On the regular Hilbert space representation of a Moyal quantization

Braunss, G.

doi:10.1063/1.530537

Cited by 7 publications

(17 citation statements)

References 32 publications

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“…The operator L f R h turns out to be trace-class whenever f, h ∈ S(R 2N ), say. This fact is known to the experts, but rarely mentioned -to the knowledge of the author, its first mention in writing is in [1]. We do a little disgression to see how it comes about.…”

Section: Vol 6 2005mentioning

confidence: 99%

Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

Gayral¹

2005

Ann. Henri Poincaré

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We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) 'quantum spaces', generalizing Moyal planes and noncommutative tori, are constructed using Rieffel's theory of deformation quantization by actions of R l . Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic and non-periodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing.

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Section: Vol 6 2005mentioning

confidence: 99%

Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

Gayral¹

2005

Ann. Henri Poincaré

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“…the latter from the series (3) and (5). So g n (x, p) = 0 if and only if p ≥ n − ⌊n/2⌋, and thus has infinite support in p similar to f n .…”

Section: Hybrid Wfsmentioning

confidence: 99%

“…Reasoning along these same lines also leads to a compact expression for Wigner transforms in terms of star products of wave functions with an intercalated momentum-space delta [5], either Dirac or Kronecker.…”

Section: Appendix B Arbitrary Functions Acting On Wfs Through Star Pmentioning

confidence: 99%

“…l! e 2ix(l−k) δ n−p,k+l (16) where the sums result from taking χ n (x) = A n e ix as given by the series in (5). Note the support in p is now finite: This particular f n (x, p) is non-zero for 0 ≤ n − 2 ⌊n/2⌋ ≤ p ≤ n. That is to say, 0 ≤ p ≤ n for n even, and 1 ≤ p ≤ n for n odd.…”

Section: Dual Wfsmentioning

confidence: 99%

“…The sums result from using the series in (5). Given the range of the sums, we must have 0 ≤ 2p ≤ k + n for a non-zero result.…”

Section: Non-diagonal Dual Wfsmentioning

confidence: 99%

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Quasi-Hermitian quantum mechanics in phase space

Curtright

Veitia

2007

Journal of Mathematical Physics

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We investigate quasi-hermitian quantum mechanics in phase space using standard deformation quantization methods: Groenewold star products and Wigner transforms. We focus on imaginary Liouville theory as a representative example where exact results are easily obtained. We emphasize spatially periodic solutions, compute various distribution functions and phase-space metrics, and explore the relationships between them.

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Quantum dynamics in phase space: Moyal trajectories 2

Braunss

2013

Journal of Mathematical Physics

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Continuing a previous paper [G. Braunss, J. Phys. A: Math. Theor. 43, 025302 (2010)10.1088/1751-8113/43/2/025302] where we had calculated ℏ2-approximations of quantum phase space viz. Moyal trajectories of examples with one and two degrees of freedom, we present in this paper the calculation of ℏ2-approximations for four examples: a two-dimensional Toda chain, the radially symmetric Schwarzschild field, and two examples with three degrees of freedom, the latter being the nonrelativistic spherically Coulomb potential and the relativistic cylinder symmetrical Coulomb potential with a magnetic field H. We show in particular that an ℏ2-approximation of the nonrelativistic Coulomb field has no singularity at the origin (r = 0) whereas the classical trajectories are singular at r = 0. In the third example, we show in particular that for an arbitrary function γ(H, z) the expression β ≡ pz + γ(H, z) is classically (ℏ = 0) a constant of motion, whereas for ℏ ≠ 0 this holds only if γ(H, z) is an arbitrary polynomial of second order in z. This statement is shown to extend correspondingly to a cylinder symmetrical Schwarzschild field with a magnetic field. We exhibit in detail a number of properties of the radially symmetric Schwarzschild field. We exhibit finally the problems of the nonintegrable Hénon-Heiles Hamiltonian and give a short review of the regular Hilbert space representation of Moyal operators.

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On the regular Hilbert space representation of a Moyal quantization

Cited by 7 publications

References 32 publications

Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

Quasi-Hermitian quantum mechanics in phase space

Quantum dynamics in phase space: Moyal trajectories 2

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