26.1 Complex structures and phase space

26.1 Complex structures and phase space

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · Lie algebra · Lie algebra representation · complexification

Quantization of phase space $M={\mathbf{R}}^{2d}$ using the Schr¨odinger representation gives a unitary Lie algebra representation ${\Gamma }_S^{\prime }$ of the Heisenberg Lie algebra ${\mathfrak{h}}_{2d+1}$ which takes the $q_j$ and $p_j$ coordinate functions on phase space to operators $-i{Q}_j$ and $-i{P}_j$ on ${\mathcal{H}}_S=L^2({\bar{\mathbf{R}}}^d)$ . This involves a choice, that of taking states to be functions of the $q_j$ , or (using the Fourier transform) of the $p_j$ . It turns out to be a general phenomenon that quantization requires choosing some extra structure on phase space, beyond the Poisson bracket.

For the case of the harmonic oscillator, we found in chapter 22 that quantization was most conveniently performed using annihilation and creation operators, which involve a diferent sort of choice of extra structure on phase space. There we introduced complex coordinates on phase space, making the choice

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

The $z_j$ were then quantized using creation operators $a_j^{†},$ the ${\overline{z}}_j$ using annihilation operators $a_j$ . In the Bargmann-Fock representation, where the state space is a space of functions of complex variables $z_j$ , we have

$$a_j=\frac{\partial }{\partial z_j},a_j^{†}=z_j$$

and there is a distinguished state, the constant function, which is annihilated by all the $a_j$

In this section we’ll introduce the notion of a complex structure on a real vector space, with such structures characterizing the possible ways of introducing complex coordinates $z_j,{\overline{z}}_j$ and thus annihilation and creation operators. The abstract notion of a complex structure can be formalized as follows. Given any real vector space $V={\mathbf{R}}^n$ , we have seen that taking complex linear combinations of vectors in $V$ gives a complex vector space $V\otimes \mathbf{C}$ , the complexification of $V.$ and this can be identified with ${\mathbf{C}}^n$ , a real vector space of twice the dimension. When $n=2d$ is even there is another way to turn $V={\mathbf{R}}^{2d}$ into a complex vector space, by using the following additional piece of information:

Definition (Complex structure)

A complex structure on a real vector space $V$ is a linear operator

$$J:V\to V$$

such that

$$J^2=-1$$

Given such a pair $(V={\mathbf{R}}^{2d},J)$ complex linear combinations of vectors in $V$ can be decomposed into those on which $J$ acts as i and those on which it acts as −$i$ (since $J^2=-1$ , its eigenvalues must be ±i), so we have

$$V\otimes \mathbf{C}=V_J^{+}\oplus V_J^{-}$$

where $V_J^{+}$ is the +$i$ eigenspace of the operator $J$ on $V\otimes \mathbf{C}$ and $V_J^{-}$ is the $-i$ eigenspace. Note that we have extended the action of $J$ on $V$ to an action on $V\otimes \mathbf{C}$ using complex linearity. Complex conjugation takes elements of $V_J^{+}$ to $V_J^{-}$ and vice-versa. The choice of J has thus given us two complex vector spaces of complex dimension $d$, $V_J^{+}$ and $V_J^{-}$ , related by this complex conjugation.

Since

$$J(v-iJv)=i(v-iJv)$$

for any $v\in V$ , the real vector space $V$ can by identified with the complex vector space $V_J^{+}$ by the map

$$v\in V\to \frac{1}{\sqrt{2}}(v-iJv)\in V_J^{+}\text{\hspace{1em}(26.1)}$$

The pair $(V,J)$ can be thought of as giving V the structure of a complex vector space, with $J$ providing multiplication by $i$. Similarly, taking

$$v\in V\to \frac{1}{\sqrt{2}}(v+iJv)\in V_J^{-}\text{\hspace{1em}(26.2)}$$

identifies V with $V_J^{-}$ , with $J$ now providing multiplication by $-i.V_J^{+}$ and $V_J^{-}$ are interchanged by changing the complex structure $\mathcal{J}to-\mathcal{J}.$

For the study of quantization, the real vector space we want to choose a complex structure on is the dual phase space $\mathcal{M}=M^{\ast }$ , since it is elements of this space that are in a Heisenberg algebra, and taken to operators by quantization. There will be a decomposition

$$\mathcal{M}\otimes \mathbf{C}={\mathcal{M}}_J^{+}\oplus {\mathcal{M}}_J^{-}$$

and quantization will take elements of ${\mathcal{M}}_J^{+}$ to linear combinations of creation operators, elements of ${\mathcal{M}}_J^{-}$ to linear combinations of annihilation operators.

The standard choice of complex structure is to take $J=J_0$ , where $J_0$ is the linear operator that acts on coordinate basis vectors $q_j,p_j$ of M by

$$J_0{q}_j=p_j,J_0{p}_j=-q_j$$

Making the choice

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

implies

$$J_0{z}_j=\frac{1}{\sqrt{2}}(p_j+i{q}_j)=i{z}_j$$

and the $z_j$ are basis elements (over the complex numbers) of ${\mathcal{M}}_{J_0}^{+}$ . The complex conjugates

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

provide basis elements of ${\mathcal{M}}_{J_0}^{-}$

With respect to the chosen basis $q_j,p_j$ , the complex structure can be written as a matrix. For the case of $J_0$ and for $d=1$ , on an arbitrary element of $\mathcal{M}$ the action of $J_0$ is

$$J_0(c_{q}q+c_{p}p)=c_{q}p-c_{p}q$$

so $J_0$ in matrix form with respect to the basis $(q,p)$ is

$$J_0\left(\genfrac{}{}{0ex}{}{c_q}{c_p}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\genfrac{}{}{0ex}{}{c_q}{c_p}\right)=\left(\genfrac{}{}{0ex}{}{-c_p}{c_q}\right)\text{\hspace{1em}(26.3)}$$

or, the action on basis vectors is the transpose

$$J_0\left(\genfrac{}{}{0ex}{}{q}{p}\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\genfrac{}{}{0ex}{}{q}{p}\right)=\left(\genfrac{}{}{0ex}{}{p}{-q}\right)\text{\hspace{1em}(26.4)}$$

Note that, after complexifying, three diferent ways to identify the original M with a subspace of $\mathcal{M}\otimes \mathbf{C}$ are:

$\mathcal{M}$ is identified with ${\mathcal{M}}_{J_0}^{+}$ by equation 26.1, with basis element $q_j$ going to $z_j$ , and $p_j$ to $i{z}_j$

$\mathcal{M}$ is identified with ${\mathcal{M}}_{J_0}^{-}$ by equation 26.2, with basis element $q_j$ going to ${\overline{z}}_j.$ , and $p_j\mathrm{to}-i{\overline{z}}_j$

$\mathcal{M}$ is identified with elements of ${\mathcal{M}}_{J_0}^{+}\oplus {\mathcal{M}}_{J_0}^{-}$ that are invariant under conjugation, with basis element $q_j$ going to $\frac{1}{\sqrt{2}}\overline{(z_j+{\overline{z}}_j)}$ and $\begin{array}{rl}{p}_j\mathrm{to}\frac{i}{\sqrt{2}}(z_j-& \end{array}$ ${\overline{z}}_j)$

Chapter contents · Next section: 26.2 Compatible complex structures and positivity · 中文版本

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