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Physics Letters A

The k-component q-coherent state representations of the quantum Heisenberg-Weyl algebra

Le‐Man Kuang, Fa-Bo Wang, Gao-Jian Zeng

Published May 1, 1993• 11 citations
Publisher
DOI
10.1016/0375-9601(93)90306-k

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A new approach to probability theory based on quantum mechanical and Lie algebraic ideas is proposed and developed. The underlying fact is the observation that the coherent states of the Heisenberg-Weyl, su (2), , and algebras in certain symmetric (bosonic) representations give the `probability amplitudes' (or the `square roots') of the well known Poisson, binomial, multinomial, negative binomial and negative multinomial distributions in probability theory. New probability distributions are derived based on coherent states of the classical algebras , and in symmetric representations. These new probability distributions are simple generalization of the multinomial distributions with some added new features reflecting the quantum and Lie algebraic construction. As byproducts, simple proofs and interpretation of addition theorems of Hermite polynomials are obtained from the `coordinate' representation of the (negative) multinomial states. In other words, these addition theorems are higher rank counterparts of the well known generating function of Hermite polynomials, which is essentially the `coordinate' representation of the ordinary (Heisenberg-Weyl) coherent state.
Interpolating coherent states for HeisenbergndashndashndashWeyl and single-photon SU(1, 1) algebras
New quantal states which interpolate between the coherent states of the Heisenberg–Weyl ( W 3 ) and SU (1, 1) algebras are introduced. The interpolating states are coherent states of a closed and symmetric algebra containing a parameter k which takes values from zero to one. The elements of the algebra are realized in terms of operator-valued functions of creation and annihilation operators. The realization is such that the operators become those of the W 3 algebra when k = 0 and of the SU (1, 1) algebra when k is unity. The overcompleteness of the interpolating coherent states is established by constructing suitable integration measures. Differential operator representations for the elements of the algebra are given along with the relevant spaces of entire functions. A nonsymmetric set of operators to realize the W 3 algebra is presented and the relevant coherent states are studied. Physical properties such as squeezing and photon statistics of the interpolating coherent states are investigated.