Transformations In Classical Mechanics Research Articles

Canonical Realizations of Lie Symmetry Groups

A general theory of the realizations of Lie groups by means of canonical transformations in classical mechanics is given. The problem is the analog to that of the characterization of the projective representations in quantum mechanics considered by Wigner, Bargmann, and others in the case of the Galilei and the Lorentz group. However, no application to particular groups is given in this paper. It turns out that the generators yτ of the infinitesimal transformations in a canonical realization of a Lie group 𝒢 satisfy relations of the form {yρ, yσ} = cρστyτ + dρσ, where cρστ are the structure constands of 𝒢 and dρσ are constants depending on the particular realization. It also turns out that any canonical realization of 𝒢 can be reduced to a fundamental typical form by means of a constant canonical transformation in the phase space of the system. This typical form allows one to obtain a complete characterization of all the possible canonical realizations of 𝒢. Once a suitable definition of ``irreducible'' canonical realization is given, a simple classification can be obtained in terms of the values of certain functions of the generators yτ (canonical invariants). Recalling the correspondence {} → [], formal analogy appears to be achieved with the quantum mechanical case. Even more, a parallel development of the outlined theory in quantum mechanics is of interest in the construction of the invariant operators (even in case of non-semisimple groups) and of complete systems of commuting observables acting within the irreducible representations.

Point Transformations in Quantum Mechanics

An isomorphism is shown to exist between the group of point transformations in classical mechanics and a certain subgroup of the group of all unitary transformations in quantum mechanics. This isomorphism is used to indicate that the quantum analogs of physically significant classical expressions can be constructed uniquely in any coordinate system. There is no ambiguity in the ordering of noncommuting quantum operators, and the method of constructing the quantum analogs is covariant under general coordinate transformations. The method is actually only applicable to systems having Lagrangians which are at most quadratic in the velocities, but this includes all systems which are presently of interest in physics. The method is applied to two intrinsically nonlinear examples, one of which is the gravitational field. The correct Hamiltonian operator for a quantized version of Einstein's gravitational theory is constructed.