Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras

Accardi, Luigi; Boukas, Andreas

doi:10.3842/sigma.2009.056

Cited by 15 publications

(18 citation statements)

References 49 publications

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“…Accardi, Lu and Volovich in [15] proved that these commutation relations effectively define a * -Lie algebra and that there exists a * -representation π F of this * -Lie algebra on a Hilbert space H (see [9] for the definition of this notion) uniquely characterized by the properties: (i) There exists a unit vector Φ ∈ H cyclic for the representation. (i) For any choice of the test functions…”

Section: Quadratic Second Quantizationmentioning

confidence: 99%

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Renormalization and central extensions

Accardi

Boukas

2012

P-Adic Num Ultrametr Anal Appl

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The stochastic limit of quantum theory [1] motivated a new approach to the renormalization program. Subsequent investigations brought to light unexpected connections with conformal field theory and some subtle relationships between renormalization and central extensions. In the present paper we review the path that has lead to these connections at the light of some recent results. , higher powers of white noise, central extensions.Dedicated to Igor V. Volovich with friendship and admiration for his scientific achievements 1. INTRODUCTION The stochastic limit of quantum theory [1] has led to a multiplicity of developments in physics and mathematics. In particular the quite nontrivial identification of classical and quantum stochastic equations with first order white noise Hamiltonian equations naturally rose the question of the meaning of higher order white noise Hamiltonian equations.Due to the identification of quantum white noise with the free Boson field (in momentum representation) this problem is equivalent to the problem of giving a meaning to nonlinear functions of the local quantum fields, i.e. to the old standing renormalization problem.The equivalence of this problem with that of constructing a continuous analogue of the * -Liealgebra of differential operators with polynomial coefficients acting on the space C ∞ (R n ; C) and of its unitary representations has been discussed in the paper [9] (continuous analogue means that the space R n ≡ {functions {1, . . . , n} → R} is replaced by a space of test functions {functions R → R }).In the present paper we will discuss the connections of this problem with that of central extensions of * -Lie-algebras.Since the simplest power higher that the first is the square, it was natural to choose this as the starting point to attack the general problem.In the case of finitely many degrees of freedom quadratic Hamiltonians are easily diagonlized by a Boglyubov transformation but, as pointed out in the paper [16], when one tries to apply this technique to the field case, a constraint appears in the form of an inequality which excludes the simplest example one would like to be able to deal with: the square of the local quantum field, i.e. of classical white noise.This remark convinced us that, for a successful attack the renormalization problem, a radically new approach to the problem was required.Such a new approach was proposed by Accardi, Lu and Volovich in the paper [15] and its basic new idea can be formulated in the following problem: first renormalize the Lie algebra structure (i.e. the commutation relations), thus obtaining a new * -Lie algebra, then construct (nontrivial) Hilbert space representations.In the same paper [15] the concrete realizability of the new approach was proved by explicitly constructing the Fock representation for the Renormalized Square of White Noise (RSWN).

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Section: Quadratic Second Quantizationmentioning

confidence: 99%

“…The equivalence of this problem with that of constructing a continuous analogue of the * -Liealgebra of differential operators with polynomial coefficients acting on the space C ∞ (R n ; C) and of its unitary representations has been discussed in the paper [9] (continuous analogue means that the space R n ≡ {functions {1, . .…”

Section: Introductionmentioning

confidence: 99%

Renormalization and central extensions

Accardi

Boukas

2012

P-Adic Num Ultrametr Anal Appl

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“…The survey [5] describes two inequivalent renormalization procedures and the more recent paper [7] shows the connection between them.…”

Section: Introduction: the C * -Non-linear Quantization Programmentioning

confidence: 99%

“…Moreover, any gaussian representation can be obtained, by means of a standard construction, from the Fock representation which is characterized by the property that the cyclic vector, called vacuum, is in the kernel of the annihilation operators. This property has been taken as an heuristic principle to define the notion of Fock state also in the higher order situations (see [5] for a precise definition).…”

Section: Introduction: the C * -Non-linear Quantization Programmentioning

confidence: 99%

“…Using this, we construct an inductive system of C * -algebras each of which is isomorphic to a finite tensor product of copies of the one-mode generalized Weyl algebra but the embeddings defining the inductive system are not the usual tensor product embeddings because they depend on the above-mentioned identifications. The identities (7) provide an intuitive bridge between our inductive construction and our goal to give a meaning to the formal expressions (5). Once a representation with good continuity properties is obtained, one can extend the test function space to more general functions thus obtaining non-linear white noise integrals.…”

Section: Introduction: the C * -Non-linear Quantization Programmentioning

confidence: 99%

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C*-Non-Linear Second Quantization

Accardi

Dhahri

2015

Ann. Henri Poincaré

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We construct an inductive system of C * -algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22 (3):1550001, 2015). We prove that the inductive limit C * -algebra is factorizable and has a natural localization given by a family of C * -sub-algebras each of which is localized on a bounded Borel subset of R. Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C * -algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras.

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A Poisson algebra on the Hida Test functions and a quantization using the Cuntz algebra

2022

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In this note, we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan–Schwinger map which has been known and used for a long time by physicists. The difference, compared to Jordan–Schwinger map, is that we use generators of Cuntz algebra $$\mathcal {O}_{\infty }$$ O ∞ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a “building blocks” instead of creation–annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization, i.e. exact conservation of Lie brackets and linearity.

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Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras

Cited by 15 publications

References 49 publications

Renormalization and central extensions

Renormalization and central extensions

C*-Non-Linear Second Quantization

A Poisson algebra on the Hida Test functions and a quantization using the Cuntz algebra

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