Exploring the Isomorphism Between SU(2) and SO(3) Groups

Thread starter ber70
Start date Jun 5, 2010
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Groups Isomorphism So(3) Su(2)

In summary, the relationship between SU(2) and SO(3) groups is that they are both mathematical groups used to describe the symmetries of physical systems. The isomorphism between these groups is important because it provides a connection between seemingly different groups and allows for the use of techniques and concepts from one group to solve problems in the other. The representations of SU(2) and SO(3) are related through the isomorphism, but may have different properties. This isomorphism can also be extended to other groups, such as the special linear group and the special orthogonal group in 3 dimensions. Some applications of this isomorphism include its use in physics, molecular and atomic structures, crystallography, and

Jun 5, 2010

ber70

I have not seen why SU(2) and SO(3) groups are isomorphic?

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Jun 5, 2010

D H

Staff Emeritus

Science Advisor

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They aren't isomorphic. SU(2) is a double cover of SO(3).

Jun 5, 2010

mrbohn1

Their Lie algebras are isomorphic, and they are locally isomorphic...but D H is right - they aren't isomorphic!

Related to Exploring the Isomorphism Between SU(2) and SO(3) Groups

1. What is the relationship between SU(2) and SO(3) groups?

The SU(2) group and the SO(3) group are both mathematical groups that are used to describe the symmetries of physical systems. The SU(2) group is a special unitary group of 2x2 complex matrices, while the SO(3) group is a special orthogonal group of 3x3 real matrices. They are isomorphic, which means that they have the same structure and can be mapped onto each other, but they have different representations.

2. Why is the isomorphism between SU(2) and SO(3) important?

The isomorphism between SU(2) and SO(3) is important because it provides a mathematical connection between two seemingly different groups. This connection allows physicists to use techniques and concepts from one group to understand and solve problems in the other group. It also helps to simplify calculations and make connections between different physical theories.

3. How are the representations of SU(2) and SO(3) related to each other?

The representations of SU(2) and SO(3) are related through the isomorphism between the two groups. This means that for every representation of SU(2), there is a corresponding representation of SO(3) and vice versa. However, the representations are not the same and may have different properties, such as different dimensions.

4. Can the isomorphism between SU(2) and SO(3) be extended to other groups?

Yes, the isomorphism between SU(2) and SO(3) can be extended to other groups. In fact, SU(2) and SO(3) are part of a larger group called the special linear group, or SL(2), which contains all 2x2 complex matrices with unit determinants. This group is also isomorphic to the special orthogonal group in 3 dimensions, or SO(3).

5. What are some applications of the isomorphism between SU(2) and SO(3)?

The isomorphism between SU(2) and SO(3) has many applications in physics, particularly in the study of symmetries and spin in quantum mechanics. It is also used in the study of molecular and atomic structures, as well as in crystallography. Additionally, the isomorphism has applications in computer graphics and computer vision, as it can be used to rotate and transform 3D objects in a virtual environment.