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1
C o n s t r u c t i o n
T o g g l e C o n s t r u c t i o n s u b s e c t i o n
1 . 1
G r o u p g e n e r a t o r s
1 . 2
H i l b e r t s p a c e
2
I s o m o r p h i s m
3
D i s c r e t e s u b g r o u p
4
S e e a l s o
5
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
T h e t a r e p r e s e n t a t i o n
1 l a n g u a g e
● 한 국 어
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
The theta representation is a representation of the continuous Heisenberg group
H
3
(
R
)
{\displaystyle H_{3}(\mathbb {R} )}
over the field of the real numbers. In this representation, the group elements act on a particular Hilbert space . The construction below proceeds first by defining operators that correspond to the Heisenberg group generators. Next, the Hilbert space on which these act is defined, followed by a demonstration of the isomorphism to the usual representations.
Group generators [ edit ]
Let f (z ) be a holomorphic function , let a and b be real numbers , and let
τ
{\displaystyle \tau }
be an arbitrary fixed complex number in the upper half-plane ; that is, so that the imaginary part of
τ
{\displaystyle \tau }
is positive. Define the operators S a and T b such that they act on holomorphic functions as
(
S
a
f
)
(
z
)
=
f
(
z
+
a
)
=
exp
(
a
∂
z
)
f
(
z
)
{\displaystyle (S_{a}f)(z )=f(z+a)=\exp(a\partial _{z})f(z )}
and
(
T
b
f
)
(
z
)
=
exp
(
i
π
b
2
τ
+
2
π
i
b
z
)
f
(
z
+
b
τ
)
=
exp
(
i
π
b
2
τ
+
2
π
i
b
z
+
b
τ
∂
z
)
f
(
z
)
.
{\displaystyle (T_{b}f)(z )=\exp(i\pi b^{2}\tau +2\pi ibz)f(z+b\tau )=\exp(i\pi b^{2}\tau +2\pi ibz+b\tau \partial _{z})f(z ).}
It can be seen that each operator generates a one-parameter subgroup:
S
a
1
(
S
a
2
f
)
=
(
S
a
1
∘
S
a
2
)
f
=
S
a
1
+
a
2
f
{\displaystyle S_{a_{1}}\left(S_{a_{2}}f\right)=\left(S_{a_{1}}\circ S_{a_{2}}\right)f=S_{a_{1}+a_{2}}f}
and
T
b
1
(
T
b
2
f
)
=
(
T
b
1
∘
T
b
2
)
f
=
T
b
1
+
b
2
f
.
{\displaystyle T_{b_{1}}\left(T_{b_{2}}f\right)=\left(T_{b_{1}}\circ T_{b_{2}}\right)f=T_{b_{1}+b_{2}}f.}
However, S and T do not commute:
S
a
∘
T
b
=
exp
(
2
π
i
a
b
)
T
b
∘
S
a
.
{\displaystyle S_{a}\circ T_{b}=\exp(2\pi iab)T_{b}\circ S_{a}.}
Thus we see that S and T together with a unitary phase form a nilpotent Lie group , the (continuous real) Heisenberg group , parametrizable as
H
=
U
(
1
)
×
R
×
R
{\displaystyle H=U(1 )\times \mathbb {R} \times \mathbb {R} }
where U (1 ) is the unitary group .
A general group element
U
τ
(
λ
,
a
,
b
)
∈
H
{\displaystyle U_{\tau }(\lambda ,a,b)\in H}
then acts on a holomorphic function f (z ) as
U
τ
(
λ
,
a
,
b
)
f
(
z
)
=
λ
(
S
a
∘
T
b
f
)
(
z
)
=
λ
exp
(
i
π
b
2
τ
+
2
π
i
b
z
)
f
(
z
+
a
+
b
τ
)
{\displaystyle U_{\tau }(\lambda ,a,b)f(z )=\lambda (S_{a}\circ T_{b}f)(z )=\lambda \exp(i\pi b^{2}\tau +2\pi ibz)f(z+a+b\tau )}
where
λ
∈
U
(
1
)
.
{\displaystyle \lambda \in U(1 ).}
U
(
1
)
=
Z
(
H
)
{\displaystyle U(1 )=Z(H )}
is the center of H , the commutator subgroup
[
H
,
H
]
{\displaystyle [H,H]}
. The parameter
τ
{\displaystyle \tau }
on
U
τ
(
λ
,
a
,
b
)
{\displaystyle U_{\tau }(\lambda ,a,b)}
serves only to remind that every different value of
τ
{\displaystyle \tau }
gives rise to a different representation of the action of the group.
Hilbert space [ edit ]
The action of the group elements
U
τ
(
λ
,
a
,
b
)
{\displaystyle U_{\tau }(\lambda ,a,b)}
is unitary and irreducible on a certain Hilbert space of functions. For a fixed value of τ, define a norm on entire functions of the complex plane as
‖
f
‖
τ
2
=
∫
C
exp
(
−
π
y
2
ℑ
τ
)
|
f
(
x
+
i
y
)
|
2
d
x
d
y
.
{\displaystyle \Vert f\Vert _{\tau }^{2}=\int _{\mathbb {C} }\exp \left({\frac {-\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy.}
Here,
ℑ
τ
{\displaystyle \Im \tau }
is the imaginary part of
τ
{\displaystyle \tau }
and the domain of integration is the entire complex plane.
Mumford sets the norm as
∫
C
exp
(
−
2
π
y
2
ℑ
τ
)
|
f
(
x
+
i
y
)
|
2
d
x
d
y
{\displaystyle \int _{\mathbb {C} }\exp \left({\frac {-2\pi y^{2}}{\Im \tau }}\right)|f(x+iy)|^{2}\ dx\ dy}
, but in this way
T
b
{\displaystyle T_{b}}
is not unitary.
Let
H
τ
{\displaystyle {\mathcal {H}}_{\tau }}
be the set of entire functions f with finite norm. The subscript
τ
{\displaystyle \tau }
is used only to indicate that the space depends on the choice of parameter
τ
{\displaystyle \tau }
. This
H
τ
{\displaystyle {\mathcal {H}}_{\tau }}
forms a Hilbert space . The action of
U
τ
(
λ
,
a
,
b
)
{\displaystyle U_{\tau }(\lambda ,a,b)}
given above is unitary on
H
τ
{\displaystyle {\mathcal {H}}_{\tau }}
, that is,
U
τ
(
λ
,
a
,
b
)
{\displaystyle U_{\tau }(\lambda ,a,b)}
preserves the norm on this space. Finally, the action of
U
τ
(
λ
,
a
,
b
)
{\displaystyle U_{\tau }(\lambda ,a,b)}
on
H
τ
{\displaystyle {\mathcal {H}}_{\tau }}
is irreducible .
This norm is closely related to that used to define Segal–Bargmann space [citation needed ] .
Isomorphism [ edit ]
The above theta representation of the Heisenberg group is isomorphic to the canonical Weyl representation of the Heisenberg group. In particular, this implies that
H
τ
{\displaystyle {\mathcal {H}}_{\tau }}
and
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
are isomorphic as H -modules . Let
M
(
a
,
b
,
c
)
=
[
1
a
c
0
1
b
0
0
1
]
{\displaystyle M(a,b,c)={\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}}}
stand for a general group element of
H
3
(
R
)
.
{\displaystyle H_{3}(\mathbb {R} ).}
In the canonical Weyl representation, for every real number h , there is a representation
ρ
h
{\displaystyle \rho _{h}}
acting on
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
as
ρ
h
(
M
(
a
,
b
,
c
)
)
ψ
(
x
)
=
exp
(
i
b
x
+
i
h
c
)
ψ
(
x
+
h
a
)
{\displaystyle \rho _{h}(M(a,b,c))\psi (x )=\exp(ibx+ihc)\psi (x+ha)}
for
x
∈
R
{\displaystyle x\in \mathbb {R} }
and
ψ
∈
L
2
(
R
)
.
{\displaystyle \psi \in L^{2}(\mathbb {R} ).}
Here, h is Planck's constant . Each such representation is unitarily inequivalent . The corresponding theta representation is:
M
(
a
,
0
,
0
)
→
S
a
h
{\displaystyle M(a,0,0)\to S_{ah}}
M
(
0
,
b
,
0
)
→
T
b
/
2
π
{\displaystyle M(0,b,0)\to T_{b/2\pi }}
M
(
0
,
0
,
c
)
→
e
i
h
c
{\displaystyle M(0,0,c)\to e^{ihc}}
Discrete subgroup [ edit ]
Define the subgroup
Γ
τ
⊂
H
τ
{\displaystyle \Gamma _{\tau }\subset H_{\tau }}
as
Γ
τ
=
{
U
τ
(
1
,
a
,
b
)
∈
H
τ
:
a
,
b
∈
Z
}
.
{\displaystyle \Gamma _{\tau }=\{U_{\tau }(1,a,b)\in H_{\tau }:a,b\in \mathbb {Z} \}.}
The Jacobi theta function is defined as
ϑ
(
z
;
τ
)
=
∑
n
=
−
∞
∞
exp
(
π
i
n
2
τ
+
2
π
i
n
z
)
.
{\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz).}
It is an entire function of z that is invariant under
Γ
τ
.
{\displaystyle \Gamma _{\tau }.}
This follows from the properties of the theta function:
ϑ
(
z
+
1
;
τ
)
=
ϑ
(
z
;
τ
)
{\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau )}
and
ϑ
(
z
+
a
+
b
τ
;
τ
)
=
exp
(
−
π
i
b
2
τ
−
2
π
i
b
z
)
ϑ
(
z
;
τ
)
{\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z;\tau )}
when a and b are integers. It can be shown that the Jacobi theta is the unique such function.
See also [ edit ]
References [ edit ]
David Mumford, Tata Lectures on Theta I (1983), Birkhäuser, Boston ISBN 3-7643-3109-7
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Theta_representation&oldid=1224495499 "
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