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.