Heat Equation from Exact Renormalization Group Equation (ERGE) at Local Potential Approximation (LPA)

TL;DR

This paper demonstrates that applying the Local Potential Approximation to Polchinski's ERG flow equations simplifies them into heat equations, revealing logarithmic interactions at fixed points for both Bosonic and Fermionic fields.

Contribution

It shows that under LPA, ERG flow equations reduce to heat equations, providing a new perspective on fixed point interactions in quantum field theories.

Findings

ERG flow equations become heat equations under LPA

Logarithmic interactions emerge at fixed points

Applicable to both Bosonic and Fermionic fields

Abstract

By simply applying the Local Potential Approximation (LPA) on the Polchinski's Exact Renormalization Group (ERG) flow equation for single Bosonic and spinless Fermionic fields, and initially considering only the coarse-graining (blocking) aspect of Wilson's Renormalization Group program. Within the LPA limit the Polchinski's ERG flow equation simplifies into a heat differential equation for both Bosonic and Fermionic fields. Solving the differential equations leads to logarithmic interactions (logarithmic vertex function) in both Bosonic and Fermionic fields at their fixed points.