Spherical Harmonics<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msubsup><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>: Positive and Negative Integer Representations of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mtext>su</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1,1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:mi>m</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>l</mml:mi><mml:mo>+</mml:mo><mml:mi>m</mml:mi></mml:math>

Abstract

The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.</jats:p