Representations of the Schrödinger algebra and Appell systems

Feinsilver, Philip; Kocik, Jerzy; Schott, René

doi:10.1002/prop.200310124

Cited by 28 publications

(28 citation statements)

References 12 publications

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“…The theorem of Stone and von Neumann (see appendix A) implies that, given the mass m, there is a unique UIR of the Heisenberg subalgebra. The authors of [47] proved that any massive representation of the Schrödinger algebra is equivalent to the following realisation of the remaining generatorŝ 5) where the operatorsL ij ,L ± andL 0 commute with all the other generators and provide a representation of o(d) ⊕ sl(2, R) with usual notations. In a sense, the latter operators correspond to the "spinning" or "internal" part of the generators while the "orbital" part is entirely built out of the translation and boost generators.…”

Section: Jhep02(2012)113mentioning

confidence: 99%

Symmetries and currents of the ideal and unitary Fermi gases

Bekaert

Meunier

Moroz

2012

J. High Energ. Phys.

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Abstract:The maximal algebra of symmetries of the free single-particle Schrödinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schrödinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N ) model, a putative bulk dual of the unitary and the ideal Fermi gases is briefly discussed.

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Section: Jhep02(2012)113mentioning

confidence: 99%

Symmetries and currents of the ideal and unitary Fermi gases

Bekaert

Meunier

Moroz

2012

J. High Energ. Phys.

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“…The lowest weight vector w 0 of the factor module V d,r /I 0 satisfies the condition 22) and the basis of V d,r /I 0 is given by…”

Section: Irreducible Modulesmentioning

confidence: 99%

“…We mention the followings: Projective representations of the Schrödinger group in 3 spatial dimension were constructed in [19]. Irreducible representations of the Schrödinger algebras up to 3 spatial dimension were investigated in [20][21][22]. The author of [23] studied the highest weight representations of the N = 2 super Schrödinger algebra in 2 spatial dimension ("exotic" algebra in the terminology of [14]).…”

Section: Introductionmentioning

confidence: 99%

Lowest weight representations of super Schrödinger algebras in one dimensional space

Aizawa¹

2011

Journal of Mathematical Physics

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Lowest weight modules, in particular, Verma modules over the N = 1, 2 super Schrödinger algebras in (1 + 1) dimensional spacetime are investigated. The reducibility of the Verma modules is analyzed via explicitly constructed singular vectors. The classification of the irreducible lowest weight modules is given for both massive and massless representations. A vector field realization of the N = 1, 2 super Schrödinger algebras is also presented.

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“…where a, b ∈ C with |a| < 2 and Φ is the Fock vacuum such that B 2 Φ, is given by (see Theorem 6.1 of [9] for c = μ(I) / 2 and a, b ∈ R; for a detailed proof see [10])…”

Section: The Schrödinger Algebra In Terms Of the Rhp W N Generatorsmentioning

confidence: 99%

“…the avoidance of the RHP W N no-go theorems, in what follows we will use the notation B 3. The Schrödinger kernel and the no-go theorem for Heis and RSW N As shown in [9], for μ(I) = 0 the operators…”

Section: The Schrödinger Algebra In Terms Of the Rhp W N Generatorsmentioning

confidence: 99%

The Schrödinger Fock kernel and the no-go theorem for the first order and Renormalized Square of White Noise Lie algebras

Accardi

Boukas

2011

Proc. Amer. Math. Soc.

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Abstract. Using the non-positive definiteness of the Fock kernel associated with the Schrödinger algebra we prove the impossibility of a joint Fock representation of the first order and Renormalized Square of White Noise Lie algebras with the convolution type renormalization δ 2 (t − s) = δ(s) δ(t − s) for the square of the Dirac delta function. We show how the Schrödinger algebra Fock kernel can be reduced to a positive definite kernel through a restriction of the set of exponential vectors. We describe how the reduced Schrödinger kernel can be viewed as a tensor product of a Renormalized Square of White Noise (sl (2)) and a First Order of White Noise (Heisenberg) Fock kernel. We also compute the characteristic function of a stochastic process naturally associated with the reduced Schrödinger kernel. The renormalized higher powers of white noiseThe quantum white noise functionals b † t (creation density) and b t (annihilation density) satisfy the Boson commutation relations, where t, s ∈ R and δ is the Dirac delta function, as well as the duality relations . In order to consider the smeared fields defined by the higher powers of b t and b † t , for a test function f and n, k ∈ {0, 1, 2, ...}, the sesquilinear formswhere dt denotes integration with respect to Lebesgue measure μ, with involution

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Representations of the Schrödinger algebra and Appell systems

Cited by 28 publications

References 12 publications

Symmetries and currents of the ideal and unitary Fermi gases

Symmetries and currents of the ideal and unitary Fermi gases

Lowest weight representations of super Schrödinger algebras in one dimensional space

The Schrödinger Fock kernel and the no-go theorem for the first order and Renormalized Square of White Noise Lie algebras

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