Characteristic length of dynamical reduction models and decay of cosmological vacuum
aa r X i v : . [ qu a n t - ph ] J un Characteristic length of dynamical reduction modelsand decay of cosmological vacuum
S.V. Akkelin ∗ October 30, 2018
Abstract
Characteristic length of mass density resolution in dynamical reduction modelsis calculated utilizing energy conservation law and viable cosmological model withdecreasing energy density of vacuum (dark energy density). The value found, ∼ − cm, numerically coincides with phenomenological spatial short-length cutoff parameterintroduced in the Ghirardi-Rimini-Weber model. It seems that our results support thegravity induced mechanism of dynamical reduction. Bogolyubov Institute for Theoretical Physics, 03680 Kiev-143, Metrologichna 14b, Ukraine
PACS:
Keywords: dynamical reduction models, characteristic length, dark energy density
The Schr¨odinger equation when applied for system that is not isolated and interacts withcomplicated ”environment” (it is typical for macro-systems) results in appearance of the en-tanglement states involved many degrees of freedom. The superposition of such states for allpractical purposes results in the same outcome of measurements as ”mixed” state describedby diagonal density matrix. However it does not mean that superposition disappears, itstill exists globally but is unobservable at either system alone. The apparent decoherence(suppression of interference) for macro-systems can be explained then as result of the entan-glement of system with its environment (for a review see, e.g., [1]). The density matrix ofthe corresponding system in this approach is just means for calculating expectation valuesor probabilities for outcomes of measurements.Another approach to decoherence of macro-systems is grounded on idea of stochasticdynamical reduction (for a review see, e.g., [2, 3]) and treats the decoherence as real processof state vector reduction that takes place because corresponding system is influenced bystochastic forces, in other words, interacts with some stochastic environment. The nature ∗ E-mail: [email protected]
1f the stochastic field is usually not specified there, except for the dynamical reductionmodels involving gravity, so called Newtonian Quantum Gravity approach [4, 5, 6, 7, 8],where stochastic field is associated with non-relativistic gravity potential field. Reasons whygravity can be treated as stochastic field at small space-time scales are known for a longtime (see Ref. [9]): inasmuch as refined length measurement requires large momentum, aprobe itself disturbs the gravity field curving space-time and distorting the interval one seeksto measure leading to limit of measurability of gravity field (metric tensor). In fact, it isconsequence of the equivalence principle.Noteworthy that spatial reduction of wave function by stochastic forces leads to mo-mentum increase because of the uncertainty principle, and, therefore, to increase of particleenergy. It results in apparent violation of the conservation laws, and the rate of increasediverges for a point-like object [7]. To alleviate the problem the spatial coarse graining ofthe mass density (cutoff on spatial mass density resolution) is used in the dynamical reduc-tion models, this cutoff parameter fixes also the distance scale (localization width) beyondwhich the wave function collapse becomes effective. The value of the short-length cutoff,10 − cm, that was proposed in Ref. [10] (see also [7]), keeps energy non conservation ratebelow experimental limits. Certainly, to explicitly satisfy the conservation laws one needsaccount for not only particle contribution, but also contribution of the stochastic field to theconserved quantities.Because, as we discussed above, the most natural candidate for stochastic field of thedynamical reduction models is gravity, and a value of the scale factor governs the rateof energy transfer from stochastic field to the matter particles, it might be expected that avalue of the scale factor is related to the rate of decay of cosmological (gravitational) vacuumand, consequently, related to decrease of the vacuum energy density with time. Note thatcosmological scenarios of the universe evolution where Λ( t ) decreases slowly with time (Λrepresents the energy density of the vacuum) are allowed by the observational cosmology(see, e.g., [11, 12]).In this paper we utilize the correlation function of fluctuations of gravity acceleration field[4] (see also [5]) and viable model of the cosmological vacuum decay [13] to estimate valueof the spatial short-length cutoff parameter of the dynamical reduction models. In Section2 the cosmological model [13] with decaying vacuum energy density is briefly reviewed. Therate of the cosmological vacuum decay found in the model is utilized in Section 3 to calculatethe characteristic length of the dynamical reduction models involving gravity. In Section 4conclusions are given and some possible consequences for biogenesis are briefly discussed.
Let us first briefly review the cosmological vacuum decay scenario recently proposed in theRef. [13] (see also [12]). We choose this model among many others phenomenological Λ( t ) The relation of the particle energy increase with the loss of vacuum gravitational energy has been alsodiscussed in Refs. [3, 14]. ~ = c = 1) R µν − g µν R = 8 πG (cid:20) T µν + Λ8 πG g µν (cid:21) , (1)where T µν is the energy-momentum tensor of ”ordinary” (nearly 3% of total energy density)and ”dark” (nearly 27%) matter, while Λ is the cosmological constant that is responsible forfamous ”dark” energy density (nearly 70%) contribution to total energy density, and alsofor acceleration of the universe expansion at the present epoch. The cosmological constantcan be treated as the energy density of the cosmological vacuum, ǫ vac . Since a vacuum hasequation of state ǫ vac = − p vac , p is pressure, then (for recent review see, e.g., [16])Λ8 πG g µν = T µνvac = ǫ vac g µν . (2)The nature of ”dark” matter is still unclear, most probably (see, e.g., [17]) some weaklyinteracting particles are responsible for the ”dark” matter energy density contribution tothe energy-momentum tensor of matter fields.According to Bianchi identities ∂ ν ( R µν − g µν R ) = 0 , (3)therefore if Λ depends on time (i.e. vacuum decays in the course of the expansion) then T µν can not be separately conserved and there is a coupling between T µν and Λ that follows fromEq. (1): u µ ∂ ν T µν = − u µ ∂ ν (cid:18) Λ8 πG g µν (cid:19) , (4)here u µ ( x ) is local 4-velocity of cosmological expansion. The assumption of isotropy andhomogeneity implies that the large scale geometry can be described by a metric of the form ds = dt − a ( t ) d r , (5)here a ( t ) is scale (expansion) factor of the universe and flat spatial sections are also assumed.Taking then perfect fluid form for T µν , T µν = ( ǫ + p ) u µ u ν − pg µν , (6)one can get ˙ ǫ + ( ǫ + p ) ∂ α u α = − ˙Λ8 πG , (7)3here the overdot denotes covariant derivative along the world lines (time comoving deriva-tive, for instance, ˙ ǫ = u α ∂ α ǫ ). For homogeneous and isotropic Friedmann-Robertson-Walkergeometry ∂ α u α = 3 H, (8)where H ( t ) = ˙ aa (9)is the Hubble parameter, it measures the rate of expansion of the universe.Then, if vacuum transfers energy to matter, the question arises where the matter storesthe energy received from the vacuum decay process. The traditional approach for the vacuumdecay process is vacuum decay into matter particles. Following to Ref. [13], one can assumethat vacuum decay results (mainly) in creation of weakly interacting ”dark” matter particlesand, so, neglect a contribution of the ”ordinary” matter to the left hand side of Eq. (7).It allows to relate the time dependence of the energy density of the vacuum with temporalevolution of the ”dark” matter. Then, assuming that ”dark” matter is pressureless, p d = 0(cold ”dark” matter model), one can get the following equation:˙ ǫ d + 3 ˙ aa ǫ d ≈ − ˙ ǫ vac . (10)Because vacuum decays into ”dark” matter the latter will dilute more slowly comparedto its standard evolution, ǫ d ∼ a − , when the vacuum energy density does not change inthe course of the expansion, ˙ ǫ vac = 0. Then making a specific ansatz for the ”dark” matterenergy density ǫ d = ǫ d a − δ a − δ , (11)where δ > ǫ d = ǫ d ( t ) and a = a ( t ) are the current values of the ”dark” matter energy density andof the scale factor of the universe respectively, we have ǫ vac = e ǫ vac + δ − δ a − δ a − δ ǫ d (12)where e ǫ vac does not depend on time.It was found from analysis of cosmological data that δ = 0 . ± .
10 [12]. Such a relativelyslow decrease of Λ( t ) (vacuum energy density) indicates, perhaps, that viable cosmologicalvariable-Λ models with cold ”dark” matter (CDM) and standard physics should not differ toodrastically from concordance ΛCDM model to be compatible with observational cosmology.In the next Section we study whether the found in the Refs. [13, 12] rate of cosmologicalvacuum decay is compatible with the rate of particle energy increase presupposed in thedynamical reduction models, and, so, whether the corresponding characteristic length canbe defined by Λ( t )CDM cosmology. 4 Characteristic length for given rate of the cosmolog-ical vacuum decay
To perform the corresponding analysis let us assume that, while main energy gain from cos-mological vacuum decay is adopted by the ”dark” matter, the cosmological vacuum couplesnot only to the ”dark” matter but to the ”ordinary” matter as well, and assume that vac-uum decay does not lead to creation of ”ordinary” matter particles but increases the meankinetic energy of the ones. Then, if corresponding kinetic energy gain attributed to all theparticles in the universe is much smaller than the total loss of vacuum energy in the universe,it can not essentially influence on the results of Ref. [13] that are briefly reviewed in theprevious Section. The next step then is to relate the energy gain of a particle, whose mass islocally smeared within the corresponding characteristic volume, with the rate of energy lossof vacuum within the same volume. It gives us, in fact, upper limit to the particle energyincrease allowing by the loss of vacuum energy.For the sake of simplicity, let us consider a energy gain of a single nucleon. Let usassume that decay of the cosmological vacuum induces stochastic gravity field that results instochastic acceleration of a particle. Then the mean non-relativistic kinetic energy inducedby the vacuum decay is E part ( t ) = m h v ( r , t ) i = m t Z t i t Z t i h g ( r , t ′ ) g ( r , t ′′ ) i dt ′ dt ′′ , (13)where g and v are stochastic acceleration and velocity field respectively, h ... i means theaveraging over the corresponding characteristic volume V c , and we assume that initially v is equal to zero, v ( t i ) = 0. To proceed we need in correlation function of fluctuations ofacceleration field, g . The analysis of measurability of the Newtonian acceleration field thatwas done in Ref. [4] (see also [5]) results in the following expression for correlation function(assuming that h g i = 0): h g ( r , t ′ ) g ( r , t ′′ ) i ∼ GV c δ ( t ′ − t ′′ ) . (14)Hereafter we will take Eq. (14) as equality. Hence we obtain from Eqs. (13) and (14) that h v ( r , t ) i = G t Z t i dt ′ V c ( t ′ ) , (15)and then the rate of the particle energy gain, dE part dt , is dE part dt = mG V c ( t ) . (16)Taking into account the energy conservation, we get finally the equation mG V c ( t ) = − V c ( t ) ˙ ǫ vac ( t ) (17)5nd, therefore, V c ( t ) = (cid:18) − mG ǫ vac ( t ) (cid:19) / . (18)One can see from Eq. (12) that ˙ ǫ vac = − δ · ǫ d · ˙ aa · a − δ a − δ . (19)Then one can conclude from the above expressions that V c ( t ) increases (in cosmological sense,i.e. very slowly) with time.Now let us estimate the present ( t = t ) value of the characteristic volume, V c ( t ). Takinginto account that ˙ ǫ vac ( t ) = − δ · H · ǫ d , (20)where H is current value of the Hubble parameter, H = H ( t ), we get for the characteristicvolume V c ( t ) = (cid:18) mG δH ǫ d (cid:19) / (21)and, therefore, for the characteristic length R c ≡ V / c R c ( t ) = (cid:18) m · G δH ǫ d (cid:19) / . (22)The total energy density of the universe is close to the critical energy density (see, e.g.[16]), ǫ crit ( t ) = H πG . Then, because ”dark” matter energy density is approximately equal to27% of the total energy density, we get ǫ d = 0 . · H πG (23)and, finally, R c ( t ) = (cid:18) π · m · G · . · δ · H (cid:19) / . (24)Here δ = 0 . H = . · cm − = 0 . · − GeV, G = 1 /M P l , M P l = 1 . · GeV, and m is assumed to be proton mass, m = 0 .
938 GeV. Also taking into account that 1 Gev − =1 . · − cm we get finally R c ( t ) = 1 . · − cm. Then the characteristic volumemultiplied by the number of particles making up ”dark” and ”ordinary” matter is muchless than the volume of the universe, justifying thereby neglect of influence of the vacuumenergy transfer to the kinetic energy of particles on temporal evolution of the vacuum energydensity. 6nterestingly enough that practically the same value, 10 − cm, was proposed in Ref. [10]for phenomenological spatial cutoff parameter of Quantum Mechanics with SpontaneousLocalizations, and is accepted now [2, 18] as low limit for the characteristic length in thedynamical reduction models. Then our finding, perhaps, supports the gravity induced mech-anism of dynamical reduction.Note that m -dependence of R c is rather weak, R c ∼ m / , and therefore R c does notchange much even for electron mass. Also this m -dependence does not spoil the fast deco-herence of the distant, r ≫ R c , superposition of macro-objects, because then decoherencetime, t dec , is t dec ∼ R c /Gm [6] and, consequently, t dec ∼ m − / resulting, as well as for m -independent R c , in extremely small value that is desired property for Schr¨odinger catstates. Summarizing this work, we point out that apparent violation of the energy conservation isnot actually shortcoming of the dynamical reduction models. Moreover, energy conservationlaw gives deep insight into the physics of the spatial cutoff parameter. Namely, we found thatthe value, ∼ − cm, of the characteristic length is, perhaps, conditioned by the presentrate of energy transfer from decaying vacuum to a particle. Because a value of the rate ofthe vacuum decay need be compatible with the observational cosmology, one can say that,in a certain sense, cosmology dictates the characteristics of dynamical reduction.One can speculate that increase of V c with cosmological time (see Eq. (18)) is relatedto the problem of appearance of life and consciousness. First of all, note that biogenesison Earth has occurred very rapidly, during 0 . +0 . − . Gyr and, therefore, appearance of lifecould be highly probable process [19]. However, while age of Earth, ≈ .
566 Gyr, is muchlower than age of the universe, ≈ . , then spatial cutoff parameter value (that regulates the”borderline” between quantum and classical worlds) is extremely important for origin andoperation of biological organisms. Therefore an appropriate value of the cutoff parameter R c fixes the cosmological time when appearance of living organisms becomes possible. Note-worthy that formation of Earth fell roughly at the same time interval when expansion of theuniverse became accelerated [22]. So, one can speculate that necessary conditions for originof a life appear only recently, during the epoch of accelerated expansion when the character-istic length also increases with acceleration. Then the observational lack of extraterrestrialintelligent life can be consequence of the ”time cutoff” for the most early emergence of a lifein the universe. For instance, since quantum system can exist in superposition of states, searches of biologically potentmolecular configurations may proceed much faster than one might expect using classical estimates, leading,thereby, to the fast biogenesis (see, e.g., recent reviews [21] and references therein). eferences [1] E. Joos, H.D. Zeh, C. Kiefer, D. Guilini, J. Kupsch, and I.-O. Stamatescu, Decoherenceand the Appearance of a Classical World in Quantum Theory , 2nd edn., Springer,Berlin (2003); E. Joos, arXiv: quant-ph/9908008; M. Schlosshauer, Rev. Mod. Phys. , 1267 (2004) [arXiv:quant-ph/0312059].[2] A. Bassi, G.C. Ghirardi, Phys. Rep. , 257 (2003) [arXiv:quant-ph/0302164]; A.Bassi, arXiv:quant-ph/0701014.[3] P. Pearle, arXiv: quant-ph/0611211; arXiv:quant-ph/0611212.[4] L. Di´osi, B. Luk´acs, Annln. Phys. , 488 (1987).[5] L. Di´osi, Phys. Lett. A , 377 (1987); L. Di´osi, B. Luk´acs, Phys. Lett. A , 331(1989)[6] L. Di´osi, Phys. Rev. A , 1165 (1989); Braz. J. Phys. , 260(2005)[arXiv:quant-ph/0412154]; J. Phys. A , 2989 (2007) [arXiv:quant-ph/0607110];arXiv:quant-ph/0703170.[7] G.C. Ghirardi, R. Grassi, A. Rimini, Phys. Rev. A , 1057 (1990).[8] R. Penrose, Gen. Rel. Grav. , 581 (1996); Phil. Trans. Roy. Soc. Lond. , 1927(1998).[9] M. Bronstein, Phys. Ztschr. der Sowjetunion, , 140 (1936); M.F.M. Osborne, Phys.Rev. , 1579 (1949); T. Regge, Nuovo Cim. , 215 (1958); F. K´ a rolyh´ a zy, Nuovo Cim.A , 390 (1966).[10] G.C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D , 470 (1986).[11] J.A.S. Lima, Braz. J. Phys. , 194 (2004) [arXiv:astro-ph/0402109].[12] J.S. Alcaniz, Braz. J. Phys. , 1109 (2006) [arXiv:astro-ph/0608631].[13] J.S. Alcaniz, J.A.S. Lima, Phys. Rev. D , 063516 (2005) [arXiv:astro-ph/0507372].[14] P. Pearle, E. Squires, Found. Phys. , 291 (1996) [arXiv:quant-ph/9503019].[15] J.M. Overduin, F.I. Cooperstock, Phys. Rev. D , 043506 (1998)[arXiv:astro-ph/9805260].[16] P. J. E. Peebles, B. Ratra, Rev. Mod. Phys. , 559 (2003) [arXiv: astro-ph/0207347]; T.Padmanabhan, Phys. Rept. , 235 (2003) [arXiv:hep-th/0212290]; T. Padmanabhan,AIP Conf.Proc. , 179 (2006) [arXiv:astro-ph/0603114].[17] D. Majumdar, arXiv:hep-ph/0703310. 818] S.L. Adler, J. Phys. A , 2935 (2007) [arXiv:quant-ph/0605072].[19] C.H. Lineweaver, T.M. Davis, Astrobiology , 293 (2002) [arXiv:astro-ph/0205014];arXiv:astro-ph/0209385.[20] K.D. Olum, Analysis , 1 (2004) [arXiv:gr-qc/0303070].[21] P.C.W. Davies, arXiv:quant-ph/0403017; Int. J. Astrobiol.2