Fundamental theories of physics

Symmetry in quantum mechanics

Klaas Landsman

Published Jan 1, 2017• 1 citation

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: \(uu^*=u^{*}u=1_H\) ; u is invertible with \(u^{-1}=u^{*}\) ; u is a surjective isometry (or, if \(\mathrm{dim}(H)<\infty \) , just an isometry); u is invertible and preserves the inner product, i.e., \(\langle u \varphi , u \psi \rangle = \langle \varphi , \psi \rangle (\varphi , \psi \in H).\) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI: 10.1007/978-3-319-51777-3_5

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Symmetry in quantum mechanics

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