What is the use of infinite-dimensional representation of group

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SUMMARY

The discussion centers on the application of infinite-dimensional representations of Lie groups in physics, particularly in the context of Hilbert spaces. It establishes that while massive fields utilize finite representations such as SO(3), massless fields relate to non-compact groups like ISO(2), where only finite representations (e.g., SO(2)) hold physical significance. The infinite-dimensional representation is crucial for providing the most general representation of physical states, especially in quantum mechanics.

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  • Understanding of Lie groups and their representations
  • Familiarity with Hilbert space concepts
  • Knowledge of finite and infinite-dimensional spaces
  • Basic principles of quantum field theory
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This discussion is beneficial for physicists, mathematicians, and students interested in quantum mechanics, representation theory, and the mathematical foundations of particle physics.

liucl78
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What is the use of infinite-dimensional representation of lie group?
Now, I know Hilbert space is infinite-dimensional, and physical states must be in Hilbert space.
However, for massive fields, the transformation group is SO(3), its unitary representation is finite.
For massless fields, the transformation group is non-compact ISO(2), but only the finite representation of ISO(2), namely SO(2) representation, has physical meanings.

I want to know where the infinite-dimensional representation is used in physics?

thanks
 
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It provides the most general possible representation of what you are trying to do.
 

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