Effective action in elliptic and hyperbolic spacetimes

FTUAM-21- IFT-UAM/CSIC-21-16 arXiv:2103.04897v2 [hep-th] 10 Mar 2021 Effective action in elliptic and hyperbolic spacetimes. Enrique Álvarez and Jesús Anero. Departamento de Fı́sica Teórica and Instituto de Fı́sica Teórica, IFT-UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain E-mail: [email protected], [email protected] Contents 1 Introduction 3 2 Hyperbolic Heat kernel 2.1 Dimensional reduction from odd to even dimensions. 4 7 3 Four dimensional hyperbolic space H4 . 8 4 Five dimensional hyperbolic space H5 8 5 Elliptic spacetimes. 9 6 Elliptic five dimensional space, E5 . 10 7 Conclusions 11 8 Acknowledgements 11 1 Introduction There are not many examples of exact effective actions, even to one loop order, and even for scalar fields. The usual approach (cf. [1] and references therein) only determines the ultraviolet divergent small proper-time DeWitt coefficients. This leaves undetermined the infrared behavior. There is however a theorem [3][4] asserting that whenever the spacetime manifold is such that its Ricci curvature is non-negative, Rµν ≥ 0, and the manifold has got maximal volume growth, then the heat kernel corresponding to the ordinary laplacian obeys √
Ω(n) (1.1) τ K (x, x′ ; τ ) = lim V τ →∞ (4π)n/2 √ where Ω(n) is the volume of the unit ball in Rn and V ( τ ) is the volume of the geodesic √ ball centered at x′ and radius τ . The asymptotic volume ratio is defined as V (r) =Θ>0 r→∞ r n lim (1.2) The fact that Θ > 0 is what qualifies for the assertion that the manifold has maximal volume growth. In fact this is a generalization of a previous theorem by Li and Yau [5] asserting that with the same hypothesis there should exist a constant C(ǫ) such that σ(x,x′ ) σ(x,x′ ) C(ǫ) − (2+ǫ/2)τ 1 √ e− (2−ǫ/2)τ ≤ K(x, x′ ; τ ) ≤ √ e C(ǫ)V ( τ ) V ( τ) (1.3) The situation improves in spacetimes with special amounts of symmetry, where we can find exact expressions for the heat kernel corresponding to the ordinary laplacian1 . In this work we shall precisely be concerned with maximally symmetric spacetimes and their euclidean counterparts. The Riemann tensor obeys Rµνρσ = R (gµρ gνσ − gµσ gνρ ) n(n − 1) (1.4) and the curvature can be positive or negative R=± n(n − 1) L2 (1.5) Elliptic spacetimes have got positive curvature. In our conventions, anti-de Sitter spacetime (AdSn ) is one such. In Poincaré coordinates Pn−1 ηij dy i dy j − L2 dz 2 2 (1.6) dsAdSn = z2 1 Related computations have been done for the Dirac operator by Camporesi [6]. See also [2] 3 (where as usual, ηij ≡ diag(1, −1, . . . , −1)). Its euclidean version is the sphere Sn , which does not admit Poincaré coordinates [7], although, as all other spacetimes considered here, does admit stereographic coordinates. Hyperbolic spacetimes have got negative curvature. De Sitter (dSn ) spacetime falls in this category. In Poincaré coordinates P − n−1 δij dy i dy j + L2 dz 2 2 dsdSn = (1.7) z2 with z is a timelike coordinate, the euclidean version reads Pn−1 δij dy i dy j + L2 dz 2 2 dsEdSn = z2 (1.8) Correlators, including the energy-momentum tensor in this family of spaces have been thoroughly analyzed in [8] under the hypothesis that those only depend on the invariant arc length, s and its derivatives. Physically this is equivalent to the assumption that the relevant vacuum enjoys all spacetime isometries. For timelike geodesics the arc length coincides with the physical proper time. We shall refrain from using this notation though because we shall use it in Schwinger’s sense later. We shall also assume that hypothesis (that is, that the the only dependence on coordinates is through the geodesic length) in the present work; this amounts to demand invariance (or proper behavior) under all conformal isometries [9]. When working with lorenztian signature the square of the arc length is not positive semidefinite; it can become zero or even negative. This is in fact the reason why J.L. Synge introduced the world function, [10], which is essentially the square of the invariant arc length. Our formulas however remain formally valid with appropiate analytic continuation. 2 Hyperbolic Heat kernel Acting on functions of the geodesic arc length [8] the laplacian in Hn reads ✷= n−1 ∂ ∂2 + 2 ∂s L tanh Ls ∂s (2.1) The corresponding heat equation reads ∂ Kp+1 (τ, s) = Dp Kp+1 (τ, s) ∂τ 4 (2.2) where Dp Kp+1 (τ, s) =  p ∂ ∂2 + s 2 ∂s L tanh L ∂s  Kp+1 (τ, s) (2.3) In the following we shall often work with in terms of a dimensionless proper time and dimensionless arc length, Ls . • In the flat limit s → 0 this reduces to p ∂ ∂ ∂2 K(τ, s) + K(τ, s) = K(τ, s) ∂s2 s ∂s ∂τ (2.4) and the canonical solution is K(τ, s) = (4πτ ) −(p+1)/2 s2 exp − 4τ   (2.5) this obeys the correct boundary condition lim K0 (τ, s) = δ n (x − x′ ) 6= δ(s) τ →∞ (2.6) • In the opposite limit, s → ∞ the heat equation reduces to ∂2 ∂ ∂ K(τ, s) + p K(τ, s) = K(τ, s) 2 ∂s ∂s ∂τ whose general solution is a wave packet composed out of r  2  π s ps p2 τ K(τ, s) = exp − − − τ 4τ 2 4 (2.7) (2.8) this does not satisfy the boundary condition at τ = 0, but this is presumably natural because our approximation is valid for large values of s only. • Let us find a recurrence relation in flat spacetime. This recurrence relation is known in the mathematical literature [11], but our proof stems directly from the heat equation. Apply the lineal approximation DpL = p ∂ ∂2 + 2 ∂s s ∂s with p completely arbitrary  ′  Kp−1 2−p ′ p − 2 ′′ 1 ′′′ L Dp = Kp−1 + 2 Kp−1 + Kp−1 3 s s s s 5 (2.9) (2.10) where K ′ = ∂K , on other hand, let us assume that the function K obeys the heat ∂s kernel equation on p − 2 dimension, and derive one more time  3  ∂ p − 2 ∂2 p−2 ∂ ∂ ′ + − 2 Kp−1 = K (2.11) 3 2 ∂s s ∂s s ∂s ∂τ p−1 obtain DpL  ′ Kp−1 s  1 ∂ ′ K s ∂τ p−1 (2.12) 1 ∂ Kp (τ, s) 2πs ∂s (2.13) = in conclusion, this implies Kp+2 (τ, s) = − Of course, in flat spacetime, where we know the full dependence of the heat kernel with the spacetime dimension Kn (s) = 1 2 e−s /4τ n/2 (4πτ ) there is another trivial recurrence relation r 1 τ ∂Kn (s) Kn+1 (s) = − s π ∂s (2.14) (2.15) • Let us try to generalize this to the hyperbolic case. Consider the expression (which is independent of the dimension p)  ′  Kp−1 1−p ′ 2−p ′ (p − 2) cosh s ′′ 1 = Kp−1 + K ′′′ Dp Kp−1 + 3 Kp−1 + 2 sinh s sinh s sinh s p−1 sinh s sinh s (2.16) if we derive again the heat equation for p − 2 dimension  3  ∂ p − 2 ∂2 p−2 ∂ ∂ ′ + − Kp−1 = K (2.17) 2 3 2 ∂s tanh s ∂s ∂τ p−1 sinh s ∂s obtain    ′  ′ Kp−1 Kp−1 1−p ′ 1 ∂ ′ (1−p)τ −(1−p)τ ∂ = e K + K =e Dp sinh s sinh s p−1 sinh s ∂τ p−1 ∂τ sinh s (2.18) this implies ′ Kp−1 = e−(1−p)τ Kp+1 sinh s 6 (2.19) because with the heat equation Dp e−(1−p)τ Kp+1
= e−(1−p)τ ∂ Kp+1 ∂τ (2.20) finally we have the recurrence relationship Kp+2 (τ, s) = − e−pτ ∂ Kp (τ, s) 2π sinh s ∂s (2.21) It is to be stressed that this relationship is a consequence of the heat equation exclusively. Independently of any boundary conditions. 2.1 Dimensional reduction from odd to even dimensions. The starting point in the recurrence relationship is the formal one-dimensional case. It can be easily checked that s2 e− 4τ K1 (τ, s) = (4πτ )1/2 (2.22) ∂ ∂2 K1 (τ, s) = K1 (τ, s) 2 ∂s ∂τ (2.23) it obeys the engineering dimensions of the heat kernel are determined by its behavior when τ = 0. From here on we can determine via the recurrence all odd dimension heat kernels. The recurrence relationship easily leads to 2 e−p τ K2p+1 (τ, s) = (2π)p  1 ∂ − sinh s ∂s p K1 (τ, s) (2.24) Here we can see why dimensional reduction from n = 2p + 1 towards n = 2p (what Jacques Hadamard [12] dubs ”the method of descent”) does not naively work in this case. The reason is clearly that n−1 (2.25) p≡ 2 is a fractional number for even n, so that we have to take a fractional power of the operator ∂ 1 D≡− (2.26) sinh s ∂ s 7 which has been worked out for example in [13] in the framework of Scrödinger’s equation, with the result for hyperbolic spaces,  1 ∂ − sinh s ∂s  n−1 2 1 f (s) = √ π Z ∞ dx s  1 ∂ − sinh x ∂x n/2 f (x) √ sinh x cosh x − cosh s (2.27) Let us examine in detail a couple of examples. Four dimensional hyperbolic space H4 . 3 Our recurrence relation leads to √ − 9τ −M 2 L2 τ Z ∞ 2 2e 4 x2 − 2τ + 2xτ coth x sinh x − x4τ K4 (τ, s) = dx e 2 (4πτ )5/2 sinh x (cosh x − cosh s)1/2 s (3.1) The effective action (related work is found in e.g. [14]) is Z ∞ dτ Vef f [φ̄] ≡ K(τ, s) τ 0 for a g 4 φ 4! (3.2) self-interaction is then given in the effective potential approximation by √ h √ 1 sinh x 8 + 4x 9 + 4M 2 L2 + 2 1/2 3 sinh x x (cosh x − cosh s) s   i √ √ g 2 2 +x2 (9 + 4M 2 L2 ) + 2x 2 + x 9 + 4M 2 L2 coth x e−x 9+4M L /2 + φ̄4 4! (3.3) 2 Vef f [φ̄] = (4π)2 where Z ∞ dx g M 2 ≡ m2 + φ̄2 2 (3.4) The minimum is still at φ̄ = 0 as long as m2 ≥ 0. 4 Five dimensional hyperbolic space H5 Following with recurrence again leads to the five-dimensional heat-kernel 2 s2 2 e−4τ −M L τ − 4τ s2 − 2τ + 2sτ coth s K5 (τ, s)s = (4πτ )5/2 sinh2 s 8 (4.1) there is no polynomial interaction of the type gφn that enjoys dimensionless coupling constant in five dimensions, since the field itself has [φ] = 3/2. The only potential (besides the mass term) with a positive dimension coupling constant corresponds to a g φ3 interaction with [g] = 1/2, which we will assume henceforth. The effective potential, in turn, reads √ 2 2   i g √ √ 1 e−s M L +4 h 2 L2 + 4 + s2 (M 2 L2 + 4) + s 1 + s M 2 L2 + 4 coth s + 2 + 2s φ̄3 M Vef f [φ̄] = (2π)2 s3 sinh2 s 3! (4.2) where the effective mass in order to compute the effective potential is M 2 → m2 + g φ̄ (4.3) and we are making the approximation that φ̄ is constant ∂µ φ̄ = 0 (4.4) The minimum is still at the origin φ̄ = 0 as long as m2 ≥ 0. 5 Elliptic spacetimes. Acting on functions of the geodesic arc length [8] the laplacian in En reads ✷= n−1 ∂ ∂2 + 2 ∂s L tan Ls ∂s (5.1) There is a quite similar recurrence in the case of positive curvature, with trigonometric functions taking the place of hyperbolic ones. We shall be brief here. Again we start with the operator of heat kernel over K  ′  Kp−1 p−1 ′ 2−p ′ (p − 2) cos s ′′ 1 = Dp Kp−1 + Kp−1 + K ′′′ (5.2) 3 Kp−1 + 2 sin s sin s sin s p−1 sin s sin s on other hand if derive again the heat equation for p − 2 dimension   3 ∂ ′ p − 2 ∂2 p−2 ∂ ∂ Kp−1 = + − K 2 3 2 ∂s tan s ∂s ∂τ p−1 sin s ∂s (5.3) obtain Dp  ′ Kp−1 sin s  p−1 ′ 1 ∂ ′ ∂ = Kp−1 + Kp−1 = e−(p−1)τ sin s sin s ∂τ ∂τ 9  e (p−1)τ ′ Kp−1 sin s  (5.4) this implies ′ Kp−1 = e−(p−1)τ Kp+1 sin s (5.5) because with the heat equation Dp e−(p−1)τ Kp+1
= e−(p−1)τ ∂ Kp+1 ∂τ (5.6) Finally we get the recurrence relationship Kp+2 (τ, s) = − epτ ∂ Kp (τ, s) 2π sin s ∂s (5.7) Let us work out an example in some detail. 6 Elliptic five dimensional space, E5 . The starting point, as well as in the hyperbolic case, is the n = 1 heat kernel which is common to both cases s2 e− 4τ K1 = (4πτ )1/2 (6.1) we use the recurrence (5.7) with the normal term in mass 2 2 s2 e4τ −M L τ − 4τ s2 − 2τ + 2sτ cot s K5 (τ, s) = (4πτ )5/2 sin2 s (6.2) The effective action is defined by √ 2 2   i g √ √ 1 e−s M L −4 h 2 L2 − 4 + s2 (M 2 L2 − 4) + s 1 + s M 2 L2 − 4 cot s + 2 + 2s Vef f [φ̄] = φ̄3 M (2π)2 s3 sin2 s 3! (6.3) where the effective mass in order to compute the effective potential is M 2 → m2 + g φ̄ (6.4) in the conclusions we explain in detaill this result, for the effective action on elliptic spacetime. 10 7 Conclusions It is remarkable that the effective action for massless scalars in elliptic spacetimes (which are compact when euclidean) is IR divergent, whereas in (non compact even when euclidean) hyperbolic spacetimes it is not. Actually, there is always a minimal value for the effective mass above which the effective action does converge. For example, for five dimensional elliptic spaces, IR divergences in the effective action disappear whenever M 2 ≡ m2 + g φ̄ ≥ 4 L2 (7.1) The reason for this divergence in the massless case might be related to the presence of antipodal points. The reason is that the laplacian in elliptic spaces ✷= ∂2 n−1 ∂ + 2 ∂s L tan Ls ∂s (7.2) is singular not only when s = 0, but also whenever s = nπL (7.3) for any admissible integer n. Usually n = 1 corresponds to the geodesic distance to the antipodal point. This fails to happen in hyperbolic spaces, where the hyperbolic tangent only vanishes when s = 0. This is the reason why Schrödinger [15] proposed already in 1957 the elliptic interpretation, where all antipodal points are identified. While suggestive, the fact that the spacetime dSn /Z2 is not orientable poses many physical problems. 8 Acknowledgements One of us (EA) is grateful for useful correspondence with José Gracia-Bondı́a. This work has been supported in part by AEI grant PID2019-108892RB-I00/AEI/10.13039/501100011033 as well as from the Spanish Research Agency (Agencia Estatal de Investigacion) through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grants agreement No 674896 and No 690575. We have also been partially supported by FPA2016-78645-P(Spain). This project has received funding /support from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska -Curie grant agreement No 860881-HIDDeN. 11 References [1] E. Alvarez, “Windows on Quantum Gravity,” [arXiv:2005.09466 [hep-th]]. Fortschritte der Physik 2020-10-06 — journal-article DOI: 10.1002/prop.202000080 [2] A. A. Bytsenko, S. D. Odintsov and S. Zerbini, “The Effective action in gauged supergravity on hyperbolic background and induced cosmological constant,” Phys. Lett. B 336 (1994) 355 doi:10.1016/0370-2693(94)90545-2 A. A. Bytsenko, S. D. Odintsov and S. Zerbini, “The Large distance limit of the gravitational effective action in hyperbolic backgrounds,” Class. Quant. Grav. 12 (1995) 1 Erratum: [Class. Quant. Grav. 12 (1995) 2355] doi:10.1088/0264-9381/12/1/002 [hep-th/9410112]. A. Bytsenko, K. Kirsten and S. Odintsov, “Selfinteracting scalar fields on space-time with compact hyperbolic spatial part,” Mod. Phys. Lett. A 8 (1993) 2011 doi:10.1142/S0217732393001720 [hep-th/9303061]. [3] Guoyi Xu. ”Large time behavior of the heat kernel” arXiv:1310.2382v1 [math.DG] [4] H. P. McKean, ”An upper bound to the spectrum of ∆ on a manifold of negative curvature, J . Diff Geom. 4 (1970) 359-366 [5] Peter Li and Shing Tung Yau, ”On the parabolic kernel of the Schrödinger operator ”, Acta Math, 156,(1983) 153. [6] R. Camporesi, “The Spinor heat kernel in maximally symmetric spaces,” Commun. Math. Phys. 148 (1992), 283-308 doi:10.1007/BF02100862 R. Camporesi and A. Higuchi, “On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces,” J. Geom. Phys. 20 (1996), 1-18 doi:10.1016/0393-0440(95)00042-9 [arXiv:gr-qc/9505009 [gr-qc]]. [7] E. Alvarez and R. Vidal, “Eternity and the cosmological constant,” JHEP 10 (2009), 045 doi:10.1088/1126-6708/2009/10/045 [arXiv:0907.2375 [hep-th]]. [8] B. Allen and C. A. Lutken, “Spinor Two Point Functions in Maximally Symmetric Spaces,” Commun. Math. Phys. 106, 201 (1986) doi:10.1007/BF01454972 B. Allen and T. Jacobson, “Vector Two Point Functions in Maximally Symmetric Spaces,” Commun. Math. Phys. 103, 669 (1986) doi:10.1007/BF01211169 H. Osborn and G. M. Shore, “Correlation functions of the energy momentum tensor on spaces of constant curvature,” Nucl. Phys. B 571 (2000), 287-357 doi:10.1016/S0550-3213(99)00775-0 [arXiv:hep-th/9909043 [hep-th]]. [9] E. Alvarez and R. Santos-Garcia, “CFT in Conformally Flat Spacetimes,” Phys. Rev. D 101, no.12, 125009 (2020) doi:10.1103/PhysRevD.101.125009 [arXiv:2001.07957 [hep-th]]. [10] Synge, J.L., Relativity: The General Theory, (North-Holland, Amsterdam, 1960). 12 [11] E.B. Davies, ”Heat kernels and spectral theory”, (cambridge University press, 2007) [12] Jacques Hadamard, ”Lectures on Cauchy’s problem in linear partial differential equations”, (Dover) [13] Jean-Philippe Anker, Vittoria Pierfelice , ”Nonlinear Schrdinger equation on real hyperbolic spaces”, Ann. I. H. Poincar AN 26 (2009) 18531869 [14] C. Schomblond and P. Spindel, “Unicity Conditions of the Scalar Field Propagator Delta(1) (x,y) in de Sitter Universe,” Ann. Inst. H. Poincare Phys. Theor. 25 (1976), 67-78 J. S. Dowker and R. Critchley, Phys. Rev. D 13 (1976), 224 doi:10.1103/PhysRevD.13.224 T. Inami and H. Ooguri, “One Loop Effective Potential in Anti-de Sitter Space,” Prog. Theor. Phys. 73 (1985), 1051 doi:10.1143/PTP.73.1051 C. P. Burgess and C. A. Lutken, Phys. Lett. B 153 (1985), 137-141 doi:10.1016/0370-2693(85)91415-7 [15] E. Schrodinger, “Expanding universes,” (Cambridge, 1956) M. K. Parikh, I. Savonije and E. P. Verlinde, “Elliptic de Sitter space: dS/Z(2),” Phys. Rev. D 67 (2003), 064005 doi:10.1103/PhysRevD.67.064005 [arXiv:hep-th/0209120 [hep-th]]. 13