Abstract
Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.
Highlights
In 1973, Nambu proposed a generalization of the classical Hamiltonian dynamics [1] that is nowadays referred to as the Nambu mechanics
Hamiltonian dynamics is the classical dynamics of the canonical doublet (q(t), p(t)), which is given by a Hamiltonian H = H(q, p) and the Poisson bracket defined by the two-dimensional Jacobian
We have shown that the Nambu mechanical structure is hidden in classical Hamiltonian dynamics and in some quantum or semiclassical dynamics
Summary
In 1973, Nambu proposed a generalization of the classical Hamiltonian dynamics [1] that is nowadays referred to as the Nambu mechanics. The Nambu multiplet is given as a function of classical variables (q, p), and the induced constraints are always trivial, i.e. set to zero [13]. We present a general procedure to find the Nambu mechanical structure in quantum or semiclassical systems of one degree of freedom with some specific examples. The resulting hidden Nambu mechanics becomes pathological because in many-degrees-of-freedom systems the Nambu bracket does not satisfy the fundamental identity, which is an important property of the Nambu bracket and corresponds to the Jacobi identity in Hamiltonian dynamics [2, 11]. The hidden Nambu mechanics in many-degrees-of-freedom systems is an example of dynamics without the canonical structure. In the last section we give our conclusions and discuss the direction of future work
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