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arXiv:2211.05534 · Progress of Theoretical and Experimental Physics

Exact renormalization group for wave functionals

Takaaki Kuwahara, Gota Tanaka, Asato Tsuchiya, Kazushi Yamashiro

Published Feb 23, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract Motivated by the construction of continuum tensor networks for interacting field theories, which are relevant in understanding the emergence of space-time in the gauge/gravity correspondence, we derive a non-perturbative functional differential equation for wave functionals in scalar field theories from the exact renormalization group equation. We check the validity of the equation using the perturbation theory. We calculate the wave functional up to the first-order perturbation and verify that it satisfies the equation.

DOI
10.1093/ptep/ptad029

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Abstract The known approximation schemes for the solution of the Wetterich exact renormalization group (RG) equation are critically reconsidered, and a new truncation scheme is proposed. In particular, the equations of the derivative expansion up to the ∂ 2 order for a scalar model are derived in a suitable form, clarifying the role of the off-diagonal terms in the matrix of functional derivatives. The natural domain of validity of the derivative expansion appears to be limited to small values of q / k in the calculation of the critical two-point correlation function, depending on the wave-vector magnitude q and the infrared cut-off scale k . The new approximation scheme has the advantage to be valid for any q / k , and, therefore, it can be auspicious in many current and potential applications of the celebrated Wetterich equation and similar models. Contrary to the derivative expansion, derivatives are not truncated at a finite order in the new scheme. The RG flow equations in the first approximation of this new scheme are derived and approximately solved as an example. It is shown that the derivative expansion up to the ∂ 2 order is just the small- q approximation of our new equations at the first order of truncation.