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
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.
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