Natural microstructures 13
transformation ρ
x
1 + |v
x
|, p of L
2
R, C with 1
+ |v
x
|, p ∈ G
a x
and vice versa. Thus v
x
∈ P
a x
determines a unitary representation ρ on L
2
R, C characterizing the collection C
a v
x
of all periodic lifts of β passing through v
x
. The unitary linear transformation ρ
ν
e
t ·
a |a|
+|v
x
|, p of L
2
R, C characterizes the circular polarized wave w on P
a
|
im β
with frequency ν 6= 0
generated by γ and vice versa. The frequency determines the equivalence class of ρ
ν
. As a consequence we have
C
OROLLARY
1. The Schr¨odinger representation ρ
ν
of G
a x
describes the transport of any piece of information
|v
x
|, p ∈ T
v
x
,
P
a
|
β
along the field line β, with R ·
a |a|
as information transmission channel.
The mechanism by which each geodesic is associated with a Schr¨odinger representation as expressed in theorem 1 is generalized for the solar field as follows cf. [12]: Let O
= E\{0}. Given i m β of an integral curve β, we consider the Heisenberg algebra R
·
a |a|
⊕ F
a |a|
equipped with the symplectic structure determined by
a |a|
. Now let γ be a geodesic on P
a
|
im β
and ψ
∈ SR, C. Then the Schr¨odinger representation ρ
sol
of the solar field on the Heisenberg group G
a x
is given by ρ
sol
z, xsψ τ s : = z · e
us ·τ ·i
· e
−
1 2
·us·vs·i
· ψτ − vs for all s in the domain of γ and any τ
∈ R.
8. Periodic lifts of β on P
a
|
β
, the metaplectic group M pF
a x
and quantization
Let ρ
x
be given as in 13, meaning that Planck’s constant is set to one. For v
x
∈ P
a x
and ˙γ
v
x
of a periodic lift γ
v
x
of β, ˙γ
v
x
= ˙γ
v
x
F
a x
+ ˙ β
hor v
x
is an orthogonal splitting of the velocity of γ
v
x
at 0. Clearly, the F
a x
-component of ˙γ
v
x
0 is ˙γ
v
x
F
a x
= p · ¯p
x
, where p is the momentum. Thus the momenta of periodic lifts of β passing through v
x
are in a one-to-one correspondence with elements in T
v
x
P
a x
. Therefore, the collection ¯
C
a x
of all periodic lifts of β on P
a
|
β
is in a one-to-one corre- spondence with T P
a x
being diffeomorphic to a cylinder via a map f : ¯ C
a x
−→ T P
a x
, say. Let
j : T P
a x
|
β
−→ F
a x
be given by j : = T ˜j where ˜j : P
a x
−→ P
a x
is the antipodal map. Thus j w
x
, λ = j w
−x
, λ = λ
for every w
x
, λ ∈ T
w
x
P
a x
with w
x
∈ P
a x
and λ ∈ R. Clearly, j is two-to-one. Setting
˙ F
a x
= F
a x
\{0}, the map j
◦ f : ¯ C
a x
−→ ˙ F
a x
is two-to-one, turning ¯ C
a x
into a two-fold covering of ˙ F
a x
. j ◦ f describes the correspondence
between periodic lifts in ¯ C
a x
and their momenta. The symplectic group SpF
a x
acts transitively on F
a x
equipped with ω
a
as symplectic structure. Therefore, the metaplectic group M pF
a x
, which is the two-fold covering of SpF
a x
, acts transitively on T P
a x
.
14 E. Binz - S. Pods - W. Schempp
Thus given u ∈ F
a x
, there is a smooth map 8
: SpF
a x
−→ F
a x
given by 8 A : = Au for all A ∈ SpF
a x
. Since j ◦ f u
w
x
= j ◦ f u
w
x
for all u
w
x
∈ T P
a
|
β
, the map 8 lifts smoothly to ˜8 : MpF
a x
−→ ¯ C
a x
such that j
◦ f ◦ ˜8 = ˜ pr
◦ 8 where
˜ pr : M pF
a x
−→ SpF
a x
is the covering map. Clearly, the orbit of M pF
a x
on ¯ C
a x
is all of ¯
C
a x
, and M pF
a x
acts on F
a x
with a one-dimensional stabilizing group cf. [14]. Now let us sketch the link between this observation and the quantization on R. SpF
a x
operates as an automorphism group on the Heisenberg group G
a x
leaving the centre fixed via Az
+ h = z + Ah ∀ z + h ∈ G
a x
. Any A
∈ SpF
a x
determines the irreducible unitary representation ρ
A
defined by ρ
A
z + h := ρ
x
z + Ah
∀ z + h ∈ G
a x
. Due to the Stone-von Neumann theorem it must be equivalent to ρ
x
itself, meaning that there is an intertwining unitary operator U
A
on L
2
R, C , determined up to a complex number of
absolute value one in C
a x
, such that ρ
A
= U
A
◦ ρ ◦U
−1 A
and U
A
1
◦U
A
2
= cocA
1
, A
2
·U
A
1
◦A
2
for all A
1
, A
2
∈ SpF
a x
. Here coc is a cocycle with value coc A
1
, A
2
∈ C\{0}. Thus U is a projective representation of SpF
a x
and hence lifts to a representation W of M
p
F
a x
. Since the Lie algebra of M
p
F
a x
is isomorphic to the Poisson algebra of homogenous quadratic polynomials, d W provides the quantization procedure of quadratic homogeneous polynomials on
R and moreover describes the transport of information in P
a
along the field line β, as described in [4].
References
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INZ
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NIATYCKI
J.
AND
F
ISCHER
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INZ
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Natural microstructures 15
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CHEMPP
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CHEMPP
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AMS Subject Classification: 53D25, 43A65.
Ernst BINZ, Sonja PODS Lehrstuhl f¨ur Mathematik I
Universit¨at Mannheim D-68131 Mannheim, GERMANY
e-mail: binzmath.uni-mannheim.de e-mail: podsmath.uni-mannheim.de
Walter SCHEMPP Lehrstuhl f¨ur Mathematik I
Universit¨at Siegen D-57068 Siegen, GERMANY
e-mail: schemppmathematik.uni-siegen.de
