Probability Distributions and Coherent States of $B_r$, $C_r$ and $D_r$ Algebras

Details Probability Distributions and Coherent States of $B_r$, $C_r$ and $D_r$ Algebras Probability Distributions and Coherent States of $B_r$, $C_r$ and $D_r$ Algebras

1997-06-05

arxiv.org/abs/hep-th/9706034v2

J.Phys.A31:901-925,1998

Physics

High Energy Physics

High Energy Physics - Theory

32 pages, LaTeX2e. Two references on generalised Mehler formula are added. A few sentences in the Abstract, Introduction, Ackn

Scientific paper

10.1088/0305-4470/31/3/006

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, $su(2)$, $su(r+1)$, $su(1,1)$ and $su(r,1)$ 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 $B_r$, $C_r$ and $D_r$ in symmetric representations. These new probability distributions are simple generalisation 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.

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