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Theoretical and mathematical physics

Coherent States from Square Integrable Representations

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

Published Aug 26, 2013• 2 citations
Publisher
DOI
10.1007/978-1-4614-8535-3_8

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Coherent states defined with respect to an irreducible ray representation u: g→ug, g∈G, of an arbitrary locally compact separable group G are examined. It is shown that the following conditions (a)–(d) are equivalent: (a) u admits coherent states, (b) u is square integrable, (c) the W*-system implemented by u is integrable, and (d) u is a subrepresentation of the left regular c-representation, where c is the respective multiplier of u. Furthermore, the group theoretical background of what is called the ‘‘P-representation of observables’’ associated with coherent states is investigated: It is shown that the P-representation (which corresponds to a covariant semispectral measure) fulfills a certain maximality requirement. The P-representation can be used to represent the quantum system in question on the Hilbert space L2(G,dg) of square-integrable functions (with respect to Haar measure dg) on the kinematical group G.
We derive a generalization of the well-known theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.