**10. A Quantum Mechanical Perspective on Emergent Structures in the Nuclear Many-Body Problem**

Nuclei are finite many-body quantum systems that self-organize to yield well-defined sizes and moments. The size of the nucleus determines the energy scale of quantization by virtue of the confinement (specific length) of nucleons (specific mass), scaled by *h*¯. Defining the nucleon position and momentum observables, *xj* and *pj*, with *j* = 1, 2, 3, and the nucleon mass *m*, this leads to the stationary states of any given nucleus via the definition of a Hamiltonian and the fundamental relationships

$$[\mathbf{x}\_{\mathbf{j}\prime} p\_{\mathbf{j}}] = i\hbar\,,\tag{9}$$

and

$$p\_j = m\ddot{x}\_j.\tag{10}$$

The consideration herein is limited to a discussion of model forms of independentparticle potentials and residual two-body interactions between the nucleons. At this point of inception, a representation of the problem for determining the eigenstates of the Hamiltonian must be chosen. This involves the use of symmetries of the Hamiltonian. If the nucleus has spherical symmetry, the handling of the independent-particle part of the Hamiltonian is greatly simplified by using the familiar representation that is a factorization into angular momentum and radial degrees of freedom. This extends to labeling states with angular momentum quantum numbers. The familiar radial confining potentials—the infinite square well and the harmonic oscillator—are solvable in closed form.

The factorization into radial and angular degrees of freedom equips us with the powerful algebra of angular momentum,

$$L\_k \equiv \mathbf{x}\_i p\_j - \mathbf{x}\_j p\_{i\prime} \tag{11}$$

and

$$[L\_i, L\_j] = i\hbar \varepsilon\_{ijk} L\_{k\prime} \tag{12}$$

where  *ijk* is the permutation symbol. The power of this algebra, the so(3) Lie algebra, is that it permits an enormous reduction of computational labor via the classification of states and operators as so(3) tensors, with their associated irreducible representations, Kronecker products, and Wigner–Eckart theorem. Spins of nucleons are simply accommodated by extension of so(3) to its isomorphic algebra, su(2). The Hamiltonian becomes block diagonal in su(2) irreps, reducing computational labor; selection rules emerge; many transitions of interest appear in ratios that depend only on Clebsch–Gordan coefficients.

But, when one factorizes the shell model problem into angular and radial parts, and arrives at the so(3) algebra of angular momentum, one does not look any further for algebraic structures in the problem. However, there is another algebra "right under our noses": the radial degree of freedom possess an su(1,1) algebra. This is a so-called dual algebra for the shell model. Details are presented in pedagogical form in [250] and in a more advanced form in [6]. This is not found in any quantum mechanics textbook. It can be used to evaluate radial matrix elements in shell model computations, and this has recently been explored [251]. The algebra is defined by

$$T\_1 \equiv \mathbf{r} \cdot \mathbf{r}, \quad T\_2 \equiv \frac{1}{2}(\mathbf{r} \cdot \mathbf{p} + \mathbf{p} \cdot \mathbf{r}), \quad T\_3 \equiv \mathbf{p} \cdot \mathbf{p}, \tag{13}$$

which, via linear combinations of the {*Ti*} and scale factors, leads to the commutator brackets recognizable as su(1,1) (see [250]). Indeed, radial matrix elements possess simple relationships including "cancellations", which reflect properties of su(1,1) irreps and a su(1,1) Wigner–Eckart theorem.

Thus, what other algebraic structures can one expect in nuclei that emerge from functions of *xi*, *pi* and Equation (9)? The clue comes from the dominance of quadrupole deformation in nuclei. One can define "quadrupole" coordinates, *xixj*: these are rank-2 symmetric Cartesian tensors and there are six of them—using *xi* = *x*, *xj* = *y*, *xk* = *z*, they are *xx*, *xy*, *xz*, *yy*, *yz*, *zz*. From these, in a straightforward manner, combinations such as *xpx*, etc., and *px px*, etc., are obtained, yielding a Lie algebra with 21 generators, called sp(3,R). Details are presented in pedagogical form in [5] and full details are presented in [6]. The Lie algebra possesses many useful subalgebras: so(3), su(3), and others which need not concern us here; however, note that the su(3) subalgebra is that of the historical Elliott model [160]. A characteristic of the majority of nuclei is that they possess a very large value for the leading sp(3,R) quantum number, N—the total number of shell model oscillator quanta carried by the sum of all the nucleons—counting the number of oscillator quanta for each nucleon partitioned across the entire occupancy of the oscillator shells of the given nucleus. For example, for 168Er, N = 814 [252]. This leads to contraction in nuclei dominated by the su(3) subalgebra, yielding a (near) rigid rotor with properties that closely match observations [253] with the use of effective charges *ep* = +*e*, *en* = 0 [254]. Contraction is a process where a Lie product, e.g., Equation (12), approaches zero asymptotically as quantum numbers become very large: for a state with angular momentum *L* = 100 and projection *mL* = +100, the cone of indeterminacy appears almost identical to a classical angular momentum vector with three sharp Cartesian components. The origin of the concept of contraction is in a paper by Inonu and Wigner [255]; and the process is often called Inonu–Wigner contraction.

Thus, how does the shell model stand in relationship to the foregoing categorization? The shell model utilizes the su(2) spin-angular momentum algebra and adopts a central potential, but one does not find use of the su(1,1) algebra. This leaves open the functional form of the central potential: an su(1,1) algebraic structure is only realized for four central potentials—the Coulomb potential, the harmonic oscillator potential and their less wellknown modifications—through augmentation with a 1/*r*<sup>2</sup> term—the Kratzer potential and the Davidson potential [256]. Furthermore, the shell model does not make use of the sp(3,R) algebra because of spin–orbit coupling. Such an interaction lies "outside" of the symplectic model and must be treated as a perturbation. While the shell structure of nuclei and the dependence of magic numbers on spin–orbit coupling appear to invalidate the symplectic model, *Q* · *Q* interactions shift shell structures by up to 100 MeV, as manifested in observed shape coexisting structure; thus, *L* · *S* and *Q* · *Q* interactions have their respective domains of influence in nuclear structure. Indeed, the dividing line of their influence epitomizes the primary focus of this contribution. Notably, where the *L* · *S* interaction

dominates, *J* emerges as a good quantum number and pairing interactions result in the emergence of seniority structure and its underlying quasispin su(2) algebra. These few mathematical structures appear to cover all the structures manifested in nuclei, observed so far, and as summarized in this contribution.

The foregoing leaves open the answer to the question posed by the title of this paper. The shell model versus the symplectic model approaches, with their respective dominance by spin–orbit coupling versus quadrupole–quadrupole coupling, each go some way to describing the structure of transitional nuclei. A shell model description can be achieved by using effective interactions. However, it should be noted that it is beginning to emerge that the effective interactions used in ab initio shell model calculations appear to be dominated by just those components that are compatible with symplectic model structures [257–259].
