**3. Quantum integrability and dynamical symmetry**

Quantum phase space defined here can be compact or noncompact depending on the finite or infinite nature of the Hilbert space. A consequence of this development is that the classical definition of integrability can in general be directly transferred to the quantum case.

*Definition 3*: (Quantum Integrability) A quantum system with *M* independent dynamical degrees of freedom, hence a 2*M*-dimensional quantum phase space, is integrable if and only if there are *M* quantum constraints of motion, or good quantum numbers, which are related to the eigenvalues of *M* non-fully degenerate observables: *<sup>O</sup>*1, *<sup>O</sup>*2, … , *On*. □

Any set of variables that commute may be put in the form of a complete set of commuting observables **C** by including certain additional observables with it. The definition then says that if the system is integrable, a complete set of commuting variables **C** can be found so that the Hamiltonian is always diagonal in the basis referred to by **C**. In the reverse sense, the definition implies that if the system is integrable, simultaneous accurate measurements of *M* non-fully degenerate observables in the energy eigenvalues can be carried out.

The link with the dynamical group structure can be developed. This specifies exactly the integrability of a quantum system. To this end the definition of dynamical symmetry is needed.

*Definition 4*: (Dynamical Symmetry) A quantum system with dynamical group *G* possesses a dynamical symmetry if and only if the Hamiltonian operator of the system can be written and presented in terms of the Casimir operators of any specific chain with *α* fixed

$$H = \mathcal{F}\left(\mathbf{C}\_{k\circ}^{a}\right) \tag{21}$$

The index of a particular subgroup chain *C<sup>α</sup> kj* the *i*-th Casimir operator of subgroup *Gα <sup>k</sup>*, *k* ¼ *s <sup>α</sup>*, … , 1, *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>l</sup> α <sup>k</sup>* and *l α <sup>k</sup>* denotes that the rank of subgroup *G<sup>α</sup> <sup>k</sup>* is *l*. It is now possible to state a theorem which gives a condition for integrability to apply.

*Proposition 1*: (Quantum Integrability) A quantum system with dynamical group *G* is said to be integrable if it possesses a dynamical symmetry of *G*.

To prove this, note that it can be broken down into two cases or subgroup classes for a given dynamical group *G* and are referred to as canonical and noncanonical.

First consider the case in which *G<sup>α</sup>* is a canonical subgroup chain of *G*. The Casimir operators of *<sup>G</sup>*, f g *CGi* and all Casimir operators *<sup>C</sup><sup>α</sup> ki* � � corresponding to the subgroups in chain *G<sup>α</sup>* form a complete set of commuting operators *C<sup>α</sup>* of any carrier irrep space *H* of *G<sup>α</sup>* so for fixed *α*,

$$\mathcal{C}^{a}: \{\mathcal{C}\_{Gi}\} \cup \{\mathcal{C}\_{ki}^{a}\} \equiv \left\{Q\_{j}, j = \mathbf{1}, \cdots, N\right\}. \tag{22}$$

When *<sup>G</sup><sup>α</sup>* is the dynamical symmetry of the system, all operators in <sup>C</sup>*<sup>α</sup>* are constants of motion

$$\left[H, Q\_j\right] = \mathbf{0}.\tag{23}$$

There are always *<sup>M</sup>* nonfully dynamical operators in <sup>C</sup>*<sup>α</sup>*. By the third definition, the system is integrable.

For a non-canonical subgroup chain *<sup>G</sup><sup>α</sup>* the number of Casimir operators f g *CGi* of *G* and all Casimir operators of *C<sup>α</sup> ki* � � of *<sup>G</sup><sup>α</sup>* is less than the number of the complete set of commuting operators C of any irrep carrier space of *G*. By definition, of any complete set of commuting operators, there must exist other commuting operators *O j* � � that commute with f g *CGi* and *<sup>C</sup><sup>α</sup> ki* � �. These have to be included in the union as well when putting together <sup>C</sup>*<sup>α</sup>*

$$\mathcal{C}^{a}: \{\mathcal{C}\_{Gi}\} \cup \{\mathcal{C}\_{ki}^{a}\} \cup \{\mathcal{O}\_{j}\} \equiv \left\{Q\_{j}: j = \mathbf{1}, \ldots, N\right\}.\tag{24}$$

When the system is characterized by the dynamical symmetry of *G<sup>α</sup>*, the operators in (24) satisfy relation (23) as well. In this case as well, there must exist *M* non-fully degenerate operators of constants of motion as in the previous case.

Based on this proposition, it can be stated that nonintegrability of a quantum system involves the breaking of the dynamical symmetry of the system. It may be concluded that dynamical symmetry breaking can be said to be a property which characterizes quantum nonintegrability. □

Let us summarize what has been found as to what quantum mechanics tells us. In a given quantum system with dynamical Lie group *G* which is of rank *l* and dimension *n*, the dimension of a complete set of commuting operators C of *G* with any particular subgroup chain is *d* ¼ *l* þ ð Þ *n* � *l =*2 in which the *l* operators are Casimirs of *G* and are fully degenerate for any given irrep of *G*. The number *M* of the non-fully degenerate operators in C for a given irrep of *G* cannot exceed *M* ≤ð Þ *n* � *l =*2. When dynamical symmetry is broken such that any of the *M* constants of the motion for the system is destroyed the system becomes nonintegrable.
