26.2 Compatible complex structures and positivity

Concept links · terms present in this machine draft; source roles are unverified: vector space · unitary group

Our interest is in vector spaces $M$ that come with a symplectic structure $Ω,$ a non-degenerate antisymmetric bilinear form. To successfully use a complex structure J for quantization, it will turn out that it must be compatible with Ω in the following sense:

Definition (Compatible complex structure)

A complex structure on $M$ is said to be compatible with Ω $i f$

$Ω (J v_{1}, J v_{2}) = Ω (v_{1}, v_{2}) (26.5)$

Equivalently, $J \in S p (2 d, R)$ , the group of linear transformations of M preserving Ω.

The standard complex structure $J = J_{0}$ is compatible with $Ω_{:}$ , since (treating the $d = 1$ case, which generalizes easily, and using equations 16.2 and 26.3)

Ω (J_{0} (c_{q} q + c_{p} p), J_{0} (c_{q}^{'} q + c_{p}^{'} p)) = ((01 - 1 0) (c_{q} c_{p}))^{T} (0 - 1 10) ((01 - 1 0) (c_{q}^{'} c_{p}^{'})) = (c_{q} c_{p}) (0 - 1 10) (0 - 1 10) (01 - 1 0) (c_{q}^{'} c_{p}^{'}) = (c_{q} c_{p}) (0 - 1 10) (c_{q}^{'} c_{p}^{'}) = Ω (c_{q} q + c_{p} p, c_{q}^{'} q + c_{p}^{'} p)

More simply, the matrix for $J_{0}$ is obviously in $S L (2, R) = S p (2, R)$

Note that elements $g$ of the group $S p (2 d, R)$ act on the set of compatible complex structures by

J \to g J g^{- 1} (26.6)

This takes complex structures to complex structures since

(g J g^{- 1}) (g J g^{- 1}) = g J^{2} g^{- 1} = - 1

and preserves the compatibility condition since, if $J \in S p (2 d, R)$ , so is $g J g^{- 1}$

A complex structure $J$ can be characterized by the subgroup of $S p (2 d, R)$ 1 that leaves it invariant, with the condition $g J g^{- 1} = J$ equivalent to the commutativity condition $g J = J g$ . For the case $d = 1$ and $J = J_{0}$ this becomes

(a c b d) (01 - 1 0) = (01 - 1 0) (a c b d)

(b d - a - c) = (- c a - d b)

which implies $b = - c$ and $a = d .$ . The elements of $S L (2, R)$ that preserve $J_{0}$ will be of the form 2

(a - b b a)

with unit determinant, so $a^{2} + b^{2} = 1$ . This is the $U (1) = S O (2)$ subgroup of $S L (2, R)$ of matrices of the form

(cos θ - sin θ sin θ cos θ) = e^{θ Z}

Other choices of $J$ will correspond to other $U (1)$ subgroups of $S L (2, R)$ , and the space of compatible complex structures conjugate to $J_{0}$ can be identified with the coset space $S L (2, R) / U (1)$ . In higher dimensions, it turns out that the subgroup of $S p (2 d, R)$ that commutes with $J_{0}$ is isomorphic to the unitary group $U (d)$ , and the space of compatible complex structures conjugate to $J_{0}$ is $S p (2 d, R) / U (d)$

Even before we choose a complex structure $^{J,}$ we can use $Ω$ to define an indefinite Hermitian form on $M \otimes C$ by:

Definition (Indefinite Hermitian form on $\mathcal { M } \otimes \mathbf { C } )

$. F or$ u _ { 1 } , u _ { 2 } \in \mathcal { M } \otimes \mathbf { C }$

$⟨ u_{1}, u_{2} ⟩ = i Ω (\overline{u}_{1}, u_{2}) (26.7)$

is an indefinite Hermitian form on $M \otimes C$

This is clearly antilinear in the first variable, linear in the second, and satisfies the Hermitian property, since

⟨ u_{2}, u_{1} ⟩ = i Ω (\overline{u}_{2}, u_{1}) = - i Ω (u_{1}, \overline{u}_{2}) = \overline{i Ω (\overline{u}_{1}, u_{2})} = \overline{⟨ u_{1}, u_{2} ⟩}

Restricting $⟨ \cdot, \cdot ⟩$ to $M_{J}^{+}$ and using the identification 26.1 of $M$ and $M_{J}^{+}$ $⟨ \cdot, \cdot ⟩$ gives a complex-valued bilinear form on $M$ . Any $u \in M_{J}^{+}$ can be written as

u = \frac{1}{2} (v - i J v) (26.8)

for some non-zero $v \in M$ , so

⟨ u_{1}, u_{2} ⟩ = i Ω (\overline{u}_{1}, u_{2}) = i \frac{1}{2} Ω (v_{1} + i J v_{1}, v_{2} - i J v_{2}) = \frac{1}{2} (- Ω (J v_{1}, v_{2}) + Ω (v_{1}, J v_{2})) + \frac{i}{2} (Ω (v_{1}, v_{2}) + Ω (J v_{1}, J v_{2})) = Ω (v_{1}, J v_{2}) + i Ω (v_{1}, v_{2}) (26.9)

where we have used compatibility of J and $J^{2} = - 1$ to get

Ω (J v_{1}, v_{2}) = Ω (J^{2} v_{1}, J v_{2}) = - Ω (v_{1}, J v_{2})

We thus can recover Ω on $M$ as the imaginary part of the form $⟨ \cdot, \cdot ⟩$

This form $⟨ \cdot, \cdot ⟩$ is not positive or negative-definite on $M \otimes C$ . One can however restrict attention to those J that give a positive-definite form on $M_{J}^{+}$

Definition (Positive compatible complex structures)

A complex structure $J$ on $M$ is said to be positive and compatible with Ω if it satisfies the compatibility condition 26.5 (i.e., is in $S p (2 d, R),$ and one of the equivalent (by equation 26.9) positivity conditions

$⟨ u, u ⟩ = i Ω (\overline{u}, u) > 0 (26.10)$
for non-zero $u \in M_{J}^{+}$ . or

$Ω (v, J v) > 0 (26.11)$
for non-zero $v \in M$

For such a $J, ⟨ \cdot, \cdot ⟩$ restricted to $M_{J}^{-}$ will be negative-definite since
$Ω (u, \overline{u}) = - Ω (\overline{u}, u)$

and complex conjugation interchanges $M_{J}^{+}$ and $M_{J}^{-}$ . The standard complex structure $J_{0}$ is positive since

⟨ z_{j}, z_{k} ⟩ = i Ω (\overline{z}_{j}, z_{k}) = i {\overline{z}_{j}, z_{k}} = δ_{j k}

and $⟨ \cdot, \cdot ⟩$ is thus the standard Hermitian form on $M_{J_{0}}^{+}$ for which the $z_{j}$ are orthonormal.

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 290、291、292、293、294、295、296、297、298、299、300、301、302、303、304

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：3b238588245d4c885cb260509b0aa88f50298babece889973a8caaad1a41ff0f

OCR 产物 SHA-256：3b238588245d4c885cb260509b0aa88f50298babece889973a8caaad1a41ff0f

数学物理阅读

书架

26.2 Compatible complex structures and positivity

26.2 Compatible complex structures and positivity

关系图谱