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.
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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. 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. From the viewpoint of Feynman diagrams, vertex functions are one-line irreducible. The structure of fixed points may then be more complicate.
Extracting precise numbers from the known terms of the series requires a summation method. Callan-Symanzik equation , Dimensional Renormalization , Gauge theories , Ising model , Quantum electrodynamics , Renormalization group , Renormalization group for non-relativistic fermions , Scaling laws.
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Riccardo Guida. Sponsored by: Dr. Namespaces Page Discussion. Views Read View source View history. Contents 1 The renormalization group: Introduction 2 Statistical field theory 3 The renormalization group: General formulation 4 The Gaussian fixed point 4.
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In chapters , renormalization properties of theories with symmetries are studied and specific applications to particle physics are emphasized. Chapters are devoted to critical phenomena. Chapters describe the role of instantons in quantum mechanics and field theory.
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