**4. Quantum phase space**

It is of interest then to develop a model for phase space for quantum mechanics which may be regarded as an analogue to classical physics. By what has been said so far, the Hilbert space *H* of the system can be broken up into a direct sum of the unitary irreps carrier spaces of *G*,

$$H = \sum\_{\Lambda} \oplus Y\_{\Lambda} H\_{\Lambda}.\tag{25}$$

In (25), the subscript Λ labels a particular irrep of Lie group *G*, Λ is the largest weight of the irrep and *Y*<sup>Λ</sup> the degeneracy of Λ in *H* with no correlations existing between various *H*Λ. The study of the dynamical properties of the system can be located on one particular irreducible subspace *H*<sup>Λ</sup> of *H*. For a quantum system with *M*<sup>Λ</sup> independent quantum dynamical degrees of freedom, the corresponding quantum phase space should be a 2*M*Λ-dimensional, topological phase space without additional constraints.

To construct the quantum phase space from the quantum dynamical degrees of freedom for an arbitrary quantum system, the elementary excitation operators can be obtained from the structures of *G* and *H*Λ. Let *a*† *i* � � be a subset of generators of *G* such that any states ∣Ψi of the system are generated for all ∣Ψi∈ *H*<sup>Λ</sup> by means of

*Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

$$|\Psi\rangle = F(a\_i^\dagger)|\mathbf{0}\rangle. \tag{26}$$

Moreover *F a*† *i* � ) is a polynomial in the operators *a*† *i* � � and <sup>∣</sup>0i<sup>∈</sup> *<sup>H</sup>*<sup>Λ</sup> is the reference state. The requirement placed on state ∣0i is that one can use a minimum subset of **<sup>g</sup>** to generate the entire subspace *<sup>H</sup>*<sup>Λ</sup> from <sup>∣</sup>0i. In this event, the collection *<sup>a</sup>*† *i* � � is called the set of elementary excitation operators of the quantum dynamical degrees of freedom. If *G* is compact, ∣0i is the lowest ∣Λ, � Λi or highest weight ∣Λ,Λi state of *H*Λ. If *G* is noncompact, it is merely the lowest state. The number of *a*† *i* � � is the same as the number of quantum dynamical degrees of freedom. Physically this has to be the case since that is how the operators are defined. Thus the set *a*† *i* � � and Hermitian conjugate f g *ai* in **g**<sup>Λ</sup> form a dynamical variable subspace *μ* of *g* so we can write

$$\mu: \{a\_i^\dagger, a\_i; i = 1, \dots, M\_\Lambda\}. \tag{27}$$

With respect to *μ* there exists a manifold whose dimension is twice that of the quantum dynamical degrees. It can be realized by means of a unitary exponential mapping of the dynamical variable operator subspace *μ*

$$\Omega = \exp\left(\sum\_{i=1}^{M\_{\Lambda}} \left(\eta\_i a\_i^\dagger - \eta\_i^\* \, a\_i\right)\right) \in \coprod \,\tag{28}$$

The *η<sup>i</sup>* are complex parameters and *i* ¼ 1, … , *M*Λ. In fact, Ω is a unitary coset representation of *G=R*, where *R*⊂ *G* is generated by the subalgebra *κ* ¼ **g** � *μ*. Thus (28) shows that *q* is isomorphic to the 2*M*Λ-dimensional coset space *G=R*, and will be denoted this way from now on. The discussion will apply just to semi-simple Lie groups whose **g** satisfies the usual Cartan decomposition **g** ¼ *κ* þ *μ* and ½ � *κ*, *κ* ⊂ *κ*, ½ � *κ*, *μ* ⊂*μ* and ½ � *μ*, *μ* ⊂*κ*. Thus *G=R* will be a complex homogeneous space with topology and a group transformation acting on *G=R* is a homomorphic mapping of *G=R* into itself.

The homogeneous space *G=R* has a Riemannian structure with metric

$$\mathbf{g}\_{\vec{\eta}} = \frac{\partial^2 \log \mathcal{K}(\mathbf{z}, \overline{\mathbf{z}})}{\partial \mathbf{z}\_i \partial \overline{\mathbf{z}}\_j} \tag{29}$$

The function Kð Þ *z*, *z* is called the Bergmann kernel of *G=R* and can be represented as

$$\mathcal{K}(\mathbf{z}, \mathbf{\bar{z}}) = \sum\_{\lambda} f\_{\lambda}(\mathbf{z}) f\_{\lambda}^{\*}(\mathbf{\bar{z}}).\tag{30}$$

The functions *f <sup>λ</sup>*ð Þ*z* in (30) constitute an orthogonal basis for a closed linear subspace <sup>L</sup><sup>2</sup> ð Þ *<sup>G</sup>=<sup>R</sup>* of *<sup>L</sup>*<sup>2</sup> ð Þ *G=R* such that

$$\int\_{G/\mathbb{R}} f\_{\lambda}(\mathbf{z}) f\_{\lambda'}^\*(\overline{\mathbf{z}}) \mathsf{K}^{-1}(\mathbf{z}, \overline{\mathbf{z}}) d\nu(\mathbf{z}, \overline{\mathbf{z}}) = \delta\_{\lambda \lambda'},\tag{31}$$

and *dν*ð Þ *z*, *z* is the group invariant measure on the space *G=R*. It will be written

$$d\nu(\mathbf{z}, \overline{\mathbf{z}}) = \zeta \left[ \det \left( \mathbf{g}\_{\overline{\mathbf{y}}} \right) \right] \prod\_{i=1}^{M\_{\lambda}} \frac{dz\_i d\overline{z}\_i}{\pi}. \tag{32}$$

In (32) *ζ* is a normalization factor given by the condition that (32) integrated over the space *G=R* is equal to one. There is also a closed, nondegenerate two-form on *G=R* which is expressed as,

$$
\mu \alpha = i\hbar \sum\_{i,j} \mathbf{g}\_{ij} d\mathbf{z}\_i \wedge d\overline{\mathbf{z}}\_j. \tag{33}
$$

Corresponding to this two form there is a Poisson bracket which is given by

$$\{f, h\} = \frac{1}{i\hbar} \sum\_{i, j} g^{ij} \left[ \frac{\partial f}{\partial \overline{z}\_i} \frac{\partial h}{\partial \overline{z}\_j} - \frac{\partial f}{\partial \overline{z}\_j} \frac{\partial h}{\partial \overline{z}\_i} \right]. \tag{34}$$

In (34) *f* and *h* are functions defined on *G=R*. By introducing canonical coordinates ð Þ **q**, **p** these quantities can be rewritten in terms of these coordinates.

#### **4.1 Phase space quantum dynamics**

Based on what has been stated about *G=R*, it would be useful to describe the quantum phase space. This means for a given quantum system a phase space representation must exist. Such a representation can be found if there exists an explicit mapping such that

$$O(T\_i) \to U(\mathbf{q}, \mathbf{p}), \quad |\Psi\rangle \to \rho(q+ip). \tag{35}$$

Here *<sup>O</sup>* is given by (19), and *<sup>ρ</sup>*ð Þ **<sup>q</sup>**, **<sup>p</sup>** <sup>∈</sup>*L*<sup>2</sup> . For a quantum system with a quantum phase space *G=R*, this mapping can be realized by coherent states. To construct coherent states of *G* and *H*<sup>Λ</sup> defined on *G=R*, the fixed state ∣0i is chosen as the initial state

$$\mathbf{g}|\mathbf{0}\rangle = \mathfrak{\Omega}\mathbf{r}|\mathbf{0}\rangle = |\Lambda, \mathfrak{\Omega}\rangle e^{i\mathfrak{p}(\mathbf{r})}, \quad \mathbf{g} \in G, \quad \mathbf{r} \in \mathbb{R}, \quad \mathfrak{\Omega} \in G/R. \tag{36}$$

Then *R* is the maximal stability subgroup of ∣0i so any **r**∈*R* acting on ∣0i will leave ∣0i invariant up to a phase factor

$$\mathbf{r}|\mathbf{0}\rangle = e^{i\boldsymbol{\rho}(\mathbf{r})}|\mathbf{0}\rangle. \tag{37}$$

The ∣Λ, Ωi are the coherent states which are isomorphic to *G=R*. Therefore,

$$|\Lambda, \mathfrak{Q}\rangle \equiv \mathfrak{Q}|0\rangle = \exp\left(\sum\_{i=1}^{M\_{\Lambda}} \left(\eta\_i a\_i^\dagger - \eta\_i^\* a\_i\right)\right)|0\rangle = \mathcal{K}^{1/2}(\mathbf{z}, \overline{\mathbf{z}}) \exp\left(\sum\_{i=1}^{M\_{\Lambda}} z\_i a\_i^\dagger\right)|0\rangle$$

$$= \mathcal{K}^{-1/2}(\mathbf{z}, \overline{\mathbf{z}})|\Lambda, \mathbf{z}\rangle. \tag{38}$$

$$\begin{aligned} \mathcal{K}(\mathbf{z}, \overline{\mathbf{z}}) &= \left\langle \mathbf{0} \middle| \exp \left( \sum\_{i=1}^{M\_{\Lambda}} \overline{\mathbf{z}}\_{i} a\_{i} \right) \exp \left( \sum\_{i=1}^{M\_{\Lambda}} z\_{i} a\_{i}^{\dagger} \right) \middle| \mathbf{0} \right\rangle = \left\langle \Lambda, \mathbf{z} \middle| \middle| \Lambda, \mathbf{z} \right\rangle = \left\langle \mathbf{0} \middle| \Lambda, \mathbf{0} \right\rangle \Big| \mathbf{0} \\ &= \sum\_{\lambda} f\_{\lambda \mathbf{\!\!\!/}}(\mathbf{z}) f\_{\lambda \mathbf{\!\!\!/}}^{\*}(\mathbf{z}). \end{aligned}$$

The Bargmann kernel was introduced in (30), and for a semisimple Lie group, the parameters *zi* are given by

*Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

$$z = \begin{cases} \eta \frac{\tan \left(\eta^\dagger \eta\right)^{1/2}}{\left(\eta^\dagger \eta\right)^{1/2}}, & \text{ $G$  compact}, \\\\ \eta \frac{\tanh \left(\eta^\dagger \eta\right)^{1/2}}{\left(\eta^\dagger \eta\right)^{1/2}}, & \text{ $G$  noncompact.} \end{cases} \tag{39}$$

Here *η* represents the nonzero *k* � *p* block matrix of the operator P*M*<sup>Λ</sup> *<sup>i</sup>*¼<sup>1</sup> *<sup>η</sup>iai* � *<sup>η</sup>* <sup>∗</sup> *<sup>i</sup> <sup>a</sup>*† *i* � �. The state <sup>k</sup>Λ, *<sup>z</sup>*<sup>i</sup> in (38) is an unnormalized form of <sup>∣</sup>Λ, <sup>Ω</sup><sup>i</sup> and *<sup>f</sup>* <sup>Λ</sup>,*<sup>λ</sup>*ð Þ*<sup>z</sup>* is the orthogonal basis of <sup>L</sup><sup>2</sup> ð Þ *G=R* the function space

$$f\_{\Lambda,\dot{\lambda}}(\mathbf{z}) = \langle \Lambda, \lambda | | \Lambda, \dot{\lambda} \rangle,\tag{40}$$

where ∣Λ, *λ*i is a basis for *H*Λ, a particular irreducible subspace of the Hilbert space. The coherent states of (38) are over-complete

$$\int\_{G/\mathbb{R}} |\Lambda, \mathfrak{Q}\rangle\langle\Lambda, \mathfrak{Q}|d\nu(z) = I. \tag{41}$$

A classical-like framework or analogy has been established in the form of a quantum phase space specified by *G* and *H*Λ. Variables which reside in this classical analogy are denoted thus ~*c*. The 2*M*Λ-dimensional quantum phase space *G=R* has all the required structures of a classical mechanical system. It is always possible a classical dynamical theory can be established in *G=R* whose motion is confined to *G=R* and is determined by the following equations of motion

$$\frac{d\tilde{\mathcal{U}}}{dt} = \{\tilde{\mathcal{U}}(q, p), \tilde{H}(q, p)\}, \qquad q, p \in \mathcal{G}/\mathbb{R}. \tag{42}$$

This equation can be replaced by Hamilton's equations

$$\frac{dq\_i}{dt} = \frac{\partial \tilde{H}(q, p)}{\partial p\_i}, \qquad \frac{dp\_i}{dt} = -\frac{\partial \tilde{H}(q, p)}{\partial q\_i}.\tag{43}$$

In (42) and (43), *H q* <sup>~</sup> ð Þ , *<sup>p</sup>* is the Hamiltonian of the system, and *U q* <sup>~</sup> ð Þ , *<sup>p</sup>* is a physical observable. A correspondence principle is implied here and requires that suitable conditions can be found such that the quantum dynamical Heisenberg equations can be written this way.

Clearly, if suitable conditions hold the phase space representation of the commutator of any two operators is equal to the Poisson bracket of the phase space representation of these two operators so that

$$\frac{1}{i\hbar}\langle\Lambda,\Omega|[A\_H,B\_H]|\Lambda,\Omega\rangle = \left\{\tilde{\mathcal{A}},\tilde{\mathcal{B}}\right\}.\tag{44}$$

Then the phase space representation of the Heisenberg equation

$$\frac{dA\_H}{dt} = \frac{1}{i\hbar}[A\_H, H\_H],\tag{45}$$

given by (42) is therefore equivalent to (43). In (45), *AH* is the Heisenberg operator

$$A\_H = UAU^{-1}, \qquad U = e^{iHt/\hbar},\tag{46}$$

and *A* is time-independent in the Schrödinger picture. The coherent state on the left of (44) is time-independent. Observables on the right side are the expectation values of the Schrödinger operators in the time-dependent coherent state. The quantum phase space maintains many of the quantum properties which are important, such as internal degrees of freedom, the Pauli principle, statistical properties and dynamical symmetry. Formally the equation of motion is classical. The phase space representation is based on the whole quantum structure of the coset space *G=R*.

Let us discuss integrability and dynamical symmetry. A quantum system with *M*<sup>Λ</sup> independent degrees of freedom is integrable if and only if the *M*<sup>Λ</sup> non-fully degenerate observables can simultaneously be measured in the energy representation. There exist non-fully degenerate observables *Ci* f g : *i* ¼ 1, … , *M*<sup>Λ</sup> � 1 which commute with each other and *H*

$$\left[\mathbf{C}\_{i}, \mathbf{C}\_{j}\right] = \mathbf{0}, \qquad \left[\mathbf{C}\_{i}, H\right] = \mathbf{0}. \tag{47}$$

It follows that in the classical limit which has been formulated,

$$\{\tilde{\mathcal{C}}\_i, \tilde{\mathcal{C}}\_j\} = \mathbf{0}, \qquad \{\tilde{\mathcal{C}}\_i, \tilde{H}\} = \mathbf{0}.\tag{48}$$

Together with the Hamilton equations, (47) also formally defines classical integrability, so quantum integrability is completely consistent with the classical theory. In the classical analogy, the group structure of the system is defined by Poisson brackets. The concept of dynamical symmetry is naturally preserved in the classical analogy, so the theorem on dynamical symmetry and integrability is also meaningful for the classical analogy. If the Hamiltonian has the symmetry *S*, then its phase space picture representation has the same symmetry. To see this, if

$$\text{SHS}^{-1} = H,\tag{49}$$

in the phase space representation, it holds that

$$
\langle \Lambda, \mathfrak{Q} | H | \Lambda, \mathfrak{Q} \rangle = \left\langle \Lambda, \mathfrak{Q} | \mathrm{SHS}^{-1} | \Lambda, \mathfrak{Q} \right\rangle = \langle \Lambda, \mathfrak{Q}' | H | \Lambda, \mathfrak{Q}' \rangle. \tag{50}
$$

To put this concisely, we write

$$
\tilde{H}(q, p) = \tilde{H}(q', p'), \tag{51}
$$

where *S*�<sup>1</sup> <sup>∣</sup>Λ, <sup>Ω</sup>i ¼ *<sup>S</sup>*�<sup>1</sup> Ω∣0i ¼ ∣Λ, Ω<sup>0</sup> <sup>i</sup>*e<sup>i</sup>φ*ð Þ *<sup>h</sup>* .
