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1)
s
r
q
r−
2
s
p
2
s
1)
s
+1
r
r/
2
(
r
+1)
/
2
N
N
+
β
r
(
−
α
r
(
−
2
s
2
s
−
1
r
=3
s
=0
r
=3
s
=1
q
r−
2s+1
p
2
s−
1
,
x
y
=
α
0
+(
β
1
I
2
+
α
1
A)
p
+
β
0
q
(16.17)
+
vec
−α
2
β
2
,
vec
−β
2
−α
2
T
p
q
p
q
+O
3
,
⊗
β
2
α
2
−
α
2
β
2
identifying the conformal transformation group, namely of type translation (parameters
α
0
and
β
0
), of type rotation (parameter
α
1
), of type dilatation (parameter
β
1
), and of type special-
conformal (parameters
α
2
and
β
2
) up to order three (O
3
), actually the six-parameter subalgebra
C
6
(2) of the infinite dimensional algebra C
∞
(2) in two dimensions
2
. Note that the
rotation parameter
α
1
operates on the antisymmetric matrix (
16.18
), while the matrices (
16.19
),
which generate the special conformal transformation, are traceless and symmetric.
A:=
01
{
q, p
}∈
R
,
(16.18)
−
10
H
1
:=
−
,
H
2
:=
−
.
α
2
β
2
β
2
α
2
β
2
−
α
2
(16.19)
−
α
2
β
2
There remains the task to determine the coecients
α
0
,β
0
,α
1
,β
1
,α
2
,β
2
etc. by means of prop-
erly chosen boundary condition.
16-2 The Oblique Reference Frame
Oblique reference frame and normal reference frame, central oblique plane, circle-reduced
meta-longitude and circle-reduced meta-pole.
In the following discussion, let us orientate a set of orthonormal base vectors
{
e
1
,
e
2
,
e
3
}
along
the principal axes of
[(
x
1
)
2
+(
x
2
)
2
]
A
−
1
+(
x
3
)
2
A
−
2
=1
,A
1
∈
R
E
A
1
,A
2
:=
{
x
∈
R
3
|
+
,A
2
∈
R
+
}
.
Against this frame of reference
{
e
1
,
e
2
,
e
3
,
O}
(consisting of the base vectors
e
i
, and the origin
O
). we introduce the oblique one
{
e
1
,
e
2
,
e
3
,
O}
by means of (
16.20
) illustrated by Fig.
16.3
.
⎡
⎤
⎡
⎤
e
1
e
2
e
3
e
1
e
2
e
3
⎣
⎦
=
R
1
(
i
)
R
3
(
Ω
)
⎣
⎦
.
(16.20)
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