Some Continuous Field Quantizations, Equivalent to the C*-Weyl Quantization

Honegger, Reinhard; Rieckers, Alfred

doi:10.2977/prims/1145475406

Cited by 13 publications

(7 citation statements)

References 24 publications

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“…for all F ∈ E, where α is any complex inner product on E. It follows from results of Binz et al (2004) and Honegger and Rieckers (2005) that this structure forms a strict quantization. And importantly for what follows, the choice of a complex inner product α does not matter at this stage because for any other complex inner product α ′ , the corresponding quantization maps are equivalent in the sense that lim…”

Section: Strict Quantization and Number Operatorsmentioning

confidence: 96%

Localizable Particles in the Classical Limit of Quantum Field Theory

Feintzeig,

Librande,

Soiffer

2021

Preprint

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A number of arguments purport to show that quantum field theory cannot be given an interpretation in terms of localizable particles. We show, in light of such arguments, that the classical → 0 limit can aid our understanding of the particle content of quantum field theories. In particular, we demonstrate that for the massive Klein-Gordon field, the classical limits of number operators can be understood to encode local information about particles in the corresponding classical field theory. Keywords quantum field theory • particle interpretation • classical limit • deformation quantization 1 IntroductionRelativistic quantum field theory underlies the modern discipline of particle physics. Practitioners use the theory to conceptualize interactions between particles and make quantitative predictions about scattering experiments. Yet a number of arguments purport to show that various features of our particle concept are incompatible with the constraints of relativistic quantum physics. 1 Building on results of Malament (1996), Halvorson and Clifton (2002) argue that in relativistic quantum theory, particles cannot be localized in spatial regions. The conclusions of such arguments leave interpreters of relativistic quantum field theory with a puzzle. How can an underlying theory that does not allow for localized particles support descriptive and explanatory practices that appear to involve localized particles?Previous investigations have focused on the issue of recovering the phenomenology of particle physics from relativistic quantum field theory. For example, Buchholz (1995) provides a way of recovering scattering theory at asymptotic times. 2 In contrast, the goal of this paper is to make a small contribution toward our understanding of the theoretical role of particles in quantum field theory. We aim to make precise a sense in which a theoretical description of particles emerges from quantum field theory through the behavior of number operators.

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Section: Strict Quantization and Number Operatorsmentioning

confidence: 96%

Localizable Particles in the Classical Limit of Quantum Field Theory

Feintzeig,

Librande,

Soiffer

2021

Preprint

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“…A normally ordered monomial in the creation and annihilation operators, introduced by means of a complex structure, is just a special real-multilinear operator expression of the fields. It is elaborated in [23], how the various operator orderings define direct field quantizations. They all lead back to a decorated Weyl quantization of the form Q w Π, (W 0 (f )) := w( , f)Π (W (f )) , ∀f ∈ E , (6.4) where the w( , f) are certain numerical factors.…”

Section: E Binz R Honegger and A Rieckersmentioning

confidence: 99%

“…They all lead back to a decorated Weyl quantization of the form Q w Π, (W 0 (f )) := w( , f)Π (W (f )) , ∀f ∈ E , (6.4) where the w( , f) are certain numerical factors. With some modifications of the foregoing arguments it is shown in [23] that these are again strict and continuous deformation quantizations, which all of them refer to the described continuous field of C*-Weyl algebras and which are equivalent, in the sense of [8], to the Weyl quantization.…”

Section: E Binz R Honegger and A Rieckersmentioning

confidence: 99%

Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization

Binz

Honegger

Rieckers

2004

Ann. Henri Poincaré

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For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space (E, σ) the family of Weyl algebras {W(E, σ)} ∈R , introduced in a previous work [1], is shown to constitute a continuous field of C*-algebras in the sense of Dixmier. Various Poisson algebras, given as abstract (Fréchet-) *-algebras which are C*-norm-dense in W(E, 0), are constructed as domains for a Weyl quantization, which maps the classical onto the quantum mechanical Weyl elements. This kind of a quantization map is demonstrated to realize a continuous strict deformation quantization in the sense of Rieffel and Landsman. The quantization is proved to be equivariant under the automorphic actions of the full affine symplectic group. The relationship to formal field quantization in theoretical physics is discussed by suggesting a representation dependent direct field quantization in mathematically concise terms.

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“…The primary example of this is the formulation of the classical → 0 limit of quantum theories in the framework of strict deformation quantization. The general theory is presented in Rieffel (1989Rieffel ( , 1993 and Landsman (1998aLandsman ( , 2007Landsman ( , 2017, and many examples are investigated in Landsman (1993aLandsman ( ,b, 1998b, Binz et al (2004), Honegger and Rieckers (2005), Honegger et al (2008), Bieliavsky and Gayral (2015), and van Nuland (2019). Under modest conditions, a strict deformation quantization determines a continuous bundle of C*-algebras, with so-called equivalent quantizations determining the same continuous bundle.…”

Section: Introductionmentioning

confidence: 99%

Extensions of Bundles of C*-algebras

Steeger,

Feintzeig

2021

Preprint

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Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the → 0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the → 0 case) on the fiber C*-algebras.

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Some Continuous Field Quantizations, Equivalent to the C*-Weyl Quantization

Cited by 13 publications

References 24 publications

Localizable Particles in the Classical Limit of Quantum Field Theory

Localizable Particles in the Classical Limit of Quantum Field Theory

Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization

Extensions of Bundles of C*-algebras

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