Loop Transfer Matrix and Loop Quantum Mechanics

TL;DR

This paper generalizes the loop transfer matrix to include self-intersection coupling, introduces a loop Fourier transform for diagonalization, and develops a loop quantum mechanics framework connecting transfer matrices with statistical system correlations.

Contribution

It extends the loop transfer matrix formalism to nonzero self-intersection coupling and establishes a loop Fourier transform for diagonalization, linking quantum mechanics and statistical correlations.

Findings

Diagonalization of 3D loop transfer matrix via loop Fourier transform.

Eigenvalues expressed through 2D statistical system correlation functions.

Extension to 4D transfer matrix describing membrane propagation.

Abstract

We extend the previous construction of loop transfer matrix to the case of nonzero self-intersection coupling constant $\kappa$. The loop generalization of Fourier transformation allows to diagonalize transfer matrices depending on symmetric difference of loops and express all eigenvalues of $3d$ loop transfer matrix through the correlation functions of the corresponding 2d statistical system. The loop Fourier transformation allows to carry out analogy with quantum mechanics of point particles, to introduce conjugate loop momentum P and to define loop quantum mechanics. We also consider transfer matrix on $4d$ lattice which describes propagation of memebranes. This transfer matrix can also be diagonalized by using generalized Fourier transformation, and all its eigenvalues are equal to the correlation functions of the corresponding $3d$ statistical system.