25.2 Complex coordinates on phase space and U(d) ⊂ Sp(2d, R)

25.2 Complex coordinates on phase space and U(d) ⊂ Sp(2d, R)

Concept links · terms present in this machine draft; source roles are unverified: unitary group · Lie algebra · unitary representation · Lie bracket · complexification

As in the $d=1$ case, annihilation and creation operators can be thought of as the quantization of complexified coordinates $z_j,{\bar{z}}_j$ on phase space, with the standard choice given by

$$z_j=\frac{1}{\sqrt{2}}(q_j-i{p}_j),{\overline{z}}_j=\frac{1}{\sqrt{2}}(q_j+i{p}_j)$$

Such a choice of $z_j,{\bar{z}}_j$ gives a decomposition of the complexified Lie algebra sp(2d, $C$) (as usual, the Lie bracket is the Poisson bracket) into three Lie subalgebras as follows:

• A Lie subalgebra with basis elements $z_j{z}_k$ . There are $\frac{1}{2}(d^2+d)$ distinct such basis elements. This is a commutative Lie subalgebra, since the Poisson bracket of any two basis elements is zero.

• A Lie subalgebra with basis elements ${\overline{z}}_j{\overline{z}}_k$ . Again, this has dimension $\frac{1}{2}(d^2+d)$ and is a commutative Lie subalgebra.

• A Lie subalgebra with basis elements $z_j{\overline{z}}_k$ , which has dimension $d^2$ . Computing Poisson brackets one finds

$$\begin{array}{rl}\{z_j{\overline{z}}_k,z_l{\overline{z}}_m\}& =z_j\{{\overline{z}}_k,z_l{\overline{z}}_m\}+{\overline{z}}_k\{z_j,z_l{\overline{z}}_m\}\\ & =-i{z}_j{\overline{z}}_m{\delta }_{kl}+i{z}_l{\overline{z}}_k{\delta }_{jm}\end{array}\text{\hspace{1em}(25.2)}$$

In this chapter we’ll focus on the third subalgebra and the operators that arise by quantization of its elements.

Taking all complex linear combinations, this subalgebra can be identified with the Lie algebra $\mathfrak{gl}(d,\mathbf{C})$ of all d by $d$ complex matrices. One can see this by noting that if $E_{jk}$ is the matrix with 1 at the j-th row and k-th column, zeros elsewhere, one has

$$[E_{jk},E_{lm}]=E_{jm}{\delta }_{kl}-E_{lk}{\delta }_{jm}$$

and these provide a basis of $\mathfrak{gl}(d,\mathbf{C})$ . Identifying bases by

$$i{z}_j{\overline{z}}_k\leftrightarrow E_{jk}$$

gives the isomorphism of Lie algebras. This $\mathfrak{gl}(d,\mathbf{C})$ is the complexification of $\mathfrak{u}(d)$ , the Lie algebra of the unitary group $U(d)$ . Elements of $\mathfrak{u}(d)$ will correspond to, equivalently, skew-adjoint matrices, or real linear combinations of the quadratic functions

$$z_j{\overline{z}}_k+{\overline{z}}_j{z}_k,i(z_j{\overline{z}}_k-{\overline{z}}_j{z}_k)$$

on $M$.

In section 16.1.2 we saw that the moment map for the action of the symplectic group on phase space is just the identity map when we identify the Lie algebra $\mathfrak{sp}(2d,\mathbf{R})$ with order two homogeneous polynomials in the phase space coordinates $q_j,p_j$ . We can complexify and identify sp $(2d,\mathbf{C})$ with complex-valued order two homogeneous polynomials which we write in terms of the complexified coordinates $z_j,{\bar{z}}_j$ . The moment map is again the identity map, and on the sub-Lie algebra we are concerned with, is explicitly given by

$$A\in \mathfrak{gl}(d,\mathbf{C})\to {\mu }_A=i\sum _{j,k}{z}_j{A}_{jk}{\overline{z}}_k\text{\hspace{1em}(25.3)}$$

We can at the same time consider the complexification of the Heisenberg Lie algebra, using linear functions of $z_j$ and ${\overline{z}}_j,$ , with Poisson brackets between these and the order two homogeneous functions giving a complexified version of the derivation action of $\mathfrak{sp}(2d,\mathbf{R})$ on ${\mathfrak{h}}_{2d+1}$ .

We have (complexifying and restricting to $\mathfrak{gl}(d,\mathbf{C})\subset \mathfrak{sp}(2d,\mathbf{C}))$ the following version of theorems 16.2 and 16.3

Theorem 25.1

The map of equation 25.3 is a Lie algebra homomorphism, $i$.e.

$$\left\{{\mu }_A,{\mu }_{A^{\prime }}\right\}={\mu }_{[A,A^{\prime }]}$$

The ${\mu }_A$ satisfy (for column vectors $z$ with components $z_1,\dots ,z_d)$

$$\{{\mu }_A,\overline{\mathbf{z}}\}=-A\overline{\mathbf{z}},\{{\mu }_A,\mathbf{z}\}=A^T\mathbf{z}\text{\hspace{1em}(25.4)}$$

Proof. Using 25.2 one has

$$\begin{array}{rl}\{{\mu }_A,{\mu }_{A^{\prime }}\}& =-\sum _{j,k,l,m}\{z_j{A}_{jk}{\overline{z}}_k,z_l{A}_{lm}^{\prime }{\overline{z}}_m\}\\ & =-\sum _{j,k,l,m}{A}_{jk}{A}_{lm}^{\prime }\{z_j{\overline{z}}_k,z_l{\overline{z}}_m\}\\ & =i\sum _{j,k,l,m}{A}_{jk}{A}_{lm}^{\prime }(z_j{\overline{z}}_m{\delta }_{kl}-z_l{\overline{z}}_k{\delta }_{jm})\\ & =i\sum _{j,k}{z}_j[A,A^{\prime }{]}_{jk}{\overline{z}}_k={\mu }_{[A,A^{\prime }]}\end{array}$$

To show 25.4, compute

$$\begin{array}{rl}\{{\mu }_A,{\overline{z}}_l\}& =\{i\sum _{j,k}{z}_j{A}_{jk}{\overline{z}}_k,{\overline{z}}_l\}=i\sum _{j,k}{A}_{jk}\{z_j,{\overline{z}}_l\}{\overline{z}}_k\\ & =-\sum _j{A}_{lj}{\overline{z}}_j\end{array}$$

and

$$\begin{array}{rl}\{{\mu }_A,z_l\}& =\{i\sum _{j,k}{z}_j{A}_{jk}{\overline{z}}_k,z_l\}=i\sum _{j,k}{z}_j{A}_{jk}\{{\overline{z}}_k,z_l\}\\ & =\sum _k{z}_k{A}_{kl}\end{array}$$

Note that here we have written formulas for $A\in \mathfrak{gl}(d,\mathbf{C})$ , an arbitrary complex d by $d$ matrix. It is only for $A\in \mathfrak{u}(d)$ , the skew-adjoint $(\overline{A^T}=-A)$ matrices, that ${\mu }_A$ will be a real-valued moment map, lying in the real Lie algebra $sp(2d,\mathbf{R})$ , and giving a unitary representation on the state space after quantization. For such A we can write the relations 25.4 as a (complexified) example of 16.22

$$\left\{{\mu }_A,\left(\genfrac{}{}{0ex}{}{\overline{\mathbf{z}}}{\mathbf{z}}\right)\right\}=\left(\begin{array}{cc}\overline{A^T}& 0\\ 0& A^T\end{array}\right)\left(\genfrac{}{}{0ex}{}{\overline{\mathbf{z}}}{\mathbf{z}}\right)$$

The standard harmonic oscillator Hamiltonian

$$h=\sum _{j=1}^d{z}_j{\overline{z}}_j\text{\hspace{1em}(25.5)}$$

lies in this $\mathfrak{u}(d)$ sub-algebra (it is the case $A=-i1)$ , and its Poisson brackets with the rest of the sub-algebra are zero. It gives a basis element of the one dimensional $\mathfrak{u}(1)$ subalgebra that commutes with the rest of the $\mathfrak{u}(d)$ subalgebra.

While we are not entering here into the details of what happens for polynomials that are linear combinations of the $z_j{z}_k$ and ${\overline{z}}_j{\overline{z}}_k$ , it may be worth noting one confusing point about these. Recall that in chapter 16 we found the moment map ${\mu }_L=-\mathbf{q}\cdot A\mathbf{p}$ for elements $L\in \mathfrak{sp}(2d,\mathbf{R})$ of the block-diagonal form

$$\left(\begin{array}{cc}A& 0\\ 0& -A^T\end{array}\right)$$

where $A$ is a real d by $d$ matrix and so in $\mathfrak{gl}(d,\mathbf{R})$ . That block decomposition corresponded to the decomposition of basis vectors of M into the two sets $q_j$ and $p_j$ . Here we have complexified, and are working with respect to a diferent decomposition, that of basis vectors $\mathcal{M}\otimes \mathbf{C}$ into the two sets $z_j$ and ${\overline{z}}_j.$ The matrices $A$ in this case are complex, skew-adjoint, and in a diferent nonisomorphic Lie subalgebra, $\mathfrak{u}(d)$ rather than $\mathfrak{gl}(d,\mathbf{R})$ . For the simplest example of this, $d=1$ , the distinction is between the R Lie subgroup of $SL(2,\mathbf{R})$ (see section 20.3.4), for which the moment map is

$$-qp=\mathrm{Im}(z^2)=\frac{1}{2i}(z^2-{\overline{z}}^2)$$

and the $U(1)$ subgroup (see section 20.3.2), for which the moment map is

$$\frac{1}{2}(q^2+p^2)=z\overline{z}$$

Previous section: 25.1 Multiple degrees of freedom · Next section: 25.3 The metaplectic representation and U(d) Sp(2d, R) · 中文版本

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