Exact RG flow equations and quantum gravity
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Springer
Received: August 29, Revised: January 18, Accepted: March 8, Published: March 21,
2017 2018 2018 2018
S.P. de Alwis Physics Department, University of Colorado, Boulder, CO 80309 U.S.A.
E-mail: [email protected] Abstract: We discuss the different forms of the functional RG equation and their relation to each other. In particular we suggest a generalized background field version that is close in spirit to the Polchinski equation as an alternative to the Wetterich equation to study Weinberg’s asymptotic safety program for defining quantum gravity, and argue that the former is better suited for this purpose. Using the heat kernel expansion and proper time regularization we find evidence in support of this program in agreement with previous work. Keywords: Models of Quantum Gravity, Renormalization Group ArXiv ePrint: 1707.09298
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)118
JHEP03(2018)118
Exact RG flow equations and quantum gravity
Contents 1 Introduction
1
2 Review and comments
4 8 10
4 The beta function equations 4.1 General considerations 4.2 The heat kernel expansion and the beta functions
12 12 14
5 Comments and conclusions
20
1
Introduction
Following the seminal work by Ken Wilson [1] many authors have discussed the formulation and consequences of continuum exact renormalization group (RG) equations for quantum field theory (QFT). Amongst these the most popular have been those of Polchinski [2] and Wetterich [3](see also [4–6]).1 The former is a differential equation in RG “time” ln Λ for the Wilsonian effective action IΛ [φ] obtained by integrating out the ultra-violet (UV) degrees of freedom down to some scale Λ. The latter is a differential equation for the so-called average effective action, obtained from the functional integral for the quantum effective action Γ[φc ] by cutting off the integral over the eigenmodes of the kinetic operator of the QFT at some infra-red (IR) scale k. This produces a functional Γk [φc ] such that its k → 0 limit gives back Γ[φc ]. It is claimed that this equation defines the evolution all the way from an “initial” UV action all the way down to the deep IR k → 0. The standard model and Einstein’s theory of gravity are usually regarded as effective field theories (EFT’s). The UV completion of these EFT’s is one of the main motivations for string theory. In the latter case it is expected that these EFT’s are valid only up to the string scale, which is typically an order of magnitude or so below the four dimensional Planck scale.2 It is commonly believed that above such a UV scale one needs to replace the EFT by string theory, with the parameters of the EFT being determined by the fundamental theory through matching conditions in the transitional region defined by the UV cutoff Λ. 1
For reviews and references to recent work see for example [7–10]. For applications to the asymptotic safety program see [11–13] and references therein. 2 If we have a large volume compactification the scale at which the EFT
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