Heading towards chaos criterion for quantum field systems

Chaos criterion for quantum field theory is proposed. Its accordance with classical chaos criterion is demonstrated in the semi-classical limit of quantum mechanics. Originally phenomenon of chaos was associated with problems of classical mechanics and statistical physics. Attempt of substantiation of statistical mechanics initiated intensive study of chaos and uncover its basic properties mainly in classical mechanics. One of the main results in this direction is a creation of KAM theory and understanding of the phase space structure of Hamiltonian systems [1]. It was clarified that the root of chaos is local instability of dynamical system [2]. Local instability leads to mixing of trajectories in phase space and thus to non-regular behavior of the system and chaos [3]. Significant property of chaos is its prevalence in various natural phenomena. It explains the importance of study of chaos and 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 [3],[4]. Semi-classical restrictions are important, because a large number of energy levels in small energy interval is needed to provide a statistics [5]. 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 the investigation of some model field configurations [6]. There are papers devoted chaos in quantum field theory [7]. But there is no generally recognized definition of chaos for pure quantum systems [8]. This fact restricts use of chaos theory in the field of elementary particle physics. At the same time it is well known that the field equations of all four types of fundamental interactions have chaotic solutions [9] and high energy physics reveals the phenomenon of intermittency [10]. To clear the role of chaos in particle physics one has to learn how chaos affects on observable quantities. To reach it, it is necessary to study chaos in quantum gauge field theories, modern theories of particle interactions. But, as it was mentioned above, it is still unclear what is chaos in quantum field theory and what phenomena is it become apparent in? Thus additional chaos criterion formulated for purposes of quantum field theory is needed. It