Abstract
The \betaβ-functions describe how couplings run under the renormalization group flow in field theories. In general, all couplings that respect the symmetry and locality are generated under the renormalization group flow, and the exact renormalization group flow is characterized by the \betaβ-functions defined in the infinite dimensional space of couplings. In this paper, we show that the renormalization group flow is highly constrained so that the \betaβ-functions defined in a measure zero subspace of couplings completely determine the \betaβ-functions in the entire space of couplings. We provide a quantum renormalization group-based algorithm for reconstructing the full \betaβ-functions from the \betaβ-functions defined in the subspace. As examples, we derive the full \betaβ-functions for the O(N)O(N) vector model and the O_L(N) \times O_R(N)OL(N)×OR(N) matrix model entirely from the \betaβ-functions defined in the subspace of single-trace couplings.
Highlights
The β-functions describe how couplings run under the renormalization group flow in field theories
Being an operator that acts on wavefunctions defined in the space of single-trace sources, Hλ generates the quantum evolution of the state associated with the renormalization group (RG) flow
In quantum RG, the conventional RG flow in the space of couplings is replaced with a quantum evolution of a wavefunction defined in the subspace of single-trace couplings, where fluctuations of the dynamical single-trace couplings encode the information about all multi-trace couplings
Summary
One of the greatest advances in modern theoretical physics is the invention of the renormalization group (RG) [1,2,3,4,5,6,7,8,9,10,11,12]. The theory includes dynamical gravity because the coupling functions for the single-trace energy-momentum tensor is nothing but a metric that is promoted to a dynamical variable in quantum RG [16] For this reason, quantum RG provides a natural framework for the AdS/CFT correspondence [18,19,20] in which the extra dimension in the bulk is interpreted as the RG scale [21,22,23,24,25,26,27,28].1. The bulk theory that governs the quantum RG flow is entirely fixed by the β-functions defined in the subspace of single-trace couplings [15, 16].
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