<<

Discrete space-time, collapse and

Gao Shan

The Scientists Work Team of Electro-Magnetic Velocity,

Chinese Institute of Electronics

LongZeYuan 24-3-501, HuiLongGuan, ChangPing District

Beijing 102208, P.R.China

E-mail: [email protected]

The relation between discrete space-time, quantum collapse and quantum gravity is analyzed.

A possible way to combine quantum and gravity in terms of the quantum collapse in discrete

space-time is proposed. It has been shown that the combination of quantum theory and general

relativity results in the existence of discrete space-time, in which there exist a minimum time

interval and a minimum length. The physical meaning of discrete space-time is analyzed. We

CERN-EXT-2004-108 16 September 2004 further argue that the discreteness of space-time may inevitably result in the collapse of wave

function. A quantum collapse model in the discrete space-time is briefly introduced. A possible

way to combine quantum and gravity is finally proposed based on the discrete space-time and

quantum collapse. We argue that it may provide a consistent framework for a fundamental theory

of quantum gravity.

Key words: discrete space-time, quantum collapse, quantum gravity

PACS: 04.60 Introduction

Quantum theory and general relativity are two most fundamental physical theories of our times. Quantum theory describes the quantum of matters in a fixed space-time, and general relativity describes the gravitation between matters in classical motion, in which space-time is dynamical and influenced by the classical motion of matters. Whereas matters generally undergo quantum motion and gravitation universally exists between matters, a theory combining quantum and gravity should be reasonably expected in order to provide a complete and consistent account of space-time and motion of matters. Such to-be-found theory has been called the theory of quantum gravity.

However, how to combine quantum and gravity turns out to be one of the hardest problems

[1]. Quantum theory and general relativity are not only incomplete severally, but also incompatible together. Each of the two theories is unable to describe the relation between space-time and quantum motion or the quantum motion in dynamical space-time. Furthermore, their views on how to describe such motion conflict with each other [2-5]. Quantum theory requires a presupposed fixed and definite space-time underlying the quantum motion of matter, but the space-time is dynamical and determined by the motion of matter in general relativity.

Concretely speaking, there exists a profound conflict between the in quantum theory and the principle of general covariance in general relativity. Quantum theory requires a definite space-time and rejects the superposition of different space-times, whereas according to general relativity, the superposition of different space-times seems to be an inevitable result of the quantum motion of matters.

Then how to combine the two most successful but incompatible theories? A natural way is to let them split the difference each other. This means that in the situations where the gravity produced by the matters in quantum motion is weak enough, the space-time will be not influenced by the quantum motion of matters and can be fixed, and the quantum theory for a fixed space-time is valid, and in the situations where gravity produced by the matters is strong enough, the motion of matters will become classical motion and space-time is still dynamical, and the general relativity for classical motion of matters is valid. In the middle situations, there exists some kind of superposition of a little different space-times in which the quantum motion of matters evolves according to quantum theory, and the space-time is continuously changed by the quantum motion of matters in a dynamical way according to general relativity. When the difference between the space-time branches in the superposition is very small, they can be physically taken as the same space-time, and this provides a definite space-time framework for the quantum evolution. When the difference between the space-time branches in the superposition becomes large enough due to the evolution of quantum motion of matters, the whole superposition collapses to one of the definite space-time branches, then the quantum motion of matters also evolves in the collapsed definite space-time. This process will ceaselessly proceed due to the above interaction between space-time and quantum motion of matters. However, this way to combine quantum and gravity demands the existence of an inherent fuzziness in the space-time and the existence of the collapse of . Concretely speaking, the space-time branches in the superposition can be physically taken as the same space-time when their difference is smaller than the inherent fuzziness. This provides a definite space-time framework for the evolution of the of matters. When the difference between the space-time branches in the superposition becomes larger than the inherent fuzziness due to the quantum evolution, the whole superposition collapses to one of the definite space-time branches. Such collapse guarantees that the new quantum state also evolves in a definite space-time. Surprisingly, the inherent space-time fuzziness and the quantum collapse may be required by the combination of quantum theory and general relativity. It has been widely demonstrated that there exists a minimum Planck space-time size, which just denotes the fuzziness or discreteness of space-time [6-15]. It has been also argued that the incompatibility between quantum theory and general relativity may require the existence of quantum collapse [2].

As we will see, the fuzziness of space-time may also result in the existence of quantum collapse.

Thus it seems that quantum theory and general relativity have been ready for their passionate combination, and the above way may be a consistent and natural one to combine quantum and gravity.

In this paper, we will analyze the above combination way in detail. In section 2 the discrete space-time is briefly introduced. The physical meaning of discrete space-time is analyzed. It is denoted that the discrete space-time may be a foundation stone of a complete theory of quantum gravity. In section 3 we argue that the discreteness of space-time may naturally result in the quantum collapse of wave function, which is the other element in the way to combine quantum and gravity. A possible quantum collapse model in the discrete space-time is briefly introduced. In section 4 we present a possible way to combine quantum and gravity based on the discrete space-time and quantum collapse. The consistency of such combining way is discussed.

Conclusions are given in the last section.

The discrete space-time

Quantum theory and general relativity are both based on the continuous space-time assumption. However, the appearance of infinity in and singularity in general relativity has implied that space-time may be not continuous but discrete. In fact, it has been widely argued that the proper combination of quantum theory and general relativity may inevitably result in the discreteness of space-time [6-15], and a complete theory of quantum gravity must be founded on such discrete space-time [15].

In the discrete space-time, there exists a minimum time interval 2TP and a minimum length G G 2 L , where T = ( h)1/ 2 , L = ( h)1/ 2 is respectively the Planck time and Planck length. P P c 5 P c 3 The physical meaning of such discrete space-time is that any space-time difference smaller than the minimum time interval and minimum length is in principle indistinguishable, i.e., the space-time with a difference smaller than the minimum sizes are physically identical. It further means that any physical existence should no longer be defined in a position at an instant, but be defined in the minimum space-time unit, especially the duration of any change should be no

shorter than the minimum time interval 2TP . In the discrete space-time, there doesn’t exist deeper space-time structure beneath the minimum space-time unit, and the point-particle picture in continuous space-time should be replaced by some kind of extending existence (e.g. string or brane) in the discrete space-time. This character will radically ensure the finiteness of the physical predictions of the theories based on such discrete space-time. In this meaning, it may be more reasonable to found a complete theory of quantum gravity directly on the discrete space-time. It should be noted that the holography principle, which is widely taken as one of the basic principles of quantum gravity theory [15-17], may also be a direct result of the physical requirement of discrete space-time.

Although the space-time is essentially discrete or fuzzy as required by the proper combination of quantum theory and general relativity, it doesn’t mean there should exist the quantum states of space-time or the quantum superpositions of different space-times. On the contrary, it may imply that the superposition of very different space-time can’t exist at all. As we know, the position and momentum of a particle can be in a superposition state, and there exists an uncertainty relation between them as a result of such superposition in the quantum theory. Besides, a bound quantum system can only possess discrete energies, and this kind of discreteness also results from the existence of quantum superposition. Thus it seems that the existence of the minimum space-time uncertainty should also result from some kind of quantum superposition of different space-times. However, the conclusion may be premature.

First, the existence of quantum superposition of some property essentially relies on the presupposition that the property can be measured in arbitrary precision, i.e., there must exist the eigenstates of the property in which the property possesses an infinitely precise value. If there exist no such eigenstates, then the superposition of them is surely meaningless. This stringent condition is not satisfied for the space-time itself, since it can’t be measured in arbitrary precision, and there exists a minimum space-time uncertainty. Secondly, the minimum uncertainty appearing in quantum theory only results from the product of the uncertainty of one pair conjugate variables such as position and momentum. The variable itself doesn’t possess such minimum uncertainty, and can have a precise value. This is very different from the uncertainty situation appearing in discrete space-time, where the minimum uncertainty is possessed by the space-time variable itself.

The intrinsic difference indicates that these two kinds of uncertainties may have different origins.

Thus the minimum uncertainty of discrete space-time may not result from the quantum superposition as in quantum theory. For example, even in the noncommutative space-time model where there exists a noncommutative relation [x u , x v ] = iθ uv [18], only the product of two lengths or the area, not the length itself, can possess the minimum uncertainty. In short, quantum superposition generally results in the minimum product of the uncertainty of the values of two conjugate variables, not the minimum uncertainty of the value of one variable.

As we will further demonstrate in the next section, the existence of a minimum space-time uncertainty or fuzziness may actually result in the non-existence of the quantum superposition of very different space-times.

Quantum collapse in discrete space-time

As we know, the most serious problem in quantum theory is the . The existing quantum theory doesn't tell us how and when the measurement result appears. The projection postulate is just a makeshift [19]. In this sense, the existing quantum theory is an incomplete description of the realistic process. Besides, mainly due to the research in [20-21], physicists have come to realize that the measurement process does not need to be related to the observer (as the orthodox view requires), but must be taken as a self-acting process of the wave function. Therefore it is natural to combine the normal linear evolution with the instantaneous collapse process to form a unified evolution process, where the normal linear evolution and the instantaneous collapse process are only two ideal approximations of the unified evolution process. The resulting theory is generally called revised , and has been widely and deeply studied in recent times [22-31].

An important problem of revised quantum dynamics is the origin of quantum collapse. It may be very natural to guess that the dynamical collapse of wave function is induced by gravity. The reasons include: (1) gravity is the only universal force being present in all physical interactions; (2) gravitational effects grow with the size of the objects concerned, and it is in the context of macroscopic objects that linear superpositions may be violated. The gravity-induced collapse conjecture can be traced to Feynman [32]. In his Lectures on Gravitation, he considers the philosophical problems in quantizing macroscopic objects and contemplates on a possible breakdown of quantum theory. He said, “I would like to suggest that it is possible that quantum fails at large distances and for large objects, …it is not inconsistent with what we do know. If this failure of is connected with gravity, we might speculatively

2 −5 expect this to happen for masses such that GM / hc = 1, of M near 10 grams” .

Penrose further strengthened the gravity-induced collapse argument [2]. He argued that the superposition of different space-time is physically improper, and the evolution of such superposition can’t be defined in a consistent way. This requires that a quantum superposition of two space-time geometries, which corresponds to two macroscopically different distributions of energy, should collapse after a very short time. Penrose’s gravity-induced collapse argument reveals the profound and fundamental conflict between the general covariance principle of general relativity and the superposition principle of quantum mechanics. According to general relativity, there exists one kind of dynamical connection between motion and space-time, i.e., space-time is determined by the motion of particles, at the same time, the motion of particles must be defined in space-time. Then when we consider the superposition state of different positions of a particle, say position A and position B, one kind of logical inconsistency appears. On the one hand, according to quantum theory, the valid definition of such superposition requires the existence of a definite space-time background, in which the position A and position B can be distinguished. On the other hand, according to general relativity, the space-time, including the distinguishability of the position A and position B, can’t be predetermined, and it must be dynamically determined by the superposition state of particle. Since the different position states in the superposition will determine different space-time, the space-time determined by the superposition state is indefinite.

Then an essential logical inconsistency between quantum theory and general relativity does appear.

The inconsistency requires that the quantum superposition of different space-time can’t exist in a precise way, and should collapse after a very short time. This conclusion only relies on the validity of general relativity in the classical domain, and is irrelevant to its validity in the quantum domain.

Thus gravity may indeed be the physical origin of collapse of wave function.

However, Penrose’s argument has one deficiency. In fact, the above logical inconsistency will require that the quantum superposition of different space-time can’t exist at all, and should collapse instantaneously. This doesn’t accord with the experimental facts. If space-time is indeed continuous, then Penrose’s argument will inevitably fail. But if space-time is discrete, and there exist a minimum time interval and a minimum length, then his argument can be reinforced and thus succeed. The key point is that two space-times with a difference smaller than the minimum sizes is the physically same space-time in the discrete space-time. The fuzziness of discrete space-time will permit the quantum superposition of two space-times with a difference smaller than the minimum sizes to form and then collapse after a finite time interval. But the quantum superposition of two space-times with a difference larger than the minimum sizes can’t exist, and should collapse instantaneously. In fact, the collapse should have completed before such quantum superposition forms. Such dynamical collapse of wave function can accord with the experimental facts.

In the following, we will give a new argument supporting the above gravity-induced collapse proposal. It will be argued that the discreteness or fuzziness of space-time, which results from the proper combination of quantum theory and general relativity, may also inevitably result in the dynamical collapse of wave function.

Consider a quantum superposition of two energy eigenstates. Assume the two energy eigenstates can be defined in the same space-time, then it can be written as follows:

ψ 1 ϕ ϕ (x,0) = [ E (x) + E (x) ] ------(1) 2 1 2 where ϕ (x) and ϕ (x) are two energy eigenstates with the energy eigenvalues E and E1 E2 1

E2 . According to the linear Schrödinger evolution, we have:

1 − − ψ iE1t / h ϕ iE2t / h ϕ (x,t) = [ e E (x) + e E (x) ] ------(2) 2 1 2 and

1 2 2 ρ(x,t) =|ψ (x, t) |2 = [ϕ (x) + ϕ (x) + 2ϕ (x)ϕ (x)cos(∆E ⋅ t/ )] ------(3) 2 E1 E2 E1 E2 h This result indicates that the probability density ρ(x,t) will oscillate with a period T = h / ∆E

∆ = − in each position of space, where E E2 E1 is the energy difference.

Now consider an extreme situation in which the energy difference ∆E is so large that it exceeds the energy E p / 2 , where E p = h / Tp is the Planck energy. Then the probability

ρ density (x,t) will oscillate with a period shorter than the time interval 2TP . But as we know,

the time interval 2TP is the minimum distinguishable size of time in the discrete space-time, and

no change can happen during the time interval shorter than the minimum time interval 2TP . Thus the energy superposition state in which the energy difference is larger than one half of the Planck energy can’t hold, and must collapse to one of the energy eigenstates.

Further analysis shows that a viable collapse model in discrete space-time can be founded in terms of the above energy-driven collapse mechanism [29][31]. The evolution equation of wave function generally contains a linear Schrödinger term and a stochastic nonlinear term resulting in the dynamical collapse of wave function. The resulting wavefunction collapse law can indeed bring the collapse results predicted by present quantum theory. According to such model, the collapse time formula is as follows: hEp τ ≈ ------(4) c (∆E)2

It should be denoted that this collapse time formula has also been guessed by Fivel in terms of the

0 analysis of K L decay [28]. Besides, Percival and Hughston also assumed the similar collapse time formula in their wavefunction collapse models [26-27].

Since different energy distributions correspond to different space-times according to general relativity, the above analysis has also shown that the superposition of very different space-times can’t hold, and must collapse to one of the definite space-times. In fact, we can also reach the same conclusion from an analysis of the difference of space-time in the discrete space-time.

Assume the above energy eigenstates be those of micro-black holes. Then the difference of the space-times corresponding to the above energy eigenstates may be characterized by the difference of the radiuses of the micro-black holes: ∆ ∆ ∆ ∆ = 2G M = 2G E = E rS 2 4 2LP ------(5) c c EP

Since there exists an inherent space uncertainty or fuzziness in the discrete space-time, which

equals to 2 LP , the space-times with a difference larger than the minimum uncertainty are physically different, and the quantum superposition of them can’t hold due to the essential logical inconsistency. In other words, the superposition of space-time can only possess a space-time uncertainty permitted by the discrete space-time. If such uncertainty limit is exceeded, the superposition will collapse to one of the definite space-times with smaller uncertainty. Then the energy superposition state of micro-black holes, in which the difference of the corresponding space-times of the energy eigenstates is larger than the minimum uncertainty, can’t hold, and must

∆ = collapse to one of the energy eigenstates with definite space-time. This requires rS 2LP , and

∆ = thus we get E E p again. The result reaffirms the equivalence between the energy-driven collapse and the gravity-induced collapse resulting from the inherent discreteness of space-time.

For a more general situation, there exists a local point-to-point mapping between two space-time branches in a quantum superposition state, and the difference of these two space-time branches can be defined as the maximum difference of local space-time distance in the unit of minimum space-time size, which can be written as follows:

= ∆ 2 2 D Max( ds / 4LP ) ------(6)

When the difference of these two space-time branches becomes so large that a local point-to-point mapping between them does no longer exist, the quantum superposition state will have collapsed to one of the branches. This guarantees that the above formula can be consistently defined. As an example, we consider the following Schwarzschild metric:

r − r ds 2 = (1− S ) 1 dr 2 + r 2 dθ 2 + r 2 sinθ 2 dφ 2 − (1− S )c 2 dt 2 ------(7) r r 2GM where r = is the Schwarzschild radius. The difference of two space-times with different S c 2 mass M or rS can be written as follows:

∆r = ∆ 2 2 ≈ 1 S 2 + 2 2 = ∆ D Max( ds / 4LP ) 2 Max[ (dr c dt )] rS / 2LP ------(8) 4Lp r

Here we assume the first rank approximation of rS and consider the minimum space-time size in the discrete space-time. This result is consistent with the above analysis.

Lastly, it should be noted that the existence of discrete space-time may also imply that the many-worlds theory is not right [33-36], and the dynamical collapse of wave function does happen. Since there exists a minimal time interval in discrete space-time, and each parallel world must solely occupy one minimal time interval at least, there must exist a maximal number of the parallel worlds during any finite time interval. Then when the number of possible parallel worlds exceeds the maximal number, they will be merged in some way, i.e., the whole wave function will collapse to a smaller state space. This may indicate the occurrence of the collapse of wave function.

A possible way to combine quantum and gravity

An immense amount of effort has been devoted to combining quantum and gravity [37]. Yet although a great deal has been learned in the course of this endeavor, there is still no satisfactory theory. The present approaches still face severe problem, both technical and conceptual [3][38]. It has been argued that it may be improper to quantize the gravitational field in a theory of quantum gravity [2-5]. The reasons include that the metric tensor may be not a fundamental field, and the gravitational field is concerned with the structure of space-time which is fundamentally classical in nature etc. If it is indeed wrong to quantize the gravitational field, then how can we combine quantum and gravity in a consistent way? In this section, we will argue that the quantum collapse in discrete space-time may provide a consistent way to combine quantum and gravity along this direction.

According to the above analysis, when the difference of the space-time geometries corresponding to the braches of a quantum superposition is larger than the minimum Planck size of space-time, the superposition state must collapse to one of the branches, and correspondingly the superposition of space-times also collapses to a definite space-time. In short, there is no quantum superposition of different space-time geometries which difference is larger than the minimum Planck size of space-time. Since the energy difference of the braches of a quantum superposition of macroscopic objects is generally much larger than the Planck energy due to the environmental influences [29][39], and the difference of the corresponding space-time geometries is also much larger than the minimum Planck size of space-time, the space-time geometry determined by the motion of macroscopic objects is always definite. The theory describing such situations is approximately the general relativity.

When the difference of the space-time geometries corresponding to the braches of a quantum superposition of microscopic particles is much smaller than the minimum Planck size of space-time, the quantum motion of microscopic particles doesn’t influence the definite background space-time determined by the macroscopic environment. Thus the quantum states of microscopic particles and their evolution can be consistently defined. The resulting theory is approximately the quantum theory in curved space-time [40-41].

When the difference of the space-time geometries corresponding to the braches of a quantum superposition of microscopic particles is close to the minimum Planck size of space-time, the collapse process of the quantum superposition will happen frequently. As a result, the definite background space-time is also influenced by the quantum collapse, and undergoes an intrinsic stochastic fluctuation. Whereas the background space-time undergoing the fluctuations is still definite in each moment, the quantum states of microscopic particles and their evolution can also be consistently defined. The resulting new theory should be a quantum theory in the stochastic curved discrete space-time. It can be formally written as follows: = π ψ Gµν 8 Tµν ( s ) ------(9) dψ ^ ^ g s = Hψ + Sψ ------(10) 00 dt s s ψ where Gµν is the definite Einstein tensor and Tµν ( s ) is one of the eigenvalues of the

ψ ψ stress-energy in the quantum state s . The quantum state s satisfies a stochastic

nonlinear evolution equation in the curved discrete space-time with the metric g µν , which

^ ^ ψ ψ contains a linear Schrödinger term H s and a stochastic nonlinear term S s resulting from the dynamical collapse of wave function [29]. A detailed analysis of the theory will be given in another paper.

It should be stressed that the quantities in the above equations are all defined in the discrete space-time, and all contain the minimum space-time fuzziness in the direct or indirect way. This means the quantum and gravity in the combination should both be revised by the discreteness of space-time, and the last theory of quantum gravity will be the combination of the revised quantum and gravity in the above way.

Lastly, we give a primary analysis about the revised quantum and gravity in the discrete space-time. The relation between the continuous space-time and discrete space-time may be written as follows:

2 Lp ∆x = ∆x (1+ ) ------(11) d c ∆ 2 xc

2 Tp ∆t = ∆t (1+ ) ------(12) d c ∆ 2 tc ∆ ∆ ∆ ∆ where xc 、 tc are the sizes in continuous space-time, and xd 、 td are the corresponding sizes in discrete space-time. As we think, both quantum theory and general relativity need to be revised in the discrete space-time according to the above relation. For example, the relativistic boost factor is revised as follows: γ = 1 d ------(13) 2 2 2L 2 − v + v p + 4 1 2 2 2 O(Lp ) c c l0 where l0 is the intrinsic length. It can be also written in energy form:

γ = 1 d ------(14) 2 2 2E 2 − v + v 0 + 1 1 2 2 2 O( 4 ) c c E p E p

where E0 is the rest energy. Then the corresponding dispersion relation is:

2E 2 2 = 2 4 + 2 2 − 0 2 2 + 1 E m0 c p c 2 p c O( 4 ) ------(15) E p E p

It should be noted that this dispersion relation is consistent with the recent analysis of the high energy gamma ray [42]. Besides, the revised quantum uncertainty relation will be:

∆x ⋅ ∆p ≥ h + 2L 2 ∆p 2 ------(16) 2 p This is just the generalized (GUP)[13].

On the whole, the above analysis has shown that the quantum and gravity can be consistently combined with the help of the quantum collapse in discrete space-time. In this way, there is no quantized gravity in the usual meaning. If the logical inconsistency between quantum theory and general relativity does exist, the above way may be the only consistent way to combine quantum and gravity.

Conclusions

In this paper, we analyze the relation between discrete space-time, quantum collapse and quantum gravity, and propose a possible way to combine quantum and gravity in terms of the quantum collapse in discrete space-time. The physical meaning of discrete space-time is analyzed.

We argue that the discreteness of space-time may inevitable result in the quantum collapse of wave function. A possible collapse model in the discrete space-time is briefly introduced. A possible way to combine quantum and gravity is proposed based on the discrete space-time and quantum collapse. Different from the usual semi-classical theory of quantum gravity, it may provide a consistent framework for a fundamental theory of quantum gravity. Certainly, the discrete property of space-time still needs to be studied, though our analysis implies that space-time may be not a dynamical entity possessing quantum properties.

References

[1] C. J. Isham, in Quantum Gravity 2: A Second Oxford Symposium, eds. C. J. Isham, R.

Penrose and D.W. Sciama, (Oxford University Press, Oxford, 1981).

[2] R. Penrose, Gen. Rel. and Grav. 28 (1996) 581-600.

[3] J. Butterfield, C. J. Isham, in Physics meets Philosophy at the Planck Scale, eds. C. Callender

and N. Huggett, (Cambridge University Press, Cambridge, 2001).

[4] J. Christian, in Physics meets Philosophy at the Planck Scale, eds. C. Callender and N.

Huggett, (Cambridge University Press, Cambridge, 2001).

[5] S. Weinstein, in Physics meets Philosophy at the Planck Scale, eds. C. Callender and N.

Huggett, (Cambridge University Press, Cambridge, 2001).

[6] A. Einstein, Journal of the Franklin Institute, 221 (1936) 378.

[7] H.S. Snyder, Phys Rev 71 (1947) 38.

[8] W. Heisenberg, Rev. Mod. Phys 29 (1957) 269.

[9] H. Salecker and E. P. Wigner, Phys Rev 109 (1958) 571.

[10] L. J. Garay, Int. J. Mod. Phys A10 (1995) 145-166.

[11] C. Rovelli and L. Smolin, Nuclear Physics B 442 (1995) 593.

[12] J. Polchinski, String Theory, (Cambridge University Press, Cambridge, 1998).

[13] R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14 (1999) 1371

[14] G. Amelino-Camelia, Nature 408 (2000) 661-664

[15] L. Smolin, Three Roads to Quantum Gravity (Weidenfeld and Nicolson and Basic Books,

London and New York, 2001).

[16] G. ’t Hooft, LANL e-print gr-qc/9310026.

[17] L. Susskind, J. Math. Phys. 36, 6377 (1995).

[18] A. Connes, Non Commutative Geometry. (Academic Press, New York, 1994).

[19] J. S. Bell, in The Ghost In the Atoms, eds P.C.W.Davis et al, (1986).

[20] B. S. DeWitt, Phys.Rev. 160, (1967) 1113.

[21] J.B.Hartle and S.W.Hawking, Phys.Rev. 28, (1983) 2960. [22] P. Pearle, Phys. Rev. A 39, (1989) 2277- 2289.

[23] L. Diosi, Phys. Rev. A, 40, (1989) 1165-1174.

[24] G.C.Ghiradi, A.Rimini and T.Weber. Phys. Rev. D, 34 (1986) 470-491

[25] G.C.Ghiradi, P.Pearle and A.Rimini. Phys. Rev. A, 42 (1990) 78-89

[26] I.C.Percival, Proc. Roy. Soc. Lond. A, 447, (1994) 189-209

[27] L.P.Hughston, Proc.Roy.Soc.Lond.A, 452, (1996) 953

[28] D.I.Fivel, Phys. Rev. A 56 (1997) 146-156.

[29] Gao Shan, Quantum Motion and Superluminal Communication (Chinese B&T Publishing

House, Beijing, 2000).

[30] S.L.Adler, Todd A. Brun, J. Phys. A 34, (2001). 4797-4809.

[31] Gao Shan, Quantum (Tsinghua University Press, Beijing, 2003).

[32] R. Feynman, Feynman Lectures on Gravitation, eds. B. Hatfield, (Reading, Massachusetts,

Addison-Wesley, 1995).

[33] H.Everett, Rev.Mod.Phys, 29, (1957) 454-462

[34] DeWitt, B. S. and N. Graham (eds): The Many-Worlds Interpretation of Quantum Mechanics,

( Princeton University Press, Princeton, 1973).

[35] D.Deutsch, Int. J. Theor. Phys. 24, (1985) 1-41

[36] D.Guilini, E. Joos, C. Kiefer, J. Kupsch, I.O. Stamatiscu, and H.D. Zeh, Decoherence and the

Appearance of a Classical World in Quantum Theory, (Springer-Verlag, Berlin, New York,

1996)

[37] C. Rovelli, LANL e-print gr-qc/0006061.

[38] L. Smolin, LANL e-print hep-th/0303185.

[39] S. L. Adler, L. P. Horwitz, LANL e-print quant-ph/9909026.

[40] L.H. Ford, LANL e-print gr-qc/9707062

[41] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, (

Chicago University Press, Chicago, 1994).

[42] F.W. Stecker, Astropart. Phys. 20 (2003) 85-90.