Quantization of Alternative Hamiltonians

D. B. Fairlie, J. Nuyts

The authors take a standard Hamiltonian, quadratic in a set of creation and annihilation operators, and demonstrate an alternative quantization by quommutators; i.e. a set of rules for transposition of these operators, different from the those of the usual canonical commutation relations which yields a new spectrum, of a type reminiscent of logarithmic trajectories.

A. Horzela

Alternative Hamiltonians of classical mechanics, i.e. , canonically inequivalent Hamiltonians leading to the same Newton equations of motion, generate different phase space descriptions of the same mechanical systems. Relation of such descriptions to different algebraic descriptions of quantum systems consistent with the same time evolution - so called Wigner quantization - is presented and illustrated on example of two particles system moving on a line.

L.-M. Duan, Guang‐Can Guo

A simple approach is proposed for the quantization of the electromagnetic field in nonlinear and inhomogeneous media. Given the dielectric function and nonlinear susceptibilities, the Hamiltonian of the electromagnetic field is determined completely by this quantization method. From Heisenberg's equations we derive Maxwell's equations for the field operators. When the nonlinearity goes to zero, this quantization method returns to the generalized canonical quantization procedure for linear inhomogeneous media [Phys. Rev. A 43, 467 (1991)]. The explicit Hamiltonians for the second-order and third-order nonlinear quasi-steady-state processes are obtained based on this quantization procedure.

A. Horzela