Group Representation for Quantum Theory by Masahito Hayashi

Author:Masahito Hayashi
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(4.43)

which can be shown by Schur’s lemma (Lemma 2.​4) because the left hand side is commutative with for an arbitrary element g of . Hence, we can construct a measurement with symmetric property by using of coherent states. Similar to the case with , the LHS of (4.43) gives a POVM, which describes the statistical behavior of the measurement outcome.

4.3.2 Irreducible Unitary Representation of

Next, we deal with an irreducible representation of the Lie group . Similar to the case of , we focus on the Lie subalgebra composed of diagonal elements of the Lie algebra . The Lie subalgebra is given by adding the one-dimensional space spanned by to , and is written as . Then, the basis of is given by

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