SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES R. TRINCHERO Abstract. Deformationsofthecanonicalspectraltriplesoverthen dimensional − 2 torus are considered. These deformations have a discrete dimension spectrum 1 consistingofnon-integervalueslessthann. The differentialalgebracorrespond- 0 ing to these spectral triples is studied. No junk forms appear for non-vanishing 2 deformation parameter. The action of a scalar field in these spaces is consid- n ered, leading to non-trivial extra structure comparedto the integer dimensional a cases, which does not involve a loss of covariance. One-loop contributions are J computed leading to finite results for non-vanishing deformation. 0 2 ] h p 1. Introduction - h The dimension ofa space isabasic concept ofparticular relevance bothinnature t a and in mathematics. Non-commutative geometry[1][2, 3, 4] provides a generaliza- m tion of classical geometry. In particular, it includes a definition of dimension that [ allows for complex non-integer values[5]. A motivation for this definition and a 1 series of very interesting examples of geometries with non-integer dimensions has v 5 been given in relation to the study of fractal sets in this geometrical setting([1],[6] 6 and references therein). 3 The motivation for this work comes from a different subject. In the realm 4 . of quantum field theory(QFT), the widely employed dimensional regularization 1 0 technique[7] provides a hint that non-integer dimensional spaces could be of rele- 2 vance there. This technique is employed in QFT as a means to regularize divergent 1 integrals appearing in perturbation theory, being preferred in the regularization of : v gauge theories since it preserves gauge invariance. The technique essentially con- i X sists in considering the analytical continuation in the number of dimensions for the r surface of a d-dimensional sphere, a quantity that appears in the calculation of a the above-mentioned integrals. The general question to be addressed in this work is whether a suitable well-defined differential geometry can be found that makes sense for non-integer dimensions and reduces to the canonical one for the integer case1. Intheaffirmative casethenaturalquestiontoaskis, whatdoesafieldtheory defined in such a space look like?. More precisely, the idea is to take a field theory defined purely in geometrical terms and repeat the construction in the deformed Date: 20 December 2011. Key words and phrases. Dimensional regularization, non-commutative geometry, non-integer dimensions. R.T. is supported by CONICET.. 1 A preliminary study of this question in the 1-dimensional case appears in [8] 1 SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 2 case. The output of that procedure is by no means obvious since, as will be seen in subsequent sections, the differential algebra is qualitatively different between the integer and non-integer case, and such a change reflects directly in the action of the field theory. For the case of the field theory of a scalar field considered in section 6, the resulting theory is of a novel type. This theory, in spite of reflecting its non-commutative origin, does not involve a breakdown of covariance, as happens in the so-called non-commutative field theories[9]. The salient features and results of this work are summarized as follows, Spectral triples are considered that differ from the canonical ones only in • the choice of the Dirac operator. The dimension spectrum of these triples consists of a discrete set of real • values less than the dimension of the canonical triple. The differential of a zero form is not a multiplicative operator. • There are no junk forms for a non-zero deformation parameter. • The action of a scalar field contains derivatives of any order and involves • an integration over the co-sphere. In spite of the "non-commutativeness" of the differential algebra, there is • no loss of covariance involved in the field theory mentioned above. The calculation of the tadpole diagram and a loop involving two free prop- • agators, show that for non-zero deformation these diagrams give a finite result, showing neither ultraviolet nor infrared singularities. The last sin- gularitiesbeingruledoutbytheappearanceofamasstermwhosecoefficient vanishes when the deformation parameter goes to zero. Ultraviolet power counting and comparison with dimensional regulariza- • tion, indicate that the perturbation theory obtained from the action men- tioned above leads to finite contributions for a non-zero deformation pa- rameter. This paper is organized as follows. Section 2 describes the spectral triple to be considered. In section 3 the corresponding dimension spectrum is computed. The differential of a 0-form is considered in section 4. Section 5 considers the calcu- lation of the action for a complex scalar field. Section 6 presents the one-loop computations and Section 7 contains conclusions and the schematic description of further research motivated by the present work. In addition, two appendices are included, Appendix A showing the absence of junk forms, and Appendix B, which contains the calculation of the Wodzicki residue involved in the definition of the above-mentioned action. 2. The Dirac operator The differential algebra derived from the canonical spectral triple involving func- tions over a manifold M reduces to the usual exterior differential algebra over M. The spectral triples to be considered in this work differ from the canonical ones only in the choice of the Dirac operator. More precisely, the triples ( , ,D ) are α A H considered, where, SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 3 isthecommutativeC∗-algebraofsmoothfunctionsoverthen-dimensional • tAorus Tn n N. ∈ is the Hilbert space of square integrable sections of a spinor bundle over • H Tn. D : is a self-adjoint linear operator to be defined below. α • H → H The usual Dirac operator over a n dimensional torus Tn is given by, D = iγ ∂ = iγ ∂ ,γ = γ† , γ γ +γ γ = 2δ ,µ,ν = 1, ,n · µ µ µ µ µ ν ν µ µν ··· this operator is not positive definite. Indeed since, D2 = ∆ = ∂ ∂ µ µ − − denoting by λ 0 an eigenvalue of D2, then √λ will be eigenvalues of D. ≥ ± In this work the usual Dirac operator will be replaced by D given below. One of α the motivations for this choice is to obtain a dimension spectrum with non-integer real values. This could be done in many ways, for example, choosing, Da = D D2 −(1−2a) ,a R, 1 > a > 0 | | ∈ 2 this operator leads to a dimension spectrum which consists in a single value given by z = n. However, it is not well-behaved in the infrared. In order to improve a its infrared properties and have the same behavior in the ultraviolet, the following operator will be considered in this work, D = D(1+D2)−α, α > 0 α the power appearing in this last equation being defined by, 1 ∞ (2.1) (1+D2)−α = dτ τα−1e−τ(1+D2) Γ(α) Z 0 Thus the Dirac operator to be considered is, 1 ∞ D = dτ τα−1D(τ) ,D(τ) = e−τ(1+D2)D α Γ(α) Z 0 this operator is self-adjoint in , with compact resolvent, and such that the dif- H ferential of any a is bounded. This last condition is ensured by the choice ∈ A α 0, as can be readily shown using the expression for the differential of section ≥ 4. Therefore, the triple fulfills all the properties required for it to be a spectral triple. 3. Dimension spectrum The definition of dimension spectrum of a spectral triple is briefly reviewed. 2 See the next section for the definition of dimension spectrum[5]. SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 4 Definition 1. [Connes-Moscovici] Discrete dimension spectrum. A spectral triple ( , ,D) has discrete dimension spectrum Sd if Sd C is discrete and for any A H 3 ⊂ element b in the algebra the function, B (3.3) ζD(z) = Tr[π(b) D −z] b | | extends holomorphically to C/Sd. The interpretation of these poles is that each of them gives the dimension of a certain piece of the whole space. In order to apply this definition to the spectral triples considered in this work, it is useful to note that, ∞ αz D −z = D −z(1+ D 2)αz = D −z D 2(αz−k) | α| | | | | | | (cid:18) k (cid:19)| | X k=0 ∞ αz (3.4) = D 2((α−12)z−k) (cid:18) k (cid:19)| | X k=0 where Newton’s binomial formula has been employed. From the definitions above it is clear that, ∞ αz 1 (3.5) ζDα(z) = ζD(2(k (α )z)) b (cid:18) k (cid:19) b − − 2 X k=0 where the binomial coefficients are given by, αz αz(αz 1) (αz k +1) αz = − ··· − , = 1 (cid:18) k (cid:19) k! (cid:18) 0 (cid:19) The zeta functions appearing in the r.h.s. of (3.5) are the ones corresponding to the canonical spectral triple. Thus, since for the canonical spectral triples the corresponding zeta functions have a single simple pole at its argument equal to n, then ζDα(z) has simple poles at, b n 2k z = − ,k = 0,1,2, 1 2α ··· − these values of z are therefore the dimension spectrum of the spectral triple con- sidered in this work. 3 Thedefinitionofthealgebra isthefollowing. Letδ denotethederivationδ :L( ) L( ) B H → H defined by, (3.1) δ(T)=[D ,T] ,T L( ) | | ∈ H The algebra is generated by the elements, B (3.2) δn(π(a)), a , n 0(δ0(π(a))=π(a)) ⊂A ≥ SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 5 4. The differential The differential of a 0-form f is given by, 1 ∞ (4.1) df = [D ,f] = dτ τα−1df(τ) α Γ(α) Z 0 (4.2) df(τ) = [D(τ),f] .D(τ) = U(τ)D ,U(τ) = e−τ(1+D2) thus when applied to an element φ of , df(τ) is given by, H df(τ)φ = [D(τ),f]φ = U(τ)[(Df)φ+fDφ] f U(τ)Dφ − = [U(τ)(Df)+[U(τ)f f U(τ)]D]φ − (4.3) = U(τ)[(Df)+[f U( τ)f U(τ)]D]φ − − the second term in the parenthesis of the r.h.s. can be expressed as, (4.4) eτ(1+D2)f(x)e−τ(1+D2) = f(x 2τ∂) − this can be easily derived using an analogy with quantum mechanics. This is done noting that e−τ(1+D2) is, up to a constant, the imaginary time evolution operator for a free particle of mass m = 1/2. Thus, df(τ) = U(τ)[(Df) [f(x) f(x 2τ∂)]D] − − − integrating the second line in (4.3) as in (4.1) leads to, df = (1+D2)−α(Df)+[(1+D2)−αf f(1+D2)−α]iγ ∂ − · which clearly shows that when α 0, df iγ ∂f, which is the corresponding → → · expression in the canonical case. It is worth remarking that, as the last equations indicate, this differential is a non-multiplicative operator for any value of α = 0. 6 As Appendix A shows, this fact plays an important role in showing the absence of junk forms. 5. The scalar field In this section the part of this space corresponding to the highest pole will be considered, i.e. for d = n . The action for a free scalar field propagating in this 1−2α space is taken to be, 1 S = < dφ,dφ > 2 4 where φ is a 0-form and the norm in the space forms is given by , (5.1) < ω,ω > = tr [ωω† D −d] ω α | | thus, S = tr [dφdφ∗ D −d] ω α − | | 4 See for example ref.[3] SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 6 where it was used that dφ† = dφ∗ and tr denotes the Diximier trace. In the ω − evaluation of this trace it is important to note that replacing d = n in (3.4) 1−2α leads to, ∞ αn (5.2) |Dα|−d = |Dα|−1−n2α = (cid:18) 1−k2α (cid:19)|D|−n−2k X k=0 Therefore S is given by, ∞ αn S = 1−2α S (cid:18) k (cid:19) k X k=0 S = tr [dφ(τ)dφ(τ′)∗ D −n−2k] k ω − | | Noting that, dφ = [U D,φ(x)], U = (1+D2)−α α α dφ∗ = [U D,φ∗(x)] α leads to, S = tr [U D,φ][DU ,φ∗] D −n−2k k ω α α | | Thus replacing the expression o(cid:8)btained in Appendix B for(cid:9)S leads to, k n (5.3) S = 2[2]VSn−1 φ(D2 + αn )(1+D2)−2αφ∗ − n(2π)n ZTn 1−2α where Vsn−1 = 2πn/2/Γ(n/2) is the area of the n 1 dimensional sphere. It is worth − noting that in spite of starting with an action involving no mass term, the fact of working on a non-integer dimensional space generates effectively such a term as shown by (5.3), with a coefficient that vanishes in the integer case(α = 0). In that case (5.3) reduces to the usual action of a mass less complex scalar field, i.e., n S = limS = 2[2]Vsn−1 1∂ φ(x)∂ φ∗(x) can α→0 (2π)n ZTn (cid:18)2 µ µ (cid:19) 6. One loop calculations As mentioned in the introduction, the dimensional regularization technique is a widely employed tool used to make sense of divergences in perturbative quan- tum field theory. These divergences appear when calculating the contribution of Feynman diagrams involving closed loops. Having obtained a field theory in a non-integer dimensional space, it is natural to perform the same calculations and see whether the analogous diagrams are divergent or not. This is the purpose of this section. Two simple diagrams are considered: the tadpole diagram and a loop involving two free propagators. In both cases the calculations below show that the corresponding diagrams give a finite result for α = 0. The comparison of the 6 results for these diagrams with dimensionally regularized ones, show that the loca- tion of poles in the complex α plane coincides with the one obtained in dimensional SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 7 regularization. A simple argument showing that this should be so is obtained by considering the ultraviolet behavior of the integrals involved, as shown in subsec- tion 6.1. In spite of these similarities, the dimensionally regularized result and the ones in this non-integer dimensional space are different. Another important point is that the appearance of the mass term in (5.3) automatically regulates possible infrared divergences. In order to compare with results of standard dimensional regularization calcula- tions, it is convenient to restore physical units in our calculations. Unlike common use in physics, where coordinates are assumed to have dimensions of length, up to this point in this work coordinates and fields have been taken to be dimensionless. Physical units, in the natural system of units where action and speed are measured in units of the Planck constant ~ and the velocity of light c, are restored by, x n−2 xP = , φP = M 2 φ M where x and φ denote the dimensionfull quantities and M is a mass scale. This P P can be derived recalling the basic requirement that the action should be dimen- sionless in natural units. To show how this works it is noted for example that, ∂ 1 ∂ 1 = (1+D2) = (M2 +D2) ∂x M ∂x ⇒ M2 P P In the subsections below it should be understood that the quantities involved are dimensionfull, although the subindices P will not be explicitly written. 6.1. The tadpole. According to(5.3)thepropagatorcorresponding tothataction is5, in terms of dimensionfull quantities6, 1 D(x y) = dnp e−ip·(x−y) − Z (p2 +m2)(M2 +p2)−α where, αn m2 = M2 1 2α − The tadpole diagram is, p 5 In the following expressions the discrete summation over the allowable momenta is replaced by an integral, this is justified in the limit where all the radius of the n-dimensional torus tend to infinity. 6In this work p2 >0 indicates the euclidean positive norm squared of the vector p, this is dif- ferentfromtheusualnotationinfieldtheorywhereafterWickrotatingtheeuclideanmomentum is considered with p2 <0. E SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 8 it corresponds to the following integral, 1 (6.1) IT(m) = dnp α Z (p2 +m2)(M2 +p2)−α It will be shown below that this integral converges for α = 0 and α < 1. A simple 6 | | argument showing that this should be so can be given comparing the ultraviolet behavior of IT(m) and of the corresponding dimensionally regularized integral α IT(m) given by, d 1 IT(m) = ddp d Z (p2 +m2) the behavior of the integrand in the ultraviolet (p ) is given by pd−1−2, as is → ∞ well known this dimensionally regularized integralconverges for anyd = 2,4,6, . 6 ··· Next the ultraviolet behavior of IT(m) is considered it goes like pn−1−2+2α which α coincides with dimensionally regularized case if d = n+2α. The important state- ment being that considering α = 0 in IT(m) is equivalent, from the point of view 6 α of the ultraviolet behavior of the integrand, to considering IT(m) for non-integer d d. Of course, the correspondence between ultraviolet behavior of the integrands does not mean equality of the corresponding integrals, as is shown by the following calculation. In order to evaluate IT(m), the integral representation of a power in (2.1) is re- α called, ∞ 1 (6.2) A−α = dτ τα−1e−τA Γ(α) Z 0 this last formula is valid only for negative α, which is not the case of interest here. In what follows, only the case α < 0 is considered. It will be shown that the final result can be analytically continued to the case α > 0. Applying the last formula to (6.1) leads to, ∞ ∞ 1 IT(m) = dnp dae−a(p2+m2) dbb−α−1e−b(M2+p2) α Z Z Γ( α) Z − 0 0 next, the following change of variables is employed, a z = a+b ,x = a = zx ,b = z(1 x) z ⇒ − the Jacobian and limits of integration in these new variables implying that, ∞ ∞ 1 dadb = dzz dx Z −Z Z 0 0 0 SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 9 which leads to, ∞ 1 1 IT(m) = − dnp dzz dx[z(1 x)]−α−1e−z[p2+(1−x)M2+xm2] α Γ( α) Z Z Z − − 0 0 in order to make the p integration the following change of variables is performed, p p˜= √zp dnp = dnp˜z−n2 → ⇒ thus, n 1 ∞ IT(m) = −π2 dx(1 x)−α−1 dzz−α−n2 e−z[(1−x)M2+xm2] α Γ( α) Z − Z − 0 0 where use was made of, dnp˜e−p˜2 = πn2 Z next employing (2.1), 1 n IT(m) = −π2 Γ(1 n α) dx(1 x)−α−1[(1 x)M2 +xm2]α+n2−1 α Γ( α) − 2 − Z − − − 0 thislastintegralcanbewrittenintermsofthehypergeometricfunction F (a,b,c,z), 2 1 i.e., IαT(m) = −πn2ΓΓ((1−α)n2α−α)(M2)α+n2−12F1(1,1−α− n2,1−α,1− Mm22) − For illustrative purposes, let us replace m2 = M2 αn by, 1−2α m˜2 = m2 +m2 0 where m2 is a constant additional mass. Taking the limit α 0 of IT(m˜) gives, 0 → α lim IT(m˜) = πn2Γ(1 n)(m2)n2−1 α→0 α − 2 0 which, upon replacing n by a complex number, is the dimensionally regularized result of this diagram for a scalar field of mass m . It is important to note that 0 the analytical properties of both results are the same, that is so because, Γ( α)α is an analytic function of α for α < 1. • − | | The hypergeometric function F (1,1 α n,1 α,1 m2) is an analytic • 2 1 − − 2 − − M2 function of α whenever its third argument is not equal to 0, 1, 2, , i.e. − − ··· for α not a positive integer. SCALAR FIELD ON NON-INTEGER DIMENSIONAL SPACES 10 6.2. A loop involving two free propagators. The contribution of the closed loop in the following Feynman diagram, p p+k is given by, 1 (6.3) IL(k,m) = dnp α Z (p2 +m2)(M2 +p2)−α((p+k)2 +m2)(M2 +(p+k)2)−α using eq. (6.2) the integral IL(k,m) can be written as follows, α ∞ 1 IL(k,m) = dnp dadbda˜d˜ba˜−α−1˜b−α−1 α Z Γ( α)2 Z × − 0 eE(p,k,a,b,a˜,˜b) where, E(p,k,a,b,a˜,˜b) = a(p2 +m2)+b((p+k)2 +m2)+a˜(M2 +p2)+˜b(M2 +(p+k)2) − h i = (a+b+a˜+˜b)p2 +(b+˜b)(k2 +2p k)+(a˜+˜b)M2 +(a+b)m2 − · h i = z (p+(1 x)k)2 + (1 x) (1 x)2 k2 +yM2 +(1 y)m2 − − − − − − (cid:2) (cid:2) (cid:3) (cid:3) where in the last equality, the following change of variables has been used, a+a˜ ˜ z = (a+b+a˜+b) , x = z ˜ ˜ a˜+b b y = , w = z z the Jacobian and limits of integration in these new variables implying that, ∞ ∞ 1 x y dadbda˜d˜b = dz dx dy ( z3) Z Z Z Z Z − 0 0 0 0 0 in terms of the variable p′ = √z(p+(1 x)k) the quantity IL(k,m) is written as − α follows, ∞ IL(k,m) = dnp′ e−p′2 dz z3−n2−2−2α α − Z Z × 0 1 x y dx dy dw[w(y w)]−(1+α)e−z[x(1−x)k2+yM2+(1−y)m2] Z Z Z − 0 0 0