Abstract
Roughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is. Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map \(u : H \rightarrow H\) satisfying one and hence all of the following equivalent conditions:
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\(uu^*=u^{*}u=1_H\);
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u is invertible with \(u^{-1}=u^{*}\);
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u is a surjective isometry (or, if \(\mathrm{dim}(H)<\infty \), just an isometry);
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u is invertible and preserves the inner product, i.e., \(\langle u \varphi , u \psi \rangle = \langle \varphi , \psi \rangle (\varphi , \psi \in H).\)
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Keywords
- Irreducible Representation
- Unitary Representation
- Unitary Irreducible Representation
- Projective Representation
- Weyl Chamber
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Landsman, K. (2017). Symmetry in quantum mechanics. In: Foundations of Quantum Theory. Fundamental Theories of Physics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-51777-3_5
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DOI: https://doi.org/10.1007/978-3-319-51777-3_5
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