The density matrix and the Wigner function formalism requires the doubling of the degrees of freedom in quantum mechanics (QM) and quantum field theory (QFT). The doubled degrees of freedom play the role of the thermal bath or environment degrees of freedom and are entangled with the system degrees of freedom. They also account for quantum noise in the fluctuating random forces in the system-environment coupling. The algebraic structure of QFT turns out to be the one of the deformed Hopf algebra. In such a frame, the trajectories in the space of the unitarily inequivalent representations of the canonical commutation relations turn out to be classical trajectories and, under convenient conditions, they may exhibit properties typical of classical chaotic trajectories in nonlinear dynamics. The quantum Brownian motion and the two-slit experiment in QM are discussed in connection with the doubling of the degrees of freedom.
Abstract The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.
It is supposed the alternative to Quantum Mechanics Axiomatic. Fluctuational Theory save the Mathematics of Quantum Mechanic without change, naming this Mathematics as Method of Indirect Computation. Fluctuational Theory is delete the axiomatic of Quantum Mechanics and replaces it by the assumption of Gravitational Noise. This assumption is connects the Method of Indirect Computation to the Classical Physics. Physical fluctuations of classical gravitational fields are mathematically expressed through geometric fluctuations of metrics of Riemann Space. Metrics Fluctuational Theory and Quantum Mechanic is describe the classical experiment of electrons interference by two different way.