3.6]: lim ǫ → ( x ± iǫ ) − γ = x − γ + + e ∓ iπγ x − γ − , (4.16)where γ = 1 , , . . . . Then, ∆ − ( K ) = [ − ( k + iǫ ) + ω ] − γ − [ − ( k − iǫ ) + ω ] − γ = [ k + m − iǫ sgn( k )] − γ − [ k + m + iǫ sgn( k )] − γ = sgn( k ) (cid:2) ( k + m − iǫ ) − γ − ( k + m + iǫ ) − γ (cid:3) (4.16) = 2 i sin( πγ ) sgn( k ) ( k + m ) − γ − = 2 i sin( πγ ) sgn( k ) Θ( − k − m ) | k + m | − γ (4.17) = 2 i sin( πγ ) sgn( k ) Θ( k − ω ) ( k − ω ) − γ . (4.18)This weight function is spread on the domain − k > m , i.e., k ∈ ( −∞ , − ω ] ∪ [ ω, + ∞ ) ,for a generic γ / ∈ N , while for γ → one recovers the Lorentz-invariant free-wave solution– 20 –ith ∆ − ( K ) = 2 πiδ ( k + m ) , where we used the Sokhotski–Plemelj formula lim ǫ → + x − iǫ = PV (cid:20) x (cid:21) + iπδ ( x ) , (4.19)and PV denotes the principal value. The propagator of the free field is − iG ( x ) , where the Green’s function G ( x ) is one of thesolutions of the equation with source K ( (cid:3) ) G ( x ) = δ D ( x ) . (4.20)To solve it in momentum space, we need a function f ( k ) := K ( − k ) ˜ G ( − k ) that instead of(4.10) yielded the delta distribution: Z Γ d D k e − ik · x f ( k ) = δ D ( x ) . (4.21)The spatial-momentum part is easy: since (2 π ) D − δ D − ( x ) = R + ∞−∞ d D − k e − i k · x , it followsthat f ( k ) = F ( k ) / (2 π ) D − for some function F . The latter is F ( k ) = sgn(Im k ) / (4 π ) [60]: Z Γ dk π e ik x sgn(Im k ) = 12 Z + ∞−∞ dk π e ik x − Z −∞ + ∞ dk π e ik x = Z + ∞−∞ dk π e ik x = δ ( x ) . Therefore, G ( x ) = 12 Z Γ d D k (2 π ) D e − ik · x sgn(Im k ) K ( − k )= 12 Z + ∞−∞ d D k (2 π ) D e − ik · x ∆ + ( K ) , (4.22)where the two terms in ∆ + given in eq. (4.11) are the causal and anti-causal Green’sfunction, respectively. In particular, for the kinetic term (4.1)
12 ∆ + ( K ) = 12 (cid:8) [ − ( k + iǫ ) + ω ] − γ + [ − ( k − iǫ ) + ω ] − γ (cid:9) = 12 (cid:8) [ k + m − iǫ sgn( k )] − γ + [ k + m + iǫ sgn( k )] − γ (cid:9) = 12 (cid:2) ( k + m − iǫ ) − γ + ( k + m + iǫ ) − γ (cid:3) (4.16) = ( k + m ) − γ + + cos( πγ )( k + m ) − γ − . (4.23)Other Green’s functions can be obtained by adding a solution to the homogeneous equa-tion. Of all the possible outcomes, one can take the causal one with Feynman prescription[58, 60]: G F ( x ) = Z Γ F d D k (2 π ) D e − ik · x K ( − k ) , (4.24)– 21 –here Γ F = Θ( x ) Γ + ∪ Θ( − x ) Γ − and Γ ± are shown in Fig. 2. The path Γ F runs frombelow the branch cut k ∈ ( −∞ , ω ) to above the cut k ∈ ( ω, + ∞ ) . One can checkthat (4.24) is a solution of eq. (4.20) when applying K ( (cid:3) ) : the right-hand side yields (2 π ) − D R Γ F d D k e − ik · x = (2 π ) − D R + ∞−∞ d D k e − ik · x = δ D ( x ) .Regardless of the specific choice of contour, the Green’s function scales as ˜ G ( − k ) = 1( k + m ) γ , (4.25)which has a branch cut − k > m corresponding to time-like vectors with ( k ) > ω (alsolight-like vectors if m = 0 ). Thus, we cannot talk about a fundamental scalar particlebecause (4.25) has branch cuts instead of poles. In other words, while in ordinary QFTthe bare propagator has poles and dressed propagators typically are non-local and containbranch cuts, in theories with fractional operators the bare propagator itself can have branchcuts. The branch cut signals the presence of a continuum of modes with momenta − m and > m . In [5], these modes were described as “quasi-particles” in the lack of a betterlabel. Perhaps, a characterization as a gas may also be a viable alternative.Doing quantum field theory with branch cuts in the bare propagator is possible: thestructure of these objects is under control and has been studied in [48, 60, 61, 63]. Theappearance of branch cuts in quantum gravity and beyond the Einstein theory is not new,either, and people learned to live peacefully with them. Examples will be given in sections6.1 and 6.2. The representation (4.3) suggests that the theory has ghosts due to the higher-derivativeoperator ( m − (cid:3) ) n , while the exponential exp( τ (cid:3) ) does not introduce any extra pole. Onemight then conclude that the quantum theory is unitary only if n = 1 , i.e., γ < . However,in general integral parametrizations of non-local operators do not give direct, transparentinformation on the spectrum in non-local theories [88, 89] and one should verify classicalstability and quantum unitarity with other means. In this particular case, it will turn outthat the unitarity bound γ < is almost correct.In this sub-section, we will recall the basics of the Källén–Lehmann representation [90–94], an essential tool to verify unitarity of a QFT at all perturbative orders. At first, wewill check only free-level unitarity, that is, the absence of ghosts in the free theory, whichlater we will extend to one-loop unitarity and, partially, to all loops.Let ˜ G ( − k ) be the Fourier transform of the exact (i.e., interacting) Green’s functionwith Feynman prescription of a generic scalar field theory on Minkowski spacetime. Ex-tending to the complex plane, assume that ˜ G ( z ∗ ) = ˜ G ∗ ( z ) (this condition holds for ourtheory) and consider a closed contour ˜Γ encircling the point z = − k and such that ˜ G ( z ) is analytic inside and on Γ . Then, by Cauchy’s integral formula the Green’s function canbe written as ˜ G ( − k ) = 12 πi I ˜Γ dz ˜ G ( z ) z + k . (4.26)– 22 – - k m M Re zIm z
Figure 3 . Contour ˜Γ (black thick curve) in the ( s = Re z, Im z ) plane for a propagator with asimple pole at z = M and a branch cut at z > m (gray thick line). Suppose that ˜ G is singular at several places on the real axis. For instance, if ˜ G ( z ) had asimple pole at z = M and a branch point at z = m > M , with a branch cut on thepositive z > m half line, the contour would be the one in Fig. 3. We can deform the contour continuously as in Fig. 4 and split it into four: a loop Γ ε of radius ε encircling the simple pole z = M , a mini-contour C ε of radius ε around thebranch point z = m , an empty contour in the region M < z < m not shown in thefigure, the paths going back and forth along the branch cut and a counter-clockwise circle Γ R of radius R .The contribution of the empty contour to (4.28) is zero by the Cauchy–Goursat theo-rem and so is the contribution of Γ R when R → ∞ if ˜ G ( z ) falls off at z → ∞ . Parametrizing z as z = m + ε exp( iθ ) with − π < θ < , the contribution of the contour around thebranch point is πi Z C ε dz ˜ G ( z ) z + k = ε π ( k + m ) Z − π dθ e iθ ˜ G ( m + ε e iθ ) + . . . , (4.27) Since ˜ G ( z ) is analytic on the contour path, the integrand in (4.26) has a simple pole at z = − k andthe residue theorem gives +2 πi ˜ G ( − k ) , with the contour ˜Γ in the counter-clockwise direction. ˜ G ( s ) is monodromic and analytic for M < s < m and there is no discontinuity when crossing the realaxis ˜ G ( s + iǫ ) = ˜ G ( s − iǫ ) , hence Im ˜ G ( s + iǫ ) = 0 in this region. – 23 – R Γ ε C ε - k M m Re zIm z
Figure 4 . Deformation of the contour ˜Γ of Fig. 3 into two disconnected contours (black thickcurves), for a propagator with a simple pole at z = M and a branch cut at z > m (gray thickline). where the ellipsis stands for higher-order terms in ε . This integral can vanish, diverge orbe finite depending on ˜ G . Assuming that it vanishes (which must be checked explicitly forany given ˜ G ), the only contributions left are those of the pole at z = M and of the pathsalong the branch cut. The latter is πi Z cut dz ˜ G ( z ) z + k = lim ǫ → + πi Z + ∞ m ds ˜ G ( s + iǫ ) − ˜ G ( s − iǫ ) k + s − iǫ = 1 π lim ǫ → + Z + ∞ m ds Im[ ˜ G ( s + iǫ )] k + s − iǫ = Z + ∞ m ds ρ ( s ) k + s − iǫ , (4.28)where s = Re z and ρ ( s ) := 1 π lim ǫ → + Im[ ˜ G ( s + iǫ )] . (4.29)The loop around the pole admits a similar expression because one can divide Γ ε into twolines above and below the real axis plus two infinitesimal half arcs that give a zero contri- With an abuse of notation, we leave the contour prescription − iǫ in (4.28) out of the limit ǫ → + taken in (4.29). – 24 –ution: πi Z Γ ε dz ˜ G ( z ) z + k = Z M + εM − ε ds ρ ( s ) k + s − iǫ . (4.30)By construction, the support of δ ( s + k ) is to the right of the lower integration limit s min = M − ε .Combining (4.28) and (4.30), the Källén–Lehmann representation is [90–92] ˜ G ( − k ) = Z M + εM − ε ds ρ ( s ) k + s − iǫ + Z + ∞ m ds ρ ( s ) k + s − iǫ . (4.31)From the representation (4.31) for the exact propagator, one can determine the spectralfunction via the Sokhotski–Plemelj formula (4.19).For the theory to be unitary, the spectral function must be positive semi-definite forall s in the integration domain: ρ ( s ) > . (4.32)As we will see in appendix B for the scalar theory T [ (cid:3) γ ] , this condition is equivalent toimpose reflection positivity, one of the Osterwalder–Schrader conditions [80, 81] requiredfor a Euclidean field theory to admit an analytic continuation to Minkowski spacetime. Ifreflection positivity is violated, there is no positive semi-definite scalar product in the spaceof functionals of the field φ and there are no unitary representations of the Poincaré group.This would signal the presence of negative-norm states (ghosts). For the ordinary interacting scalar field theory (2.1), where K = (cid:3) , the full propagator hasthe singularities shown in Fig. 3, a simple pole at z = M and a branch cut at z > m =4 M , where m is the lowest mass in the multi-particle spectrum.In the presence of interactions, the first contribution is the renormalized free part, whilethe second encodes multi-particle states. Here we will be interested only in the free partwithout interactions. The free propagator in Minkowski momentum space is ˜ G ( − k ) = 1 k + M ⇒ ˜ G ( s + iǫ ) = − s − M + iǫ , (4.33)where ǫ > . The determination of the free spectral function is an almost tautologicalexercise because we can already read off the sign of the residue from the propagator, butwe will do it anyway because we are interested in the parallelism with the fractional case.In fact, in the ordinary case one can check the absence of ghosts from the rule-of-the-thumb“the sign of the residue (of the propagator) must be positive,” but in the fractional casethere is no residue to begin with.A quick way to find ρ ( s ) is to multiply and divide (4.33) by k + M + iǫ , so that fromeq. (4.29) ρ ( s ) = lim ǫ → + π Im (cid:20) − s − M − iǫ ( M − s ) + ǫ (cid:21) = lim ǫ → + π ǫ ( s − M ) + ǫ = δ ( s − M ) , (4.34) In [93], the spectral function is defined as the interacting part of our ρ , while the latter coincides withthe free + interacting ρ of [94]. – 25 –here we used the representation of the Dirac delta distribution as the limit of the Poissonkernel. When s = M , the limit when ǫ → + is zero, while when s = M it diverges as / ( πǫ ) . This is the behaviour of the Dirac distribution.The same result can be reached from the Schwinger representation of (4.33) in Lorentziansignature: ˜ G ( − k ) = i Z + ∞ dτ e − iτ ( k + M − iǫ ) ⇒ ˜ G ( s + iǫ ) = i Z + ∞ dτ e − iτ ( M − s − iǫ ) . (4.35)From (4.35), ρ ( s ) (4.29) = 1 π lim ǫ → + Z + ∞ dτ e − τǫ cos[ τ ( s − M )]= lim ǫ → + π ǫ ( s − M ) + ǫ = δ ( s − M ) . Thus, the spectral distribution is positive semi-definite and singular at s = M , correspond-ing to the k = ω mass pole. Integrating between M − ε and M + ε , one recovers theGreen’s function (4.33): Z M + εM − ε ds δ ( s − M ) k + s − iǫ = 1 k + M − iǫ . In this case, there is a neat correspondence between positivity of the pole residue andpositivity of the spectral function (absence of ghosts).
Let us repeat the procedure of section 4.1.6 for the fractional causal Green’s function ˜ G ( − k ) = 1( k + m ) γ , ⇒ ˜ G ( s + iǫ ) = 1( m − s − iǫ ) γ , (4.36)where the prescription − iǫ is such that (4.36) yields the Feynman propagator in the limit γ → . This function has a branch cut at s > m and no isolated poles (Fig. 5). First ofall, we check for which γ the piece of contour around z = m gives a vanishing contributionin the limit ε → + . Plugging eq. (4.36) into (4.27), πi Z C ε dz ˜ G ( z ) z + k = ε − γ e − iπγ π ( k + m ) Z − π dθ e iθ (1 − γ ) + O ( ε − γ )= − ε − γ (1 − γ )( k + m ) sin( πγ ) π + O ( ε − γ ) , (4.37)which implies γ < . (4.38)Then, the Källén–Lehmann representation of the Green function is ˜ G ( − k ) = Z + ∞ m ds ρ ( s ) k + s − iǫ . (4.39)– 26 – R C ε m Re zIm z
Figure 5 . Contour ˜Γ (black thick curve) in the ( s = Re z, Im z ) plane for a propagator with abranch cut at z > m (gray thick line). The first method to compute the spectral function, analogous to (4.34), is to write ˜ G ( s + iǫ ) = ( m − s − iǫ ) − γ = exp (cid:2) − γ Ln( m − s − iǫ ) (cid:3) = exp h − γ ln p ( s − m ) + ǫ − iγ Arg( m − s − iǫ ) i , (4.40)where Ln and Arg are the principal value of the complex logarithm and of the argument(the phase of a complex number). Then, ρ ( s ) = − lim ǫ → + π s − m ) + ǫ ] γ/ sin (cid:2) γ Arg( m − s − iǫ ) (cid:3) . In the presence of a branch cut on the real axis, the principal value Arg is evaluated quadrantby quadrant separately, depending on the sign of the real part m − s and of the imaginarypart ǫ . In our case, m − s < and − ǫ , which implies that Arg( m − s − iǫ ) = arctan (cid:18) ǫm − s (cid:19) − π , – 27 –o that ρ ( s ) = − lim ǫ → + π s − m ) + ǫ ] γ/ sin (cid:20) γ arctan (cid:18) ǫm − s (cid:19) − πγ (cid:21) (4.41) = sin( πγ ) π s − m ) γ . (4.42)The same result comes from the expression ˜ G ( s + iǫ ) = e i πγ Γ( γ ) Z + ∞ dτ τ γ − e − iτ ( m − s − iǫ ) , (4.43)where we generalized the Schwinger representation (4.35) by using the definition of thegamma function Γ( γ ) = R + ∞ dx x γ − e − x , valid for γ > but that can be analyticallycontinued to all γ = 0 , − , − , . . . . Taking the imaginary part and integrating in τ , ρ ( s ) (4.29) = 1 π Γ( γ ) lim ǫ → + Z + ∞ dτ τ γ − e − τǫ × n sin (cid:16) πγ (cid:17) cos[ τ ( m − s )] − cos (cid:16) πγ (cid:17) sin[ τ ( m − s )] o = 1 π Γ( γ ) lim ǫ → + Z + ∞ dτ τ γ − e − τǫ sin h πγ τ ( s − m ) i = lim ǫ → + π [( s − m ) + ǫ ] γ/ sin (cid:18) πγ − γ arctan m − sǫ (cid:19) = sin( πγ ) π s − m ) γ , where we used the fact that lim ǫ → + arctan[( m − s ) /ǫ ] = − π/ , since m − s < . Thelast line is (4.42); integrating it in (4.39), one recovers (4.36). Note that, just like the limit s → m , the limit γ → does not commute with the limit ǫ → + and one can recover thedelta distribution (4.34) only from (4.41), not from (4.42) (a similar non-commutation ruleholds, for instance, in causal sets [48]).The spectral function (4.42) is real ( s > m ) and is positive definite when sin( πγ ) > .This condition and inequality (4.38) fix the allowed range for γ > to < γ < , while for γ < one has − < γ < − , − < γ < − , and so on: − n < γ < − n , n ∈ N . (4.44)We excluded equalities in order to get a non-trivial function ρ = 0 but later they will beremoved anyway imposing one-loop finiteness. In appendix B, we recover the unitarityconstraint (4.44) with a different method based on reflection positivity.The lowest interval ( n = 0 ) < γ < (4.45)is the only one for which the spectral dimension of spacetime is positive definite, according toeq. (4.5). Note that the ranges (4.44) correspond to a theory with modifications dominatingin the IR. – 28 – .1.8 One-loop renormalization: vertices and vacuum diagram In section 2.4, we have found that the theory is power-counting renormalizable for γ > D/ ,corresponding to γ > in four dimensions. The power-counting argument holds in acut-off regularization scheme and one should wonder whether loop integrals contain otherdivergences than those seen in this scheme. Here we compute two one-loop diagrams forthe φ theory, the vacuum diagram and the self-energy, to show that they are finite.Each internal leg contributes a factor − i ˜ G ( − k ) , each loop corresponds to an inte-gration in d D k and each bare vertex is identical to the ordinary lowest-order N -particleamplitude: ... k k k k N = V ( k , . . . , k N ) = iλ N Z d D x (2 π ) D e ik tot · x = iλ N δ D ( k tot ) , (4.46)where k µ tot := P Nn =1 k µn . In the following, N = 3 and we use the standard notation forscalar-field diagrams: dashed lines and, for vertices, thick dots as in (4.46). Some of theintegrals involving (4.36) can be found as limits of the formulæ in [9, appendix A].Vacuum diagrams have no external legs. At one loop, the only contribution is theone-point function, a propagator closing on itself: k = i A = Z d D k − i ( k + m − iǫ ) γ . (4.47)To compute it, we analytically continue to Euclidean momentum space and use the Eu-clidean version of the Schwinger representation (4.43), with τ = − iσ : i A = 1Γ( γ ) Z d D k Z + ∞ dσ σ γ − e − σ ( k + m ) = Ω D Γ( γ ) Z + ∞ dσ σ γ − Z + ∞ dk k D − e − σ ( k + m ) = π D Γ( γ ) Z + ∞ dσ σ γ − − D e − m σ = π D Γ (cid:0) γ − D (cid:1) Γ( γ ) ( m ) D − γ , (4.48)where Ω D = 2 π D/ / Γ( D/ is the surface of the unit D -ball coming from integration of thesolid angle in momentum polar coordinates, d D k = d Ω D dk k D − . Note that integration in σ and k commute.When γ = 1 , equation (4.48) agrees with the ordinary scalar-field one-point functionfound in dimensional regularization [95]. In that case, the result diverges like Γ( − in D = 4 and a subtraction scheme must be enforced. In our case, the result is finite if γ − D = − n = 0 , − , − , . . . , (4.49)– 29 –hich corresponds to γ = 2 , , , − , − , . . . in four dimensions. These values are alreadyexcluded by the lower bound (2.12) for power-counting renormalizability. The self-energy (or bubble) one-particle-irreducible diagram ∼ V R ˜ G is the one-loopcorrection to the two-point function: k k’pp+k = i Π( k, k ′ ) = iδ D ( k − k ′ ) ˜Π( k ) , (4.50)where the outer segments are external legs, not propagators, and ˜Π( k ) := λ i Z d D p p + m − iǫ ) γ [( k + p ) + m − iǫ ] γ (4.51) p → ip → λ Z d D p p + m ) γ [( k + p ) + m ] γ = λ ( γ ) Z d D p Z + ∞ dσ dσ ( σ σ ) γ − e − σ ( p + m ) − σ [( k + p ) + m ] = λ ( γ ) Z + ∞ dσ dσ ( σ σ ) γ − e − ( σ + σ ) m Z d D p e − σ p − σ ( k + p ) , where we analytically continued to Euclidean momentum space. Using the Feynman pa-rametrization x := σ /y and y := σ + σ (so that σ = xy and σ = y (1 − x ) ), the lastexponent in the above expression reads σ p + σ ( k + p ) = y [ xp + (1 − x )( k + p ) ] = y [ p + (1 − x ) k ] + yx (1 − x ) k , and, calling p ′ µ = p µ + (1 − x ) k µ , we get ˜Π( k ) = λ ( γ ) Z dx Z + ∞ dy [ y x (1 − x )] γ − [ x (1 − x )] γ − e − y [ m + x (1 − x ) k ] Z d D p ′ e − yp ′ = λ π D ( γ ) Z dx [ x (1 − x )] γ − Z + ∞ dy y γ − − D e − y [ m + x (1 − x ) k ] = λ π D ( γ ) Γ (cid:18) γ − D − (cid:19) Z dx [ x (1 − x )] γ − [ x (1 − x ) k + m ] D − γ , (4.52)where the last step holds only for γ − D/ − > but it can be analytically continuedto γ − D/ − < if the gamma function at the numerator does not diverge (we willcome back to this important point later). One can verify that in the limit γ → and D → − ε one obtains, after subtracting the /ε divergence, the standard bubble diagramin four-dimensional canonical scalar field theory [96].– 30 –alling z := 4 x (1 − x ) , ˜Π( k ) = λ π D γ Γ ( γ ) Γ (cid:18) γ − D − (cid:19) Z dz (1 − z ) − z γ − (cid:18) zk + m (cid:19) D − γ = λ γ π D Γ (cid:0) (cid:1) Γ (cid:0) γ − D − (cid:1) Γ( γ ) Γ (cid:0) γ + (cid:1) ( m ) D − γ × F (cid:18) γ, γ − D − γ + 34 ; − k m (cid:19) , (4.53)valid for γ > and analytically continuable to γ < . F ( a, b ; c ; z ) = P + ∞ n =0 [( a ) n ( b ) n / ( c ) n ] × z n /n ! is the hypergeometric function, where ( a ) n = Γ( a + 1) / Γ( a + 1 − n ) . The expressionin Lorentzian momenta is obtained from (4.53) under the replacement k → κ − iǫ , with k in Lorentzian signature.The self-energy (4.53) is finite provided Γ(2 γ − D/ − does not diverge. Thus, γ = D − n , n ∈ N . (4.54)In D = 4 dimensions, γ = 3 − n , , , , − , . . . , (4.55)where negative values are excluded by the positivity bound (4.6). In particular, ˜Π divergesin the double limit γ → , D → , the standard four-dimensional theory.For a massless theory, one can integrate eq. (4.52) with m = 0 to get ˜Π( k ) = λ π D +12 D +2 − γ Γ (cid:0) D + 1 − γ (cid:1) Γ (cid:0) γ − D − (cid:1) Γ ( γ )Γ (cid:0) D + − γ (cid:1) ( k ) D − γ , (4.56)if γ < D/ . This range already excludes the divergence points γ = D/ n comingfrom the first Γ in the numerator, where n ∈ N , so that the finiteness condition is again(4.54). In this sub-section, we will write the full quantum propagator − i ˜ G Dyson at all orders inperturbation theory as a Dyson series of one-particle-irreducible diagrams. Consider adiagram consisting in the one-loop self-energy (4.50) to which a bare propagator in k replacesthe left external leg. Integrating over the internal momentum k makes this a truncatedvertex, to which one can attach any other diagram with outgoing momentum k : k’k = B = Z d D k − i ( k + m − iǫ ) γ i Π( k, k ′ )= 1( k ′ + m − iǫ ) γ ˜Π( k ′ ) = ˜ G ˜Π . (4.57)– 31 –he iteration of this diagram gives the Dyson series for the full quantum Green’s function: ˜ G Dyson = ˜ G + B ˜ G + B ( B ˜ G ) + · · · = (1 − B ) − ˜ G = 1˜ G − − ˜Π= 1( k + m − iǫ ) γ − ˜Π( k ) . (4.58)Having already established a set of conditions to make ˜Π finite, we can extend the above con-clusions on renormalizability to all loop levels. This expression does not include higher-loopirreducible diagrams and their level of divergence should be studied separately. However,there are good chances that these diagrams do not change the picture because not only theyare composed of finite bubble diagrams, buy they also have a larger number of external legsthan one-particle-irreducible diagrams, which reduces the superficial degree of divergence.Therefore, we expect also these higher-loop irreducible diagrams to be finite.From eqs. (4.29) and (4.58), we can check one-loop unitarity, which we will do only inthe massless case m = 0 . Assuming Im ˜Π = 0 , we can neglect the iǫ term in eq. (4.58) andwrite Im ˜ G Dyson = Im ˜Π( k )[( k + m ) γ − Re ˜Π( k )] + [Im ˜Π( k )] , (4.59)so that we have that ρ ( s ) > if, and only if, lim ǫ → + Im ˜Π( − s − iǫ ) > . (4.60)The calculation is the same as the one leading to eq. (4.42) but with m = 0 , γ replacedby γ − D/ − and an overall coefficient that can be read off from (4.56): ρ ( s ) = λ π D +12 D +2 − γ Γ (cid:0) D + 1 − γ (cid:1) Γ (cid:0) γ − D − (cid:1) Γ ( γ )Γ (cid:0) D + − γ (cid:1) sin (cid:2) π (cid:0) γ − D − (cid:1)(cid:3) π s γ − D − = λ π D γ Γ (cid:0) D + 1 − γ (cid:1) Γ ( γ )Γ ( D + 3 − γ ) 1 s γ − D − . (4.61)Recalling that (4.56) was calculated for γ < D/ , this expression is positive semi-definiteif, and only if, Γ( D + 3 − γ ) > , i.e., γ < D + 34 , D + 4 + 2 n < γ < D + 5 + 2 n , n = 0 , , . . . , (cid:22) D − (cid:23) . (4.62)For example, in D = 4 dimensions one gets n = 0 , and D = 4 : γ ∈ (cid:18) −∞ , (cid:19) ∪ (cid:18) , (cid:19) ∪ (cid:18) , (cid:19) . (4.63)One could extend the unitarity range by analytically continue (4.56) to γ > D/ , but itis not necessary. In fact, in order for the perturbative theory to be well-defined the unitarityrange should be the same at all orders. Comparing the free-level range (4.44) with (4.62),we see that their intersection coincides with the range (4.44). Therefore, we take (4.44) as the unitarity range of the theory and conjecture that the unitarity range at higher loopswill always contain (4.44). – 32 – .1.11 Range of γ Comparing the bounds (2.12) coming from power-counting renormalizability and (4.44)from unitarity, one might be induced to conclude that it is not possible to make a fun-damental quantum field theory with the fractional d’Alembertian because one encounterseither infinitely many divergences in the UV or instabilities and negative-norm states.However, we also showed that the power-counting argument does not tell the wholestory and one can get unitarity and one-loop finiteness, and possibly even finiteness at allorders, if γ falls in one of the ranges (4.44) and does not pick any of the values (4.55). Forexample, if γ is positive in order to have a positive spectral dimension according to (4.6),then the four-dimensional fractional theory is one-loop finite in the intervals < γ < / and / < γ < . T [ (cid:3) + (cid:3) γ ] To get a multi-scale geometry with explicit scaling, we must combine the operator (4.1)with at least one length scale ℓ ∗ . There are at least two ways in which one can do that.• Combine operators with different masses as done by Trinchero [45–47]: K = K T := ( (cid:3) − m )( E ∗ − (cid:3) ) γ − , (4.64)where m is an arbitrary mass scale that can be much smaller than the characteristicenergy E ∗ = E Pl ℓ Pl /ℓ ∗ . The IR limit (momentum-energy scales ≪ E ∗ ) of this operatoris K T ≃ E γ − ∗ ( (cid:3) − m ) , k ≪ E ∗ , (4.65)and redefining the scalar as φ eff = E γ − ∗ φ , one gets the standard scalar field theorywith ordinary dimensionality [ φ eff ] = ( D − / . In the UV limit (momentum-energyscales ≫ E ∗ ), K T ≃ ( − (cid:3) ) γ , k ≫ E ∗ , (4.66)and the theory becomes effectively massless.• Take a sum of fractional d’Alembertians with the same mass, in line with the paradigmof multi-fractal geometries where scale-dependent correlation functions are describedby a discrete set of critical exponents [5, 69]. Dimensional flow is realized here byexplicit multi-scaling [70], i.e., the fundamental scales of the geometry appear directlyin the coefficients of the sum of operators. The minimum to get a non-trivial dimen-sional flow is one fundamental scale and two exponents, 1 and γ = 1 . Therefore, wepropose K = ℓ − γ ) ∗ ( (cid:3) − m ) + ( m − (cid:3) ) γ , (4.67)where one could naturally identify ℓ ∗ = m − (we do not do it in order to allow forthe massless case). With this definition, the propagator of the free theory diverges at − k = m at all scales. Note that (4.67) is similar to the definition of distributed-order fractional derivatives D := R γ dγ ′ µ ( γ ′ ) ∂ γ ′ [97–104], where one integrates over a– 33 –arameter γ ′ with a weight µ ( γ ′ ) . If µ ( g ′ ) ≃ ℓ − γ ) ∗ δ ( γ ′ −
1) + δ ( γ ′ − γ ) , we get (4.67).To get a realistic dimensional flow, it is not necessary to consider a more complicated µ ( γ ′ ) .Plugging the kinetic term (4.67) into the action (2.3) with v = 1 , the action for the theory T [ (cid:3) + (cid:3) γ ] is S = Z d D x (cid:26) φ h ℓ − γ ) ∗ ( (cid:3) − m ) + ( m − (cid:3) ) γ i φ − V ( φ ) (cid:27) , (4.68)where, again, the potential includes only non-linear interactions.The equation of motion δS/δφ = 0 from the action (4.68) is calculated using the integralrepresentation (4.3) and reads h ℓ − γ ) ∗ ( (cid:3) − m ) + ( m − (cid:3) ) γ i φ − V ′ ( φ ) = 0 . (4.69)Free-field solutions can be found according to the general scheme presented in section 4.1.3and the final result therein is valid also for this theory in the UV limit. The propagator inthe UV is the one calculated in section 4.1.4. In the theory T [ (cid:3) + (cid:3) γ ] with the measure choice (2.5), dimensional flow is the same de-scribed in section 3.3.3. The Hausdorff dimension is constant, eq. (3.31), while the spectraldimension varies according to one of the following alternatives.• When γ > , the fractional operator in (4.67) dominates the dynamics in the UV,eq. (3.39), and the renormalization calculations of section (4.1) apply to the UV limitof the theory.• When γ < , the spectral dimension is anomalous in the IR and standard in the UV,eq. (3.40), and renormalizability is not improved.• Extending the operator (4.67) to three operators with three different exponents andtwo scales ℓ < ℓ , K = ( (cid:3) − m ) + ℓ γ − ( m − (cid:3) ) γ + ℓ γ − ( m − (cid:3) ) γ , γ < < γ , (4.70)we get the three-regime dimensional flow of eq. (3.42). γ Considering the multi-fractional extension of the Green’s function (4.36), G ( − k ) = 1 ℓ − γ ) ∗ ( k + m − iǫ ) + ( k + m − iǫ ) γ , (4.71)with the method discussed in sections 4.1.5 and 4.1.7 one can show that ρ ( s ) = 1 π sin( πγ ) ℓ − γ ) ∗ ( s − m ) − γ + ( s − m ) γ − ℓ − γ ) ∗ cos( πγ )( s − m ) , (4.72)– 34 –hich is positive definite in the range (4.44), in particular in the interval < γ < (4.45),and it reproduces (4.42) in the UV regime ℓ ∗ k ≪ . In the IR regime ℓ ∗ k ≫ , one doesnot recover the delta (4.34) unless one takes the limit ǫ → + after approximating theexpression to the Poisson kernel.Concerning one-loop renormalization, Since we are interested in the fate of UV diver-gences, we can consider only the integer or the fractional part of the multi-scale propagator(4.71), depending on whether γ < or γ > , respectively. The integer sector is a standardQFT, while the fractional sector was studied in sections 4.1.8–4.1.10.The conclusions laid out in section 4.1.11 hold for a theory with a single fractionaloperator as a kinetic term. However, in the multi-fractional case (4.68) having γ < meansthat the fractional limit we studied in sections 4.1.8–4.1.10 corresponds to the IR limit ofthe theory, not the UV one. Therefore, the multi-fractional theory (4.68) can never beone-loop finite and unitary at the same time. T [ (cid:3) γ ( ℓ ) ] In this version of the multi-scale dynamics with fractional d’Alembertian, one can makethe parameter γ scale dependent [68, 69] or coordinate dependent [105]. In the first case,which is simpler, the kinetic operator becomes K = K γ ( ℓ ) = ( m − (cid:3) ) γ ( ℓ ) , (4.73)where we can choose the profile γ ( ℓ ) given in (3.46). When computing the propagator orFeynman diagrams, one may find the Green function of (4.73) much simpler to handle thanthat of (4.67).The action gets an overall extra integration over the probed scale as in (3.43), S = 1 ℓ ∗ Z + ∞ dℓ w ( ℓ ) Z d D x (cid:20) φ ( m − (cid:3) ) γ ( ℓ ) φ − V ( φ ) (cid:21) , (4.74)where w ( ℓ ) is a one-parameter weight. The equation of motion is ( m − (cid:3) ) γ ( ℓ ) φ − V ′ ( φ ) = 0 . (4.75)Its solutions in the free case and the propagator were discussed in, respectively, sections4.1.3 and 4.1.4, where we have to replace γ → γ ( ℓ ) . This is the same flow as in T [ ∂ γ ( ℓ ) ] :• With the profile (3.46) for any positive γ , one gets the dimensional flow (3.39).• With the profile (3.47) for any positive γ , one recovers (3.40).• With the profile (3.48) for any positive γ and γ , we get the flow (3.42): a UV regimewhere renormalizability can improve, a mesoscopic one where standard classical andquantum field theory is recovered, and an IR or ultra-IR one which may be relevantfor cosmology. – 35 – .3.3 Unitarity, renormalization and range of γ The calculation done in section 4.1.7 holds also for T [ (cid:3) γ ( ℓ ) ] . The theory is unitary if γ fallswithin one of the bounds (4.44) at all scales ℓ .The results of sections 4.1.8–4.1.10 hold, since in any Feynman diagram all exponents γ ( ℓ ) coming from different internal lines are evaluated at the same scale ℓ .In the theory (4.74), the limit of fractional exponent γ ( ℓ ) → γ is in the UV by con-struction assuming (3.46), even when γ < . Therefore, we can export the results of section4.1.11 on the range of γ preserving unitarity and renormalizability to T [ (cid:3) γ ( ℓ ) ] . Having studied in depth the scalar QFT with fractional derivative operators, we turn tothe gravitational sector, limiting the discussion to the definition of the action, the classicalequations of motion and some of the expected quantum properties.
The gravitational theory with multi-fractional derivatives can be written down in a straight-forward manner noting that, according to the paradigm, the whole integro-differential struc-ture is supplanted by a multi-fractional structure. This means, in particular, that the can-didate action should resemble the Einstein–Hilbert action where the integration measureis the same (see eq. (2.5)) and all ordinary derivatives are replaced by fractional or multi-fractional derivatives. Contrary to the scalar case (3.23) and (3.24), we are not at libertyof choosing between the fractional derivatives generalizing even- or odd-order derivatives,since in the gravitational action the equivalent of first-order derivatives appear isolated fromone another. Therefore, we must select D γ − , defined in eq. (3.22), or its multi-fractional gen-eralization (3.34). We omit the subscript − from now on. T [ ∂ γ ] The fractional Levi-Civita connection, Ricci tensor and Ricci scalar feature the fractionalderivatives D γµ given by eq. (3.22) with the − sign choice: ˜Γ ρµν := 12 g ρσ (cid:0) D γµ g νσ + D γν g µσ − D γσ g µν (cid:1) , (5.1) R ( γ ) µν := D γσ ˜Γ σµν − D γν ˜Γ σµσ + ˜Γ τµν ˜Γ σστ − ˜Γ τµσ ˜Γ σντ , (5.2) R ( γ ) := g µν R ( γ ) µν . (5.3)Note that this and the following are metric theories with an added non-dynamical structurethat implies that covariance and diffeomorphism invariance are no longer equivalent [106].In other words, ordinary diffeomorphism invariance is deformed and local inertial framesare described by fractional Lorentz transformations [38].The gravitational action is S = 12 κ Z d D x p | g | h R ( γ ) − i , (5.4)– 36 –here κ = 8 πG is proportional to Newton’s constant, g is the determinant of the metricand Λ is a cosmological constant.In appendix C.1, we calculate the equations of motion for this and the following theoriesin a general form. For T [ ∂ γ ] , adding a matter action S m to (5.4) we have R ( γ ) µν − g µν R ( γ ) + Λ g µν + O µν = κ T µν , (5.5)where O µν is a tensor defined in appendix C.1 and T µν := − p | g | δS m δg µν (5.6)is the matter energy-momentum tensor.Taken as a physical theory, T [ ∂ γ ] would be such that γ = 1 − ε . in order to respectall gravitational phenomenology without violating any experimental constraint. However,as we had occasion to say, we have no theoretical reason justifying such a fine tuning of a γ very close to, but different from, one. Therefore, we regard (5.4) as a starting point tounderstand the next theories rather than a physical model by itself. T [ ∂ + ∂ γ ] The theory with multi-fractional derivatives can be defined in two ways, which we might callnaive and natural. The naive one involves a fractional and an ordinary sector independentlyadded to the action. The fractional sector corresponds to the connection ˜Γ ρµν and Riccitensor R ( γ ) µν of the theory T [ ∂ γ ] , while the ordinary sector is described by the usual Levi-Civita connection and curvature tensors: Γ ρµν := 12 g ρσ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) , (5.7a)Riemann tensor: R ρµσν := ∂ σ Γ ρµν − ∂ ν Γ ρµσ + Γ τµν Γ ρστ − Γ τµσ Γ ρντ , (5.7b)Ricci tensor: R µν := R ρµρν , (5.7c)Ricci scalar: R := R µν g µν , (5.7d)Einstein tensor: G µν := R µν − g µν R . (5.7e)Then, the action is (5.4) plus the Einstein–Hilbert action: S = 12 κ Z d D x p | g | h R + ℓ γ − ∗ R ( γ ) − i . (5.8)The equations of motion are h R µν + R ( γ ) µν i − g µν h R + R ( γ ) i + Λ g µν + O µν = κ T µν . (5.9)In contrast, the natural definition of T [ ∂ + ∂ γ ] realizes multi-scaling within each deriva-tive instead of at the level of action operators. Defining the fractional Levi-Civita connectionwith the multi-fractional derivatives (3.34) D − (with omitted − subscript), ¯Γ ρµν := 12 g ρσ ( D µ g νσ + D ν g µσ − D σ g µν ) , (5.10)– 37 –e construct the fractional generalization of the covariant derivative of a rank-2 tensor, ¯ ∇ σ A µν := D σ A µν − ¯Γ ρσµ A ρν − ¯Γ ρσν A µρ . (5.11)Similarly, for a contravariant vector ¯ ∇ σ A µ = D σ A µ + ¯Γ µσν A ν . (5.12)By reverse engineering, one should be able to determine the transformation laws of vectorsand tensors under coordinate transformations in the multi-fractional geometry defined byfractional calculus. Note that angles and lengths are not parallel transported by the Levi-Civita connection and the metric is not covariantly constant, ∇ σ g µν = 0 . Still, there is afractional notion of metric compatibility, ¯ ∇ σ g µν = 0 , (5.13)which, just like the standard one, allows the Minkowski metric in local inertial framesbecause constants are in the kernel of the fractional derivatives we are using (eq. (3.9) with m = 1 ). If we further impose < γ < , (5.14)then the Minkowski metric is the only admissible one in local frames, since the kernel ofthe fractional derivatives becomes trivial. Violation of integer-order metric compatibilityis common in multi-fractional spacetimes, in particular, in the theories T v with weightedderivatives and T q with q -derivatives [106].The multi-fractional generalization of the Ricci tensor is R µν := D σ ¯Γ σµν − D ν ¯Γ σµσ + ¯Γ τµν ¯Γ σστ − ¯Γ τµσ ¯Γ σντ , (5.15)while the Ricci scalar is R := g µν R µν . The gravitational action in D topological dimensionsreads S = 12 κ Z d D x p | g | ( R − . (5.16)Due to the multi-scaling of the derivatives, at the level of local inertial frames there isno simple analogue of fractional Lorentz transformations [38]. Also, contrary to the naiveversion (5.8), the action (5.16) hides mixed terms of derivative order γ .Variation of the action (5.16) with respect to the metric yields, in the presence of matterwith action S m , to the multi-fractional generalization of Einstein’s equations (appendix C.1) R µν − g µν R + Λ g µν + O µν = κ T µν . (5.17) T [ ∂ γ ( ℓ ) ] This is the analogue of the action (5.4) with variable-order fractional derivatives D γ ( ℓ ) µ : S = 12 κ ℓ ∗ Z + ∞ dℓ w ( ℓ ) Z d D x p | g | h R ( γ ( ℓ )) − i . (5.18)– 38 –he theory does not have unnecessary mixed-order terms and may allow for a relativelysimple variable-order form of fractional Lorentz invariance in local frames [38, section 2.3].The equations of motion stemming from (5.18) are (5.5) with the replacement γ → γ ( ℓ ) ,without integration on ℓ . In fact, in a unified theory also the right-hand side is madeof matter fields with multi-fractional dynamics, whose action S m also contains fractionalexponents γ ( ℓ ) and the ℓ integration as in (5.18), with same weight. Therefore, the equalityin (5.5) must hold for all ℓ in the integration range. In a compact notation, R ( γ ( ℓ )) µν − g µν R ( γ ( ℓ )) + Λ g µν + O ( ℓ ) µν = κ T ( ℓ ) µν . (5.19)Of the three versions of gravity with multi-fractional derivatives, the easiest but alsothe less elegant is perhaps (5.8), while (5.18) can keep some simplicity and admit a richersymmetry structure. All three versions share the difficulty of the complicated compositionrule (3.7) and Leibniz rule (3.11) for fractional derivatives, that make the derivation andmanipulation of the equations of motion tricky, especially in the calculation of the tensor O µν , as detailed in appendix C.1. In this section, we construct a covariant gravitational theory with fractional Laplace–Beltrami operators with the same quantum properties discovered for the scalar QFT insection 4.Just like in the scalar-field case we proposed the fractional d’Alembertian as a way topreserve ordinary Lorentz invariance, so do we wish to preserve ordinary diffeomorphisminvariance and local ordinary Lorentz invariance in the gravity case. The action we lookfor will then be made of some functions F i ( (cid:3) ) of the covariant d’Alembertian acting onthe standard curvature tensors. Spacetime geometry is described by the usual Levi-Civitaconnection and curvature tensors (5.7). Angles and lengths are parallel transported andthe compatibility equation ∇ σ g µν = 0 holds as usual.Our starting point is the generic gravitational action S = 12 κ Z d D x p | g | [ R −
2Λ + R F ( (cid:3) ) R + G µν F ( (cid:3) ) R µν + R µνρσ F ( (cid:3) ) R µνρσ ] , (5.20)plus a matter action S m , where the Ricci tensor and Ricci scalar are defined by the ordinaryexpressions (5.7). This action is written in the so-called Einstein basis (Riemann tensor,Ricci tensor and Ricci scalar) but one could also adopt the Weyl basis, where the Riemanntensor in the last term is replaced by the Weyl tensor.The Einstein–Hilbert term R is necessary to recover the limit of general relativity, whilethe second-order curvature invariants are the simplest non-trivial covariant operators whereone can insert the functions F i ( (cid:3) ) .The equations of motion from the action (5.20) can be found in [107, 108] for all F i = 0 and a series representation of the variation of non-local operators, and in [88] for F = 0 = F and an integral representation of the variation of non-local operators. Thelatter will be very convenient in the case of fractional operators, so that in the following– 39 –e extend the calculation of [88] to F = 0 , although we will check a posteriori that thisgeneralization is not necessary. We still keep F = 0 for simplicity, since this term does notplay any role in the unitarity of the theory.We sketch the derivation of the equations of motion in appendix C.2. The final resultis κ T µν = (1 + F (cid:3) ) G µν + Λ g µν +2[ g µν (cid:3) − ∇ ( µ ∇ ν ) ] F R + g µν ∇ σ ∇ τ F G στ − ∇ σ ∇ ( µ F G ν ) σ +( G µν + R µν ) F R − g µν G στ F R στ + 2 G σ ( µ F G ν ) σ + 12 ( G µν F R + R F G µν )+ ϑ µν ( R, R ) + Θ µν ( R αβ , G αβ ) , (5.21)where ϑ µν ( R, R ) :=
R δ F δg µν R , Θ µν ( R αβ , G αβ ) := G ρσ δ F δg µν R ρσ . (5.22)At this point, the choice of form factors F , determines the theory and, in particular,the terms (5.22) and the kinetic operator K of the graviton. Split the metric into Minkowskibackground and a perturbation, g µν = η µν + h µν . (5.23)Working in the transverse traceless gauge ∂ µ h µν = 0 = h µµ , (5.24)which can always be selected in any covariant theory on a D -dimensional Minkowski back-ground [109, 110], the on-shell graviton is the transverse traceless part h ij of the pertur-bation. Finding the linearized equation of motion for h µν in vacuum is easy, since allsecond-order curvature terms in eq. (5.21) vanish on Minkowski background. The lineariza-tion of the Ricci tensor can be read off from eq. (C.10a) with δg µν = h µν , so that by virtueof (5.24) δ (1) R µν = − (cid:3) h µν / and δ (1) R = 0 . Setting Λ = 0 in (5.21), the modified waveequation for the graviton in vacuum is [1 + F ( (cid:3) ) (cid:3) ] (cid:3) h µν = 0 , (5.25)where (cid:3) = (cid:3) η is the d’Alembertian in Minkowski spacetime. Unitarity can be evinced fromthe choice of kinetic term K ( (cid:3) ) = (cid:3) + F ( (cid:3) ) (cid:3) , which does not involve F . Therefore,without loss of generality we can set F = 0 . (5.26) T [ (cid:3) γ ] The analogue of the operator (4.1) is F ( (cid:3) ) = ( − ℓ ∗ (cid:3) ) γ − − (cid:3) ⇒ ( − (cid:3) ) γ h µν = 0 . (5.27)– 40 –he action and equations of motion for the theory with such form factor are S = 12 κ Z d D x p | g | (cid:20) R −
2Λ + G µν ( − ℓ ∗ (cid:3) ) γ − − (cid:3) R µν (cid:21) , (5.28)and κ T µν = ( − ℓ ∗ (cid:3) ) γ − G µν + Λ g µν + g µν ∇ σ ∇ τ ( − ℓ ∗ (cid:3) ) γ − − (cid:3) G στ − ∇ σ ∇ ( µ ( − ℓ ∗ (cid:3) ) γ − − (cid:3) G ν ) σ − g µν G στ ( − ℓ ∗ (cid:3) ) γ − − (cid:3) R στ + 2 G σ ( µ ( − ℓ ∗ (cid:3) ) γ − − (cid:3) G ν ) σ + 12 (cid:20) G µν ( − ℓ ∗ (cid:3) ) γ − − (cid:3) R + R ( − ℓ ∗ (cid:3) ) γ − − (cid:3) G µν (cid:21) + Θ µν ( R αβ , G αβ ) , (5.29)where Θ µν can be written explicitly for a suitable integral or series representation of theform factor [88].Since the free equation of motion (5.27) of the graviton is the same as the free mass-less version of the equation of motion (4.4), we expect the unitarity and renormalizabilityanalysis carried out for the scalar field theory of sections 4.1.7–4.1.10 to hold also for thegravitational theory defined by the action (5.28). Concerning unitarity, γ can take valuesonly within the intervals (4.44), reduced to (4.45) ( < γ < ) if we insist in having a geom-etry with well-defined spectral dimension. Regarding renormalizability, the power-countingargument of section 2.4 does not hold and we have to check order by order in perturbationtheory. At one loop, the fractional operator does not introduce divergences provided (4.54)holds; in four dimensions and in the above unitarity interval, it means that γ = 1 / .Exponential or asymptotically polynomial form factors have been the main election innon-local quantum gravity, a unitary and super-renormalizable perturbative QFT of gravity[111–113]. There, the action is (5.20) and the form factor in (5.25) is F ∝ [exp H( ℓ ∗ (cid:3) ) − / (cid:3) , where H( − z ) is an entire function, so that the graviton equation of motion (5.25)is exp[ − H( ℓ ∗ (cid:3) )] (cid:3) h µν = 0 and no extra poles or branch cuts are introduced. Exponentialform factors correspond to H( − z ) = z n , n = 1 , , . . . . Asymptotically polynomial formfactors are more complicated but they all have the asymptotic limit exp H( − z ) ∼ | z | n deg forsome integer power n deg . This limit is very similar to what we want to obtain in fractionalgravity, the only difference being that instead of having | z | n deg asymptotically we have z γ − from the start, where γ is non-integer. There is also another similitude between the twotheories. In the sense of the integral representation (4.3), the fractional d’Alembertian is asort of weighting of the exponential form factor with a fractional measure. Assuming γ < , ( − (cid:3) ) γ − = 1Γ(1 − γ ) Z + ∞ dτ τ − γ e τ (cid:3) . (5.30)Although also our theory is non-local and quantum, we will keep a different naming toavoid confusion. Quantum gravity with fractional operators, or fractional gravity in short,– 41 –oes one step further than non-local quantum gravity inasmuch as its non-local operatorsare non-analytic. In this sense, fractional gravity is more difficult and makes a more radicaldeparture than non-local quantum gravity from standard QFT. Still, it is fascinating thatwe ended up with the same Lagrangian asymptotically in the UV, except for the value ofthe powers in the derivative operators (integer n deg in asymptotically polynomial non-localquantum gravity, non-integer γ in fractional gravity). T [ (cid:3) + (cid:3) γ ] The analogue of the multi-fractional operator (4.67) is F ( (cid:3) ) = ℓ ∗ ( − ℓ ∗ (cid:3) ) γ − ⇒ (cid:2) (cid:3) + ℓ − ∗ ( − ℓ ∗ (cid:3) ) γ (cid:3) h µν = 0 . (5.31)From the integral representation (4.3) with γ < , ( − (cid:3) ) γ − = 1Γ(2 − γ ) Z + ∞ dτ τ − γ e τ (cid:3) . (5.32)Using this formula, the function Θ µν is calculated in appendix C.2 for the fractional operator(5.31): Θ µν ( R αβ , G αβ ) = ℓ γ − ∗ Γ(2 − γ ) Z + ∞ dτ τ − γ Z τ dq ¯Θ µν [ e q (cid:3) R αβ , e ( τ − q ) (cid:3) G αβ ] , (5.33)where ¯Θ µν is second-order in derivatives and is given in (C.15b). To summarize, for thetheory with form factor (5.31) the action (5.20) simplifies to S = 12 κ Z d D x p | g | (cid:2) R −
2Λ + ℓ ∗ G µν ( − ℓ ∗ (cid:3) ) γ − R µν (cid:3) , (5.34)while the equations of motion (5.21) read κ T µν = [1 − ( − ℓ ∗ (cid:3) ) γ − ] G µν + Λ g µν + ℓ ∗ g µν ∇ σ ∇ τ ( − ℓ ∗ (cid:3) ) γ − G στ − ℓ ∗ ∇ σ ∇ ( µ ( − ℓ ∗ (cid:3) ) γ − G ν ) σ − ℓ ∗ g µν G στ ( − ℓ ∗ (cid:3) ) γ − R στ + 2 ℓ ∗ G σ ( µ ( − ℓ ∗ (cid:3) ) γ − G ν ) σ + 12 ℓ ∗ [ G µν ( − ℓ ∗ (cid:3) ) γ − R + R ( − ℓ ∗ (cid:3) ) γ − G µν ] + Θ µν ( R αβ , G αβ ) , (5.35)with Θ µν given by (5.33).At the quantum level, the theory should enjoy the same properties and problems ofits scalar-field counterpart, due to the equivalence between the graviton linearized equation(5.31) and the free massless version of the equation of motion (4.69). In this case, theresults of sections 4.1.7–4.1.10 do not apply to the UV of the theory T [ (cid:3) + (cid:3) γ ] because thefractional operator dominates at low energies/low curvature. Therefore, the exclusion points(4.54) guarantee the absence of IR rather than UV divergences. Although a calculationwith the full multi-scale kinetic term would say the final word, the theory is probably non-renormalizable just like Einstein gravity, since the (cid:3) kinetic term dominates at short scales.Still, it can be used as an interesting generator of classical and quantum modifications ofgravity at large scales, with cosmological applications such as in the problems of dark matterand dark energy (see section 6.2). – 42 – .2.3 Theory T [ (cid:3) γ ( ℓ ) ] The analogue of the operator (4.73) with scale-dependent exponent is eq. (5.27) with γ → γ ( ℓ ) : F ( (cid:3) ) = ( − ℓ ∗ (cid:3) ) γ ( ℓ ) − − (cid:3) , ( − (cid:3) ) γ ( ℓ ) h µν = 0 . (5.36)The action and equations of motion for the theory with such form factor can be madeexplicit in a similar way and are S = 12 κ ℓ ∗ Z + ∞ dℓ w ( ℓ ) Z d D x p | g | " R −
2Λ + G µν ( − ℓ ∗ (cid:3) ) γ ( ℓ ) − − (cid:3) R µν , (5.37)and eq. (5.29) with γ → γ ( ℓ ) , where we chose the weight w such that R + ∞ dℓ w ( ℓ ) = ℓ ∗ .As we discussed in section 5.1.3, the equations of motion are valid at any given ℓ .The case with scale-dependent γ ( ℓ ) represented by the form factor and graviton equa-tion (5.36) may be more promising as a fundamental theory than the theories T [ (cid:3) γ ] (unitaryand one-loop finite but with no multi-scaling) and T [ (cid:3) + (cid:3) γ ] (never unitary and renormal-izable at the same time). The unitarity and one-loop finiteness set γ ( ℓ ) ∈ (0 , / ∪ (1 / , corresponds to a class of theories well behaved in the UV, as originally desired, while main-taining the possibility to find a non-trivial cosmological imprint. In this section, we state what of the above is new and what was known from the literature,recapitulating past results to the best of our knowledge. • The mathematical properties of the fractional d’Alembertian K ( (cid:3) ) = (cid:3) γ acting on ascalar field have been studied extensively [53–63]. Using Caffarelli–Silvestre extensiontheorem [114], it was recently shown that the D -dimensional action of the masslessfree scalar field φ ( x ) with K ( (cid:3) ) = (cid:3) γ is equivalent to the ( D + 1) -dimensional actionof a scalar Φ( x, y ) with an extra fictitious spatial direction with fractional unilateralmeasure [115]: S = 12 Z d D x φ ( x ) (cid:3) γ φ ( x )= 2 γ − Γ( γ ) γ Γ( − γ ) Z d D x Z + ∞ dy y − γ ∂ M Φ( x, y ) ∂ M Φ( x, y ) , (6.1)where M = ( µ, y ) = 0 , , . . . , D − , D and lim y → y − γ ∂ y Φ( x, y ) = 0 . This corre-spondence allows for a smooth quantization and expands to the realm of fractionaloperators the notion, valid for non-local quantum gravity with exponential or asymp-totically polynomial operators [88, 89], that it is possible to recast non-local systemsas higher-dimensional local systems. The infinite number of initial conditions of non-local dynamics translate into field boundary conditions along the extra direction.– 43 –owever, (6.1) is surprising also for another reason, unnoticed in [115]: modulo theoverall constant, the last line of eq. (6.1) coincides with the action of a free masslessscalar field in the multi-fractional theory T with normal derivatives and unilateralfractional measure in the y direction [3, 38–40]. This suggests a relation between T [ (cid:3) γ ] and T we will comment upon in the conclusions.• The canonical quantization of a free scalar theory with K ( (cid:3) ) = (cid:3) ( − (cid:3) ) − α with <α < has been carried out in [58, 60].• Trinchero [45–47] considered the unitarity and renormalization properties of a Eu-clidean theory with operator (4.64), L = 12 φ ( (cid:3) − m )( E ∗ − (cid:3) ) − α φ − λφ , α = 1 − γ (6.2)with m = E ∗ and α > ( γ < ).• A non-local kinetic term K = ( (cid:3) − m ) p E ∗ − (cid:3) can arise for a scalar field in κ -Minkowski non-commutative spacetime [116]. The propagator and its branch cutswere studied in [117].• The canonical quantization of a free scalar was generalized to an arbitrary K ( (cid:3) ) withbranch cuts in [60] and later in [48–52]. In [62] (reviewed in [48]), it was shown thatonly massless states appear asymptotically in the free quantum theory and, if onlybranch cuts are present, then the only asymptotic state is the vacuum. The Huygens’principle for this class of models was discussed in [48, 51, 55, 57]. A class of operators K ( (cid:3) ) gives rise to a model which does not admit a variational principle and such thatthe modes on the branch cut cannot be detected through scattering experiments,since they never appear as in-states in non-zero amplitudes [49]. Since these modesdo not interact, they can serve as a dark-matter candidate [49]. The interpretation ofthe continuum of modes of the branch cut as infinitely many local scalars has beenstudied in [52].• A non-minimally coupled scalar field with fractional Laplacian ∆ / was introducedto preserve detailed balance in Hořava–Lifshitz gravity in the matter sector [85].Our scalar theory differs from the above proposals not only in the form of the kinetic op-erator but also in the motivation and the focus. Regarding the justification, some previousproposals share an interest in toy models of quantum gravity, but while [45–47] were mo-tivated by non-commutative geometry and [48–52] by causal sets, our action (4.68) hasbeen built from basic considerations of multi-fractal geometry [5, 69, 70]. This change inperspective also accounts for the different stress in the geometrical interpretation of thetheory, here more centered on dimensional flow. Moreover, our focus is on unitarity andrenormalizability, while previous works mainly studied the canonical quantization and theunitarity of their models, except [45, 47] where some Feynman diagrams were calculatedfor the theory (6.2). – 44 –ausal sets, non-commutative geometry and multi-fractal geometry are all connectedwithin the bigger scheme of quantum gravity [5, 118], which explains how independent rea-sonings led to operators belonging to the same mathematical class of fractional derivatives.The fractional proposals of [45–52] appeared after the multi-fractal and multi-fractional the-ories, where fractional operators were invoked in quantum gravity as early as [39, 41, 42],but all of them were influenced by previous studies on the fractional d’Alembertian. Several models studied two limited aspects of gravity with fractional derivatives withoutembedding them in a fundamental theory: Newtonian gravity and cosmology.• The linearized equations of motion and some general properties of ghost-free large-distance modifications of gravity were studied in [119]. Here the graviton kinetic termis augmented by a fractional d’Alembertian, so that, up to other trace h µµ and second-order derivative terms, [ (cid:3) + r γ − (cid:3) γ ] h µν ∼ κ T µν , where r c is a cosmological scaleand γ < . Using the spectral representation reviewed in section 4.1.5, the unitarityconstraint γ > was evinced. The resulting range < γ < corresponds to theunitarity constraint (4.45). No non-linear gravitational action was proposed.• Newton’s potential was derived from an ad hoc fractional Poisson equation [120–123].As an application, under the hypothesis that the matter distribution of galaxies be-haves as a fractal medium with non-integer dimension, solving a Poisson equationwith fractional Laplacian ( − ∆) γ one can describe the properties of such matter dis-tribution with a fractional version of Newtonian gravity and account for the observedgalaxy rotation curves without invoking dark matter, if the fractional exponent γ isclose to / [122, 123]. • The ordinary time derivatives of the Friedmann equations of homogeneous and isotropiccosmology were replaced by fractional derivatives in [130–133]. In particular, in [133]it was shown that supernovæ data on the late-time acceleration of the universe canbe explained by fractional Friedmann equations with derivative order close to / .These equations were not obtained from the symmetry reduction of any background-independent non-linear gravitational action. A source of confusion in the literature may be that models called “fractional” indiscriminately refer tomodifications of the integral and/or the differential structure of gravitational dynamics. Here we strictlyrefer to models with fractional derivatives, not with fractional integration measure, which are reviewed in[106, section 1.2]. A similar result can be obtained from a Poisson equation originated from a Gauss theorem with frac-tional measure [124–126]. The ensuing Laplacian operator is made of ordinary derivatives but has anomalousdimension and is the one appearing in the diffusion equations proposed for fractal media [127–129]. Thismodel falls into the category of scenarios mentioned in footnote 6 that we do not consider here but, never-theless, we notice that a theory that could easily accommodate Varieschi’s model and extend it to a matterdistribution with scale-dependent dimension is the multi-fractional theory T with ordinary derivatives[3, 39, 40]. – 45 – Another attempt to explain dark energy was made using thermodynamical argumentsbased on a Schrödinger equation with fractional Laplacian [134].It would be important to check whether these phenomenological results, interesting per se but lacking a robust theoretical motivation, can be obtained in the gravitational theoriesproposed here, without violating experimental constraints on gravity at sub-galactic andcosmological scales. Apart from the class of multi-fractional theories T γ , there are not many other proposals fora non-linear action of gravity with fractional operators.• A non-minimal scalar-tensor theory with fractional Laplacians ∆ / and ∆ / wasformulated in order to preserve detailed balance in the matter sector in the firstformulation of Hořava–Lifshitz gravity [85]. Detailed balance is a condition imposedon the action that suppresses the proliferation of operators at the quantum level. Theensuing action is complicated and is made of operators of order 6 in spatial derivatives.It contains covariant derivatives of the spatial Ricci tensor R ij and of a scalar φ , aswell as the fractional terms ∆ / φ , ∆ / φ and ∆ / R ij . The theory is unitary and isargued to be renormalizable but it has other problems because Lorentz invariance isnot recovered in the IR [85].• The generalization of the Ricci tensor and the Einstein equations to fractional deriva-tives was written down in [135], where the fractional Poisson equation was also derived,thus justifying the starting point of [120, 121] one step further. However, the modifiedEinstein equations [135] R ( γ ) µν − g µν R ( γ ) = κ T µν (6.3)were not derived from the variation of an action. In [135], the Ricci tensor is eq. (5.2)with fractional Levi-Civita connection (5.1), where the mixed fractional derivative D γµ is replaced everywhere by the Riemann–Liouville fractional derivative rl ∂ γµ [66, 67].Although the Ricci tensor looks similar, the dynamics is not because the derivativeoperator is different and, in particular, the Riemann–Liouville derivative of a constantis not zero. This can create serious problems when defining local inertial frames, therole of Minkowski metric, the covariant conservation of the metric, and so on [42].Also, the expressions of the Levi-Civita connection and Ricci tensor were used in [135]to define the equations of motion (6.3), while in the case of the theory T [ ∂ γ ] we havebuilt the Levi-Civita connection and Ricci tensor according to the paradigm of multi-scale spacetimes, then we defined the action (5.4), then we derived the equations ofmotion (5.5), which include the contribution of the operator O µν missing in (6.3).Also, in our case we regard T [ ∂ γ ] as a sort of toy model for the multi-fractionalversions of the theory, unless γ ≈ .• The same Ricci tensor and Einstein equations of the previous bullet were constructedby Vacaru in [136] using instead Caputo fractional derivative ∂ γµ . This proposal is– 46 –uch more rigorous than the previous one since it relies on the formalism of Finslergeometry [137] and, in particular, a structure of fractional manifold built with frac-tional differentials similar to those later employed in [42]. Once again, the Einsteinequations were not derived from an action, although this should be possible using thefractional Euler–Lagrange equations found in [138, 139]. A preliminary attempt toquantization was sketched in [140] but at a very formal level. Overall, this theory ismore a subject of mathematical physics than of observation-oriented QFT or quantumgravity. Finally, we compare the theories with multi-fractional operators in the class T γ with otherquantum gravities. The choice (2.5) for the spacetime measure draws the theory closerto those quantum gravities, most of them summarized in [79], where the spectral dimen-sion d s of spacetime (governed by the type of kinetic term) varies with the probed scalewhile the Hausdorff dimension d h (governed by the spacetime measure) does not (sections3.3.3, 3.4.3, 4.2.2 and 4.3.2). String field theory, asymptotic safety, causal dynamical trian-gulations (CDT), non-local quantum gravity, most non-commutative spacetimes, Hořava–Lifshitz gravity and a few others fall into this category ( d h = D ); notable exceptions, wherealso d h varies, are cyclic-invariant theories on κ -Minkowski spacetime and the set of mu-tually related discretized theories made of group field theory (GFT), spin foams and loopquantum gravity (LQG).When γ > , the spectral dimension decreases at short scales, as in Stelle gravity [141],string field theory [142, 143], asymptotic safety [31], CDT [144], non-local quantum gravity[111], Hořava–Lifshitz gravity [35] and GFT/spin foams/LQG [7, 145, 146]. Therefore,combining constancy of d h and decrease of d s towards the UV, the theories with mostsimilar dimensional flow with respect to ours are string field theory, asymptotic safety,CDT, non-local quantum gravity and Hořava–Lifshitz gravity, among others. A specialcase is the limit γ → + ∞ for all directions µ . The spectral dimension vanishes in the UV,as in non-local quantum gravity and in the discreteness-effects scale range of GFT/spinfoams/LQG.When γ < , the spectral dimension increases at short scales as in κ -Minkowski space-time with bicross-product Laplacian [147], for which d uvs = 6 . To have eq. (3.39) reproducethis value in D = 4 dimensions, it should be γ = 2 / .In the limit γ → + , one would approximate the dimensional flow of κ -Minkowskispacetime with relative-locality Laplacian [147] as well as Padmanabhan’s model of non-local black holes [148], where d s diverges in the UV.In the same limit γ → + , the action (5.34) of the theory T [ (cid:3) + (cid:3) γ ] reproduces, up toa Ricci tensor-Ricci tensor term, the action of a phenomenological non-local model of IRmodifications of gravity, dubbed RR model [149]: L = R − m R (cid:3) R . (6.4)This model, which is not of quantum gravity, was proposed to explain dark energy andwith a possibly strong cosmological imprint in the propagation of gravitational waves, but– 47 –t is ruled out because it violates the bounds on the time variation of the effective Newton’sconstant [150].
In this paper, we studied the classical and quantum properties of scalar and gravitationaltheories with fractional kinetic terms, respecting or violating Lorentz invariance.The theories with fractional derivatives, labelled T [ ∂ γ ] , T [ ∂ + ∂ γ ] and T [ ∂ γ ( ℓ ) ] do nothave Lorentz symmetry and they are technically difficult due to the presence of fractionalderivatives. Here we have defined basic aspects of their classical and quantum dynamics.The theories labelled T [ (cid:3) γ ] , T [ (cid:3) + (cid:3) γ ] and T [ (cid:3) γ ( ℓ ) ] are Lorentz invariant and arethose that we worked out more extensively. We showed that, in general, it is difficult tochoose a value of γ accommodating both unitarity and renormalizability. This is actuallyimpossible when the kinetic operator is the sum of integer and fractional d’Alembertians asin T [ (cid:3) + (cid:3) γ ] , although the quantum theory displays an interesting IR finiteness and large-scales modifications with, hopefully, cosmological applications. When the kinetic operatoris a single fractional d’Alembertian, the theory T [ (cid:3) γ ] can be made unitary and one-loopfinite, with evidence that it may be finite at all loops.The agenda for the future is filled with many items. On the theoretical side, fractionalgravitational theories need a full exploration ranging from classical (e.g., cosmological)solutions to perturbative renormalizability, in the cases where this is possible. Early- andlate-time cosmology would deserve to be studied, with emphasis on dark energy. We haveseen with eq. (6.4) that, in the limit γ → + , the theory T [ (cid:3) + (cid:3) γ ] resembles somerealizations of IR non-local gravity that can generate late-time acceleration [150]. Weare unaware whether the extra Ricci tensor-Ricci tensor term in (5.34) or the fact that γ is not exactly zero can evade the Lunar Laser Ranging bound violated by the RR model ofnon-local IR gravity. Nevertheless, multi-scale dynamics with dimensional flow is known tosustain cosmological acceleration in all the other multi-fractional theories, either at early orat late times, or both [5, 151]. Therefore, we expect the theories with fractional operatorsto have similar phenomenology.Also the propagation of signals is an open problem. The Huygens principle statesthat the Green’s function has support on the light cone, i.e., signals propagate with thespeed of light. Results with non-local operators with branch cuts indicate that the Huygensprinciple is violated for certain powers of the d’Alembertian and that propagation happensalso inside the light cone [48, 57]. Whether this occurs in one of the special intervals selectedfor unitarity and renormalizability remains to be seen.There is still much work to do to understand the quantum properties of the class oftheories T γ . The failure of power-counting renormalizability as a tools to estimate diver-gences and the question about perturbative renormalizability are non-trivial and expose theadded difficulties of fractional operators with respect to analytic non-local quantum gravity[152, 153]. In higher-order derivative theories, (integer γ ), we know that divergences onlyscale as powers of the cut-off energy Λ UV and divergence counting can be done analytically.If γ is rational, divergences become rational powers of Λ UV and one deals with Puiseux– 48 –eries (series of powers with negative and fractional exponents), in which case one couldstill show convergence analytically. In fact, sub-leading divergences can be expressed by afinite multiplicity of Λ UV powers. We already calculated one-loop diagrams for the scalarQFT T [ (cid:3) γ ] with analytic methods, even for irrational γ . From two loops on, one mighthave to give up analytic methods and recur to numerical methods.It would be interesting to explore more in detail the existence of a duality between thefractional theory T [ (cid:3) γ ] (single fractional d’Alembertian) and the multi-fractional theory T (normal derivatives but multi-scale measure), as mentioned in section 6.1. Speakingabout these two scenarios, they have the potential of offering a robust embedding for thepromising galaxy-rotation-curve models, alternative to dark matter, proposed by Giusti[122, 123] (anomalous Poisson equation with fractional Laplacian, possibly related to thetheory T [ (cid:3) γ ] or, more realistically, T [ (cid:3) + (cid:3) γ ] ) and Varieschi [124–126] (anomalous Poissonequation with ordinary derivatives, possibly related to the theory T ). If a duality betweenthese theories actually existed, it could also explain why the modified Newtonian potentialin the models by Giusti and Varieschi is so similar in some range in the effective dimensionof matter distribution [124]. Acknowledgments
The author thanks F. Briscese, L. Modesto, L. Rachwał and especially G. Nardelli for usefulcomments.
A Properties of fractional derivatives
The Liouville derivative (3.1), ∞ ∂ γ f ( x ) := 1Γ( m − γ ) Z x −∞ dx ′ ( x − x ′ ) γ +1 − m ∂ mx ′ f ( x ′ ) , m − γ < m , (A.1)obeys the following properties.1. Limit to ordinary calculus: lim γ → n ∞ ∂ γ = ∂ n , γ = n ∈ N . (A.2)2. Linearity: ∞ ∂ γ [ c f ( x ) + c g ( x )] = c ( ∞ ∂ γ f )( x ) + c ( ∞ ∂ γ g )( x ) . (A.3)3. Commutation: ∞ ∂ γ ∞ ∂ β = ∞ ∂ β ∞ ∂ γ = ∞ ∂ γ + β , ∀ γ, β > . (A.4)4. Kernel: ∞ ∂ γ x β = ( − γ Γ( β + 1)Γ( β + 1 − γ ) sin( πβ )sin[ π ( β − γ )] x β − γ = ( − γ Γ( γ − β )Γ( − β ) x β − γ . (A.5)– 49 –quation (A.5) vanishes for β = 0 , , , . . . , m − and is ill-defined for β = γ . Inparticular, the Liouville fractional derivative of a constant is zero and the kernel of aLiouville derivative with < γ < is trivial: ∞ ∂ γ , < γ < . (A.6)5. Eigenfunctions: ∞ ∂ γ e λx = λ γ e λx . (A.7)6. Leibniz rule: ∞ ∂ γ ( f g ) = + ∞ X j =0 Γ(1 + γ )Γ( γ − j + 1)Γ( j + 1) ( ∂ j f )( ∞ ∂ γ − j g ) . (A.8)7. Integration by parts: Z + ∞−∞ dx f ∞ ∂ γ g = Z + ∞−∞ dx ( ∞ ¯ ∂ γ f ) g . (A.9)With minor changes, the above formulæ hold also for the Weyl derivative (3.2), ∞ ¯ ∂ γ f ( x ) := 1Γ( m − γ ) Z + ∞ x dx ′ ( x ′ − x ) γ +1 − m ∂ mx ′ f ( x ′ ) , m − γ < m . (A.10)1. Limit to ordinary calculus: lim γ → n ∞ ¯ ∂ γ = ( − n ∂ n , γ = n ∈ N . (A.11)2. Linearity: ∞ ¯ ∂ γ [ c f ( x ) + c g ( x )] = c ( ∞ ¯ ∂ γ f )( x ) + c ( ∞ ¯ ∂ γ g )( x ) . (A.12)3. Commutation: ∞ ¯ ∂ γ ∞ ¯ ∂ β = ∞ ¯ ∂ β ∞ ¯ ∂ γ = ∞ ¯ ∂ γ + β , ∀ γ, β > . (A.13)4. Kernel: ∞ ¯ ∂ γ x β = Γ( β + 1)Γ( β + 1 − γ ) sin( πβ )sin[ π ( β − γ )] x β − γ = Γ( γ − β )Γ( − β ) x β − γ . (A.14)Equation (A.14) vanishes for β = 0 , , , . . . , m − and is ill-defined for β = γ . Inparticular, the Weyl fractional derivative of a constant is zero and the kernel of aWeyl derivative with < γ < is trivial: ∞ ¯ ∂ γ . (A.15)5. Eigenfunctions: ∞ ¯ ∂ γ e λx = ( − λ ) γ e λx . (A.16)– 50 –. Leibniz rule: ∞ ¯ ∂ γ ( f g ) = + ∞ X j =0 Γ(1 + γ )Γ( γ − j + 1)Γ( j + 1) ( ∂ j f )( ∞ ¯ ∂ γ − j g ) . (A.17)7. Integration by parts: eq. (A.9).The reader can find the proofs of all these statements in [42, 64–67]. B Alternative proof of unitarity
In this appendix, we check unitarity of the real scalar field theory T [ (cid:3) γ ] of section 4.1 usinga different route than the spectral decomposition of section 4.1.5. Namely, we will showthat the scalar product of field functionals in the Euclidean version of the theory is positivedefinite only for the set of ranges (4.44). Our calculation is similar to that in [46] for thetheory (6.2) with kinetic operator (4.64).We work in Euclidean position space with coordinates x , x , . . . , x D and define a re-flection operation R with respect to the x D = 0 plane: R x µ = ( − δ µD x µ , i.e., R x = x for µ = 1 , . . . , D − and R x D = − x D . Let F [ φ ] = Z d D x ϕ ( x ) φ ( x ) be a linear functional of the field φ , where ϕ is a test function with support in x D > .We denote the complex conjugate and reflected functional as F ∗ R [ φ ] = R d D x ϕ ∗ (R x ) φ ( x ) .Reflection positivity states that the expectation value of F R [ φ ] F [ φ ] defined through thepath integral is positive semi-definite: hF R [ φ ] F [ φ ] i := 1 R D φ e − S [ φ ] Z D φ e − S [ φ ] F ∗ R [ φ ] F [ φ ] > , where D φ is the functional measure. This expectation value can be calculated as in standardQFT by introducing a current J φ in the action and taking the second-order functionalderivative δ/δ J . The result is hF R [ φ ] F [ φ ] i = Z d D x d D y ϕ ∗ ( x ) G (R x − y ) ϕ ( y ) =: (R ϕ, ϕ ) , (B.1)where G is the Green’s function, which in our case is G ( x ) = Z d D k (2 π ) D e − ik · x ( k + m ) γ , (B.2)where k = P Dµ =1 k µ . The test functions ϕ are by construction square integrable in thisscalar product. For the sake of the calculation, one can take a “charge” distribution ϕ ( x ) = N X i =1 q i δ D [ x − x ( i ) ] , q i ∈ C . – 51 –herefore, calling α k [ x ( i ) ] := q i exp[ i k · x ( i ) ] and r ij := x ( i ) D + y ( j ) D and using definition (4.14), (R ϕ, ϕ ) = X i,j q ∗ i q j Z d D x d D y δ D [ x − x ( i ) ] G (R x − y ) δ D [ y − y ( j ) ]= X i,j q ∗ i q j G [R x ( i ) − y ( j ) ]= X i,j q ∗ i q j Z d D k (2 π ) D e − ik · [R x ( i ) − y ( j ) ] ( k + m ) γ = X i,j Z + ∞−∞ d D − k (2 π ) D − α ∗ k [ x ( i ) ] α k [ y ( j ) ] Z + ∞−∞ dk D π e ik D r ij ( k D + ω ) γ =: X i,j Z d D − k (2 π ) D − α ∗ k [ x ( i ) ] I ij R α k [ y ( j ) ] , (B.3)so that (R ϕ, ϕ ) > ⇐⇒ I ij R > . (B.4)At this point, we analytically continue the Euclidean momentum k D to a complex momen-tum and perform the above integration making use of a contour Γ E in the (Re k D , Im k D ) plane, where E stands for Euclidean. The integrand has branch points on the imaginaryaxis at k D = ± iω and branch cuts ramifying from those points to ± i ∞ , respectively. Since r ij > , to get an exponential suppression exp( − Im k D r ij ) we choose Γ E in the Im k D > half plane: it runs along the real axis from −∞ to + ∞ and makes a quarter arc counter-clockwise up to the branch cut ( iω, i ∞ ) , where the contour follows the cut down to thebranch point and up again in the fourth quadrant, where it closes with another quarter arccounter-clockwise from the imaginary positive semi-axis to the negative real semi-axis (Fig.6). Since the integrand does not have any singularity inside the contour Γ E , integrationover Γ E gives zero. The contributions of the arcs at infinity is also zero thanks to theexponential suppression, so that Z Γ E = Z R + Z C ε + Z cut + Z arcs = Z R + Z C ε + Z cut , implying Z R = − Z C ε − Z cut = − Z C ε − (cid:18)Z iω + εi ∞ + ε + Z i ∞− εiω − ε (cid:19) . Integration on the circle C ε surrounding the branch point iω can be done by a change ofvariable k D − iω = ε exp( iθ ) , where − π/ < θ < π/ : − Z C ε dk D π e ik D r ij ( k D + ω ) γ = − iε Z − π π dθ e iθ π e i ( εe iθ + iω ) r ij [ εe iθ (2 iω + εe iθ )] γ = ε − γ e − ωr ij πi (2 iω ) γ Z − π π dθ e iθ (1 − γ ) + O ( ε − γ )= ε − γ e − ωr ij π (1 − γ )(2 ω ) γ sin( πγ ) + O ( ε − γ ) . (B.5)– 52 – E C ε Re k D - i ω i ω Im k D Figure 6 . Contour Γ E (black thick curve) in the (Re k D , Im k D ) plane. In the limit ε → , this contribution diverges for γ > , is finite when γ = 1 (ordinarycase, no branch cut) and it vanishes for γ < . Barring the standard case, we are forced tochoose γ < .Having established that integration around the branch point gives zero, we are left tocalculate the contribution of the branch cut. Changing variable into k D = iρ = e iπ/ ρ inthe first quadrant and k D = iρ ′ = e − iπ/ ρ ′ in the second quadrant, integration along thecut yields I ij R = − Z cut dk D π e ik D r ij ( k D + ω ) γ = − i Z ω ∞ dρ π e − ρr ij ( e iπ ρ + ω ) γ − i Z ∞ ω dρ ′ π e − ρ ′ r ij ( e − iπ ρ ′ + ω ) γ = − i Z ∞ ω dρ π e − ρr ij e iπγ − e − iπγ ( ρ − ω ) γ = sin( πγ ) π Z ∞ ω dρ e − ρr ij ( ρ − ω ) γ . (B.6)The integrand is positive on a positive range, so that this expression is positive semi-definiteif, and only if, sin( πγ ) > , i.e., when n γ n + 1 with n ∈ Z . Since γ < , theonly ranges guaranteeing unitarity except the standard case are γ < , − γ − , − γ − , and so on, that is, the range (4.44).– 53 –he same result can be obtained by performing the integral in (B.6) explicitly usingformula 8.432.3 of [154], valid if γ < : I ij R = 1 √ π Γ( γ ) (cid:18) ωr ij (cid:19) − γ K − γ ( ωr ij ) , (B.7)where K / − γ is the modified Bessel function of the second kind, which is always positive.The sign of I ij R is thus determined by the one of Γ( γ ) , which is positive when γ > or − n γ − n + 1 with n ∈ N and γ = 1 . This is exactly the same condition obtainedfrom semi-positivity of the spectral function, eq. (4.32). C Equations of motion in fractional gravity
C.1 Derivation of (5.17)
The only non-trivial part of the proof of (5.17) relies on the vanishing of a boundary term.The rest is as in standard general relativity, but we will repeat it here for completeness.Recall that δg στ = − g σµ g τν δg µν , (C.1a) δ ( g αµ g βν ) A αβ B µν = − δ ( g αµ g βν ) A αβ B µν , (C.1b) δ p | g | = − g µν p | g | δg µν . (C.1c)The Levi-Civita connection (5.10) is defined with the multi-fractional derivative D − but itsvariation can be written for any derivative operator, including D γ and D γ ( ℓ ) : δ ¯Γ ρµν = 12 δg ρσ ( D µ g νσ + D ν g µσ − D σ g µν ) + 12 g ρσ ( D µ δg νσ + D ν δg µσ − D σ δg µν ) (C.1a) = − g ρα g σβ δg αβ ( D µ g νσ + D ν g µσ − D σ g µν ) + 12 g ρσ ( D µ δg νσ + D ν δg µσ − D σ δg µν )= − g ρα δg αβ ¯Γ βµν + 12 g ρσ ( D µ δg νσ + D ν δg µσ − D σ δg µν )= 12 g ρσ (cid:16) D µ δg νσ + D ν δg µσ − D σ δg µν − βµν δg σβ (cid:17) = 12 g ρσ (cid:0) D µ δg νσ − ¯Γ ρµσ δg νρ + D ν δg µσ − ¯Γ ρνσ δg µρ −D σ δg µν + ¯Γ ρσµ δg ρν + ¯Γ ρσν δg µρ − βµν δg σβ (cid:17) (5.11) = 12 g ρσ (cid:0) ¯ ∇ µ δg νσ + ¯ ∇ ν δg µσ − ¯ ∇ σ δg µν (cid:1) . (C.2)The same expression can be inferred by noting that δ ¯Γ ρµν is a tensor, so that its formin a local inertial frame, δ ¯Γ ρµν = η ρσ ( D µ δg νσ + D ν δg µσ − D σ δg µν ) / , can be immediatelypromoted to an arbitrary frame where the Minkowski metric is replaced by the genericmetric g ρσ and derivatives are replaced by covariant derivatives.From definition (5.15), one can check that the variation of the fractional Ricci tensoris δ R µν = ¯ ∇ σ δ ¯Γ σµν − ¯ ∇ ν δ ¯Γ ρµρ . (C.3)– 54 –ince p | g | δ ( p | g |R ) (C.1c) = δg µν (cid:18) R µν − g µν R (cid:19) + g µν δ R µν , (C.4)one obtains the fractional Einstein equations (5.17) with δg µν O µν := g µν δ R µν (5.13) = ¯ ∇ σ (cid:0) g µν δ ¯Γ σµν − g µσ δ ¯Γ ρµρ (cid:1) =: ¯ ∇ σ A σ (C.5) (5.12) = D σ A σ + ¯Γ σσν A ν (5.10) = D σ A σ + 12 g ρσ D ν g ρσ A ν , (C.6)where in the last line we used ¯Γ σσν = g ρσ D ν g ρσ / .In ordinary calculus, this quantity would be identically zero in the absence of bound-ary or on any boundary where δg µν = 0 , or it could be cancelled by the York–Gibbons–Hawking boundary terms. The argument would be the following. For any matrix M , thelogarithm-trace formula ln(det M ) = tr(ln M ) holds. Taking the first derivative on bothsides, (det M ) − ∂ det M = tr( M − ∂M ) . For the metric M = g µν , in the sense of variationsthis reads δg = g g µν δg µν = − g g µν δg µν , which yields (C.1c), while in the sense of spacetimederivatives it implies p | g | ∂ σ p | g | = 12 ∂ σ ln g = 12 g µν ∂ σ g µν . (C.7)Therefore, in the ordinary case the integrand in eq. (C.6) would be a total derivative, ∇ σ A σ = ∂ σ A σ + 1 p | g | ∂ σ p | g |A ν = 1 p | g | ∂ σ (cid:16)p | g |A ν (cid:17) , (C.8)which vanishes at infinity: Z d D x p | g | δg µν O D = ∂µν = Z d D x p | g | ∇ σ A σ = Z d D x ∂ σ (cid:16)p | g |A ν (cid:17) = 0 , (C.9)so that O D = ∂µν = 0 in the Einstein equations. However, there is no simple fractional analogueof eqs. (C.7) and (C.8) because the fractional derivative of a composite function f [ g ( x )] stems from the highly non-trivial Leibniz rule (3.11) [65]. It may still be possible that,once the rule for the D derivative of a composite function was found, and taking a non-trivial fractional integration measure with scaling α = γ , one could obtain a simple or evenvanishing expression for the operator O µν . We will not solve this mathematical problemhere. C.2 Derivation of (5.21) and (5.33)
To derive the equations of motion of the theory (5.20) with F = 0 , we make use of basicvariation formulæ (C.1a), (C.1b), (C.1c) and δR µν = ∇ α ∇ ( µ δg ν ) α − h (cid:3) δg µν + g αβ ∇ ( µ ∇ ν ) δg αβ i , (C.10a) δR = δg µν R µν + g µν δR µν (C.10a) = (C.1a) ( R µν + g µν (cid:3) − ∇ µ ∇ ν ) δg µν , (C.10b)– 55 –here A ( µ B ν ) := ( A µ B ν + A ν B µ ) / . The variation of the action (5.20) with F = 0 = F can be found step by step in [88, appendix E]: κ p | g | δ ( p | g | L ) δg µν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F =0 F =0 = G µν + Λ g µν − g µν G στ F R στ + 2 G σµ F G νσ + F (cid:3) G µν + g µν ∇ σ ∇ τ F G στ − ∇ σ ∇ µ F G νσ + 12 ( G µν F R + R F G µν ) + Θ µν ( R αβ , G αβ ) + O ( ∇ ) , (C.11)where O ( ∇ ) are total derivative terms and we used definition (5.22). Here we only need toadd to this equation the variation of the F term: p | g | δ (cid:16)p | g | R F R (cid:17) (C.1c) = − δg µν g µν R F R + 2 δR F R + Rδ F R + O ( ∇ ) (5.22) = (C.10b) − δg µν g µν R F R + 2 δg µν ( R µν + g µν (cid:3) − ∇ µ ∇ ν ) F R + δg µν ϑ µν ( R, R ) + O ( ∇ ) , (C.12)where the variation symbol δ always and only applies to the first operator on its right.Combining eqs. (C.11) and (C.12), we obtain (5.21).We now specialize to the form factor (5.31) in the integral representation (5.32): F ( (cid:3) ) = ℓ γ − ∗ ( − (cid:3) ) γ − = ℓ γ − ∗ Γ(2 − γ ) Z + ∞ dτ τ − γ e τ (cid:3) . (C.13)Using Duhamel identity for the exponential of an operator O , δe τ O = Z τ dq e q O ( δ O ) e ( τ − q ) O , (C.14)for two generic symmetric rank-2 tensors A αβ and B αβ we have A αβ δ F B αβ = ℓ γ − ∗ Γ(2 − γ ) Z + ∞ dτ τ − γ A αβ δe τ (cid:3) B αβ = ℓ γ − ∗ Γ(2 − γ ) Z + ∞ dτ τ − γ Z τ dq A αβ e q (cid:3) ( δ (cid:3) ) e ( τ − q ) (cid:3) B αβ = ℓ γ − ∗ Γ(2 − γ ) Z + ∞ dτ τ − γ Z τ dq e q (cid:3) A αβ ( δ (cid:3) ) e ( τ − q ) (cid:3) B αβ + O ( ∇ ) . Recalling that [88] A αβ ( δ (cid:3) ) B αβ = δg µν ¯Θ µν ( A αβ , B αβ ) + O ( ∇ ) , (C.15a) ¯Θ µν ( A αβ , B αβ ) := −∇ µ A αβ ∇ ν B αβ + 14 g µν ∇ ρ ( A αβ ∇ ρ B αβ + B αβ ∇ ρ A αβ )+ 14 g µν ∇ ρ ( A αβ ∇ ρ B αβ − B αβ ∇ ρ A αβ ) + ∇ α ( A µβ ∇ α B βν − B βν ∇ α A µβ )+ ∇ β ( B µα ∇ ν A αβ − A αβ ∇ ν B µα ) + ∇ α ( A µβ ∇ ν B βα − B βα ∇ ν A µβ ) , (C.15b) We correct a minor typo (a missing δg µν in the penultimate term) of eq. (E.4) of [88] and also in thelast line of (D.13) of the same paper (the αβ indices in Θ µν should be bottom and up instead of top anddown. – 56 –e get eq. (5.33): R αβ δ F G αβ = ℓ γ − ∗ Γ(2 − γ ) δg µν Z + ∞ dτ τ − γ Z τ dq ¯Θ µν [ e q (cid:3) R αβ , e ( τ − q ) (cid:3) G αβ ] + O ( ∇ )= δg µν Θ µν ( R αβ , G αβ ) + O ( ∇ ) . (C.16) References [1] G. ’t Hooft,
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