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The Lie Algebra of SO(1,3) is similar with that of SU(2)+SU(2) or of SO(3)+SO(3). But how do we know SO(1,3) really decomposite to SU(2)+SU(2)?
In order to prove that SO(1,3) and SU(2)+SU(2) are equivalent, we can use the fact that they both have the same Lie algebra. This means that they have the same generators, which are the matrices that represent the group's transformations. By showing that the generators for both groups are the same, we can prove that the two groups are equivalent.
SO(1,3) is the special orthogonal group in four dimensions, and is used to represent the symmetries of space and time in special relativity. On the other hand, SU(2)+SU(2) is used to describe the symmetries of spin in quantum mechanics. By proving their equivalence, we are essentially showing that the symmetries of space and time and the symmetries of spin are related, which has important implications in theoretical physics.
The mathematical proof involves showing that the generators of SO(1,3) and SU(2)+SU(2) satisfy the same commutation relations. This can be done by explicitly calculating the commutation relations for the generators and showing that they are equal for both groups. This proves that the two groups have the same Lie algebra, and therefore are equivalent.
Yes, there are several real-world applications of this equivalence. One example is in the study of black holes, where the symmetries of space and time and the symmetries of spin play a crucial role. Another example is in quantum field theory, where the equivalence helps to understand the fundamental symmetries of nature.
While there is no direct experimental evidence for this equivalence, it is supported by the success of theories that use both SO(1,3) and SU(2)+SU(2) symmetries, such as special relativity and quantum mechanics. Additionally, the mathematical proof of their equivalence has been verified by numerous physicists and is widely accepted in the scientific community.