Lectures on the functional renormalization group method

Abstract

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum eld theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed brie®y as an alternative way to arrive at the renormalization group equation. The scaling laws and xed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scalar theory. The ®attening of the e¬ective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

Fate of nonpolynomial interactions in scalar field theory

“…However in this regime, the analysis for the effective average potential (Legendre effective potential), including (5) and also (4), will differ from that for the general cutoff profile Polchinski's equation (6). We will exploit this later on.…”

“…The Gaussian fixed point corresponds to the solution V = 0 (U = 0) in all three cases (4)(5)(6). Linearising about the fixed point and separating variables, we find for such infinitesimally small perturbations V (z, t) = w(z) e λt and U (y, t) = w(y) e λt ,…”

“…[2][3][4][5][6][7][8][9]). Without the security of perturbation theory, it is clearly important if possible to make rigorous conclusions about such nonperturbative RG properties.…”

On the renormalization of theories of a scalar chiral superfield

“…It is curious that, despite the success of the ERG in addressing nonperturbative problems in Quantum Field Theory (QFT) (see [3,[18][19][20][21][22][23] for reviews) and despite the fact that some of the most penetrating insights into supersymmetric theories utilize the Wilsonian effective action (including the nonrenormalization theorems [14] and the Seiberg-Witten solution [24,25]) applications of the ERG to supersymmetric theories are rather limited, both in number and in scope [26][27][28][29][30][31][32][33][34][35][36][37] (see also the note added at the end of the paper). It is hoped, then, that the concrete results that this paper provides will lead to a development of this-surely fruitful-area.…”

Analytic continuation of functional renormalization group equations

“…By allowing for unconventional approximations and expansion schemes this made many interesting and partly non-perturbative investigations of quantum and statistical field theories possible. The field of applications ranges from cold atomic gases via QCD to Quantum Gravity, for reviews see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”