Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

Honegger, Reinhard

doi:10.3842/sigma.2008.047

Cited by 4 publications

(3 citation statements)

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“…(See p. 334 of Binz et al 8 or Eq. 2.15, p. 11 of Honegger et al 23 ) and so can serve as the domain of a quantization map. We note that in the special case where E = R 2n , the phase space is the dual E ′ = R 2n and so W(R 2n , 0) ∼ = AP (R 2n ).…”

Section: Strict Quantization and The Weyl Algebramentioning

confidence: 99%

“…This follows because although the field operators are unbounded and the norm is not defined on them, the relevant differences of operators are bounded and so the conditions are meaningful exactly as stated. In what follows, we understand the Poisson bracket to be defined as in Honegger et al 23 , Eq. 2.15, p. 11; cf.…”

Section: Next We Proceed By Induction Suppose [φ (F ) φ (G)mentioning

confidence: 99%

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Classical Limits of Unbounded Quantities by Strict Quantization

Browning,

Feintzeig,

Gates-Redburg

et al. 2020

Preprint

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This paper extends the tools of C*-algebraic strict quantization toward analyzing the classical limits of unbounded quantities in quantum theories. We introduce the approach first in the simple case of finite systems. Then we apply this approach to analyze the classical limits of unbounded quantities in bosonic quantum field theories with particular attention to number operators and Hamiltonians. The methods take classical limits in a representation-independent manner and so allow one to compare quantities appearing in inequivalent Fock space representations.

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Section: Strict Quantization and The Weyl Algebramentioning

confidence: 99%

Section: Next We Proceed By Induction Suppose [φ (F ) φ (G)mentioning

confidence: 99%

Classical Limits of Unbounded Quantities by Strict Quantization

Browning,

Feintzeig,

Gates-Redburg

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…A further argument is the intuition that mechanical theories are stabilized by deforming them whenever possible. In this spirit Weyl's formalism (see for instance [16], or [17] for a recent paper) describes quantum mechanics as a deformation of classical mechanics. By the deformation the theory becomes less reducible.…”

Section: Prefacementioning

confidence: 99%

Reducible Quantum Electrodynamics. I. The Quantum Dimension of the Electromagnetic Field

Naudts

2015

Preprint

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In absence of currents and charges the quantized electromagnetic field can be described by wave functions which for each individual wave vector are normalized to one. The resulting formalism involves reducible representations of the Canonical Commutation Relations. The corresponding paradigm is a space-time filled with two-dimensional quantum harmonic oscillators. Mathematically, this is equivalent with two additional dimensions penetrated by the electromagnetic waves.

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The classical limit of a state on the Weyl algebra

Feintzeig

2018

Journal of Mathematical Physics

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This paper considers states on the Weyl algebra of the canonical commutation relations over the phase space R 2n . We show that a state is regular iff its classical limit is a countably additive Borel probability measure on R 2n . It follows that one can "reduce" the state space of the Weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical. lemmas to characterize the countably additive Borel measures on R 2n in terms of algebraic structure, and I apply Thm. 1 to the purely classical system with phase space R 2n . Finally, in Section V, I prove the main result and discuss its significance. The results of this paper involve little, if any, mathematical novelty. I hope, however, that the perspective I provide on the construction of new quantum theories is of interest. II. PRELIMINARIESThe Weyl algebra over R 2n is formed by deforming the product of the C*-algebra AP (R 2n ) of complex-valued almost periodic functions on R 2n . The C*-algebra AP (R 2n ) is generatedfor all y ∈ R 2n , where · is the standard inner product on R 2n . Polynomials (with respect to pointwise multiplication, addition, and complex conjugation) of functions of the form W 0 (x) for x ∈ R 2n are norm dense in AP (R 2n ) with respect to the standard supremum norm. 59The Weyl algebra over R 2n , denoted W h (R 2n ), for h ∈ (0, 1] is generated from the same set of functions by defining a new multiplication operation. The symbol W h (x) ∈ W h (R 2n ) is used now to denote the element W 0 (x) as it is considered in the new C*-algebra. Define the non-commutative multiplication operation on W h (R 2n ) by W h (x)W h (y) := e ih 2 σ(x,y) W h (x + y)for all x, y ∈ R 2n , where σ is the standard symplectic form on R 2n given byfor a, b, a ′ , b ′ ∈ R n and · is now the standard inner product on R n . The Weyl algebra is the norm completion in the minimal regular norm 9,51,66 of polynomials of elements of the form W h (x) for x ∈ R 2n with respect to the non-commutative multiplication operation.It is this C*-algebra (sometimes known as the CCR algebra, or the Weyl form of the CCRs) that is often used to model the physical magnitudes, or observables, of a quantum mechanical system constructed from a classical system with phase space R 2n . One can then take the positive, normalized linear functionals, or states, on the C*-algebra of physical magnitudes to model the physically realizable states of the quantum system. 11,12,34,38

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Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

Cited by 4 publications

References 69 publications

Classical Limits of Unbounded Quantities by Strict Quantization

Classical Limits of Unbounded Quantities by Strict Quantization

Reducible Quantum Electrodynamics. I. The Quantum Dimension of the Electromagnetic Field

The classical limit of a state on the Weyl algebra

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