Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations (original) (raw)

Renormalization group made clearer

I attempt to explain the use of renormalization group in quantum field theory from an elementary point of view. I review an elementary quantum-mechanical problem involving renormalization as a pedestrian example of a theory which is inherently ill-defined without a cutoff. After introducing a cutoff, one usually obtains a perturbative expansion that becomes invalid when the cutoff is removed. The renormalization group approach is treated as a purely mathematical technique (the Woodruff-Goldenfeld method) that improves the behavior of non-uniform perturbative expansions. By means of renormalization, one derives a perturbative expansion which is uniform in the cutoff, and therefore valid in the limit of infinite cutoff. I illustrate the application of this method to singular perturbation problems in ordinary differential equations.

Renormalizing the renormalization group pathologies

Physics Reports-review Section of Physics Letters, 2001

We review the status of the “pathologies” of the Renormalization Group (RG) encountered when one tries to define rigorously the RG transformation as a map between Hamiltonians. We explain their origin and clarify their status by relating them to the Griffiths’ singularities appearing in disordered systems; moreover, we suggest that the best way to avoid those pathologies is to use the contour representation rather than the spin representation for lattice spin models at low temperatures. Finally, we outline how to implement the RG in the contour representation.

Renormalization group invariance of quantum mechanics

Physics Letters B, 2000

We propose a framework to renormalize the nonrelativistic quantum mechanics with arbitrary singular interactions. The scattering equation is written to have one or more subtraction in the kernel at a given energy scale. The scattering amplitude is the solution of a nth order derivative equation in respect to the renormalization scale, which is the nonrelativistic counterpart of the Callan-Symanzik formalism. Scaled running potentials for the subtracted equations keep the physics invariant for a sliding subtraction point. An example of a singular potential, that requires more than one subtraction to renormalize the theory is shown. q

Regularity Properties and Pathologies of Position- Space Renormalization-Group Transformations: Scope and Limitations of Gibbsian Theory

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Our main results apply to local (in position space) RG maps acting on systems of bounded spins (compact single-spin space). Regarding regularity, we show that the RG map, defined on a suitable space of interactions (=formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce, and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension d~>3, these pathologies occur in a full neighborhood {fl>/~0, Ihl < ~(/~)} of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d~>2, the pathologies occur at low temperatures for arbitrary magnetic field strength. Pathologies may also occur in the critical region for Ising models in dimension d>~4. We discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems. In addition, we discuss critically the concept of Gibbs measure, which is at the heart of present-day classical statistical mechanics. We provide a careful, and, we hope, pedagogical, overview of the theory of Gibbsian measures as well as (the less familiar) non- Gibbsian measures, emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations. We give a rather complete catalogue of the known examples of such occurrences. The main message of this paper is that, despite a well-established tradition, Gibbsianness should not be taken for granted.

Elements of the Continuous Renormalization Group

Progress of Theoretical Physics Supplement, 1998

These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the rôle of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalisability, and a discussion of ultraviolet renormalons. * ) Typical small parameters that are sometimes useful are small coupling, i.e. perturbation theory, or 1/N where N is the number of components of a field, or ǫ = 4 − D where D is the space-time dimension. typeset using PTPT E X.sty <ver.0.8> * ) in some approximation scheme, e.g. derivative expansion * ) k should be chosen to be an integer. A function of the scale can be included in front of the monomial. 9), 27)

Renormalization group functional equations

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling ' functions and lead to exact functional relations for the local flow  functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of ' are sometimes not true fixed points under continuous changes in scale and zeroes of  do not necessarily signal fixed points of the flow but instead may only indicate turning points of the trajectories.

Exact Renormalization Group for Point Interactions

Renormalization is one of the deepest ideas in physics, yet its exact implementation in any interesting problem is usually very hard. In the present work, following the approach by Glazek and Maslowski in the flat space, we will study the exact renormalization of the same problem in a nontrivial geometric setting, namely in the two dimensional hyperbolic space. Delta function potential is an asymptotically free quantum mechanical problem which makes it resemble non-abelian gauge theories, yet it can be treated exactly in this nontrivial geometry.

Renormalization group in abstract QFT

Physics Letters B, 2000

The basics of RG equations for generic partition functions are briefly reviewed, keeping in mind an application to the Polyakov-de Boer-Verlindes description of the holomorphic RG flow. 1 RG versus (generalized) Laplace equations The notion and properties of the (generalized) renormalization group (RG) [1, 2, 3] acquire a new attention last years, primarily because of the interest to its hidden (quasi)integrable sturcture (related to Whitham dynamics [4], see [5] and references therein) and to its implicit occurence in the phenomena like AdS/CFT correspondence [6] (these studies are focused on the so called holomorphic RG flows [7, 8, 9, 10]). The purpose of this note is to give a concise survey of the basics of abstract RG theory, separating generic features from the pecularities of particular models. One of our goals is to explicitly formulate the controversies between the different concepts, which people are now trying to unify. It is a resolution of these controversies, which should provide a real understanding. The basic notion of modern quantum field/string theory is partition function (exponentiated effective action, the generating function for all the correlation functions in the theory in all possible vacua), resulting from functional integration over fields with an action, formed by a complete set of operators. Importantly, there are two different notions of completeness, see sect.2.2 below for the definitions. The exact RG a la J.Polchinski [2], possessing a formulation in terms of the diffeomorphisms in the moduli space M of theories [11], requires a strong completeness. It guarantees that the linear differential equations emerge for the partition function, and exponentiation of the corresponding vector field provides one-parametric families (flow lines) in Diff M describing the RG flow. At the same time, in most interesting examples only a weak completeness is assumed * : it is the one which is supposedly enough for integrability etc [12]. However, as natural for integrable systems, the weak completeness implies that partition function (interpreted as an element of some Hilbert space, see [11]) satisfies some differential equation in M, which are rarely linear in time (coupling constant) derivatives. The typical example is the Casimir (generalized Laplace) equation for the zonal spherical functions, if of the second order it is an ordinary Laplace equation. The other avatars of the same equation are the W-(in particular Virasoro) constraints in matrix models [13, 14, 15, 16] and the Hamilton-Jacobi equations for the large-N Yang-Mills partition functions [9], studied in the context of the AdS/CFT correspondence [6, 7, 10]. Unfortunately, non-linear differential equations do not possess an RG-like interpretation, at least naively. They are rather associated with the huge group DopM of all differential operators on M, and there is no obvious way to associate them with the elements of a much smaller diffeomorphism group Diff M. Perhaps surprisingly, an old result from the matrix models theory [17] implies that some intermediate notion can exist: despite matrix model provides only a weakly complete partition function, an explicit transformation of time-variables (coupling constants) is known, which (almost) eliminates the dependence on the size N of the matrix-what one would expect the exact RG to provide. A possible explanation of this phenomenon is that the entire set of the Ward identities (i.e. the full power of integrability) was used in the calculation of [17], not just a single Virasoro constraint L2 (which is a counterpart of conventional Polchinski's equation). In refs.[10] a seemingly artificial trick was suggested to resolve the controversy between the linear (in coupling-constants derivatives) notion of RG and the quadratic (at best) form of the Laplace/Virasoro/Hamilton-Jacobi equation. Namely, it was suggested to decompose the effective action into the contributions from over and from below the normalization point µ. The problem with this idea is that generically,

Renormalization group improving the effective action

Arxiv preprint hep-th/ …, 1998

The existence of fluctuations together with interactions leads to scale-dependence in the couplings of quantum field theories for the case of quantum fluctuations, and in the couplings of stochastic systems when the fluctuations are of thermal or statistical nature. In both cases the effects of these fluctuations can be accounted for by solutions of the corresponding renormalization group equations. We show how the renormalization group equations are intimately connected with the effective action: given the effective action we can extract the renormalization group equations; given the renormalization group equations the effects of these fluctuations can be included in the classical action by using what is known as improved perturbation theory (wherein the bare parameters appearing in tree-level expressions are replaced by their scale-dependent running forms). The improved action can then be used to reconstruct the effective action, up to finite renormalizations, and gradient terms.