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The c and a-Theorems and the local renormalisation group

Author: Graham Shore
Publisher: Cham, Switzerland : Springer, 2017.
Series: SpringerBriefs in physics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
The Zamolodchikov c-theorem has led to important new insights in the understanding of the Renormalisation Group (RG) and the geometry of the space of QFTs. The present primer introduces and reviews the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local RG (LRG). The idea of renormalisation with position-dependent couplings, running under local Weyl scaling,  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Shore, Graham.
C and a-Theorems and the local renormalisation group.
Cham, Switzerland : Springer, 2017
(OCoLC)969829800
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Graham Shore
ISBN: 9783319540009 3319540009
OCLC Number: 981125728
Description: 1 online resource (vii, 102 pages) : illustrations (some color).
Contents: Abstract; 1 Introduction; 1.1 Scale, Conformal and Weyl Invariance; 1.2 Renormalisation Group Flows and the Zamolodchikovc-Theorem; 1.3 Local Renormalisation Group; 2 Renormalisation and the Conformal Anomaly; 2.1 Renormalisation and Local Couplings; 2.2 Trace Anomaly; 2.3 Renormalisation of Two-Point Green Functions; 3 The Local Renormalisation Group and WeylConsistency Conditions; 3.1 Diffeomorphism and Anomalous Weyl Ward Identities; 3.2 Renormalisation Group and Green Functions; 3.3 Weyl Consistency Conditions; 3.4 Local RGE and Weyl Consistency Conditions. 4 c-Theorem in Two Dimensions4.1 c-Theorem and the Spectral Function ; 4.2 c-Theorem and Renormalisation Group Flow; 4.3 c-Theorem in Position Space; 4.4 c-Theorem and Dispersion Relations; 5 Local RGE and Weyl Consistency Conditions in Four Dimensions; 5.1 Local RGE, Trace Anomaly and Weyl Consistency Conditions; 5.2 Renormalisation Group and Anomalous Ward Identities for Green Functions; 5.3 Weyl Consistency Conditions; 6 c, b and a-Theorems in Four Dimensions; 6.1 Spectral Functions and the c and b 'Theorems'; 6.2 Renormalisation Group Flow for the c and b 'Theorems' 6.3 Weyl Consistency Conditions and the a-Theorem7 Local RGE and Maximally Symmetric Spaces; 7.1 Renormalisation Group, Weyl Consistency Conditions and the a-Theorem; 7.2 RG Flow of on Maximally Symmetric Spaces; 7.3 Zamolodchikov Functions and the Search for a C-Theorem on the Sphere; 7.3.1 Geometry of Constant Curvature Spaces; 7.3.2 Two-Point Green Functions and Conservation Identities; 7.3.3 The Zamolodchikov C-Function on the Sphere; 7.3.4 Dimensional Analysis and RG Flow of the C-Function; 8 The Weak a-Theorem and 4-Point Green Functions; 9 Global Symmetries and Limit Cycles. 9.1 Global Symmetries and Weyl Consistency Conditions9.2 Limit Cycles; 10 Summary and Outlook; A Counterterms for Renormalised Green Functions; B RGEs for Two-Point Green Functions; C Zamolodchikov Functions in n Dimensions; References.
Series Title: SpringerBriefs in physics.
Responsibility: Graham Shore.

Abstract:

The Zamolodchikov c-theorem has led to important new insights in the understanding of the Renormalisation Group (RG) and the geometry of the space of QFTs. The present primer introduces and reviews the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local RG (LRG). The idea of renormalisation with position-dependent couplings, running under local Weyl scaling, is traced from its early realisations to the elegant modern formalism of the LRG. The key rôle of the associated Weyl consistency conditions in establishing RG flow equations for the coefficients of the trace anomaly in curved spacetime, and their relation to the c-theorem and four-dimensional a-theorem, is explained in detail. A number of different derivations of the c-theorem in two dimensions are presented and subsequently generalised to four dimensions. The obstructions to establishing monotonic C-functions related to the trace anomaly coefficients in four dimensions are explained. The possibility of deriving an a-theorem for the coefficient of the Euler-Gauss-Bonnet density is explored, initially by formulating the QFT on maximally symmetric spaces. Then the formulation of the weak a-theorem using a dispersion relation for four-point functions is presented. Finally, the application of the LRG to the issue of limit cycles in theories with a global symmetry is described, shedding new light on the geometry of the space of couplings in QFT.

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