Quantum Instantons and Quantum Chaos.pdf

a r X i v : q u a n t - p h / 9 9 1 0 1 1 6 v 3 5 F e b 2 0 0 1 Quantum Instantons and Quantum Chaos H. Jiraria, H. Kro?gera?, X.Q. Luob,c, K.J.M. Moriartyd and S.G. Rubine aDe?partement de Physique, Universite? Laval, Que?bec, Que?bec G1K 7P4, Canada bCCAST (World Laboratory), P.O. Box 8730, Bejing 100080, China cDepartement of Physics, Zhongshan University, Guangzhou 510275, China dDepartment of Mathematics, Statistics and Computer Science, Dalhousie University, Halifax, N.S. B3H 3J5, Canada eMoscow Engineering Physics Institute, Center for Cosmo-Particle Physics ”Cosmion”, Moscow, Russia February 1, 2008 Abstract We suggest a closed form expression for the path integral of quantum transition am- plitudes to construct a quantum action. Based on this we propose rigorous definitions of both, quantum instantons and quantum chaos. As an example we compute the quantum instanton of the double well potential. ?Corresponding author, Email: hkroger@phy.ulaval.ca 0 1. Introduction Instantons and chaos both play an important role in modern science. Chaos occurs in physics, chemistry, biology, physiology, meteorology, economy etc. Both concepts are de- fined by classical physics. On the other hand, for instanton solutions occuring in quantum field theory of gauge theories, quantum effects are very important. So far quantum effects have been computed mostly perturbatively (WKB). The chaotic behavior of quantum liq- uids and quantum dots requires a quantum mechanical description. However, the concept of quantum chaos has eluded yet a rigorous definition, albeit a quantitative computation. In this work we propose how to define quantum instantons and quantum chaos. As an example, we compute quantitatively the quantum instanton for a 1-dim quantum system. Chaotic phenomena were found in a number of quantum systems. For example, the hydrogen atom in a strong magnetic field shows strong irregularities in the spectrum [1]. Irregular patterns were observed in the wave functions o