Fractional Quantum Field Theory: From Lattice to Continuum

Vasily E Tarasov

doi:10.1155/2014/957863

Abstract

An approach to formulate fractional field theories on unbounded lattice space-time is suggested. A fractional-order analog of the lattice quantum field theories is considered. Lattice analogs of the fractional-order 4-dimensional differential operators are proposed. We prove that continuum limit of the suggested lattice field theory gives a fractional field theory for the continuum 4-dimensional space-time. The fractional field equations, which are derived from equations for lattice space-time with long-range properties of power-law type, contain the Riesz type derivatives on noninteger orders with respect to space-time coordinates.

Highlights

Fractional calculus and fractional differential equations [1, 2] have a wide application in mechanics and physics
The spatial fractionalorder derivatives have been actively used in the spacefractional quantum mechanics suggested in [3, 4], the quantization of fractional derivatives [5], the fractional Heisenberg and quantum Markovian equations [6, 7], the fractional theory of open quantum systems [8, 9], the quantum field theory and gravity for fractional space-time [10, 11], and the fractional quantum field theory at positive temperature [12, 13]
A connection between the dynamics of lattice system with long-range properties and the fractional continuum equations is proved by using the transform operation [18,19,20] and it has been applied for the media with spatial dispersion law [21, 22], for the fields in fractional nonlocal materials [23, 24], for fractional statistical mechanics [25], and for nonlinear classical fields [26]

Summary

Introduction

Fractional calculus and fractional differential equations [1, 2] have a wide application in mechanics and physics. We can consider space-time fractional differential equations in the quantum field theory. The fractional Laplace and d’Alembert operators of the by Riesz type are a base in the construction of the fractional field theory in multidimensional spaces As it was shown in [18,19,20], the continuum equations with fractional derivatives of the Riesz type can be directly connected to lattice models with long-range properties. The first step of the procedure is regularization that consists in introducing a space-time lattice This regularization allows us to give an exact definition of the path integral since the lattice has the denumerable number of degrees of freedom.

Fractional Field Theory on Continuum Space-Time

Fractional Field Theory on Lattice Space-Time

Continuum Fractional Field Theory from Lattice Theory

Conclusion