Quasiclassical asymptotic behavior for Wigner's 3j symbols

Borodin, K. S.; Kroshilin, A. E.; Толмачев, В. В.

doi:10.1007/bf01036473

Cited by 14 publications

(9 citation statements)

References 4 publications

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“…While we do not expect that this will add to the punchline of fast scrambling, analyzing the full form of the evolution of the entropy may help identify other interesting timescales that are typically at work in thermalization processes [35]. And while we were forced to fall back on a numerical analysis of the final entropy expression, there is the potential for a successful analytical analysis: the asymptotics of the structure constants for N → ∞ can be worked out using Edmonds' relations and their extensions [37]. The analogue of the thermodynamic limit in our system is indeed the regime N → ∞; and hence techniques used in traditional statistical systems can perhaps be used to tame the complicated entropy expression.…”

Section: Discussionmentioning

confidence: 99%

From black hole to qubits: evidence of fast scrambling in BMN theory

Pramodh

Sahakian

2015

J. High Energ. Phys.

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BMN Matrix theory admits vacua in the shape of large spherical membranes. Perturbing around such vacua, the setup provides for a controlled computational framework for testing information evolution in Matrix black holes. The theory realizes excitations in the supergravity multiplet as qubits. These qubits are coupled to matrix degrees of freedom that describe deformations of the spherical shape of the membrane. Arranging the ripples on the membrane into a heat bath, we use the qubit system as a probe and compute the associated Feynman-Vernon density matrix at one loop order. This allows us to trace the evolution of entanglement in the system and extract the characteristic scrambling timescale. We find that our numerical analysis is consistent with this time scaling logarithmically with the entropy of the qubit system, in tune with suggestions by Sekino and Susskind.

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Section: Discussionmentioning

confidence: 99%

From black hole to qubits: evidence of fast scrambling in BMN theory

Pramodh

Sahakian

2015

J. High Energ. Phys.

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“…comes from the asymptotic expansion of the 3j symbol. 44 Defining a 2 = i 2 + 2 − 2i cos θ 1 and b 2 = j 2 + 2 − 2j cos θ 2 to transform the a and b integrals into θ 1 and θ 2 integrals, and then renaming i, j and as p 1 , p 2 and q, we obtain…”

Section: B the Thermodynamic Limitmentioning

confidence: 99%

Thinking outside the box: The uniform electron gas on a hypersphere

Loos

Gill

2011

The Journal of Chemical Physics

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“…We refer to Borodin et al (1978, , which allows us to generalize the Wigner symbol to real-valued arguments. Equation (33) holds only for 1 , 2 , 3 1, which, in the quantum-mechanical context of Borodin et al (1978), originates from the use of expressions that are valid for large angular momenta, i.e. the quasi-classical limit, only.…”

Section: Comparison Of Covariancesmentioning

confidence: 99%

“…To allow for a comparison between (31) and our approach based on (6), we need to establish a relation between the square of the Wigner symbol and (8). We refer to Borodin et al (1978, see also references therein) who compute approximation formulae of the Wigner symbol in the context of the quasi-continuous limit of quantum states with high angular momenta. The base of their derivation is formed by the exact relation…”

Section: Comparison Of Covariancesmentioning

confidence: 99%

“…where D ℓ mm ′ denotes the m × m ′ element of the Wigner D matrix, which in turn is a function of the three Euler angles ϕ, θ, and ψ. Making use of a quasi-classical approximation of the D ℓ mm ′ , Borodin et al (1978) compute expressions for the general Wigner symbol in the limit of large and continuous angular frequencies. From these results we extract the approximation…”

Section: Comparison Of Covariancesmentioning

confidence: 99%

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Bispectrum covariance in the flat-sky limit

Joachimi

Shi²,

Schneider³

2009

A&A

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Aims. To probe cosmological fields beyond the Gaussian level, three-point statistics can be used, all of which are related to the bispectrum. Hence, measurements of CMB anisotropies, galaxy clustering, and weak gravitational lensing alike have to rely upon an accurate theoretical background concerning the bispectrum and its noise properties. If only small portions of the sky are considered, it is often desirable to perform the analysis in the flat-sky limit. We aim at a formal, detailed derivation of the bispectrum covariance in the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane approach. Methods. We define an unbiased estimator of the bispectrum, which takes the average over the overlap of annuli in Fourier space, and compute its full covariance. The outcome of our formalism is compared to the flat-sky spherical harmonic approximation in terms of the covariance, the behavior under parity transformations, and the information content. We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding. Results. Contrary to foregoing work, we find a difference by a factor of two between the covariances of the Fourier-plane and the spherical harmonic approach. We argue that this discrepancy can be explained by the differing behavior with respect to parity. However, in an exemplary analysis it is demonstrated that the Fisher information of both formalisms agrees to high accuracy. Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol, which leads to convenient approximation formulae for the covariances of both approaches.

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Quasiclassical asymptotic behavior for Wigner's 3j symbols

Cited by 14 publications

References 4 publications

From black hole to qubits: evidence of fast scrambling in BMN theory

From black hole to qubits: evidence of fast scrambling in BMN theory

Thinking outside the box: The uniform electron gas on a hypersphere

Bispectrum covariance in the flat-sky limit

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