Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL GROUPS
This group has one parameter: the angle . As we stated earlier, groups enter
physics because we can carry out transformations on physical systems and the
physical systems often are invariant under the transformations. Here x 2 ‡ y 2 is
left invariant.
We now introduce the concept of a generator and show that rotations of SO…2†
are generated by a special 2 2 matrix ~ 2 , where
~ 2 ˆ 0 i
:
i 0
Using the Euler identity, e i ˆ cos ‡ i sin , we can express the 2 2 rotation
matrices R…† in exponential form:
cos sin
~R…† ˆ
ˆ ~I 2 cos ‡ i~ 2 sin ˆ e i~ 2
;
sin cos
where ~I 2 is a 2 2 unit matrix. From the exponential form we see that multiplication
is equivalent to addition of the arguments. The rotations close to the
identity element have small angles 0: We call ~ 2 the generator of rotations for
SO…2†.
It has been shown that any element g of a Lie group can be written in the form
g… 1 ; 2 ; ...; n †ˆexp
X iˆ1
i i F i
!:
For n parameters there are n of the quantities F i , and they are called the
generators of the Lie group.
Note that we can get ~ 2 from the rotation matrix ~R…† by di€erentiation at the
identity of SO…2†, that is, 0: This suggests that we may ®nd the generators of
other groups in a similar manner.
For n ˆ 3 there are three independent parameters, and the set of 3 3 orthogonal
matrices with determinant ‡1 also forms a group, the SO…3†, its general
member may be expressed in terms of the Euler angle rotation
R…; ; † ˆR z 0…0; 0;†R y …0;;0†R z …0; 0;†;
where R z is a rotation about the z-axis by an angle , R y a rotation about the y-
axis by an angle , and R z 0 a rotation about the z 0 -axis (the new z-axis) by an
angle . This sequence can perform a general rotation. The separate rotations can
be written as
0
cos 0
1
sin
0
cos sin
1
0
B ~R y …† ˆ@
0 1 0
C
A;
B ~R z …† ˆ@
sin cos
C
0 A;
sin 0 cos
0 o 1
451