Such matrices will have square−1 and give a positive complex structure when
θ′=π 2 , so of the form
(
icosht −eiθsinht
−e−iθsinht −icosht)
TheU(1) subgroup ofSU(1,1) preservingJ 0 will be matrices of the form
(
eiθ 0
0 e−iθ)
Using the matrix from equation 24.9, the subgroupR⊂SL(2,R) of equation
24.7 takesJ 0 to
(
coshr sinhr
sinhr coshr
)(
−i 0
0 i)(
coshr −sinhr
−sinhr coshr)
=i(
−cosh 2r sinh 2r
−sinh 2r cosh 2r)
26.6 Complex structures and Bargmann-Fock quan-
tization for arbitraryd
Generalizing fromd= 1 to arbitraryd, the additional piece of structure in-
troduced by the method of annihilation and creation operators appears in the
following ways:
- As a choice of positive compatible complex structureJ, or equivalently a
decomposition
M⊗C=M+J⊕M−J - As a division of the quantizations of elements ofM ⊗Cinto creation
(coming fromM+J) and annihilation (coming fromM−J) operators. There
will be a correspondingJ-dependent definition of normal ordering of an
operatorOthat is a product of such operators, symbolized by :O:J. - As a distinguished vector| 0 〉J, the vector inHannihilated by all annihi-
lation operators. Writing such a vector in the Schr ̈odinger representation
as a position-space wavefunction inL^2 (Rd), it will be a Gaussian function
generalizing thed= 1 case of equation 26.20, withτ now a symmetric
matrix with positive-definite imaginary part. Such matrices parametrize
the space of positive compatible complex structures, a space called the
“Siegel upper half-space”. - As a distinguished subgroup ofSp(2n,R), the subgroup that commutes
withJ. This subgroup will be isomorphic to the groupU(n). The Siegel
upper half-space of positive compatible complex structures can also be
described as the coset spaceSp(2n,R)/U(n).