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The c and a-theorems and the Local Renormalisation Group

Graham Shore

SpringerBriefs in Physics

Swansea University Author: Graham Shore

DOI (Published version): 10.1007/978-3-319-54000-9

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 o...

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Published in: SpringerBriefs in Physics
ISBN: 9783319539997 9783319540009
Published: Cham, Switzerland Springer International Publishing 2017
Online Access: http://www.springer.com/gp/book/9783319539997
URI: https://cronfa.swan.ac.uk/Record/cronfa26479
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spelling 2022-01-07T13:14:20.0204386 v2 26479 2016-02-20 The c and a-theorems and the Local Renormalisation Group 28a24f55687c82d6f3ee378ead3cf234 Graham Shore Graham Shore true false 2016-02-20 FGSEN 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. Book SpringerBriefs in Physics Springer International Publishing Cham, Switzerland 9783319539997 9783319540009 Mathematical physics, Quantum Field Theories, String Theory, Mathematical Physics, Mathematical Methods in Physics, Physics, Renormalization group, SCIENCE Energy, Mechanics General, Physics, General, Science Quantum Theory, Local Weyl scaling, Position-dependent couplings in QFT, Zamolodchikov c-theorem, Global symmetry and limit cycles, Weyl consistency condition. 5 5 2017 2017-05-05 10.1007/978-3-319-54000-9 http://www.springer.com/gp/book/9783319539997 arXiv version: https://arxiv.org/abs/1601.06662 COLLEGE NANME Science and Engineering - Faculty COLLEGE CODE FGSEN Swansea University 2022-01-07T13:14:20.0204386 2016-02-20T16:26:42.6016565 Faculty of Science and Engineering School of Biosciences, Geography and Physics - Physics Graham Shore 1 0026479-01032017212824.pdf RIS-ctheorem-Springerv2.pdf 2017-03-01T21:28:24.7600000 Output 723155 application/pdf Accepted Manuscript true 2017-03-01T00:00:00.0000000 false eng
title The c and a-theorems and the Local Renormalisation Group
spellingShingle The c and a-theorems and the Local Renormalisation Group
Graham Shore
title_short The c and a-theorems and the Local Renormalisation Group
title_full The c and a-theorems and the Local Renormalisation Group
title_fullStr The c and a-theorems and the Local Renormalisation Group
title_full_unstemmed The c and a-theorems and the Local Renormalisation Group
title_sort The c and a-theorems and the Local Renormalisation Group
author_id_str_mv 28a24f55687c82d6f3ee378ead3cf234
author_id_fullname_str_mv 28a24f55687c82d6f3ee378ead3cf234_***_Graham Shore
author Graham Shore
author2 Graham Shore
format Book
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isbn 9783319539997
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doi_str_mv 10.1007/978-3-319-54000-9
publisher Springer International Publishing
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Biosciences, Geography and Physics - Physics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Biosciences, Geography and Physics - Physics
url http://www.springer.com/gp/book/9783319539997
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description 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.
published_date 2017-05-05T03:31:46Z
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