Towards chaos criterion in quantum field theory

Chaos criterion for quantum field theory is proposed. Its correspondence with classical chaos criterion in semi-classical regime is shown. It is demonstrated for real scalar field that proposed chaos criterion can be used to investigate stability of classical solutions of field equations. Phenomenon of chaos attracts much attention in various fields of physics. Originally it was associated with problems of classical mechanics and statistical physics. Substantiation of statistical mechanics initiated intensive study of chaos and uncovered its basic properties mainly in classical mechanics. One of the main results in this direction was a creation of KAM theory and understanding of the phase space structure of Hamiltonian systems [1],[2]. It was clarified that the root of chaos was local instability of dynamical system [3]. Local instability leads to mixing of trajectories in phase space and thus to non-regular behavior of the system and chaos [4]-[6]. Significant property of chaos is its prevalence in various natural phenomena. It explains a large number of works in this field. Large progress is achieved in understanding of chaos in semi-classical regime of quantum mechanics via analysis of the spectral properties of the system [6],[7]. Semi-classical restrictions are important, because a large number of energy levels in small energy interval is needed to provide a certain statistics [8]. Investigation of the stability of classical field solutions faces difficulties caused by infinite number of degrees of freedom. That is why authors often restrict their consideration by investigation of some model field configurations [9]. There are papers devoted chaos in quantum field theory [10]. But there is no generally recognized definition of chaos for quantum fields [11]. This fact restricts use of chaos theory in field of elementary particle physics. At the same time it is well known that field equations of all four types of fundamental interactions have chaotic solutions [12] and high energy physics reveales the phenomenon of intermittency [13]. The aim of this work is to propose chaos criterion applied for both quantum mechanical and quantum field systems. The paper is structured as follows. At first we generalize classical Toda criterion of local instability [14]-[16] for systems with any finite number of degrees of freedom. Then using analogy with statistical mechanics we formulate chaos criterion for quantum mechanics and