Operational Dynamic Modeling Transcending Quantum and Classical Mechanics

Abstract

We introduce a general and systematic theoretical framework for operational dynamic modeling (ODM) by combining a kinematic description of a model with the evolution of the dynamical average values. The kinematics includes the algebra of the observables and their defined averages. The evolution of the average values is drawn in the form of Ehrenfest-like theorems. We show that ODM is capable of encompassing wide-ranging dynamics from classical non-relativistic mechanics to quantum field theory. The generality of ODM should provide a basis for formulating novel theories.

Wigner representation of the rotational dynamics of rigid tops

“…[46,47]). One practical and formally justified [48,49] axiomatic basis for algorithm (II) postulates the equations of motion for averages of certain physical quantities [38,50]. Another possible starting point is Feynman's path integral representations of the time evolution [12].…”

Hybrid classical-quantum formulations ask for hybrid notions

“…In all these approaches one starts with initially separated purely classical and quantum sectors and then makes them interact in order to analyze the outcome. Without pretending to be exhaustive, we can classify these approaches in the following categories: (1) approaches that try to maintain the use of quantum states (or density matrices) to describe the quantum sector and trajectories for the classical sector [2,3], (2) those that first formulate the classical sector as a quantum theory [4][5][6] and then work with a formally completely quantum system [7][8][9][10][11], (3) conversely, those that first formulate the quantum sector as a classical theory [12] and then work with a formally completely classical system [13][14][15][16], and (4) approaches that take the quantum and the classical sectors to a common language and then extend it to a single framework in the presence of interactions, for instance, using Hamilton-Jacobi statistical theory for the classical sector and Madelung representation for the quantum sector [17][18][19] or modeling classical and quantum dynamics starting from Ehrenfest equations [20]. This classification is not sharp and in some cases is subject to interpretation, but it may be useful as a way to organize the possible procedures and conceptual viewpoints in the enterprise of constructing a hybrid theory.…”

Dirac open-quantum-system dynamics: Formulations and simulations

“…(121) In the classical limit, we understand the situation when the operators of the momentap µ and coordinatesx µ commute [44,137,138]. Following the Hilbert phase space formalism [44,89], we separate the commutative and noncommutative parts of the Dirac generator D by introducing the algebra of classical observables…”

“…To study the quantum-to-classical transition, it is instrumental to put both mechanics on the same mathematical footing [24,25,28,31,[38][39][40][41][42][43][44][45]. This is achieved by the Wigner quasi-probability distribution W (x, p) [46], which is a phase-space representation of the density operatorρ.…”