On the Theory of Quantization for Particle Dynamics in Non-canonical Formalism

Pei Wang

Abstract We develop a theory of random non-Hermitian action that, after quantization, describes the stochastic nonlinear dynamics of quantum states in Hilbert space. Focusing on fermionic fields, we propose both canonical quantization and path integral quantization, demonstrating that these two approaches are equivalent. Using this formalism, we investigate the evolution of a single-particle Gaussian wave packet under the influence of non-Hermiticity and randomness. Our results show that specific types of non-Hermiticity lead to wave packet localization, while randomness affects the central position of the wave packet, causing the variance of its distribution to increase with the strength of the randomness.

John Cardy

Ebbe Thue Poulsen

L. Bányai, M. Bundaru

We give here a field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This approach illustrates however the proper use of the Euler-Lagrange equations and the canonical formalism that fail if one tries to build up the many-body quantum theory starting from the classical theory of point-like particles interacting with electromagnetic fields.

Manan Sengupta

The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.