Quantum Mechanics of Complex Systems, I

A. J. Leggett

R. S. Kaushal

Aishwarya Parthasarathi

The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.

C.-D. Yang, Shiang-Yi Han

The complex-valued quantum mechanics considers quantum motion on the complex plane instead of on the real axis, and studies the variations of a particle complex position, momentum and energy along a complex trajectory. On the basis of quantum Hamilton-Jacobi formalism in the complex space, we point out that having complex-valued motion is a universal property of quantum systems, because every quantum system is actually accompanied with an intrinsic complex Hamiltonian originating from the equation. It is revealed that the conventional real-valued quantum mechanics is a special case of the complex-valued quantum mechanics in that the eigen-structures of real and complex quantum systems, such as their eigenvalues, eigenfunctions and eigen-trajectories, are invariant under linear complex mapping. In other words, there is indeed no distinction between Hermitian systems, PT-symmetric systems, and non PT-symmetric systems when viewed from a complex domain. Their eigen-structures can be made coincident through linear transformation of complex coordinates.

Walter E Thirring, Evans M. Harrell

Andrea Cejnarová

It seems to be a priori impossible to formulate any general theory or model that encompasses all of the properties of complexity. So, one must make do with partial solutions. A possible approach we propose is to take inspiration from quantum theory, since there seems to be a strong analogy between complex systems and quantum systems. Although we do not propose any literal application of quantum mechanical formalism to complexity, we suggest that the language of quantum mechanics is already so well developed - and for a much wider spectrum of problems than most theories - that it can serve as a model for complexity theory. There are many problems common to both complex systems and quantum systems and we suggest that it might be useful to test the applicability of aspects of the “language” of quantum mechanics to a general complex system. What we suggest here is an interdisciplinary talk led between the natural sciences and philosophy, which we believe is the only way in which to deal with complexity “as such”.