From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation

Stoimenov, Stoimen; Henkel, Malte

doi:10.48550/arxiv.1509.00434

Cited by 5 publications

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“…[11,13,14,15,17,18,20,21,22,30,44,49] as well as in classical non-equilibrium dynamics, see e.g. [53,19,16,26,29,51,54,3]. As shown in table 1, for d = 1 and d = 2 space dimensions, the Lie group of meta-conformal transformations is infinite-dimensional.…”

Section: Introductionmentioning

confidence: 99%

“…, N) is not necessarily satisfied. Meta-conformal invariance arises as a dynamical symmetry of the simple equation Sϕ(t, r) = −µ∂ t + ∂ r ϕ(t, r) = 0 of ballistic transport, which distinguishes a single preferred direction [51], with coordinate r , from the transverse direction(s), with coordinate r ⊥ . This is sketched in figure 1.…”

Section: Introductionmentioning

confidence: 99%

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Boundedness of meta-conformal two-point functions in one and two spatial dimensions

Henkel,

Kuczynski,

Stoimenov

2020

Preprint

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Meta-conformal invariance is a novel class of dynamical symmetries, with dynamical exponent z = 1, and distinct from the standard ortho-conformal invariance. The meta-conformal Ward identities can be directly read off from the Lie algebra generators, but this procedure implicitly assumes that the co-variant correlators should depend holomorphically on time-and space coordinates. Furthermore, this assumption implies un-physical singularities in the co-variant correlators. A careful reformulation of the global meta-conformal Ward identities in a dualised space, combined with a regularity postulate, leads to bounded and regular expressions for the co-variant two-point functions, both in d = 1 and d = 2 spatial dimensions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Boundedness of meta-conformal two-point functions in one and two spatial dimensions

Henkel,

Kuczynski,

Stoimenov

2020

Preprint

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“…The isomorphism of (4) with the conformal Lie algebra conf( 2) is seen as follows [18,23]: write The meta-conformal Lie algebra (4) acts as a dynamical symmetry on the linear advection equation [26] Sφ(t, r) = (−µ∂ t + ∂ r )φ(t, r) = 0 (5) in the sense that a solution φ of Sφ = 0, with scaling dimension x φ = x = γ/µ, is mapped onto another solution of the same equation. Hence the space of solutions of Sφ = 0 is meta-conformal invariant [18] (extended to Jeans-Poisson systems in [31]). This follows from, with…”

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confidence: 99%

“…To carry out the back-transformation, we distinguish the cases λ > 0 and λ < 0. If one has λ > 0, we have from (29,31), with…”

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confidence: 99%

Meta-conformal invariance and the boundedness of two-point correlation functions

Henkel

Stoimenov

2016

J. Phys. A: Math. Theor.

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The covariant two-point functions, derived from Ward identities in direct space, can be affected by consistency problems and can become unbounded for large time-or spaceseparations. This difficulty arises for several extensions of dynamical scaling, for example Schrödinger-invariance, conformal Galilei invariance or meta-conformal invariance, but not for standard ortho-conformal invariance. For meta-conformal invariance in (1 + 1) dimensions, which acts as a dynamical symmetry of a simple advection equation, these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in the Cartan sub-algebra. This provides a canonical interpretation of meta-conformally covariant two-point functions as correlators. Galilei-conformal correlators can be obtained from meta-conformal invariance through a simple contraction. In contrast, by an analogus construction, Schrödinger-covariant two-point functions are causal response functions. All these two-point functions are bounded at large separations, for sufficiently positive values of the scaling exponents.

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“…Although this has not yet been done explicitly, we expect that the techniques reviewed here can be readily extended to several physically distinct representations of these algebras, see e.g. [45,23,70,74] for examples. 13…”

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confidence: 99%

Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

Henkel

2015

Symmetry

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Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into account the different scaling behaviour of space and time, will be reviewed. Particular attention will be given to the causality properties as they follow for co-variant n-point functions. These are important for the physical identification of n-point functions as responses or correlators.

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From conformal invariance towards dynamical symmetries of the collisionless Boltzmann equation

Cited by 5 publications

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Boundedness of meta-conformal two-point functions in one and two spatial dimensions

Boundedness of meta-conformal two-point functions in one and two spatial dimensions

Meta-conformal invariance and the boundedness of two-point correlation functions

Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

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