Nonlinear models in 2 + ε dimensions

Friedan, Daniel

doi:10.1016/0003-4916(85)90384-7

Cited by 659 publications

(602 citation statements)

References 29 publications

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“…We must emphasis that k will not stop at k c , it could continuously go to infinity, since at large k the behavior of δ Z changes as the effective dimension approaches 2 but rather 4, and δ Z increases as log k but rather k 2 . We know that the NLSM near d = 2 becomes perturbative renormalizable [30][31][32] which is a positive feature for a good behavior of our model at UV. And finally the increasing rate slows down and stop at the UV fixed point, where the δ Z vanishes as the renormalization condition imposes, leaving the finite bare λ.…”

Section: Quantum Behaviormentioning

confidence: 72%

The Cosmological Constant Problem and Quantum Spacetime Reference Frame

Luo¹

2017

Preprint

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We generalize the idea of quantum clock time to quantum spacetime reference frame via physical realization of a reference system by quantum rulers and clocks. Omitting the internal degrees of freedom (such as spins) of the physical rulers and clocks, only considering their metric properties, the spacetime reference frame is described by a bosonic non-linear sigma model. We study the quantum behavior of the system under approximations, and obtain (1) a cosmological constant valued (2/π)ρc0 (ρc0 the critical density at near current epoch) which is very close to the observations; (2) an effective Einstein-Hilbert term in the effective action; (3) the ratio of variance to mean-squared of spacetime interval tends to a universal constant 2/π in the infrared region. This effect is testable by observing a linear dependence between the inherent quantum variance and mean-squared of the redshifts from cosmic distant spectral lines. The proportionality is expected to be the observed percentage of the dark energy. We also generalize the equivalence principle to be valid for all quantum phenomenon.

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Section: Quantum Behaviormentioning

confidence: 72%

The Cosmological Constant Problem and Quantum Spacetime Reference Frame

Luo¹

2017

Preprint

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“…But, as we shall see later on, this appears as insufficient to reabsorb the various divergences. Thus, we also allow for a finite deformation of the classical metric and torsion potential g ij + h ij = G ij to describe its quantum extension : of course, thisà la Friedan [19] extension of the notion of renormalisability involves a priori an infinite number of new parameters. Let us emphasize that we shall consider only finite deformations.…”

Section: The Two-loop Order Bare Actionmentioning

confidence: 99%

“…We know that, when expressed as functions ofβ(τ ) andγ(τ ) , equations (17,18,19) remain unchanged, up to the substitutions discussed in Section 3 [equation (10)] :…”

Section: Remarksmentioning

confidence: 99%

Dualised σ-models at the two-loop order

Bonneau

Casteill

2001

Nuclear Physics B

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We adress ourselves the question of the quantum equivalence of non abelian dualised σ-models on the simple example of the T-dualised SU (2) σ-model. This theory is classically canonically equivalent to the standard chiral SU (2) σ-model. It is known that the equivalence also holds at the first order in perturbations with the same β functions. However, this model has been claimed to be non-renormalisable at the two-loop order. The aim of the present work is the proof that it is -at least up to this order -still possible to define a correct quantum theory. Its target space metric being only modified in a finite manner, all divergences are reabsorbed into coupling and fields (infinite) renormalisations.

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“…In particular, two-dimensional non-linear sigma models with target space a Riemannian manifold with metric g are perturbatively renormalizable and their betafunction coincides with the Ricci curvature tensor at one loop [36], [20]. Thus, the Ricci flow describes changes of the metric g (viewed as generalized coupling) under changes of the logarithm of the world-sheet length scale in quantum theory provided that the curvature is small.…”

Section: Introductionmentioning

confidence: 99%

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

Bakas

Kong

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kähler manifold. Di¤ering from Fateev's examples most of our examples are non-collapsed. The construction of this paper, di¤erent from the ad hoc anastz of Fateev, is systematic, generalizing (as well as unifying) the previous constructions of Einstein metrics by Bourguignon-Karcher, Jenson, and Ziller in the sense that the existence problem to a backward nonlinear parabolic PDE is reduced to the study of nonlinear ODE systems. The key step of solving the reduced nonlinear ODE system is to find suitable monotonic and conserved quantities under Ricci flow with symmetry. Besides supplying new possible singularity models for Ricci flow on spheres and projective spaces, our solutions are counter-examples to some folklore conjectures on ancient solutions of Ricci flow on compact manifolds. As a by-product, we infer that some nonstandard Einstein metrics on spheres and complex projective spaces are unstable fixed points of the Ricci flow.

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Nonlinear models in 2 + ε dimensions

Cited by 659 publications

References 29 publications

The Cosmological Constant Problem and Quantum Spacetime Reference Frame

The Cosmological Constant Problem and Quantum Spacetime Reference Frame

Dualised σ-models at the two-loop order

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

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