We use an extended quantum field theory (EQFT) hep-th/9911168 to explore possible observational effects of the spacetime. It is shown that as it was expected the spacetime foam can provide quantum bose fields with a cutoff at very small scales, if the energy of zero - point fluctuations of fields is taken into account. It is also shown that EQFT changes the behaviour of massless fields at very large scales (in the classical region). We show that as $rgg 1/mu $ the Coulomb and Newton forces acquire the behaviour $sim 1/r$ (instead of $1/r^{2}$)

Kirillov, AA

Turaev, D

Gravitation and Cosmology

English

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arXiv:hep-th/0108060v1  9 Aug 2001Corrections to the Newton and Coulomb potentials caused by effectsof spacetime foamA.A. KirillovInstitute for Applied Mathematics and Cybernetics10 Uljanova Str., Nizhny Novgorod, 603005, Russiae-mail: kirillov@unn.ac.ruD. TuraevWeierstrass Institute for Applied Analysis and StochastisityMohrenstrasse 39, D-10117, Berlin, Germanye-mail: turaev@wias-berlin.deAbstractWe use the extended quantum field theory (EQFT) [1] to explore possible observational effectsof the spacetime. It is shown that as it was expected the spacetime foam can provide quantum bosefields with a cutoff at very small scales, if the energy of zero - point fluctuations of fields is takeninto account. It is also shown that EQFT changes the behaviour of massless fields at very largescales (in the classical region). We show that as r ≫ 1/µ the Coulomb and Newton forces acquirethe behaviour ∼ 1/r (instead of 1/r2).It is commonly believed that quantum gravity effects (spacetime foam) should provide a cutoff forquantum field theory. An extension of the quantum field theory (EQFT) which allows to account forthe spacetime foam effects was suggested by one of us in Ref. [1]. This extension involves a variablenumber of Bose fields and introduces a new operator N (k) which is the density of the number offields in the space of modes. The standard QFT represents a particular sector of the extended theorywhen N (k) = 1. In this letter we show that this operator causes a modification of Green functionsD̃ (k) = D (k) N (k) in all internal lines of the diagram technique. At very small scales k ≪ k∗ itindeed can provide a cutoff, if we take into account the energy of zero-point fluctuations of fields.From the other side, in the case of massless fields this operator causes essential modification at largescales k ≪ µ (µ is the Fermi energy, e.g., see Ref. [1]).First we note, that from a phenomenological view point the operator N (k) reflects the topologyof the physical space. In the modern Universe topology changes are strongly suppressed. Thereexist severe experimental restrictions which come from oscillation experiments (e.g., see Ref. [2] andreferences therein). This means that now there acts a superselection rule and the operator N (k)represents a constant of motion and may be considered as an ordinary function. However in the earlyUniverse, during the quantum stage, topology changes should take place and properties of the functionN (k) are formed then (e.g., there may appear some excess of field modes [1]).Consider as an example the modification of quantum electromagnetic field. The operator of theelectromagnetic interaction is described by the term V = e∫jµAµd3x (where e is the electron chargeand jµ is the current density of sources). In the extended theory the number of fields is a variableAaµ (a = 0, 1, ...), one can say that there exist photons of different ”sorts”. However these sorts areindistinguishable (there exists the identity principle for fields) and, therefore, the interaction termshould include an additional summation over the sorts of photons1 V = e∑a∫jµAaµdx.1 The same summation does appear in the total Hamiltonian describing free photons.1The Green function for photons in the coordinate representation has the form (e.g., see Ref. [3])Dµν(x − x′)= i〈T(Aµ (x) Aν(x′))〉(1)where the averaging out is taken over the vacuum state and the symbol T denotes the chronologicalordering. The Fourier transform of this function in the Feinman gauge is expressed via scalar functionD (k)Dµν (k) = gµνD (k) . (2)In the standard quantum theory in the first approximation D (k) = 4π/k2 = 4π/(ω2 − k2)(herekµ = (ω,k)). While considering photons of a particular ”sort” the same is also valid in the extendedtheory Da (k) = 4π/k2. However, it is obvious, that in the perturbation theory all diagrams willinclude additional summation over sorts of photons. Such a summation can be carried out explicitlyand, therefore, it is sufficient to consider a modified Green functionD̃ (k) =∑aDa (k) =4πN (k)k2. (3)The nontrivial fact here is that EQFT admits a nontrivial ground state in which the number of fieldsN (k) contains a dependence on the wave vector k [1].From a formal point of view the expression (3) breaks the Lorentz symmetry. This however, doesnot mean that the extended theory is not Lorentz invariant one. The symmetry breaks due to aparticular choice of the field ground state Φ0 which, was shown to possess finite energy and particlenumber densities [1]. Such particles are dark and cannot be directly observable, but should contributeto the dark matter. Therefore, the state Φ0 distinguishes a particular prefered system of coordinates.Since in all internal lines of the diagram technique the function D̃ (k) gets multiplied by e2, onecan try to interpreet the modification (3) as if a point-like charged particle acquires a dispersione2 (k) = e2N (k) . However, such an interpretation should be used somewhat cautiously. Indeed, thiskind of a renoramlization of the charge is valid only for virtual photons, while in processes involvingreal photons (external photon lines) this renormalization does not work.As it was shown in Ref. [1] the field ground state Φ0 can be characterized by occupation numbersof the typeNk,n = θ (µ − (nω + ∆ (k))) , (4)where Nk,n is the number of field modes in the quantum state (k, n) (n is the number of photons in thegiven mode), θ (x) is the Heaviside step function, µ is the chemical potential, and we added the term∆ (k) which is the minimal energy of the field modes (the vacuum spectral energy density). We notethat the current state of the field theory does not allow to fix the form of ∆ (k) and in what follows weshall not specify it. The standard definition ∆ (k) = 12ω results in the infinite energy density in QFTand requires a renormalization, while in EQFT this choice produces too small value for the cutoff (seebelow). However we may expect that ∆ (k) ∼ ω > 0 is an increasing function. Thus, for the modespectral density we getNk =∞∑n=0θ (µk − nω) =[1 +µkω], (5)where [x] denotes the integer part of the number x and µk = µ − ∆ (k). Equation (5) shows, inparticular, that Nk = 0 as µk < 0 and, therefore, there appears a cutoff k∗ whose value is the solutionof the equation µ−∆ (k∗) = 0. The standard picture of the electromagnetic field is valid in the wavenumber range (ω = ωk =√k2)ωk > µk > 0 (6)where we have Nk = 1. In what follows, for the sake of simplicity, we set µk = µ (thereby neglectingthe existence of the field energy of zero-point fluctuations and of the respective cutoff). Then, (5)2reads Nk = 1 + [µ/ω] and in the range ω < µ we find the correction to the standard Green functionD̃ (k) − D (k) = 4πk2[µω]. (7)We note that the consideration above remains also valid in the case of the linearized gravitationalfield hµν (gravitons) with the replacement of the equation (2) by Dµν,αβ (k) =12(gµαgνβ + gµβgνα)D (k)and the same function D (k) as in (2).Consider now the correction to the Coulomb (or Newton) laws. Since the number of fields is avariable the Coulomb (or Newton) potential V should be replaced by the effetive potential V =∑a Vawhich contains the sum over all sorts of photons (gravitons). Consider a rest point-like particle. Thenthe Fourier transform of the correction δV (k) to the standard Coulomb potential V (k) = 4πeZ / |k|2takes the formδV (k) =4πeZ|k|2[µ|k|], (8)where Z denotes the value of the electric charge of the rest particle (in the case of gravity one shoulduse the obvious replacement eZ → −Gm). In the coordinate representation this potential is given bythe integralδV (r) =12π2∞∫0(δV (ω) ω3) sin (ωr)ωrdωω. (9)Since δV (ω) vanishes for ω > µ, the upper limit in this integral is ω = µ. At the low point ω = 0 thisintegral is divergent. However, we note that the interaction between particles can exist only on scalesless than the horizon size ℓh. Thus, as the lowerst limit we must take ω ∼ 1/ℓh ∼ H where H is theHubble constant. This integral can be presented in the formδV (r) = σN∗∑n=1µ/n∫Hsin (ωr)ωrdω, (10)where σ = 2eZ/π and N∗ = µ/H. In the range µr ≪ 1 this correction produses a constant shift(which gives a finite contribution to the electronagnetic rest mass of the particle δm (eZ) = eZδV )δV ∼ σµN∗∑n=11n∼ σµ ln( µH). (11)In the opposite asymptotics µr ≫ 1 (but Hr ≪ 1) we find the estimateδV ∼ −σµ ln (Hr) . (12)The expression (12) shows essential deviations from the Newtons and Coulomb laws at scalesr > r0 ∼ 1/µ. In particular, at these scales the Newton and Coulomb forces acquire the behaviour1/r (instead of 1/r2). We note that the value r0 can be very large and, therefore, to carry out a directobservation of the correction to the Coulomb potential is impossible (at macroscopic scales the numberof positive and negative charges is equal and the potential vanishes long before reaching the scale r0).However in the case of gravity the situation is different, for the gravitational potential accumulatesand the correction (12) has to leave an imprint in astronomical observations. And indeed, there existsan indication that such a behaviour really takes place. As is well known observations show that theleading contribution to the distribution of matter gives the so-called dark matter which should have anexotic non-baryonic form and is not directly observable (e.g., see [4]). There are several observationswhich provide evidence for dark matter. One is measurements of the rotational velocity of galaxies as afunction of the radial distance from the center, the so-called rotation curve (see e.g. [4, 5]). According3to the standard Newton dynamics the rotation curve of a disk with an internal mass-distribution thatfollows the observed brightness law has to show a Keplerian r−1/2 behaviour at large radii. Howevermeasurements [5] show that v (r) = vm stays constant, which implies that the total mass containedwithin a radius r, M (r), varies in proportion to r . Indeed, according to the standard Newtons lawthe acceleration of a body in a circular orbit of the radius r is a = GM (r) /r2 = v2 (r) /r , which givesM (r) = v2mG−1r . This can be interpreted as if the mass per unit luminosity M/L increases withradius and, therefore, a large fraction of the total mass of a galaxy is in the form of a non-luminous,dark component located at large radii. However, if we take into account the correction (12), we findthat for r > r0 v2 (r) = v2m ∼ 2µGM/π is consistent with the light distribution (M/L ∼ const). It isnot clear yet which fraction of the dark matter can be explained by the correction (12) to the Newtonspotential (we recall that the ground state Φ0 itself unavoidably predicts the existence of dark matter[1]). However, this allows to give a previous rough estimate for the characteristic scale r0 & 1 Kpcand so the parameter µ is really small. To get a more precise estimate requires the further and morethorough confrontation with observations.We note that the idea of a modification of gravity at large scales is not new, e.g. see critics ofdifferent approaches and the list of references in Ref. [6]. Some approaches use potentials close to(12) as an empirical demand to the modification of the Newton’s law (or, equivalently, introduce anadditional force), e.g., see [7] and have no any fundamental theoretical background. On the contrary,in the extended quantum field theory the correction (12) is inevitable consequence of the masslessnature of the gravitational field.References[1] A.A. Kirillov, JETP 88 1051 (1999) [Zh. Eksp. Teor. Fiz. 115 1921 (1999)], hep-th/9911168.[2] G.L. Fogli, E. Lisi, A. Marrone, G. Scioscia, Phys.Rev. D60 (1999) 053006 (hep-ph/9904248);T.Lisi, A.Marrone, D. Montanino, Phys.Rev.Lett. 85 (2000) 1166 (hep-ph/0002053).[3] V.B. Berestezky, E.M. Lifshitz, L.P. Pitaevsky, Quantum Elektrodynamics, Moscow, Nauka,(1980).[4] G.Borner, The Early Universe: Facts and Fiction, Springer-Verlag, New York (1992); J.A. Peacock,Cosmological Physics, Cambridge University Press, Cambridge (1998).[5] C. Carignan, K.C. Freeman, Astrophys. J. 294 (1985) 494.[6] D. Scott, M. White, J.D. Cohn, and E. Pierpaoli, astro-ph/0104435 (2001).[7] K.G. Begeman, A.H. Broeils, and R.H. Sanders, Mon. Not. Roy. Astr. Soc., 249 (1991) 523; R.H.Sanders, Astrophys. J. 473 (1996) 117; astro-ph/0011439 (2000).4

Corrections to the Newton and Coulomb potentials caused by effects of spacetime foam

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Corrections to the Newton and Coulomb potentials caused by effects of spacetime foam

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