**5.2** *SU*ð Þ **2 spin system**

The phase space structure of a spin system will be constructed and as well the phase-space distribution and classical analogy.

Since the dynamical group of the spin system is *SU*ð Þ2 and the Hilbert space is described by the states *<sup>V</sup>*<sup>2</sup>*j*þ<sup>1</sup> ¼ j f g *<sup>j</sup>*, *<sup>m</sup>*<sup>i</sup> where *<sup>m</sup>* ¼ �*j*, � *<sup>j</sup>* <sup>þ</sup> 1, … , *<sup>j</sup>* and *<sup>j</sup>* is an integer or half-integer, the fixed state is ∣*j*, � *j*i. This is the lowest weight state of *V*<sup>2</sup>*j*þ<sup>1</sup> . Thus the elementary excitation operator of the spin system is *J*<sup>þ</sup> and the explicit form of ∣ *j*, *m*i is

$$|j,m\rangle = \frac{1}{(j+m)!} \binom{2j}{j+m}^{-1/2} (J\_+)^{j+m} |j,-j\rangle. \tag{69}$$

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

Any state <sup>∣</sup>Ψi ¼ <sup>P</sup>*<sup>j</sup> <sup>m</sup>*¼�*<sup>j</sup> <sup>f</sup> <sup>m</sup>*∣*j*, *<sup>m</sup>*i∈*V*2*j*þ<sup>1</sup> can be generated by a polynomial of *<sup>J</sup>*<sup>þ</sup> acting on ∣*j*, � *j*i. This means the number of quantum dynamical degrees of freedom is equal to the number of elementary excitation operators. The quantum phase space can be found by mapping *μ* : *J*þ, *J*� � � to the coset space *SU*ð Þ<sup>2</sup> *<sup>=</sup>U*ð Þ<sup>1</sup> by means of ð Þ! *<sup>η</sup>J*<sup>þ</sup> � *<sup>η</sup>J*� exp ð Þ *<sup>η</sup>J*<sup>þ</sup> � *<sup>η</sup>J*� where *<sup>η</sup>* <sup>¼</sup> ð Þ *<sup>ϑ</sup>=*<sup>2</sup> *<sup>e</sup>*�*i<sup>φ</sup>*, 0≤*<sup>ϑ</sup>* <sup>≤</sup>*π*, 0≤*φ*<sup>≤</sup> <sup>2</sup>*π*. The coset space *SU*ð Þ2 *=U*ð Þ1 is isomorphic to a two-dimensional sphere. The coherent states of *SU*ð Þ2 *=U*ð Þ1 are well known

$$\begin{aligned} \vert j\Omega \rangle &= \exp\left(\eta \mathcal{I}\_{+} - \overline{\eta} \mathcal{I}\_{-}\right) \vert j, -j \rangle = \left(\mathbb{1} + \left|\mathbf{z}\right|^{2}\right)^{-j} \exp\left(\mathbf{z} \mathcal{I}\_{+}\right) \vert j, -j \rangle = \left(\mathbb{1} + \left|\mathbf{z}\right|^{2}\right)^{-j} \vert j\mathbf{z} \rangle, \\\ z &= \tan \frac{\theta}{2} e^{-i\boldsymbol{\rho}}. \end{aligned} \tag{70}$$

The generalized Bargmann kernel on *<sup>S</sup>*<sup>2</sup> is Kð Þ¼ *<sup>z</sup>*, *<sup>z</sup>* <sup>1</sup> <sup>þ</sup> j j *<sup>z</sup>* <sup>2</sup> � �<sup>2</sup>*<sup>j</sup>* . Then the metric *gij* and measure are given by

$$\mathbf{g}\_{\dot{\mathbf{q}}} = \delta\_{\dot{\mathbf{q}}} \frac{\mathbf{2j}}{\left(\mathbf{1} + \left|\mathbf{z}\right|^2\right)^2}, \qquad d\boldsymbol{\nu} = \frac{\mathbf{1}}{\pi} (2\dot{\mathbf{j}} + \mathbf{1}) \frac{d\mathbf{z} d\overline{\mathbf{z}}}{\mathbf{1} + \left|\mathbf{z}\right|^2)^2}. \tag{71}$$

Given the canonical coordinates

$$\frac{1}{\sqrt{4j\hbar}}(q+ip) = \frac{z}{\sqrt{1+|x|^2}} = \sin\left(\frac{\theta}{2}\right)e^{-iq},\tag{72}$$

there obtains the bracket

$$\left\{\bar{F}\_1, \bar{F}\_2\right\} = \frac{\partial \bar{F}\_1}{\partial q} \frac{\partial \bar{F}\_2}{\partial p} - \frac{\partial \bar{F}\_1}{\partial p} \frac{\partial \bar{F}\_2}{\partial q},\tag{73}$$

where *<sup>q</sup>*<sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> <sup>≤</sup> <sup>4</sup>*j*ℏ, which implies the phase space of a spin system is compact. The phase space representation of the state <sup>∣</sup>Ψi∈*V*<sup>2</sup>*j*þ<sup>1</sup> is for *<sup>f</sup>* <sup>∈</sup>*L*<sup>2</sup> *<sup>S</sup>*<sup>2</sup> � �,

$$f(z) = \langle \Psi | j\Omega \rangle = \sum\_{n=0}^{\infty} f\_n \binom{2j}{j+m}^{1/2} z^{j+m},\tag{74}$$

The phase space representation of an operator *B* ¼ *B Ji* ð Þ is

$$
\tilde{B}(z,\overline{z}) = \langle j\Omega | B | j\Omega \rangle. \tag{75}
$$

When the operator *B* in (75) is chosen to be one of the three operators *J*þ, *J*� or *J*0, the results are

$$\begin{aligned} \tilde{J}\_{+} &= \left< j\mathfrak{Q} | J\_{+} | j\mathfrak{Q} \right> = \frac{2j\overline{z}}{\mathbf{1} + |z|^{2}}, \quad \tilde{J}\_{-} = \left< j\mathfrak{Q} | J\_{-} | j\mathfrak{Q} \right> = \frac{2jz}{\left(\mathbf{1} + |z|^{2}\right)^{2}}, \\\\ \tilde{J}\_{0} &= \left< j\mathfrak{Q} | J\_{0} | j\mathfrak{Q} \right> = j \frac{|z|^{2} - \mathbf{1}}{\mathbf{1} + |z|^{2}}. \end{aligned} \tag{76}$$

These can also be given in terms of *q*, *p* by using (72). The algebraic structure of *SU*ð Þ2 in the phase space representation is given by the Poisson bracket

$$i\hbar\_{\mathfrak{c}}\{\tilde{J}\_{-},\tilde{J}\_{+}\} = -2\tilde{J}\_{0}, \qquad i\hbar\_{\mathfrak{c}}\{\tilde{J}\_{0},\tilde{J}\_{\pm}\} = \pm \tilde{J}\_{\pm}. \tag{77}$$

The classical analogy of an observable *B Ji* ð Þ is given by the following expression

$$
\tilde{B}(q, p) = \langle j, \Omega | B(J\_i) | j, \Omega \rangle. \tag{78}
$$

The classical limit is found by taking *j* ! ∞ and the classical Hamiltonian function is

$$
\tilde{H}\_{\mathcal{C}}(q, p) = H(\langle j, \Omega | J\_i | j, \Omega \rangle) = H(\tilde{f}\_+, \tilde{f}\_-, \tilde{f}\_0). \tag{79}
$$
