Quantum chaos in the framework of complex probability processes.   Thermodynamics of nonrelativistic vaccum
  The problem of random motion of 1D quantum reactive harmonic oscillator (QRHO) is formulated in terms of a wave functional regarded as a complex probability process in an extended space. In the complex stochastic differential equation (SDE) for this process the variables are separated with the help of the Langevin-type model SDE introduced in the functional space. The complete positive Fokker-Plank measure of the functional space is obtained. The average wave function of roaming QRHO is obtained by means of functional integration over the process with the complete Fokker-Plank measure in the functional space. The local and averaged transition matrices of roaming QRHO were constructed. The thermodynamics of nonrelativistic vacuum is investigated in detail and expressions for the internal energy, Helmholtzian energy and entropy are obtained. The oscillator's ground state energy, its shift and broadening are calculated. 
A. V. Bogdanov and A. S. Gevorkyan