Authors: Phong Dang | J. P. Draayer | Feng Pan | T. Dytrych | Daniel Langr | David Kekejian | Kevin S. Becker | Nicholas J. Thompson
Source: The European Physical Journal Plus (article, 2024-10-25)
Field: Chemistry — Spectroscopy

Coupling and recoupling coefficients for Wigner’s U(4) supermultiplet symmetry Abstract A novel procedure for evaluating Wigner coupling coefficients and Racah recoupling coefficients for U(4) in two group–subgroup chains is presented. The canonical $$\rm U(4) \supset U(3)\supset U(2)\supset U(1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> <mml:mo>⊃</mml:mo> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> <mml:mo>⊃</mml:mo> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> <mml:mo>⊃</mml:mo> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> coupling and recoupling coefficients are applicable to any system that possesses U(4) symmetry, while the physical $$\mathrm{U(4)} \supset \textrm{SU}_{\mathrm{S}}(2) \otimes \textrm{SU}_{\mathrm{T}}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊃</mml:mo> <mml:msub> <mml:mtext>SU</mml:mtext> <mml:mi>S</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊗</mml:mo> <mml:msub> <mml:mtext>SU</mml:mtext> <mml:mi>T</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> coupling coefficients are more specific to nuclear structure studies that utilize Wigner’s supermultiplet symmetry concept. The procedure that is proposed sidesteps the use of binomial coefficients and alternating sum series and consequently enables fast and accurate computation of any and all U(4)-underpinned features. The inner multiplicity of a ( S , T ) pair within a single $$\mathrm U(4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> irreducible representation is obtained from the dimension of the null space of the $$\mathrm{SU(2)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>SU</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> raising generators, while the resolution for the outer multiplicity follows from the work of Alex et al. on $$\mathrm U(N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the canonical and the physical group–subgroup chains of U(4). Materials science Advanced NMR Techniques and Applications ; Lanthanide and Transition Metal Complexes ; Quantum Chromodynamics and Particle Interactions