**2. Quantum degrees of freedom**

The time evolution of a system in classical mechanics in time is usually represented by a trajectory in phase space and the dynamical variables are functions defined on this space. The dimension of phase space is twice the number of degrees of freedom, and a point represents a physical state. The space is even-dimensional and it is endowed with a symplectic Poisson bracket structure. Dynamical properties of the system are described completely by Hamilton's equations within this space.

For a quantum system, on the other hand, the dynamical properties are discussed in the setting of a Hilbert space. Dynamical observables are self-adjoint operators acting on elements of this space. A physical state is represented by a ray of the space, so the Hilbert space plays a role similar to phase space for a classical system. The Hilbert space cannot play the role of a quantum phase space since its dimension does not in general relate directly to degrees of freedom. Nor can it be directly reduced to classical phase space in the classical limit. Let us define first the quantum degrees of freedom as well as giving a suitable meaning to quantum phase space.

Suppose H is a Hilbert space of a system characterized completely by a complete set of observables denoted C. Set C is composed of the basic physical observables, such as coordinates, momenta, spin and so forth, but excludes the Hamiltonian. The basis vectors of the space can be completely specified by a set of quantum numbers which are related to the eigenvalues of what are usually referred to as the fully nondegenerate commuting observables C<sup>0</sup> of C. A fully degenerate operator or observable *O* ∈ C has for some constant *λ* the action

$$O|\psi\_i\rangle = \lambda |\psi\_i\rangle, \qquad |\psi\_i\rangle \in \mathcal{H}.\tag{18}$$

*Definition 1*: (Quantum Dynamical Degrees of Freedom) Let C : *O <sup>j</sup>*j *Oi*, *O <sup>j</sup>* � � <sup>¼</sup> 0; *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>* � � be a complete set of commuting observables of a quantum system. A basis set of its Hilbert space H can be labeled completely by *M* numbers *α<sup>i</sup>* f g : *i* ¼ 1, … , *M* called quantum numbers which are related to the eigenvalues of the non-fully degenerate observables *Oi* f g : *i* ¼ 1, … , *M M*ð Þ ≤ *N* , a subset of C. Then the number *M* is defined to be the number of quantum dynamical degrees of freedom. □

Since the members of C are provided by the system, not including the Hamiltonian, it depends only on the structure of the system's dynamical group **G**. Thus the number of quantum dynamical degrees of freedom based on this definition is unique for a given system with a specific Hilbert space H.

The physical and mathematical considerations for defining the dimension of the nonfully degenerate operator subset C<sup>0</sup> of C, not the dimension of C itself, as the number of quantum dynamical degrees of freedom is as follows. In a given H, all fully degenerate operators in C are equivalent to a constant multiple of the identity operator guaranteeing the irreducibility of H. The expectation values of any fully degenerate operator is a constant and contains no dynamical information.

A given quantum system generally has associated with it a well-defined dynamical group structure due to the fact that the mathematical image of a quantum system is an operator algebra **g** in a linear Hilbert space. This was seen in the case of hydrogen. It comes about from the mathematical structure of quantum mechanics. The dynamical group *G* with algebra **g** is generated out of the basic physical variables, with the corresponding algebraic structure defined by the commutation relations.

The Hamiltonian H and all transition operators f g *O* can be expressed as functions of a closed set of operators

$$H = H(T\_i), \qquad O = O(T\_i), \qquad \left[T\_i, T\_j\right] = \sum\_k C\_{ij}^k T\_k. \tag{19}$$

The *Ck ij* in (19) are called the structure constants of algebra **g**. The Hilbert space is decomposed into a direct sum of the carrier spaces of unitary irreducible (irrep) representations of the group. Consequently, the dynamical symmetry properties of the system can be restricted to an irreducible Hilbert space which acts as one of the irrep carrier spaces of *G*.

From group representation theory, it will be given that a total of *σ* subgroup chains exist for a given group

$$G^a = \left\{ G\_{\mathfrak{s}^a}^a \supset G\_{\mathfrak{s}^{a-1}}^a \supset \dots \supset G\_1^a \right\}, \qquad a = 1, \dots, \sigma. \tag{20}$$

For each subgroup chain *G<sup>α</sup>* of *G*, there is a complete set of commuting operators **C** which specifies a basis set of its irreducible basis carrier space H, so the dimension of **C** for all subgroup chains of *G* is the same. A subgroup chain of dynamical group *G* serves to determine the *M* quantum dynamical degrees of freedom for a given quantum system with Hilbert space H an irrep carrier space of *G*.

*Definition 2*: For a quantum system with *M* independent quantum dynamical degrees of freedom the quantum phase space is defined to be a 2*M*-dimensional topological space. The space is isomorphic to the coset space *G=R* with explicit symplectic structure. Here *G* is the dynamical group of the system and *R*⊂ *G* is the maximal stability subgroup of the Hilbert space. □
