A naive way to combine quantum and gravity

Gao Shan

The Scientists Work Team of Electro-Magnetic Wave Velocity,

Chinese Institute of Electronics

LongZeYuan 24-3-501, HuiLongGuan, ChangPing District

Beijing 102208, P.R.China

E-mail: [email protected]

We propose a naïve way to combine quantum and gravity in terms of the quantum collapse in

discrete space-time. It is denoted that the combination of quantum theory and general relativity

requires the existence of discrete space-time. The physical meaning of discrete space-time is

analyzed. We further argue that the discreteness of space-time may naturally result in the quantum

collapse of wave function. A possible collapse model in the discrete space-time is briefly

introduced. A natural way to combine quantum and gravity is finally proposed based on the

EXT-2004-001 01/01/2004 discrete space-time and quantum collapse. We argue that it may provide a consistent framework

for a fundamental theory of quantum theory.

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 motion of matter in a fixed space-time, and general relativity describes the gravitation between classical matters or the interaction between the dynamical space-time and the classical motion of matter. 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 a 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 interaction 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 space-time structure underlying the quantum motion of matter, but the space-time structure is dynamical and determined by the motion of matter in general relativity.

Concretely speaking, there exists a profound conflict between the superposition principle in quantum theory and the principle of general covariance in general relativity. Quantum theory rejects the superposition of different space-time, whereas according to general relativity, the superposition of different space-time 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-time 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 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 sequentially evolve 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. 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. When the difference between the space-time branches in the superposition becomes larger than the inherent fuzziness, the whole superposition collapses to one of the definite space-time branches. Surprisingly, the inherent space-time fuzziness is a natural result of the proper combination of quantum theory and general relativity [6-15]. It has been widely demonstrated that there exists a minimum Planck space-time size, which just denotes the fuzziness or discreteness of space-time. 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 collapse model in the discrete space-time is briefly introduced. In section 4 we present a natural way to combine quantum and gravity based on the discrete space-time and quantum collapse. The consistency of such 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 quantum field theory 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 c3 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 structures 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 in 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], is just 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-time structures. On the contrary, it may imply that the superposition of very different space-time can’t exist at all. As we know, in the quantum theory 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. 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, and the states of the property with different values are in principle indistinguishable. 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, what the combination of quantum theory and general relativity requires is only the existence of a minimum space-time uncertainty or fuzziness, not the existence of a serious of discrete values of space-time size. Thus it doesn’t further require that there should exist the quantum superpositions of different space-time structures which may result in such discrete spectrum. In fact, the existence of such discrete space-time with a minimum fuzziness but without quantum superposition may be more natural in comparison with continuous space-time. A minimum unit in discrete space-time just corresponds to a point in continuous space-time.

Lastly, 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 can only result in the minimum product of the uncertainty of the values of two conjugate variables, not the minimum uncertainty of the value of a variable. As we will further demonstrate in the next section, the existence of a minimum space-time uncertainty or fuzziness can actually result in the non-existence of the quantum superposition of very different space-time structures. Quantum collapse in discrete space-time

As we know, the most serious problem in quantum theory is the measurement problem. 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 quantum cosmology [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 quantum dynamics, 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 mechanics fails at large distances and for large objects, …it is not inconsistent with what we do know. If this failure of quantum mechanics 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 structure, 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 structures, the space-time structure 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 dynamical collapse of wave function.

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. It can be written as follows:

1 ψ (x,0) = [ϕE (x) +ϕE (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 −iE1t / h −iE2t / h ψ (x,t) = [ e ϕE (x) + e ϕE (x) ] ------(2) 2 1 2 and 1 ρ(x,t) =|ψ (x, t) |2 = [ϕ 2 (x) + ϕ 2 (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

density ρ(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 unit 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, in which no change occurs during the time interval shorter than the minimum time unit.

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:

1 hE p τ ≈ ------(4) c 2 (∆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 meson decay [28]. Besides, Percival and Hughston also assumed the same collapse time formula in their wavefunction collapse models [26-27].

Since different energy distributions correspond to different space-time structures according to general relativity, the above analysis has also shown that the superposition of very different space-time structures can’t hold, and must collapse to one of the definite space-time structure.

Considering consistency, it is implicitly assumed that the different energy eigenstates in the superposition are defined in the approximately same space-time in the above analysis. In fact, we can also reach the same conclusion from the analysis of the difference of space-time structure in the discrete space-time. Assume the above energy eigenstates be those of micro-black holes. Then the difference of the space-time structures 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-time structures 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 structures 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-time structure with smaller uncertainty.

Then the energy superposition state of micro-black holes, in which the difference of the corresponding space-time structures 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

structure. This requires ∆l = 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.

It should be denoted that the difference of space-time structures may be generally defined by considering the simple Schwarzschild metric. The metric can be written as follows:

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 ------(6) r r 2GM where r = is the Schwarzschild radius. The difference of space-time structures can be S c 2 represented by the difference of the metrics ∆ds 2 , which can be written as follows:

∆r ∆ds 2 = S (dr 2 + c 2 dt 2 ) ------(7) r Here we assume the first rank approximation of rS . Thus considering the minimum uncertainty of

space-time, the comparable difference of space-time structures may be defined as ∆rS . This general result is consistent with the above analysis.

Lastly, it should be denoted 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 naive 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 endeavour, 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 possible way to combine quantum and gravity.

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 uncertainty of space-time, the superposition state must collapse to one of the branches, and correspondingly the superposition of space-times also collapse to a definite space-time structure. In short, there is no quantum superposition of different space-time geometries which difference is larger than the minimum unit 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 unit of space-time, the space-time geometry determined by the motion of the macroscopic objects is always definite. This also provides a definite space-time background for the quantum motion of microscopic particles. The resulting theory 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 uncertainty of space-time, the quantum motion of microscopic particles doesn’t influence the definite background space-time determined by the 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 uncertainty 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 theory should be a quantum theory in the stochastic curved space-time with a minimum uncertainty. It can be generally formulated as follows:

Gµν = 8πTµν (ψ s ) ------(8)

where Gµν is the definite Einstein tensor and Tµν (ψ s ) is the eigenvalue of the stress-energy

operator in the quantum state ψ s . The quantum state ψ s satisfies a stochastic nonlinear

evolution equation in the curved discrete space-time with the metric g µν , which contains a linear

Schrödinger term and a stochastic nonlinear term resulting in 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 uncertainty or 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. The revision of gravity due to the discreteness of space-time seems to be plain. It requires that the Einstein tensor Gµν be defined in the discrete space-time, and contains the minimum space-time fuzziness. The general relativity can then be formulated in the discrete space-time. On the other hand, the revision of quantum due to the discreteness of space-time may have important results. For example, the de Broglie relation may be revised as follows:

2 h Lp p λ = + ------(9) p h

2 h Tp E T = + ------(10) E h

The direct inference of these relations is λ ≥ 2Lp and T ≥ 2Tp , which is consistent with the discreteness of space-time. Furthermore, the revised uncertainty relation will be:

2 2 ∆x ⋅ ∆p ≥ h + Lp ∆p ------(11)

One of the results of the revised uncertainty relation is ∆x ≥ 2L p , which is also consistent with

2 the discreteness of space-time. When assuming p = mvg and E = mc , the revised dispersion relation is as follows:

2 2 2 2 4 2 2 4 E − p c = m0 c (1+ 3LP p ) + O(Lp ) ------(12)

This relation is consistent with the recent analysis of the high energy gamma ray observations [42].

Considering the above revision of quantum due to the discreteness of space-time, the linear

∂ 2 Schrödinger term will contain the first rank revision term L 2 . p ∂x 2

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 propose a natural 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 naïve way to combine quantum and gravity is proposed based on the discrete space-time and quantum collapse. Different from the usual semi-classical quantum gravity theory, 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.

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