Scalar models of p-adic quantum field theory and hierarchical models

Lerner, É. Yu.; Missarov, M. D.

doi:10.1007/bf01018683

Cited by 46 publications

(29 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is possible to reformulate the HM as a scalar model on the 2-adic line [65]. We give here a presentation [66] that does not require a detailed knowledge of the p-adic numbers.…”

Section: Scalar Models On Ultrametric Spacesmentioning

confidence: 99%

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Meurice¹

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract.We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigorous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and other RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large-N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review article is to present a configuration space counterpart, suitable for lattice formulations, of other RG equations formulated in momentum space (sometimes abbreviated as ERGE).

show abstract

“…It is possible to reformulate the HM as a scalar model on the 2-adic line [65]. We give here a presentation [66] that does not require a detailed knowledge of the p-adic numbers.…”

Section: Scalar Models On Ultrametric Spacesmentioning

confidence: 99%

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Meurice¹

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Some of the results presented in the previous sections can derived more elegantly if we had started immediately with the reformulation of Ref. [7]. In this section, we briefly explain how this can be done.…”

Section: Remarks Concerning the 2-adic Formulation Of The Hierarchicamentioning

confidence: 93%

“…As proposed in Ref. [7], where the sites are associated with 2-adic fractions which can be transformed into each other using addition and multiplication. However, it is possible to obtain the results that will be needed later using only a few basic results in arithmetic modulo 2 n .…”

Section: The Hierarchical Ising Model and Its Poincaré Groupmentioning

confidence: 99%

The high-temperature expansion of the hierarchical Ising model: From Poincaré symmetry to an algebraic algorithm

Meurice

1995

Journal of Mathematical Physics

View full text Add to dashboard Cite

We show that the hierarchical model at finite volume has a symmetry group which can be decomposed into rotations and translations as the familiar Poincaré groups. Using these symmetries, we show that the intricate sums appearing in the calculation of the high-temperature expansion of the magnetic susceptibility can be performed, at least up to the fourth order, using elementary algebraic manipulations which can be implemented with a computer. These symmetries appear more clearly if we use the 2-adic fractions to label the sites. We then apply the new algebraic methods to the calculation of quantities having a random walk interpretation. In particular, we show that the probability of returning at the starting point after m steps has poles at D = −2, −4, .... − 2m , where D is a free parameter playing a role similar to the dimensionality in nearest neighbor models.

show abstract

“…Scalar field theories over the reals with bi-local kinetic terms were introduced in [1], and the recent work [2] provides a useful point of entry into the extensive literature. Similar field theories over the p-adic numbers were considered in [3] as a continuum description of Dyson's hierarchical model [4]. A unifying point of view on the bi-local O(N ) vector model was provided in [5], showing that the standard large N development can be framed in terms that are largely independent of whether the theory is formulated over the reals or the p-adics.…”

Section: Introductionmentioning

confidence: 99%

“…A conservative expectation is that once non-local terms are allowed in a field theory, they proliferate and the theory becomes non-renormalizable. Theories with purely quadratic bi-local kinetic terms, as studied in [1,3] (as well as many subsequent works) avoid such problems through a non-renormalization theorem: If we write…”

Section: Introductionmentioning

confidence: 99%

Non-local non-linear sigma models

Gubser

Jepsen

et al. 2019

J. High Energ. Phys.

View full text Add to dashboard Cite

We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes.

show abstract

Scalar models of p-adic quantum field theory and hierarchical models

Cited by 46 publications

References 7 publications

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

The high-temperature expansion of the hierarchical Ising model: From Poincaré symmetry to an algebraic algorithm

Non-local non-linear sigma models

Contact Info

Product

Resources

About