The Principles of Canonical Mechanics

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.

Reinhard Hentschke

Newton C. A. da Costa, Francisco Antônio Dória

Hideki Kamo, Toshiharu Kawai

Schwinger’s variational principle is formulated for the multi-dimensional quantum system which corresponds to the classical system described by the Lagrangian Lc(\dotx,x)=(M/2)gij(x)\dotxi\dotxj-v(x). The c-number variations of coordinates and time are sufficient to give the laws of quantum mechanics. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relations are derived from this principle and there is no inconsistency. An appropriate choice of the Lagrangian operator is essential in our formulation. It is shown that an arbitrary point transformation is entitled to be called a canonical transformation. Ambiguities in the quantal Lagrangian are also discussed.

Toshiharu Kawai

Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.

Peter Bongaarts