Identification of the one-mode quadratic Heisenberg group with the projective group PSU(1,1) and holomorphic representation

Rebeï, Habib; Rguigui, Hafedh; Riahi, Anis; Alhussain, Ziyad A.

doi:10.1142/s021902572050023x

Cited by 6 publications

(3 citation statements)

References 7 publications

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“…In fact, in this case this ∗–Lie algebra coincides with a (necessarily trivial) central extension of

(see [ 31 ]). The associated quantum mechanics have been developed in a series of papers by several authors (see [ 32 ] and references therein). The classical random variables in this class are exactly the three non-standard Meixner families of Meixner–Pollaczek probability measures (Gamma, negative binomial (or Pascal), and Meixner), which have been widely studied in classical probability theory.…”

Section: Feedback For Physicsmentioning

confidence: 99%

New Challenges for Classical and Quantum Probability

Accardi

2022

Entropy

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The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.

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“…In fact, in this case this ∗–Lie algebra coincides with a (necessarily trivial) central extension of

Section: Feedback For Physicsmentioning

confidence: 99%

New Challenges for Classical and Quantum Probability

Accardi

2022

Entropy

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“…The paper [38] is devoted to this problem and shows that this algebra can be identified to a kind of non-commutative sl(2, C). In the case of 1 degree of freedom the problem was solved in [23], (see also [55]) where the group manifold as well as the composition law of the quadratic Heisenberg group, denoted QHeis(1), was constructed and recently it was discovered [56] that QHeis(1) is isomorphic as a Lie group to the projective SU (1, 1) and in addition the holomorphic representation of this group was constructed.…”

Section: Commutation Relationsmentioning

confidence: 99%

Emergence of Quantum Theories from Classical Probability: Historical Origins, Developments, and Open Problems

Accardi

2020

josa

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“…The structure of this composition law was simplified in [5] and recently a substantial step forward in this direction has been achieved in [12] with the identification of the one-mode quadratic Heisenberg group with the projective group P SU (1, 1) and an explicit realization of its holomorphic representation. The multidimensional extension of these results is essential to extend the presently available results concerning the vacuum distributions of the Virasoro fields which, in their truncated form (see [4]), are elements of the (non-homogeneous) quadratic algebra.…”

Section: Introductionmentioning

confidence: 99%

The $n$-dimensional quadratic Heisenberg algebra as a "non--commutative" $\rm{sl}(2,\mathbb{C})$

Accardi,

Boukas,

2021

Preprint

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We prove that the commutation relations among the generators of the quadratic Heisenberg algebra of dimension n ∈ N, look like a kind of non-commutative extension of sl(2, C) (more precisely of its unique 1dimensional central extension), denoted heis 2;C (n) and called the complex n-dimensional quadratic Boson algebra. This non-commutativity has a different nature from the one considered in quantum groups. We prove the exponentiability of these algebras (for any n) in the Fock representation. We obtain the group multiplication law, in coordinates of the first and second kind, for the quadratic Boson group and we show that, in the case of the adjoint representation, these multiplication laws can be expressed in terms of a generalization of the Jordan multiplication. We investigate the connections between these two types of coordinates (disentangling formulas). From this we deduce a new proof of the expression of the vacuum characteristic function of homogeneous quadratic boson fields.

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Identification of the one-mode quadratic Heisenberg group with the projective group PSU(1,1) and holomorphic representation

Cited by 6 publications

References 7 publications

New Challenges for Classical and Quantum Probability

New Challenges for Classical and Quantum Probability

Emergence of Quantum Theories from Classical Probability: Historical Origins, Developments, and Open Problems

The $n$-dimensional quadratic Heisenberg algebra as a "non--commutative" $\rm{sl}(2,\mathbb{C})$

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