Discrete space-time, quantum collapse and quantum 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] 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 motion 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 superposition principle 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 wave function. 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 quantum state 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 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 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.