Renormalization Theorem Research Articles

Spectral stability and semiclassical measures for renormalized KAM systems**The author has been supported by La Caixa, Severo Ochoa ICMAT, International PhD Programme, 2014, and MTM2017-85934-C3-3-P (MINECO, Spain).

An exact semiclassical version of the classical KAM theorem about small perturbations of vector fields on the torus is given. Moreover, a renormalization theorem based on counterterms for some semiclassical systems that are close to being completely integrable is obtained. We apply these results to characterize the sets of semiclassical measures and quantum limits for sequences of L2-eigenfunctions of these systems.

On Cheeger and Sobolev differentials in metric measure spaces

Recently, Gigli developed a Sobolev calculus on non-smooth spaces using module theory. In this paper it is shown that his theory fits nicely into the theory of differentiability spaces initiated by Cheeger, Keith and others. A relaxation procedure for L^p -valued subadditive functionals is presented and a relationship between the module generated by a functional and the one generated by its relaxation is given. In the framework of differentiability spaces, which includes so called PI- and RCD( K,N )-spaces, the Lipschitz module is pointwise finite dimensional. A general renorming theorem together with the characterization above shows that the Sobolev spaces of differentiability spaces are reflexive.

Structure of rapidity divergences in multi-parton scattering soft factors

We discuss the structure of rapidity divergences that are presented in the soft factors of transverse momentum dependent (TMD) factorization theorems. To provide the discussion on the most general level we consider soft factors for multi-parton scattering. We show that the rapidity divergences are result of the gluon exchanges with the distant transverse plane, and are structurally equivalent to the ultraviolet divergences. It allows to formulate and to prove the renormalization theorem for rapidity divergences. The proof is made with the help the conformal transformation which maps rapidity divergences to ultraviolet divergences. The theorem is the systematic form of the factorization of rapidity divergences, which is required for the definition of TMD parton distributions. In particular, the definition of multi parton distributions is presented. The equivalence of ultraviolet and rapidity divergences leads to the exact relation between soft and rapidity anomalous dimensions. Using this relation we derive the rapidity anomalous dimension at the three-loop order.

Power type ξ-asymptotically uniformly smooth and ξ-asymptotically uniformly flat norms

For each ordinal ξ and each 1<p<∞, we offer a natural, isomorphic characterization of those spaces and operators which admit an equivalent ξ-p-asymptotically uniformly smooth norm. We also introduce the notion of ξ-asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent ξ-asymptotically uniformly flat norm. Given a compact, Hausdorff space K, we prove an optimal renorming theorem regarding the ξ-asymptotic smoothness of C(K) in terms of the Cantor–Bendixson index of K. We also prove that for all ordinals, both the isomorphic properties and isometric properties pass from Banach spaces to their injective tensor products. We study the classes of ξ-p-asymptotically uniformly smooth, ξ-p-asymptotically uniformly smoothable, ξ-asymptotically uniformly flat, and ξ-asymptotically uniformly flattenable operators. We show that these classes are either a Banach ideal or a right Banach ideal when assigned an appropriate ideal norm.

On super fixed point property and super weak compactness of convex subsets in Banach spaces

For a nonempty convex set C of a Banach space X, a self-mapping on C is said to a linear (respectively, affine) isometry if it is the restriction of a linear (respectively, affine) isometry defined on the whole space X. By means of super weakly compact set theory established in the recent years, in this paper, we first show that a nonempty closed bounded convex set of a Banach space has super fixed point property for affine (or, equivalently, linear) isometries if and only if it is super weakly compact; and the super fixed point property and the super weak compactness coincide on every closed bounded convex subset of a Banach space under equivalent renorming sense. With the application of Fabian–Montesinos–Zizler's renorming theorem, we finally show that every strongly super weakly compact generated Banach space can be renormed so that every weakly compact convex set has super fixed point property.

A Hardy inequality with a point-singular weight inside a domain

Sobolev spaces with weights taking infinite values at some interior points of a two-dimensional domain are considered. For functions from these spaces, a Hardy inequality is obtained. Embedding theorems for weighted Lebesgue spaces and equivalent renorming theorems are proved.

A generalization of a renorming theorem by Lin and a new nonreflexive space with the fixed point property which is nonisomorphic to [formula omitted

Given 1<p≤∞,we construct a Banach space Xp hereditarily asymptotically isometric to l1, so that for ni,mk∈N it contains an isometric copy of (∑i=1∞∑k=1mklpni)1 and Xp∗ contains a subspace isomorphic to c0(Γ), where Γ has cardinality c, and use a generalization of a renorming theorem of Lin to renorm Xp in order to have the fixed point property. Also, we give new renormings with the FPP in some known spaces. In particular we prove that (c0,‖⋅‖α)∗ has the fixed point property, where ‖(xi)‖α=supi|xi|+α∑i|xi|2i,α>0. This gives an example of a space without the FPP whose dual (an isomorph of l1) has the FPP.

Every weakly compact set can be uniformly embedded into a reflexive Banach space

Based on an application of the Davis-Figiel-Johnson-Pelzyśki procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell-Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.

Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions

In a seminal paper of almost 20 years ago, R.J. DiPerna and P.-L. Lions initiated the theory of renormalized solutions to study the well-posedness of Ordinary Differential Equations and Transport Equations with discontinuous coefficients. In a recent work L. Ambrosio solved the long-standing open problem of extending this theory to BV coefficients, the most common functional-analytic closure of classical functions with jump discontinuities. Besides its intrinsic interest, Ambrosio's Theorem has been used to solve relevant problems in Partial Differential Equations and it opened the way to a series of new questions.

Explicit results for the anomalous three point function and non-renormalization theorems

Two-loop corrections for the <VVA> correlator of the singlet axial and vector currents in QCD are calculated in the chiral limit for arbitrary momenta. Explicit calculations confirm the non-renormalization theorems derived recently by Vainshtein and Knecht et.al. We find that as in the one-loop case also at the two loops the <VVA> correlator has only 3 independent form-factors instead of 4. From the explicit results we observe that the two-loop correction to the correlator is equal to the one-loop result times the constant factor C_2(R) alpha_s/pi in the MSbar scheme. This holds for the full correlator, for the anomalous longitudinal as well as for the non- anomalous thansversal amplitudes. The finite overall alpha_s dependent constant has to be normalized away by renormalizing the axial current according to Witten's algebraic/geometrical constraint on the anomalous Ward identity. Our observations, together with known facts, suggest that in perturbation theory the <VVA> correlator is proportional to the one-loop term to all orders and that the non- renormalization theorem of the Adler-Bell-Jackiw anomaly carries over to the full correlator.

On topological Kadec norms

We present a topological analogue of the classic Kadec Renorming Theorem, as follows. Let Open image in new window be two separable metric topologies on the same set X. We prove that every point in X has an Open image in new window -neighbourhood basis consisting of sets that are Open image in new window -closed if and only if there exists a function φ: X→ℝ that is Open image in new window -lower semi-continuous and such that Open image in new window is the weakest topology on X that contains Open image in new window and that makes φ continuous. An immediate corollary is that the class of almost n-dimensional spaces consists precisely of the graphs of lower semi-continuous functions with at most n-dimensional domains.

New aspects of Mott transition in intermediately correlated one-dimensional Hubbard systems

Using the quantum Monte-Carlo method, we have theoretically studied the metal–insulator transition of the one-dimensional half-filled Hubbard model, and calculated the momentum-specified Lehmann spectra, as well as the total density of states (DOS). In the intermediate range of the on-site Coulomb repulsion, we have newly found that the DOS has coherent peaks near the Fermi level, in addition to the so-called upper and lower Hubbard band peaks. The dispersion relation near the Fermi level is also found to be very close to a original one-body type cosine band. However, this one-body nature disappears rapidly as the wave-number (≡ k) goes to the bottom of the band, k→0. Thus, the electronic structure cannot be described by any simple mass renormalization theorem independent of k, but the one-body nature is shown to be greatly reconstructed.

Local uniform convexity and Kadec‐Klee type properties in K‐interpolation spaces I: General Theory

We present a systematic study of the interpolation of local uniform convexity and Kadec‐Klee type properties in K‐interpolation spaces. Using properties of the K‐functional of J.Peetre, our approach is based on a detailed analysis of properties of a Banach couple and properties of a K‐interpolation functional which guarantee that a given K‐interpolation space is locally uniformly convex, or has a Kadec‐Klee property. A central motivation for our study lies in the observation that classical renorming theorems of Kadec and of Davis, Ghoussoub and Lindenstrauss have an interpolation nature. As a partiular by‐product of our study, we show that the theorem of Kadec itself, that each separable Banach space admits an equivalent locally uniformly convex norm, follows directly from our approach.

Deformed superspace, Script N = 1/2 supersymmetry &amp; (non)renormalization theorems

We consider a deformed superspace in which the coordinates \theta do not anticommute, but satisfy a Clifford algebra. We present results on the properties of N=1/2 supersymmetric theories of chiral superfields in deformed superspace, taking the Wess-Zumino model as the prototype. We prove new (non)renormalization theorems: the F-term is radiatively corrected and becomes indistinguishable from the D-term, while the Fbar-term is not renormalized. Supersymmetric vacua are critical points of the antiholomorphic superpotential. The vacuum energy is zero to all orders in perturbation theory. We illustrate these results with several examples.

Almost Every Real Quadratic Map Is Either Regular or Stochastic

In this paper we complete a program to study measurable dynamics in the real quadratic family. Our goal was to prove that almost any real quadratic map Pc : z t- x2 + c, c c [-2,1/4], has either an attracting cycle or an absolutely continuous invariant measure. The final step, completed here, is to prove that the set of infinitely renormalizable parametric values c c [-2,1/4] has zero Lebesgue measure. We derive this from a Renormalization Theorem which asserts uniform hyperbolicity of the full renormalization operator. This theorem gives the most general real version of the Feigenbaum-Coullet-Tresser universality, simultanuously for all combinatorial types.

Szlenk indices and uniform homeomorphisms

We prove some rather precise renorming theorems for Banach spaces with Szlenk index wo. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

Three-point Green function of the stress-energy tensor in the AdS-CFT correspondence

We compute the 3-point function of the stress-energy tensor in the d-dimensional CFT from the AdS_{d+1} gravity. For d=4 the coefficients of the three linearly independent conformally covariant forms entering the 3-point function are exactly the same as given by the free field computations in the ${\cal N}=4$ SYM just as expected from the known renormalization theorems. For d=3 and d=6 our results give the value of the corresponding 3-point function in the theories of strongly coupled ${\cal N}=8$ superconformal scalar and (2,0) tensor multiplets respectively.

Diffusive stability of rolls in the two–dimensional real and complex swift–hohenberg equation

We show the nonlinear stability of small bifurcating stationary rolls u , , for the Swift–Hohenberg–equation on the domain R2. In Bloch wave representation the linearization around a marginal stable roll uε,x, has continuous spectrum up to 0 with a locally parabolic shape at the critical Bloch vector 0. Using an abstract renormalization theorem we show that small spatially localized integrable perturbations decay diffusively to zero. Moreover we estimate the size of the domain of attraction of a roll uε,x, in terms of its modulus and Fourier wavenumber. To explain the method we also treat the nonlinear stability of stationary rolls for the complex Swift–Hohenberg equation on R2

A general non renormalization theorem in the extended antifield formalism

In the context of algebraic renormalization, the extended antifield formalism is used to derive the general forms of the anomaly consistency condition and of the Callan-Symanzik equation for generic gauge theories. A local version of the latter is used to derive sufficient conditions for the vanishing of beta functions associated to terms whose integrands are invariant only up to a divergence for an arbitrary non trivial non anomalous symmetry of the Lagrangian. These conditions are independent of power counting restrictions and of the form of the gauge fixation.

Sum theorems for monotone operators and convex functions

In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.