Strict deformation quantization of locally convex algebras and modules

Lechner, Gandalf; Waldmann, Stefan

doi:10.1016/j.geomphys.2015.09.013

Cited by 20 publications

(28 citation statements)

References 29 publications

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“…In particular, the task is to establish smoothness of the oscillatory integral in a suitable locally convex topology and to demonstrate that all derivatives are polynomially bounded (see [And13, Theorem 1] for a similar proof). After this is established, we build on results of [LW11] where smoothness and polynomial boundedness are the requirements needed in order to prove the well-definedness of the warped convolutions integral. Now for the spatial part a simpler route will be chosen.…”

Section: Qft-moyal-weyl From Deformationmentioning

confidence: 99%

Relativistic corrections to the Moyal-Weyl spacetime

Much

2015

Journal of Mathematical Physics

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Section: Qft-moyal-weyl From Deformationmentioning

confidence: 99%

Relativistic corrections to the Moyal-Weyl spacetime

Much

2015

Journal of Mathematical Physics

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“…Note that Theorem 3.10 also shows that ω zb depends holomorphically on z ∈ in so far as ∋ z → ω zb (X) ∈ is holomorphic for all X ∈ S • (V ). This is the analog of statements in [12,13] in the Rieffel setting. Proposition 3.30 Let · be a continuous antilinear involution of V and Λ a continuous Hermitian bilinear forms on V .…”

Section: )mentioning

confidence: 75%

Convergent star products for projective limits of Hilbert spaces

Schötz

Waldmann

2018

Journal of Functional Analysis

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Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.

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“…The former definition is rigorously defined for a certain bounded operators, see [BLS11]. However, since we are dealing with unbounded operators we use the results of [LW11] in order to prove that the deformation of the Coleman-Mandula operator is well-defined. Hence, let us consider the deformed operator A Θ as follows Definition 3.2.…”

Section: Mathematical Properties Of the Deformationmentioning

confidence: 99%

A Poincaré covariant noncommutative spacetime

Much

Vergara

2018

Int. J. Geom. Methods Mod. Phys.

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We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman, Mandula [CM67] (see also [MPS16]) as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincaré invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea how to obtain, in a physical and mathematical well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the whole Poincaré group. Discussion 16Conventions 0.1. We use d = n + 1, for n ∈ N and the Greek letters are split into µ, ν = 0, . . . , n. Moreover, we use Latin letters for the spatial components which run from 1, . . . , n and we choose the following convention for the Minkowski scalar product of d-dimensional vectors, a · b = a 0 b 0 + a k b k = a 0 b 0 − a · b. Furthermore, we use the common symbol S (R n,d ) for the Schwartz-space and H for a Hilbert space. * amuch@matmor.unam.mx † vergara@nucleares.unam.mx 2

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Strict deformation quantization of locally convex algebras and modules

Cited by 20 publications

References 29 publications

Relativistic corrections to the Moyal-Weyl spacetime

Relativistic corrections to the Moyal-Weyl spacetime

Convergent star products for projective limits of Hilbert spaces

A Poincaré covariant noncommutative spacetime

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