Interaction Hierarchy. Gonihedric String and Quantum Gravity

TL;DR

This paper demonstrates that Regge gravity and gonihedric string theories can be represented as superpositions of simpler random surface and path theories, revealing new connections and proposing an alternative linear action for quantum gravity.

Contribution

It introduces a novel representation of Regge gravity and gonihedric string as superpositions of simpler theories and proposes an alternative linear action for high-dimensional quantum gravity.

Findings

Partition functions are products of Feynman path integrals with specific actions.

Interaction depends on overlapping sizes of paths or surfaces.

Constructed a spin system with the same partition function as quantum gravity.

Abstract

We have found that the Regge gravity \cite{regge,sorkin}, can be represented as a $superposition$ of less complicated theory of random surfaces with $Euler~character$ as an action. This extends to Regge gravity our previous result \cite{savvidy}, which allows to represent the gonihedric string \cite{savvidy1} as a superposition of less complicated theory of random paths with $curvature$ action. We propose also an alternative linear action $A(M_{4})$ for the four and high dimensional quantum gravity. From these representations it follows that the corresponding partition functions are equal to the product of Feynman path integrals evaluated on time slices with curvature and length action for the gonihedric string and with Euler character and gonihedric action for the Regge gravity. In both cases the interaction is proportional to the overlapping sizes of the paths or surfaces on the neighboring time slices. On the lattice we constructed spin system with local interaction, which have the same partition function as the quantum gravity. The scaling limit is discussed.