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Towards a quantum universe Jaume Giné1 2 1 0 2 n a J 5 ] Abstract Inthisshortreviewwestudythestateofthe scale,atwhichthequantumeffectsmustbeconsidered. h artofthegreatproblemsincosmologyandtheirinterre- Thequantumequations,asSchrödingerandDiracones, p - lationships. Thereconciliationoftheseproblemspasses are not scale invariants, due to the presence of h. The n undoubtedly through the idea of a quantum universe. question that naturally arises is whether it is really a e g physical constant at any scale. Keywords Cosmology, Gravitation theory, Quantum . The invariance under discrete scale transformations s mechanics,Generalrelativity,largenumbers,cosmolog- c appear from one of the curious features between par- i ical constant s ticles physics and cosmology. These features are the y possibility of obtaining cosmological large numbers, as h mass M radius R and age T of the universe, scaling p 1 Introduction U U [ upthetypicalvaluesofmassm,sizerandlifetimetap- 38−40 The great challenge of contemporary physics is to rec- pearing in particle physics, by the scale factor 10 . 1 v oncile quantum mechanics, applied at micro cosmos, The scale relations are T/t ∼ RU/r ∼ (MU/m)1/2 ∼ 3 andgeneralrelativityapplied,ingeneral,atmacrocos- λ=1038−40. From here we can scale h in order to ob- 1 mos. General relativity and classical electrodynam- tain the new constant of the new scale invariance of 8 H ics equations are invariant under a scale transforma- quantummechanics. Fromasimpledimensionalanaly- 1 tion of time intervals and distances, provided we scale siswehave ∼λ3h. Thepossiblemeaningofthisnew . 1 H too the correspondent coupling factors. In particu- constant is that /(2π) is the angular momentum 0 H H lar, the scale invariance of general relativity was ap- of a rotating universe and this explanation is close to 2 1 plied to the strong gravity Salam & Strathdee (1977, the Gödel’s spin, with the Kerr limit for the spin, and : 1978); Caldirola et al. (1978); Sivaram & Sinha (1979) withtheMuradian’sRegge-likerelationforgalaxiesand v i thattriestoderivethehadronpropertiesfromascaling clusters, see Carneiro(1998) and references therein. In X downofgravitationaltheory,treatingparticleasblack- fact this new constant is ∼ 10120h and is what is r hole type solutions. Lastyears,inseveralworks,it was call in Alfonso-Faus (2008H) the cosmological Planck’s a suggestedthatalsoquantummechanicsmustbeinvari- constant. With this new Planck’s constant no large ant under discrete scale transformations, see Carneiro numbers appear at the cosmological level. In Carneiro (1998). All suggest that these two irreconcilable theo- (1998) it is also described an intermediate scale invari- ries, the gravity defined by the General relativity and anceofquantizationrelatedtotheangularmomentaof the quantum mechanics, can be applied to any scale. stars and close to the Kerr limit for a rotating black Should therefore be complementary theories that ex- 30 hole with mass around 10 kg. All these ideas suggest plain the same physical reality. treating the universe as a single particle, as we shall However, the introduction of the Planck’s constant seelater. Infactasacosmologicalquantumblackhole. h in the quantum mechanics defines a very particular In the following sections we will see that several scaling laws can explain some of the present cosmological JaumeGiné problems. Departament deMatema`tica,UniversitatdeLleida, Av. JaumeII,69. 25001Lleida,Spain 2 2 The Large number coincidence problem We reproduce here the arguments. From the scale relations (that do not constitute a coincidence problem) Hermann Weyl (1917, 1919) speculated that the ob- 1/2 2 2/3 served radius of the universe might also be the hypo- MU ∼ mP ∼ MU , theticalradiusofaparticlewhoseenergym c2isequals (cid:18)m (cid:19) (cid:18)m (cid:19) (cid:18)m (cid:19) h n n P tothegravitationalself-energyoftheelectronGm2/r , where mh is the mass of the hypothetical particlee, mee wheremP isthePlanckm3ass,weobtainM3 U ∼m4P/m3n andrethemassandtheradiusoftheelectron. Thiswas where MU =Ωm(4π/3)RUρc =(4π/3)RUρm is the observable mass of the universe. The Hubble parameter thebeginningofthelargenumbercoincidenceproblem. in a universe with zero curvature is related with the Hence, we have averagetotal energy density ε by R r m m c2r e2 U ∼ h = e = e = e = 1042, 8πGε re re mh Gc2mre2e Gme 4πε0Gm2e ≈ H2 = 3c2 , (2) where we have that re = e2/(4πε0mec2) and rh = and during the matter-dominance the total energy is e2/(4πε0mhc2). Thiscoincidencewasfurtherdeveloped ε=c2ρm. Therefore the mass of the universe is equals by Eddington(1931)who relatedthe aboveratioswith 4π R3H2 N, the estimated number of charged particles in the M = R3ρ = U . (3) universe. U 3 U m 2G 4πε0eG2 m2e ∼√N ≈1042. (T3a)kginivgesintthoeascccaoliunngtltahwat H ∼ c/RU ∼ 1/T equation Eddington obtained the most intriguing relation be- GMU ∼c2RU. (4) tween the present number of baryons in the universe, Expression (4) was obtained by Whintrow (1946), known as the Eddington number, and the squared ra- Whintrow & Randall(1951),Sciama(1953),Brans & Dicke tiooftheelectrictothegravitationalforcebetweenthe (1961); Dicke (1961) and also by Assis (1989, 1999) in proton and the electron. differentcontexts. Anotherformtoobtainequation(4) F e2 cT is applying the classical Mach’s principle by requiring e = ∼ 1040, Fg 4πε0Gmemp re ≈ that the self-energy of a body is given by the gravitational energy of interaction of a body with the whole where mp is the proton mass and T is the age of the universe: universe. This coincidence between large numbers can GmM also be expressed in the alternative form mc2 = U. R U ~2H0 ∼Gm3nc, (1) Substituting this scaling law (4) in the expression 4 3 M =m /m and remembering that the Planck mass where mn is the nucleon mass, H ≡ a˙/a is the Hub- is Um =P ~c/nG we have ble parameter and H0 is its present value and a(t) is P p thescalefactor,seeMena Marugan & Carneiro(2002). c2R m4 ~2c2 This approximate identity is called the Eddington- U = P = , G m3 G2m3 Weinberg relation. TheHubble parameteris notacon- n n stantandvaries asthe inverseofthe cosmologicaltime and from here the Eddington-Weinberg relation (1). t inthe standardFriedmann-Robertson-Walker(FRW) cosmology. ThisfactledDirac(1937,1978)tospeculate the hypothesis that Newton’s constant G must depend 3 The Cosmic coincidence problem ontimeasH,i.e. G∼1/t,sothatrelation(1)remains always valid. This fact is incompatible with the ex- In an expanding universe with scale factor a(t), where perimental bounds that exist on time variation of G, t is the cosmological time, Λ is a constant while the see Damour et al. (1988); Mena Marugan & Carneiro matter density ρ decreaseswith a3. However,the ob- m (2002); Williams et al. (1996). Hence, the coincidence servedenergydensityofmatterc2Ω ρ issoclosetothe m c (1) is only valid in this epoch. In Funkhouser (2006) it vacuum energy density attributed to the cosmological was resolvedthe large number coincidence problem us- ing scalinglawsfromthe standardcosmologicalmodel. 3 constant Λ, given by ε = 3Λc2/(8πG). This coinci- from (5) we obtain the scaling laws vac dence is known as the cosmic coincidence problem and G2m2 G2M2 may be expressed as c2Λ∼ n ∼ U. (7) λ4 R4 n U 3Λ ρ =Ω ρ ∼ . m m c 8πG This scaling law says that the energy density associ- atedto the cosmologicalconstant may be scaled to the As inthe caseofthe largenumber coincidence,this co- gravitational energy of the nucleon mass confined to a incidence occuronly in this epoch. We are goingto see sphere whose radius is the Compton wavelength of the thatthecosmiccoincidenceproblemisaconsequenceof nucleonand to the gravitationalenergy of the universe the large number coincidence and due to the fact that of mass M and whose radius is R . This is the gen- U U we are in the era of vacuum-dominance. If we assume eralizationof the Zel’dovich (1967) equation (5) to the that the presentevolutionof the universe is dominated cosmologicallevel by the cosmologicalconstantΛ,as corroboratedbyob- servationTegmark et al.(2001), we cansetH0 ∼Λ1/2. G2M6c2 Λ∼ U , (8) The continuous transition from the matter-dominance 4 H given by equation (2) to our era of vacuum-dominance withtheintroductionofthe cosmologicalPlanck’scon- gives the cosmic coincidence stant satisfying R = /(M c) and the generaliza- U U H H 8πGρ tion of the Eddington-Weinberg relation (1) H2 ∼ m ∼Λ. 0 3 HH0 ∼GMU3c, (9) 4 The Cosmological constant problem assuming that the present evolution of the universe is dominated by the cosmological constant Λ and then If Λ originates from the vacuum quantum fluctuations, we have H0 ∼ Λ1/2. These generalizations are also its theoretically expected value has order of l−2 where obtained in Alfonso-Faus (2008, 2011). Moreover the p l ~G/c3 10−35m is the Planck length, see cosmological constant problem is solved with the in- p We≡inbeprg (1989)≈. That is, 122 orders of magnitude troductionof the cosmologicalPlanck’sconstant be- H greater than the observed value Λ 10−52m−2, see cause now Λc originates from the cosmologicalvacuum ≈ quantumfluctuations,hasthevalueoforderL−2where Tegmark et al.(2001). Thishugediscrepancyisknown p as the cosmologicalconstantproblem and it is anopen Lp G/c3 1026m is the cosmological Planck ≡ H ≈ −52 −2 problem nowadays, see for instance Weinberg (1989); length,pand we obtain Λc 10 m which agrees ≈ ’t Hooft & Nobbenhuis (2006). with the observed value. In fact, this cosmological However we can get for the cosmological constant Planck length Lp is of order of the radius of the uni- Λ one scaling law that also explains the cosmic coinci- verse RU. Hence we have RU2 ∼ HG/c3, that taking dence, see Funkhouser (2006, 2008). Putting the con- into account RU = /(MUc) we reobtain the equal- H dition H0 ∼ Λ1/2 in the large number coincidence (1) ity (4) that relates MU with RU. In resume we have we have the following identities that define the cosmological scale R =GM /c2, the cosmologicalCompton wave- Λ∼ G2m6nc2. (5) length λ¯Uc = /U(MUc), the new cosmological constant ~4 Λ ∼ L−2 ∼HR−2 and it is satisfied that Λ = c3/G. c p U cH Equation(5)isessentiallythesamescalinglawderived Hence we have two important scales, the micro scale byZel’dovich(1967),fromconsiderationsoffieldtheory called Planck scale and the macro scale given by the andempiricalarguments. This formto deriveequation cosmological scale that suggest the scale relativity in- (5)wasfirstmadebyMatthews(1998),whotakesrela- variance introduced by Nottale (1992). tion(1),aswellasthepresentdominanceofthecosmo- logicalconstantoverthedensityofmatter. Takinginto 5 The critical acceleration coincidence accountthattheComptonwavelengthofthenucleonis λ =h/(m c) and the scale relation n n The observedmotions of clusters of galaxiesand mate- M 1/2 R rialwithingalaxiesmaybe interpretedto indicate that U ∼ U, (6) (cid:18)m (cid:19) λ the laws of dynamics deviate from Newtonian models n n at accelerations smaller than some critical acceleration 4 a0 10−10ms−2, see Milgrom (1983). The Hubble ac- implies equation (1), see Carneiro (2002), without any ≈ celeration cH0 is of the same order only in this epoch. additional assumption as the dominance of the cosmo- This coincidence a0 ∼cH0 is well known from the first logicalconstantΛ. Hence,inthiscasetherelation(1)is works of Milgrom (1983) see also Funkhouser (2006). valid for any cosmologicaltime. The derivative respect This coincidence is justified in Giné (2009) and Giné to the time of the relation (1) gives H˙ = 0, because (2011) by different arguments. Substituting H0 ∼Λ1/2 the variation of G is incompatible with the observa- the coincidence takes the form a0 ∼ cΛ1/2 and taking tions. Now, we recall the definition of the deceleration 2 into account the scaling law (7) we obtain parameterq = aa¨/a˙ anditsrelationwiththeHubble − parameter Gm a0 ∼ λ2n. H˙ = (1+q)H2. n − Hence, the critical acceleration is scaled to the charac- Therefore,thestrongversionofthecosmologicalholog- teristic gravitational acceleration of the nucleon mass raphy principle implies that q 1 in order to obtain at its Compton length. Moreover, taking into account H˙ = 0. This value of the de≈ce−leration parameter is the scale relation (6) we have that also found in the context of the modified Newtonian theory (MOND) in Giné (2010), when we evaluate the GM a0 ∼ R2U. (10) recessional acceleration ar(t) = −qHvr for the objects U recedingfromusataratefasterthanthespeedoflight and compare with the value of the constant accelera- Hence, the critical acceleration is scaled to the char- acteristic gravitationalacceleration of any body in our tion a0 = H0c. In this case the Hubble law is applied for close distances assuming the same behavior at first universe due to the all the rest of the mass of the uni- order for largest observable distance. verse. This interpretation of the critical acceleration appearsinGiné(2009)inthe contextofa implementa- tion ofthe inertia Mach’stheory. InTank (2010,2011) 7 The quantum universe it is found that identity (10) is invariant at any scale because is satisfied by the hadrons, the electrons, the We have seen the existence of several scaling laws that nucleus, the globularclusters,the galaxies,the clusters explain some of the present cosmological problems. ofgalaxies,the universeas a whole andothersphysical However,theoriginofthedarkenergyanddarkmatter situations. are still open problems. In Alfonso-Faus (2011) it is given a necessary and sufficient condition for an object of any mass m to be 6 The cosmic acceleration problem a quantum black hole generalizing the results obtained forthe cosmologicalscale. This generalizationis estab- ThestandardcandleobservationsoftypeIasupernovae lishedbythefollowingidentitiesthatdefineaquantum give a cosmic acceleration with a positive rate, which black hole for each m and a new scale. The first is impliestheintroductionofthecosmologicalconstantin r =Gm/c2, where r is the gravitationalradius, the the cosmological models. Hence, the expansion of the m m generalizedComptonwavelengthλ¯ =h /(mc)∼r , universeisaccelerating,seeRiess et al.(1998). Thisac- m m m where h is the generalized Planck’s constant, the celeration states the cosmic acceleration problem. The m Λ ∼ r−2 and it is satisfied that Λ h = c3/G. This question is what causes this acceleration? m m m m generalization is also justified by the described inter- It is clear that the introduction of the cosmologi- mediate scale invariance of quantization for a rotat- cal constant give as a consequence that the universe is ing black hole with certain mass, see Carneiro (1998). accelerating. However, what is the nature of this cos- Hence in Alfonso-Faus (2011); Fullana & Alfonso-Faus mological constant introduced? (2011) is adopted the idea that the universe is a quan- Thereareessentiallytwowaysofintroducingthecos- tum black hole and therefore it is possible to define, mological constant or the dark energy. The first one is following the Hawking (1975) formulation, the entropy changinggravitationwithf(R)gravitymodels,Scalar- of the universe as a quantum black hole tensormodels,braneworldmodels,etc. Thesecondone is changing matter with the quintessence, K-essence, 2 4πk R B 2 U 122 tachyons, Chaplygin gas, phantom field, etc. S = ~c GM =πkB(cid:18) l (cid:19) ≈10 kB, p We have seen that the Eddington-Weinberg relation (1) is only valid in this epoch. However, the strong which is in accordancewith the currentvalue found by version of the cosmological holography principle also Egan & Lineweaver(2010). In Fullana & Alfonso-Faus 5 (2011) it is computed the conjugate black hole of the The following exciting papers papers can shed light universe that is identified with the quantum of the on the nature of the dark matter and the solution of gravitational potential field and the bit. Besides, the the dark matter problem. In Villata (2011) showed information-entropy relation, based on the bit, the that,fromCPTinvarianceofthe generalrelativity,the Padmanabhan (2010a,b) proposal that gravity has an sign of the gravitational force between matter and an- entropic or thermodynamic origin, and the Verlinde timatter is reversed(anti-gravity). This is a controver- (2011)interpretationofgravityasanemergingentropic sial result which is being analyzed and discussed, see force, gives a hope to unify gravity with quantum the- for instance Cabbolet (2011); Cross (2011) and Villata ory. (2011b). The idea of Alfonso-Faus & Fullana reinforces the Basedin the anti-gravity(thata particleandits an- relationships between the constants of atomic physics tiparticle have the gravitational charge of the opposite andtheconstantsoftheUniverse,aswehavedescribed sign) Hajdukovic (2011a,b) consider that the quantum in the text, see also Hajdukovic (2010a); Dinculescu vacuum may be considered as a fluid of virtual gravi- (2009). In Hajdukovic (2010a) three interesting rela- tational dipoles. In such a way that when we place a tions are presented. The first one connects the Comp- gravitational mass in a quantum vacuum will induce ton wavelength of a pion and the dark energy density a polarization of the quantum vacuum, in the same of the universe; the second one connects the Compton way that a charge induces polarization in a surround- wavelengthof a pion and the mass distribution of non- ing dielectric medium. In the case of gravitation, we baryonic dark matter in a galaxy;the third one relates would expect to find more virtual particles close to the mass of a pion to fundamental physical constants a gravitating object, and more anti-particles at much and cosmological parameters which has as particular greaterdistance. Thiswouldmeanthat,inagalaxyfor case the Eddington-Weinberg relation (1) but for the example, the apparent gravitational attraction of the pionmass. Theimportanceofthe pions(insteadofthe body is an increasing function of distance out to some nucleon mass) is due to “virtual” pions, which are, ac- critical value. Following this hypothesis, Hajdukovic codingtoquantumfieldtheory,aninherentpartofvac- present the first indications that dark matter may not uumfluctuationsandasasimpleparticles(quarkpairs) existand that the phenomena for whichit wasinvoked dominate the quantum vacuum. We recall that pions might be explained by the gravitational polarization are the subatomic particles that describe the interac- of the quantum vacuum by the known baryonic mat- tion between nucleons. Under this scenario, each nu- ter. The best developed alternative to particle dark cleon is continuously emitting and reabsorbing virtual matter is the Modified Newtonian Dynamics (MOND) pions, which surround it like a swarm. Moreover cor- Milgrom (1983), but we witness a violation of the fun- rectvalueofmasstoputintheidentity(5)accordingto damentallawofgravityandhasstillfundamentalprob- the observedvalue of Λ is about 1/20times the proton lems with the observationaldata, see for instance Giné mass or about 80 times the electron mass and is about (2010, 2011b) and references therein. However in the one third the pion mass, see Santos (2010). Therefore Hajdukovicmodel the distributionofvacuum polariza- the pions must dominate the quantum vacuum fluctu- tion will depend on the distribution of matter, so the ations that contribute to the value of the cosmological apparentextra accelerationtowards the center of mass constant. In Dinculescu (2009) it is derived the values will vary fromone object to another, and as a function of the baryon density parameter, the Hubble constant, of position within the object, see Hajdukovic (2010b). thecosmicmicrowavebackgroundtemperatureandthe Moreoverthe consequencesof the model canbe tested, helium mass fraction in excellent agreement with the see Hajdukovic (2011b), where some phenomena par- the most recent observationaldata. tiallyexplainedbydarkmatterandtheoriesofmodified Following the idea of a quantum vacuum fluctua- gravity are understood in the framework of the grav- tions, with virtual particles flashing in and out of ex- itational polarization. Moreover the theory presented istence, in Santos (2010, 2011) it is showed that the in Hajdukovic (2011a,b) is not a support to MOND vacuum fluctuations effectively supplies a vacuum en- although there is a critical gravitational filed which ergypressurewhichisoftherightorderofmagnitudeto corresponds to the maximal gravitational polarization explain dark energy. The key idea of the Santos works density. isthetwo-pointcorrelationfunctionofvacuumfluctua- tionsgivesthecorrectcontributionofDarkenergy,and The final conclusion is that is needed a quantum this relies upon the disappearance of the correlation gravitational theory with a quantum granulation of within the Plancklength which solvesthe cosmological space-time and in this new framework the presented constant problem. papers have given us grounds to hope that both dark 6 energyanddarkmatterwillfindtheirnaturalexplana- tion as simply naturally-arising quantum vacuum phenomena. 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