Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length. We obtain explicit expressions for this fractional Laplacian matrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional derivative) on the finite periodic string. In this approach we introduce two material parameters, the particle mass μ and a frequency Ωα. The requirement of finiteness of the the total mass and total elastic energy in the continuum limit (lattice constant h → 0) leads to scaling relations for the two parameters, namely μ ∼ h and View the MathML source. The present approach can be generalized to define lattice fractional calculus on periodic lattices in full analogy to the usual ‘continuous’ fractional calculus

Fractional Derivative Regularization in QFT

In this paper, we propose new regularization, where integer-order differential operators are replaced by fractional-order operators. Regularization for quantum field theories based on application of the Riesz fractional derivatives of non-integer orders is suggested. The regularized loop integrals depend on parameter that is the order alpha>0 of the fractional derivative. The regularization procedure is demonstrated for scalar massless fields in phi^4-theory on n-dimensional pseudo-Euclidean space-time.Comment: 15 pages, LaTe

Fractional Path Integrals and its degeneration to Dimensional Regularization

In this work we study particles propagate in a fractional path and use fractional derivatives to extend the dynamic dimension of Quantum Field Theory. we construct the Lagrangian of fractional scalar, vector and spinor fields to obtain their propagators by path integral. Then we compute the typical tree level and one loop diagrams which correspond to QED cases. The calculations show the dimension dependence of amplitudes. Additionally, in one loop calculation we obtain results which are consistent with dimensional regularization as the dimension approaches to the Standard Model value. Therefore, the fractional Path Integrals can be regarded as an equivalent theoretical representation for regularizing the divergence in the normal Quantum Field Theory. We also derive the equation of motion for scalar, vector and spinor particles propagate in fractal paths and discuss the corresponding gauge symmetry, where we find a special non-local gauge transformation.Comment: 17page

A fractional generalization of the classical lattice dynamics approach

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n-dimensional periodic and infinite lattice in n=1,2,3 dimensions. The present model which is based on Hamilton's variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels. The fractional lattice Laplacian matrix contains for α=2 the classical local lattice approach with well known continuum limit of classic local standard elasticity, and for other integer powers to gradient elasticity. We also present a generalization of the fractional Laplacian matrix to n-dimensional cubic periodic (nD tori) and infinite lattices. We show that in the continuum limit the fractional Laplacian matrix yields the well-known kernel of the Riesz fractional Laplacian derivative being the kernel of the fractional power of Laplacian operator. In this way we demonstrate the interlink of the fractional lattice approach with existing continuous fractional calculus. The developed approach appears to be useful to analyze fractional random walks on lattices as well as fractional wave propagation phenomena in lattices

Fractional random walk lattice dynamics

We analyze time-discrete and time-continuous 'fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in n  =  1, 2, 3,.. dimensions. The fractional random walk dynamics is governed by a master equation involving fractional powers of Laplacian matrices ${{L}^{\frac{\alpha}{2}}}$ where $\alpha =2$ recovers the normal walk. First we demonstrate that the interval $0<\alpha \leqslant 2$ is admissible for the fractional random walk. We derive analytical expressions for the transition matrix of the fractional random walk and closely related the average return probabilities. We further obtain the fundamental matrix ${{Z}^{(\alpha )}}$ , and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix ${{Z}^{(\alpha )}}$ relates fractional random walks with normal random walks. We show that the matrix elements of the transition matrix of the fractional random walk exihibit for large cubic n-dimensional lattices a power law decay of an n-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the n-dimensional space, the transition matrix and return probabilities of the fractional random walk are dominated for large times t by slowly relaxing long-wave modes leading to a characteristic ${{t}^{-\frac{n}{\alpha}}}$ -decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen