Dynamical Symmetry of the Magnetic Monopole

The relationship between relativistic quantum current conservation laws in a curved-space background and the corresponding "good quantum numbers," i.e., operators that commute with the fundamental wave operator in a first-quantized field theory, is considered. It is shown that under favorable circumstances (such as vanishing Ricci curvature) the existence of such an operator for scalar fields is automatically implied by the existence of the corresponding constant for particle trajectories in the classical limit, that is to say, by the existence of a Killing vector or a "Killing tensor" in the first- and second-order cases, respectively. Thus the fourth constant of the motion for a scalar quantum field in the Kerr metric background arises automatically from the Killing tensor defining the fourth constant of the classical motion. Another application is to the Runge-Lenz constants in the nonrelativistic hydrogen atom problem. The "Schiff conjecture" concerning the relationship between classical mechanics and first-quantized field theory in connection with the equivalence principle is discussed in passing.

Killing Tensor Quantum Numbers and Conserved Currents in Curved Space

A very general realization of the so(2, 1) algebra, which easily follows from the basic commutation relations that are satisfied by any pair of mutually conjugate generalized coordinates and momenta, is constructed. Using special cases of this general realization, and closely following the well known derivation of the eigenvalue spectra of the angular-momentum operators, based on the so(3) algebra, we derive the energy spectrum for the N-dimensional isotropic harmonic oscillator, and for both the nonrelativistic and the relativistic N-dimensional hydrogen atom. Special attention is given to a simple derivation of the form of these Hamiltonians in terms of the so(2, 1) algebra generators. In particular, the usually exploited tilting transformation is avoided, and the whole derivation is presented in an extremely simple and straightforward way. The present approach stresses the similarity and mutual relationship between the systems studied and, in addition to introducing some novel techniques and providing considerable insight into the overall structure of these problems, also has a definite pedagogical value.

An algebraic approach to bound states of simple one‐electron systems

A detailed study of the group of symmetries of the time‐dependent free particle Schrodinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well‐defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x2 is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrodinger equation, several of which are new.

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Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxx −c/x2 U = 0

O(4,2) dynamical symmetry of the Kaluza-Klein monopole

The relativistic Kepler problems in Dirac and Klein‐Gordon forms are solved by dynamical group methods for particles having both electric and magnetic charges (dyons). The explicit forms of the O(4, 2)‐algebra and two special O(2, 1)‐algebras (which coincide in the symmetry limit) are given, and a new group‐theoretical form of the symmetry breaking is pointed out. The Klein‐Gordon O(2, 1)‐algebra also solves the dynamics in the case of very strong coupling constants (attractive singular potential), if the principal series of representations is used instead of the discrete series.

https://materias.df.uba.ar/ft2a2013c2/files/2013/10/HydrogenWithAlgebra.pdf

SO(4, 2)‐Formulation of the Symmetry Breaking in Relativistic Kepler Problems with or without Magnetic Charges

The two common applications of O(4,2) as a conformal group on external space‐time coordinates and as a dynamical group on internal relative coordinates are combined into a unified algebraic structure for composite systems. A method is given for obtaining from this structure infinite‐component wave equations and a discrete, linearly increasing mass spectrum.

G. L. Bornzin

Papers

SO(4, 2)‐Formulation of the Symmetry Breaking in Relativistic Kepler Problems with or without Magnetic Charges

Unification of the external conformal symmetry group and the internal conformal dynamical group