Aspects of the map from exact RG to holographic RG in AdS and dS

Abstract

In earlier work, the evolution operator for the exact RG equation was mapped to a field theory in Euclidean AdS. This gives a simple way of understanding AdS/CFT. We explore aspects of this map by studying a simple example of a Schrödinger equation for a free particle with time-dependent mass. This is an analytic continuation of an ERG like equation. We show for instance that it can be mapped to a harmonic oscillator. We show that the same techniques can lead to an understanding of dS/CFT too.

Hamiltonian and Wheeler-DeWitt Equation

“…Since this note was originally written more than 10 years ago, there have been a number of recent studies [33][34], [35], [36], [37], [38], [39], [40] [41], that need to be mentioned. Some of them [33] [34] view the holographic setting in a similar way by splitting to UV part and IR part, although they are based on standard quantization and consider the gravity action after it is integrated over quantum fluctuation.…”

“…Some of them [33] [34] view the holographic setting in a similar way by splitting to UV part and IR part, although they are based on standard quantization and consider the gravity action after it is integrated over quantum fluctuation. A series of studies [36], [37], [38], [39], [40], [41] start from the exact RG equation and express transition amplitudes between wave functions of different scale as Dx(t)e −S bulk , where x(t) denotes fields at different scale t, and regard S bulk as the bulk action. They also mention that some of their derivation corresponds to alternative quantization [37], and the relationship between Holographic exact RG and Stochastic Quantization [40].…”

“…A series of studies [36], [37], [38], [39], [40], [41] start from the exact RG equation and express transition amplitudes between wave functions of different scale as Dx(t)e −S bulk , where x(t) denotes fields at different scale t, and regard S bulk as the bulk action. They also mention that some of their derivation corresponds to alternative quantization [37], and the relationship between Holographic exact RG and Stochastic Quantization [40]. The goal of relating the exact RG to the bulk quantum gravity is the same, although they are working on deriving the bulk theory without using explicitly holographic setting.…”