Everything about Su 2 totally explained
In
mathematics, the
special unitary group of degree
n, denoted SU(
n), is the
group of
n×
n unitary matrices with
determinant 1. The group operation is that of
matrix multiplication. The special unitary group is a
subgroup of the
unitary group U(
n), consisting of all
n×
n unitary matrices, which is itself a subgroup of the
general linear group GL(
n,
C).
The SU(n) groups find wide application in the
standard model of
physics, especially SU(2) in the
electroweak interaction and SU(3) in
QCD.
The simplest case, SU(1), is a
trivial group, having only a single element. The group SU(2) is
isomorphic to the group of
quaternions of
absolute value 1, and is thus
diffeomorphic to the
3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we've a
surjective homomorphism from SU(2) to the
rotation group SO(3) whose
kernel is
However there may be better choices for
A for certain dimensions which exhibit more behaviour under restriction to subrings of
C.
Example
A very important example of this type of group is the picard modular group SU(2,1;
Z[
i]) which acts (projectively) on
complex hyperbolic space of degree two, in the same way that SL(2,
Z) acts (projectively) on real
hyperbolic space of dimension two. In 2003 Gábor Francsics and Peter D. Lax computed a fundamental domain for the action of this group on
, see
(External Link
).
Another example is SU(2,1;
C) which is isomorphic to SL(2,
R).
Important Subgroups
In physics the special unitary group is used to represent bosonic symmetries. In theories of
symmetry breaking it's important to be able to find the subgroups of the special unitary group. Important subgroups of SU(n) that are important in
GUT physics are, for p>1, n-p>1:
»
For completeness there are also the
orthogonal and
symplectic subgroups:
» »
Since the
rank of SU(n) is n-1 and U(1) is 1 a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. SU(n) is a subgroup of various other lie groups:
» » » » » »
There are also the identities
SU(4)=O(6),
SU(2)=O(3)=USp(2) and
U(1)=O(2) .
One should finally mention that SU(2) is the "covering double group" of SO(3), a relation that plays an important role in the theory of rotations of 2-spinors in non-relativistic quantum mechanics.
Further Information
Get more info on 'Su 2'.
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