Wavelets and Renormalization Group in Quantum Field Theory Problems
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ELEMENTARY PARTICLES AND FIELDS Theory
Wavelets and Renormalization Group in Quantum Field Theory Problems∗ M. V. Altaisky** Space Research Institute, Russian Academy of Scienees, Moscow, 117997 Russia Received April 17, 2018
Abstract—Using continuous wavelet transform it is possible to construct a regularization procedure for scale-dependent quantum field theory models, which is complementary to the functional renormalization group method in the sense that it sums up the fluctuations of larger scales in order to get the effective action at small observation scale. The standard RG results for φ4 model are reproduced. The fixed points of the scale-dependent theory are studied in the one-loop approximation. DOI: 10.1134/S1063778818060029
1. INTRODUCTION Renormalization group (RG) has entered quantum field theory as a group of infinitesimal reparametrizations of the S matrix emerging after the cancellation of the ultraviolet divegences [1]. The RG method has become known in quantum electrodynamics since Gell-Mann and Low have shown the charge distribution surrounding a test charge in vacuum does not at small distances depend on a coupling constant, except for a scale factor, i.e., possesses a kind of self-similarity, that enables one to express a “bare” charge at small scale using the measured value at large scale [2]. RG can be considered as a method of treating physical problems with a large number of degrees of freedom, not taking all those at once, but treating them successively scale-by-scale [3, 4]. This resulted in an elegant theory of critical phenomena and was later generalized to many other stochastic systems [5]. Same idea of separating the fluctuations of different scales has been implemented, basically in experimental data processing, in a quite different way: using wavelets. This was first done in geophysics [6, 7], and then spread over all possible data, from face recognition and medical imaging to high energy physics and cosmology [8]. The only interference of the RG and the wavelet method seems to be the the lattice regularization in quantum field theory, which can be performed either by standard lattice methods, or by using the discrete wavelet basis [9, 11]. The connections between these two seemingly different methods ∗ **
The text was submitted by the author in English. E-mail: [email protected]
are still missing. The text below is an endeavor to fill this gap partially. 2. DIVERGENCES IN QUANTUM FIELD THEORY The fundamental problem of the quantum field theory is the problem of divergences of Feynman graphs. The infinities appearing in perturbation expansion of Feynman integrals are treated by different regularization methods, from maximal momentum cutoff and Pauli–Villars regularization, to expansion, and renormalization group methods, see e.g. [12] for a review. We restrict ourselves with a simple example of the scalar φ4 field theory in Rd , which illustrates all main problems and approaches related to the problem of divergences in the quantum field theory, see e.g. [12, 13]. The Euclidean scalar field theory
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