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Journal of Nonlinear Mathematical Physics

Non-linear Schrödinger Equations, Separation and Symmetry

George Svetlichny

Published Jan 1, 1995• 6 citations
Publisher

Abstract

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property).We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers.Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries.For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number.A Lie-algebra approach for non-linear theories is developed.

DOI
10.2991/jnmp.1995.2.1.1

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We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property).We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers.Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries.For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number.A Lie-algebra approach for non-linear theories is developed.
We investigate the symmetry properties of hierarchies of non-linear Schroedinger equations (introduced by Doebner and Goldin, and Goldin and Svetlichny), which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers. Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries. For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number. A Lie-algebra approach for non-linear theories is developed.
Hierarchies of nonlinear Schrödinger equations were investigated for multiparticle systems, satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor. Such a hierarchy defines a nonlinear derivation on tensor products of the single-particle wave-function space, and satisfies a certain homogeneity property characterized by two new universal physical constants. A canonical construction of hierarchies is derived that allows the introduction, at any particular ‘‘threshold’’ number of particles, of truly new physical effects absent in systems having fewer particles. In particular, if single quantum particles satisfy the usual (linear) Schrödinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrödinger equation without violating the separation property. Examples of Galilean-invariant hierarchies are given.
We investigate hierarchies of nonlinear Schrödinger equations for multiparticle systems satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor.Such a hierarchy defines a nonlinear derivation on tensor products of the single-particle wave-function space, and satisfies a certain homogeneity property characterized by two new universal physical constants.A canonical construction of hierarchies is derived that allows the introduction, at any particular "threshold" number of particles, of truly new physical effects absent in systems having fewer particles.In particular, if single quantum particles satisfy the usual (linear) Schrödinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrödinger equation without violating the separation property.Examples of Galileian-invariant hierarchies are given.
Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU(n). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU(n). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU(n). Numerical results on the number of dimensions support the developed theory.
The global rotational degrees of freedom in the Schr\"odinger equation for an N-body system are completely separated from the internal ones. After removing the motion of the center of mass, we find a complete set of (2l+1)(2\mathcal{l}+1)(2l+1) independent base functions with angular momentum l.\mathcal{l}.l. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.