Dual pairing of symmetry and dynamical groups in physics

This article reviews many manifestations and applications of dual representations of pairs of groups primarily in atomic and nuclear physics. Examples are given to show how such paired representations are powerful aids in understanding the dynamics associated with shell-model coupling schemes and in identifying the physical situations for which a given scheme is most appropriate. In particular, they suggest model Hamiltonians that are diagonal in the various coupling schemes. The dual pairing of group representations has been applied profitably in mathematics to the study of invariant theory. Parallel applications to the theory of symmetry and dynamical groups in physics are shown to be equally valuable. In particular, the pairing of the representations of a discrete group with those of a continuous Lie group or those of a compact Lie group with those of a noncompact Lie group makes it possible to infer many properties of difficult groups from those of simpler groups. This review starts with the representations of the symmetric and unitary groups, which are used extensively in the many-particle quantum mechanics of bosonic and fermionic systems. It gives a summary of the many solutions and computational techniques for solving problems that arise in applications of symmetry methods in physics and which result from the famous Schur-Weyl duality theorem for the pairing of these representations. It continues to examine many chains of symmetry groups and dual chains of dynamical groups associated with several coupling schemes in atomic and nuclear shell models and the valuable insights and applications that result.

Figure 1

Irreps of $\mathrm{O}(N)$ in the Hilbert space of the $N$-dimensional harmonic oscillator shown as horizontal lines. Equivalent $\mathrm{O}(N)$ irreps are placed in a common column and connected by the su(1,1) raising operator ${\widehat{S}}_{+}$. Equivalent $\mathrm{O}(N)$ irreps are distinguished by the $\mathrm{u}(1)\subset \mathrm{su}(1,1)$ quantum number $\mu$. The set of $\mathrm{O}(N)$ irreps at a constant level $n$ comprise a $\mathrm{U}(N)$ irrep $\{n\}$ with $n=v+2\mu$. The carrier space of this $\mathrm{U}(N)$ irrep $\{n\}$ is ${\mathbb{H}}^{(n)}$, the space spanned by the states with $n$ harmonic-oscillator quanta.

Figure 2

The spectrum of USp(10) irreps in the $j=9/2$ shell labeled by values of the seniority quantum number $v$. The combined states of all USp(10) irreps having a common value of the particle number $n$ span the U(10) irrep $\{1^n\}$. The combined USp(10) irreps having a common value of the seniority $v$, connected by dotted lines, span an $\mathrm{SU}(2{)}_{\mathrm{qs}}\times \mathrm{USp}(10)$ irrep.

Figure 3

Low-lying energy levels of some $N=50$ isotones. The four parameters of a seniority-conserving interaction between protons in a $g_{9/2}$ shell were chosen to fit the energies of the lowest $J=0$, 2, 4, and 8 states shown by light (theory) lines for each nucleus. The remaining energy levels shown by heavy (theory) lines were then predicted for these interactions. From 146.

Figure 4

The spectrum of $\mathrm{USp}(2j+1)$ irreps, shown as lines, for $j=3/2$. Each $\mathrm{USp}(2j+1)$ irrep, labeled by $⟨\kappa ⟩$, contains states with the angular-momentum values $J$ as shown on the right side of the figure. These irreps are also labeled by the SO(5) quantum numbers: seniority $v$, and reduced isospin $t$. The basis states for each SO(5) irrep are labeled by isospin $T$ (shown above each level), and nucleon number $n$. Note that all the USp(4) irreps corresponding to lines at the same horizontal level and connected by dotted lines share common values of $v$ and $t$, and common distributions of $J$ values. They combine to span an $\mathrm{SO}(5)\times \mathrm{USp}(2j+1)$ irrep with $j=3/2$. SO(5) lowest-weight states lie in the leftmost USp(4) irreps (shown as thick lines). Thus, it is shown that the many-nucleon states of a $j=3/2$ shell fall into six distinct $\mathrm{SO}(5)\times \mathrm{USp}(2j+1)$ irreps.

Figure 5

Weight diagrams for the states of $\mathrm{SO}(5)\simeq \mathrm{USp}(4)$ irreps whose lowest-weight states have no more than four particles. States are characterized by dots and labeled by their $\mathrm{U}(2{)}_T\supset \mathrm{U}(1)$ quantum numbers, i.e., the nucleon number $n$, isospin $T$, and component of isospin $T_0$. Weights with a multiplicity of two are denoted by dots with circles around them. For example, the $n=4$ states of the $⟨00⟩$ irrep contain two states of $n=4$ and $T_0=0$; one has isospin $T=0$ and the other $T=2$.