24.4 SU(1, 1) and Bogoliubov transformations

24.4 SU(1, 1) and Bogoliubov transformations

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · eigenvalue · complexification

Changing bases in complexified phase space from $q,p$ to z, $z$ changes the group of linear transformations preserving the Poisson bracket from the group $SL(2,\mathbf{R})$ of real 2 by 2 matrices of determinant one to an isomorphic group of complex 2 by 2 matrices. We have

Theorem

The group $SL(2,\mathbf{R})$ is isomorphic to the group $SU(1,1)$ of complex $2by2$ matrices

$$\left(\begin{array}{cc}\alpha & \beta \\ \overline{\beta }& \overline{\alpha }\end{array}\right)$$

such that

$$|\alpha {|}^2-|\beta {|}^2=1$$

Proof

The equations for $\overline{z},z$ in terms of $q,p$ imply that the change of basis between these two bases is

$$\left(\genfrac{}{}{0ex}{}{\overline{z}}{z}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& i\\ 1& -i\end{array}\right)\left(\genfrac{}{}{0ex}{}{q}{p}\right)$$

The matrix for this transformation has inverse

$$\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ -i& i\end{array}\right)$$

Conjugating by this change of basis matrix, one finds

$$\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ -i& i\end{array}\right)\left(\begin{array}{cc}\alpha & \beta \\ \overline{\beta }& \overline{\alpha }\end{array}\right)\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& i\\ 1& -i\end{array}\right)=\left(\begin{array}{cc}\mathrm{Re}(\alpha +\beta )& -\mathrm{Im}(\alpha -\beta )\\ \mathrm{Im}(\alpha +\beta )& \mathrm{Re}(\alpha -\beta )\end{array}\right)\text{\hspace{1em}(24.11)}$$

The right hand side is a real matrix, with determinant one, since conjugation doesn’t change the determinant. □

Note that the change of basis 24.11 is reflected in equations 24.3 and 24.8, where the matrices on the right hand side are the matrix $Z$ and G respectively, but transformed to the $\overline{z},z$ basis by 24.11.

Another equivalent characterization of the group $SU(1,1)$ is as the group of linear transformations of ${\mathbf{C}}^2$ , with determinant one, preserving the indefinite Hermitian inner product

$${⟨\left(\genfrac{}{}{0ex}{}{c_1}{c_2}\right),\left(\genfrac{}{}{0ex}{}{c_1^{\prime }}{c_2^{\prime }}\right)⟩}_{1,1}={\overline{c}}_1{c}_1^{\prime }-{\overline{c}}_2{c}_2^{\prime }$$

One finds that

$${⟨\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\left(\genfrac{}{}{0ex}{}{c_1}{c_2}\right),\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\left(\genfrac{}{}{0ex}{}{c_1^{\prime }}{c_2^{\prime }}\right)⟩}_{1,1}={⟨\left(\genfrac{}{}{0ex}{}{c_1}{c_2}\right),\left(\genfrac{}{}{0ex}{}{c_1^{\prime }}{c_2^{\prime }}\right)⟩}_{1,1}$$

when $\gamma =\overline{\beta },\delta =\overline{\alpha }$ and $|\alpha {|}^2-|\beta {|}^2=1$

Applied not to $\overline{z},z$ but to their quantizations $a,a^{†}$ , such $SU(1,1)$ transformations are known to physicists as “Bogoliubov transformations”. One can easily see that replacing the annihilation operator a by

$$a^{\prime }=\alpha a+\beta a^{†}$$

leads to operators with the same commutation relations when $|\alpha {|}^2-|\beta {|}^2=1$ ， since

$$[a^{\prime },(a^{\prime }{)}^{†}]=[\alpha a+\beta a^{†},\overline{\alpha }{a}^{†}+\overline{\beta }a]=(|\alpha {|}^2-|\beta {|}^2)1$$

By equation 24.11 the $SO(2)\subset SL(2,\mathbf{R})$ subgroup of equation 24.1 appears in the isomorphic $SU(1,1)$ group as the special case $\alpha =e^{i\theta },\beta =0$ , so matrices of the form

$$\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)$$

Acting with this subgroup on the annihilation and creation operators just changes a by a phase (and $a^{†}$ by the conjugate phase).

The subgroup 24.7 provides more non-trivial Bogoliubov transformations, with conjugation by $U_{g_{\lambda }}$ giving (see equation 24.9) annihilation and creation operators

$$a_r=a\cosh r+a^{†}\sinh r,a_r^{†}=a\sinh r+a^{†}\cosh r$$

For $r\ne 0.$ , the state

$$|0{⟩}_r=e^{\frac{r}{2}(a^2-(a^{†}{)}^2)}|0⟩$$

will be an eigenstate of neither H nor the number operator $N$ , and describes a state without a definite number of quanta. It will be the ground state for a quantum system with Hamiltonian operator

$$H_r=a_r^{†}{a}_r+\frac{1}{2}=(\cosh (2r)a^{†}a+\frac{1}{2}\sinh (2r)(a^2+(a^{†}{)}^2)+{\sinh }^{2}r+\frac{1}{2}$$

Such quadratic Hamiltonians that do not commute with the number operator have lowest energy states $|0{⟩}_{⟨}$ with indefinite number eigenvalue. Examples of this kind occur for instance in the theory of superfluidity.

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