The definition of the renormalized parameters implicitly or explicitly requires the introduction of a new distance scale known as renormalization scale. This procedure is called renormalization. The renormalized theory resulting from this process is generically no longer related to a Gaussian field distribution, which in this example corresponds to a non-interesting free field theory.

The relation between the renormalized parameters in different parametrizations of the same renormalized theory corresponding to different values of the renormalization scale was historically called renormalization group Stueckelberg and Peterman and the linear partial differential equations for the correlators of the renormalized theory, resulting from variations of the parameters under an infinitesimal change of the renormalization scale, were called differential renormalization group equations.

Landau's theory of critical phenomena in macroscopic continuous phase transitions and thus with divergent correlation length at the critical temperature or mean field theory , are interpreted in modern language as corresponding to Gaussian distribution or perturbed Gaussian distribution "quasi-Gaussian distribution". However, for systems with short-range interactions, in space dimensions two and three, quasi-Gaussian models do not describe correctly the universal properties of phase transitions at large distance near the critical temperature.

Following Kadanoff's idea, Wilson introduced a more general notion of renormalization group. In a continuum space but not on the lattice , infinitesimal changes of scales are possible and lead to differential equations for the effective Hamiltonian, see 1. The basin of attraction of a given fixed point in the space of Hamiltonians is called a universality class. It can be verified that the Gaussian theory provides the simplest example of a fixed point, thus called Gaussian fixed point. It is now understood that both the field theory renormalization group and Wilson's general formulation are related.

The field renormalization group is the asymptotic form of the general renormalization group in a neighbourhood of the Gaussian fixed point. Even if the initial statistical model is defined in terms of random variables associated to the sites of a space lattice, and taking only a finite set of values like, e. To construct an RG flow, the basic idea is to integrate in the field integral recursively over short distance degrees of freedom.

These assumptions summarize the characteristic properties of these equations and are at the origin of many renormalization group properties. However, with the Hamiltonian 2 , the Gaussian model 6 has a problem: too singular fields contribute to the corresponding field integral in such a way that correlation functions at coinciding points are not defined. Therefore, it is necessary to modify the Gaussian model at short distance to restrict the field integration to more regular fields, continuous to define powers of the field, satisfying differentiability conditions to define expectation values of the field and its derivatives taken at the same point.

The transformation 3 generates the linearized RG flow at the Gaussian fixed point. If what follows, we assume that initially the statistical system is very close to the Gaussian fixed point. The RG flow is then first governed by the linear flow. Therefore, we implement first the corresponding RG transformation. This defines the critical domain. To determine the large distance behaviour of correlation functions, it becomes necessary to construct a general renormalization group. This leads to functional equations that, in general, cannot be solved analytically. To discuss dimensional continuation, it is convenient to introduce the Fourier representation of correlation functions.

A contribution to perturbation theory represented graphically by a Feynman diagram takes, in Fourier representation, the form of a product of propagators integrated over a subset of momenta. While for the theory of critical phenomena, dimensional continuation is sufficient since it allows exploring the neighbourhood of dimension four, for practical calculations restricted to leading order at large distance, an additional step is useful.

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The resulting perturbative contributions become meromorphic functions with poles, in particular at dimensions at which large momentum, and low momentum in the critical theory, divergences appear. This method of regularizing large momentum divergences is called dimensional regularization and is extensively used in quantum field theory.

The perturbative renormalization group, as it has been developed in the framework of the perturbative expansion of quantum field theory, relies on the so-called renormalization theory. These equations can be further generalized to deal with an external field a magnetic field for magnetic systems and the corresponding induced field expectation value magnetization for magnetic systems. An RG equation for the equation of state follows. Equation 10 is asymptotically symmetric between initial and renormalized correlations.

RG equations can be solved by the method of characteristics. For practical RG calculations, it is more convenient to deal with vertex functions than connected correlation functions.

The generating functional of vertex functions is obtained from the generating functional of connected functions by a Legendre transformation that generalizes the relation between free energy and thermodynamic potential. Worcester, D. Lippert, J. Acta , 8 Google Scholar. Gaber, B. Yellin, N. Pink, D. Lee, A.

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