Abstract
Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation- dependent notion of on-shell symmetry is introduced. The difference in associating the time-derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric.
Highlights
Focusing on simple examples, I illustrate some general features that apply to a vast class of theories including non-relativistic Schrodinger equations in 1 + d dimensions, the more general invariant equations associated with l-conformal Galilei algebras, the D = 1 Lagrangianconformal models, together with several extended supersymmetric versions of these theories
The several key features discussed in this paper can be extended to investigate the dynamics of more complicated systems
The algebra/superalgebra duality involving a finite number of firstorder and second-order differential operators can be constructed only for Schrodinger equations in 1 + d-dimensions, and from l-extended conformal Galilei algebras, with halfinteger l
Summary
See [1], we can prove that, for three special cases of the potential, the invariance algebra of the equation (2), in terms of first-order differential operators, is given by the Schrodinger algebra The algebra/superalgebra symmetry with higher differential operators The second-order differential operators w1, w0, w−1, obtained by taking the anticommutators of w±, can be constructed: w+1 = {w+, w+}, w0 = {w+, w−}, w−1 = {w−, w−} Their explicit form, in the three respective cases above, is given by i) the constant case, w+1 = 2∂x2, w0. In all three cases (constant, linear and quadratic), the second-order differential operator Ω is a generator belonging to the enlarged Schrodinger algebra (either eSch or sSch).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
