Dirac representation of the $SO(3,2)$ group and the Landau problem

Abstract

By systematically studying the infinite degeneracy and constants of motion in the Landau problem, we obtain a central extension of the Euclidean group in two dimension as a dynamical symmetry group, and $Sp(2,\mathbb{R})$ as the spectrum generating group, irrespective of the choice of the gauge. The method of group contraction plays an important role. Dirac’s remarkable representation of the $SO(3,2)$ group and the isomorphism of this group with $Sp(4,\mathbb{R})$ are revisited. New insights are gained into the meaning of a two-oscillator system in the Dirac representation. It is argued that because even the two-dimensional isotropic oscillator with the $SU(2)$ dynamical symmetry group does not arise in the Landau problem, the relevance or applicability of the $SO(3,2)$ group is invalidated. A modified Landau–Zeeman model is discussed in which the $SO(3,2)$ group isomorphic to $Sp(4,\mathbb{R})$ can arise naturally.

Appendix

In the matrix representation of Hamiltonian (36), the eigenvalues and eigenvectors were calculated in [4] in terms of the monomials $(x, y, p_x, p_y)$. The eigenvectors are

$$ \begin{aligned} \, & u=[(\omega_{ \scriptscriptstyle{\mathrm L} }^2+\omega^2)m]^{1/2}(x+iy)+im^{-1/2}(p_x+ip_y), \\ & v=[(\omega_{ \scriptscriptstyle{\mathrm L} }^2+\omega^2) m]^{1/2}(x-iy)+im^{-1/2}(p_x-ip_y). \end{aligned}$$

(70)

Linear combinations of the products of these quantities constitute the constants of motion

$$ K=\frac{u^{ \scriptscriptstyle{R} }v^*+u^{*R}v}{\sqrt{R}(u u^*)^{(R-1)/2}},\qquad L=i\frac{u^{ \scriptscriptstyle{R} }v^*-u^{*R}v}{\sqrt{R}(uu^*)^{(R-1)/2}},\qquad D=\frac{uu^*-Rvv^*}{R}.$$

(71)

These constants of motion together with the Hamiltonian $H_{ \scriptscriptstyle{\mathrm Z} }$ satisfy the $SU(2)$ Lie algebra.

In the limit case $\omega\to 0$, expressions (43) can be obtained from the general expressions (70). For convenience, the authors of [4] let the constants of motion be denoted as

$$ S=\frac{m^{1/2}}{4i}(v-v^*),\qquad Q=\frac{m^{1/2}}{4}(v+v^*).$$

(72)

In this paper, $(S,Q)$ correspond to Eqs. (26).