Canonical Formalism of Dirac and Variational Principle

I.Yu. Krivsky, В. М. Симулик, T.M. Zajac +1 more

The derivation of the Dirac and Maxwell equations from the first principles of relativistic canonical quantum mechanics (RCQM) is demonstrated. The formalism of the RCQM is constructed on the basis of a self-consistent system of axioms. The important conclusion, that an arbitrary physical and mathematical information, which contains in the model of RCQM, directly and unambiguously is translated into the corresponding information in the model given by the Dirac equation, is formulated.

Bernard H. Lavenda

Edgard Elbaz

Reiji Sugano

Since the ordinary c-number variation principle does not lead, for a non-linear Lagrangian, to the result consistent between the Lagrangian and Hamiltonian formalisms, Schwinger's variation principle is reformulated for a type of Lagrangian L=½\dotqigij(q)\dotqj-u(q) by means of a q-number variation. The canonical momentum, the Hamiltonian, the canonical commutation relations and equation of motion are derived. Also the Euler-Lagrange equation is obtained, which is consistent with the canonical equation of motion. These consequences are exactly the same as those of previous papers, but different from the ordinary ones in the Euler-Lagrange equation and the Hamiltonian.

N. Emir Anuar

We present a formalism for which a dissipative system is given by a variational principle. The formalism applies to dynamical systems where its trajectory is monotonic. Subsequently, we derive its Lagrangian and Hamiltonian. From the Hamiltonian, we quantize canonically the classical particle in a viscous media. We study the free quantum particle in a viscous media and the dissipative quantum tunnelling. It is found that the dissipation influences tunnelling probability by a factor that closely resembles the result of Caldeira and Leggett.