Loop transfer matrix and loop quantum mechanics

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.