Research Article Fractional Quantum Field Theory: from Lattice to Continuum

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Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 957863, 14 pages http://dx.doi.org/10.1155/2014/957863

Research Article Fractional Quantum Field Theory: From Lattice to Continuum

Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

Correspondence should be addressed to Vasily E. Tarasov; [email protected]

Received 23 August 2014; Accepted 8 October 2014; Published 30 October 2014

Academic Editor: Elias C. Vagenas

Copyright © 2014 Vasily E. Tarasov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

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.

1. Introduction of the Riesz type can be directly connected to lattice models with long-range properties. A connection between the Fractional calculus and fractional differential equations1 [ , dynamics of lattice system with long-range properties and 2] have a wide application in mechanics and physics. The the fractional continuum equations is proved by using the theory of integrodifferential equations of noninteger orders transform operation [18–20] and it has been applied for the is powerful tool to describe the dynamics of systems and media with spatial dispersion law [21, 22], for the fields in processes with power-law nonlocality, long-range memory, fractional nonlocal materials [23, 24], for fractional statistical and/or fractal properties. Recently the spatial fractional- mechanics [25], and for nonlinear classical fields26 [ ]. order derivatives have been actively used in the space- A characteristic feature of continuum quantum field fractional quantum mechanics suggested in [3, 4], the quanti- theories, which are used in high-energy physics, is ultraviolet zation of fractional derivatives [5], the fractional Heisenberg divergences. The divergences arise in momentum (wave- and quantum Markovian equations [6, 7], the fractional the- vector) space from modes of very high wave number, that ory of open quantum systems [8, 9], the quantum field theory is, the structure of the field theories at very short distances. and gravity for fractional space-time [10, 11], and the fractional quantum field theory at positive temperature [12, 13]. In the narrow class of quantum theories, which are called Fractional calculus allows us to take into account fractional “renormalizable,”the divergences can be removed by a singu- power-law nonlocality of continuously distributed systems. lar redefinition of the parameters of the theory. This process Using the fractional calculus, we can consider space-time is called the renormalization [27], and it defines a quantum fractional differential equations in the quantum field theory. field theory as a nontrivial limit of theory with an ultraviolet The fractional-order Laplace and d’Alembert operators have cut-off. The renormalization requires the regularization of been suggested by Riesz in [14]in1949forthefirsttime. the path integrals in momentum space. These regularized Then noninteger powers of d’Alembertian are considered in integrals depend on parameters such as the momentum cut- different works (for example, see Section 28 in1 [ ]and[15– off, the Pauli-Villars masses, and the dimensional regulariza- 17]). The fractional Laplace and d’Alembert operators of the tion parameter, which are used in the correspondent regular- by Riesz type are a base in the construction of the fractional ization procedure. This regularized integration is ultraviolet field theory in multidimensional spaces. As it was shown in finite. In some sense, we can say that the regularization pro- [18–20], the continuum equations with fractional derivatives cedure consists in the introduction of a momentum cut-off. 2 Advances in High Energy Physics

4 In quantum field theory, the path integral approach is pseudo-Euclidean space-time R1,3. Let us consider the classi- very important to describe processes in high energy physics cal field equation

[28]. The path integrals are well-defined for systems with a 2 denumerable number of degrees of freedom. In field theory, (◻+𝑀 )𝜑(𝑥) =𝐽(𝑥) , (1) we are dealing with the case of an innumerable number of ◻ 𝜑(𝑥) degrees of freedom, labeled by the space-time coordinates at where is the d’Alembert operator, is a real field, and 𝑥∈R4 𝑥 least. To give the path integrals a precise meaning, we can dis- 1,3 is the space-time vector with components 𝜇,where cretize space and time; that is, we can introduce a space-time 𝜇 = 0, 1, 2,. 3 This field equation follows from stationary lattice. The introduction of a lattice space-time corresponds action principle, 𝛿𝑆[𝜑],wheretheaction =0 𝑆[𝜑] has the to a special form of regularization of the path integrals. In form the lattice field theory, the momentum space integrals will be 1 𝑆 [𝜑] =− ∫ 𝑑4𝑥𝜑 (𝑥) (◻+𝑀2) 𝜑 (𝑥) . cut off at a momentum of the order of the inverse lattice con- 2 (2) stants. The lattice regularization can be considered as a natu- ral introduction of a momentum cut-off. The lattice renorma- In the quantum theory the generalized coordinates 𝜑(𝑥) ̇ lization procedure can be carried out for path integrals in and momenta 𝜑(𝑥)̇ become the operators, Φ(𝑥) and Φ(𝑥), momentum space. The first step of the procedure is regu- that satisfy the canonical commutation relations. The Green larization that consists in introducing a space-time lattice. functions are This regularization allows us to give an exact definition of 󵄨 󵄨 𝐺 (𝑥 ,...,𝑥 ) = ⟨Ω 󵄨𝑇 {Φ (𝑥 ) ⋅⋅⋅Φ(𝑥 )}󵄨 Ω⟩ , the path integral since the lattice has the denumerable num- 𝑠 1 𝑠 󵄨 1 𝑠 󵄨 (3) ber of degrees of freedom. Moreover the existence of the where |Ω⟩ denotes the ground state “physical vacuum” of momentum cut-off is not surprising in the lattice field the fields and 𝑇{ } denotes the time-ordered product of the theories. In the expression of the Fourier integral for lattice operators Φ(𝑥). These Green functions have a path integral fields, the momentum integration with respect to wave-vector representation [28]intheform components 𝑘𝜇 (𝜇 = 1, 2, 3,) 4 is restricted by the Brillouin zone 𝑘𝜇 ∈[−𝜋/𝑎𝜇,𝜋/𝑎𝜇],where𝑎𝜇 arethelatticeconstants. ∫𝐷𝜑(𝜑(𝑥 )⋅⋅⋅𝜑(𝑥 )) 𝑒𝑖𝑆[𝜑] 𝐺 (𝑥 ,...,𝑥 )= 1 𝑠 , The second step of the renormalization is a continualization 𝑠 1 𝑠 𝑖𝑆[𝜑] (4) procedure that removes the lattice structure by a continuum ∫𝐷𝜑𝑒 limit, where the lattice constants 𝑎𝜇 tend to zero. In this step of the renormalization process the momentum cut-off where ∫𝐷𝜑 is the sum over all possible configurations of is removed by the continuum limit. the field 𝜑(𝑥). The effects arising from quantum fluctuations At the present time, fractional-order generalization of the aredefinedbythosecontributionstotheintegral(4)that lattice field theories has not been suggested. Lattice approach come from field configurations which are not solutions of to the fractional field theories was not previously considered. the classical field equation1 ( ) and hence do not lead to a In this paper, we propose a formulation of fractional field stationary action. theory on a lattice space-time. The suggested theory can be considered as a lattice analog of the fractional field theories, 2.2. From Pseudo-Euclidean to Euclidean Space-Time. Let us and as a fractional-order analog of the lattice quantum field consider the analytic continuation of (4) to imaginary times theories. For simplification, we consider the free scalar fields. (see Section 7.4 in [29]) such that 0 It allows us to demonstrate a number of important properties 𝑥 󳨀→ − 𝑖 𝑥 . (5) in details. The lattice analogs of the fractional-order differen- 4 tial operators are suggested. We prove that continuum limit of We will use 𝑥𝐸 to denote the Euclidean four-vector. It 4 thesuggestedlatticetheorygivesthefractionalfieldtheory allows us to consider the Euclidean space-time R instead of 4 with continuum space-time. The fractional field equations the pseudo-Euclidean space-time R1,3. contain the Riesz type derivatives on noninteger orders with The Euclidean action 𝑆𝐸[𝜑𝑐] is obtained from the action respect to space-time coordinates. In Section 2,thefractional (2) by using the three steps: field theory on continuum space-time is considered for scalar 0 0 fields and fractional-order differential operators are defined. (1) firstly, the replacement 𝑥 →−𝑖𝑥4,where𝑥 appears In Section 3, the fractional-order lattice differential oper- explicitly, ators of noninteger orders are considered and the lattice (2) the use of the real valued field 𝜑𝑐(𝑥𝐸)=𝜑𝑐(𝑥,⃗ 4 𝑥 ) fractional field theory for lattice space-time is proposed. In instead of 𝜑(𝑥,⃗ 𝑡),where𝜑𝑐(𝑥,⃗ 4 𝑥 ) is not obtained from Section 4, the lattice-continuum transformation of the lattice 𝜑(𝑡, 𝑥)⃗ by substituting 𝑥4 for 𝑡,notingthat𝜑𝑐(𝑥𝐸)= fractional theories is discussed. A short conclusion is given in 𝜑𝑐(𝑥,⃗ 4 𝑥 ) is a real field which is a function of the Section 5. Euclidean variable 𝑥𝐸, −𝑖 2. Fractional Field Theory on (3) multiplying the resultant expression by . Continuum Space-Time As a result, this leads to the expression 1 2.1. Scalar Field in Pseudo-Euclidean Space-Time. For sim- 4 2 𝑆𝐸 [𝜑𝑐]= ∫ 𝑑 𝑥𝐸𝜑𝑐 (𝑥𝐸)(−◻𝐸 +𝑀 )𝜑𝑐 (𝑥𝐸), (6) plification, we consider scalar fields in the 4-dimensional 2 Advances in High Energy Physics 3

where ◻𝐸 denotes the 4-dimensional Laplacian 2.3. Continuum Fractional Derivatives of the Riesz Type. To formulate a fractional generalization of the quantum field 4 𝜕2 ◻ = ∑ . theory, we define fractional-order derivatives with respect to 𝐸 2 (7) 𝑥 𝜇 = 1, 2, 3, 4 𝜇=1 𝜕𝑥𝐸𝜇 dimensionless Euclidean coordinates 𝜇,where . ± 𝛼 These derivatives will be denoted by D𝐶 [ 𝜇 ],where𝛼 is the To formulate a fractional generalization of the field the- order of the derivative; 𝜇 denotes the coordinate 𝑥𝜇 with ory, we should use the physically dimensionless space-time respect to which the derivative is taken, 𝐶 marks that the coordinates. It allows us to have the same physical dimensions derivative is used for continuum field theory (𝐿 will be used forallotherphysicalvaluesasintheusual(nonfractional) for lattice operators), and + and − denote the even and odd field theories with dimensionless coordinates. In this paper types of the derivatives. all the quantities will be done physically dimensionless to + 𝛼 simplify our consideration. For this aim, we scale the mass Definition 1. Continuum fractional derivatives D𝐶 [ 𝜇 ] of the 𝛼>0 parameter 𝑀,thecoordinates𝑥𝐸,andthefield𝜑𝑐 according Riesz type and noninteger order are defined by the to their physical dimension. As seen from (6) the quantities equation 𝜑 𝑀 𝑐 and have the physical dimension that is the inverse + 𝛼 1 1 𝑚 𝑥 D [ ]𝜑𝐶 (x) = ∫ (Δ 𝜑) (x) 𝑑𝑧𝜇, length, and it is obviously 𝐸 that has the dimension of length, 𝐶 𝜇 󵄨 󵄨𝛼+1 𝑧𝜇 𝑑1 (𝑚, 𝛼) R1 󵄨 󵄨 since the action is dimensionless. Therefore we can define the 󵄨𝑧𝜇󵄨 dimensionless quantities 𝑀𝐶, x, and 𝜑𝐶 by the replacement: (0<𝛼<𝑚) , −1 −1 −1 x =𝑙0 𝑥𝐸,𝑥𝜇 =𝑙0 𝑥𝐸𝜇,𝜑𝐶 =𝑙0 𝜑𝑐, (14) (8) 𝑚 𝛼 𝑀 =𝑙−1𝑀. where is the integer number that is greater than and the 𝐶 0 (Δ𝑚 𝜑)(x) 𝑚 operators 𝑧𝜇 areafinitedifference[1, 2] of order of 4 We will use x (instead of 𝑥𝐸) to denote the Euclidean four- afunction𝜑𝐶(x) with the vector step z𝜇 =𝑧𝜇e𝜇 ∈ R for the 𝑥 𝜇 = 1, 2, 3, 4 4 vector with components 𝜇,where .Asaresult, point x ∈ R . The centered difference 𝑆 [𝜑 ] this leads to the expression for the Euclidean action 𝐸 𝐶 is 𝑚 given by the equation 𝑚 𝑛 𝑚! 𝑚 (Δ 𝑧 𝜑) (x𝜇)=∑(−1) 𝜑(x −( −𝑛)𝑧𝜇e𝜇). 𝜇 𝑛! (𝑚−𝑛)! 2 1 𝑛=0 𝑆 [𝜑 ]= ∫ 𝑑4x𝜑 (x) (−◻ +𝑀2 )𝜑 (x) , 𝐸 𝐶 2 𝐶 𝐸,𝐶 𝐶 𝐶 (9) (15) The constant 𝑑1(𝑚, 𝛼) is defined by where ◻𝐸,𝐶 denotes the 4-dimensional Laplacian for dimensionless variables x of continuum space-time, such that 𝜋3/2𝐴 (𝛼) 𝑑 (𝑚, 𝛼) = 𝑚 , (16) 1 2𝛼Γ (1+𝛼/2) Γ ((1+𝛼) /2) (𝜋𝛼/2) 4 𝜕2 sin ◻ = ∑ . 𝐸,𝐶 2 (10) where 𝜇=1 𝜕𝑥𝜇 [𝑚/2] 𝑚! 𝑚 𝛼 𝐴 (𝛼) =2∑ (−1)𝑗−1 ( −𝑗) The Green functions (3), which are continued to imagi- 𝑚 (17) 𝑗=0 𝑗! (𝑚 − 𝑗)! 2 nary times, have the path integral representation for the centered difference (15). −𝑆𝐸[𝜑𝐶] ∫𝐷𝜑𝐶 (𝜑𝐶 (x1)⋅⋅⋅𝜑𝐶 (x𝑠)) 𝑒 ⟨𝜑𝐶 (x1)⋅⋅⋅𝜑𝐶 (x𝑠)⟩ = , 𝐸 −𝑆 [𝜑 ] The constants 𝑑1(𝑚, 𝛼) are different from zero for all 𝛼> ∫𝐷𝜑𝑒 𝐸 𝐶 𝑚 0 in the case of an even 𝑚 and centered difference (Δ 𝑢) (11) 𝑖 (see Theorem 26.1 in1 [ ]). Note that the integral (14)doesnot 𝑚>𝛼 where we use the notation for the Euclidean Green function depend on the choice of . Therefore, we can always 𝑚 for physically dimensionless quantities (8). choose an even number so that it is greater than parameter 𝛼 In the imaginary time formulation of quantum field the- , and we can use the centered difference15 ( )forallpositive 𝛼 ory, the Green functions look like the correlation functions real values of . Using (14), we can see that the continuum fractional used in statistical mechanics. The partition function has the + 𝛼 derivative D𝐶 [ 𝜇 ] istheRieszderivativethatactsonthefield form 1 𝜑𝐶(x) with respect to the component 𝑥𝜇 ∈ R of the vector −𝑆 [𝜑 ] 𝑍=∫ 𝐷𝜑 (x) 𝑒 𝐸 𝐶 , 4 + 𝛼 𝐶 (12) x ∈ R ;thatis,theoperatorD𝐶 [ 𝜇 ] can be considered as a partial fractional derivative of Riesz type. where the integration measure 𝐷𝜑𝐶 is formally defined by The Riesz fractional derivatives for even 𝛼=2𝑚,where 𝑚∈N, are connected with the usual partial derivative of 𝐷𝜑𝐶 = ∏𝑑𝜑𝐶 (x) . integer orders 2𝑚 by the relation x (13) 2𝑚 𝜕2𝑚𝜑 (x) D+ [ ] 𝜑 (x) = (−1)𝑚 𝐶 . Most of the variables of the system can be expressed in terms 𝐶 𝜇 𝐶 𝜕𝑥2𝑚 (18) of the partition function or its derivatives. 𝜇 4 Advances in High Energy Physics

+ 2𝑚 + 𝛼 The fractional derivatives D𝐶 [ 𝜇 ] for even orders 𝛼 are local (c) For 0<𝛼<1, the fractional operator D𝐶 [ 𝜇 ] is + 1 operators. Note that the Riesz derivative D𝐶 [ 𝜇 ] cannot be defined by the equation considered as a derivative of first order with respect to 𝑥𝜇; 𝛼 that is, D− [ ]𝜑 (x) 𝐶 𝜇 𝐶 1 𝜕𝜑 (x) D+ [ ] 𝜑 (x) ≠ 𝐶 . 𝜕 𝐶 𝜇 𝐶 (19) 𝜕𝑥𝜇 = ∫ 𝑅1−𝛼 (𝑥𝜇 −𝑧𝜇)𝜑(x +(𝑧𝜇 −𝑥𝜇) e𝜇)𝑑𝑧𝜇, 𝜕𝑥𝜇 R1 + 1 For 𝛼=1the operator D𝐶 [ 𝜇 ] is nonlocal like a “square (0<𝛼<1) , root of the Laplacian.” Note that the Riesz derivatives for odd (24) orders 𝛼=2𝑚+1,where𝑚∈N,arenonlocaloperatorsthat 2𝑚+1 2𝑚+1 cannot be considered as usual derivatives 𝜕 /𝜕𝑥 . where e𝜇 is the basis of the Cartesian coordinate system, the An important property of the Riesz fractional derivatives function 𝑅𝛼(𝑥) is the Riesz kernel that is defined by is the Fourier transform F of these operators in the form 𝛾−1 (𝛼) |𝑥|𝛼−1 𝛼 =2𝑛+1,𝑛∉ N, 𝛼 󵄨 󵄨𝛼 { 1 + 󵄨 󵄨 𝑅 𝑥 = F (D𝐶 [ ]𝜑𝐶 (x)) (k) = 󵄨𝑘𝜇󵄨 (F𝜑) (k) . (20) 𝛼 ( ) { 𝜇 󵄨 󵄨 −1 𝛼−1 {−𝛾1 (𝛼) |𝑥| ln |𝑥| 𝛼=2𝑛+1,𝑛∈N, (25) Property (20) is valid for functions 𝜑𝐶(x) from the space of infinitely differentiable functions with compact support. It and the constant 𝛾1(𝛼) has the form also holds for the Lizorkin space (see Section 8.1 in [1]). Let us consider the continuum fractional derivative − 𝛼 𝛾1 (𝛼) D𝐶 [ 𝜇 ] of the Riesz type that has the property 2𝛼𝜋1/2Γ (𝛼/2) { 𝛼 =2𝑛+1,̸ − 𝛼 󵄨 󵄨𝛼 { F (D [ ]𝜑 (x)) (k) =𝑖 (𝑘 ) 󵄨𝑘 󵄨 (F𝜑) (k) Γ ((1−𝛼) /2) 𝐶 𝜇 𝐶 sgn 𝜇 󵄨 𝜇󵄨 = { (21) { 𝛼 [𝛼−1] (−1)(1−𝛼)/22𝛼−1𝜋1/2Γ( )Γ(1+ )𝛼=2𝑛+1, (𝛼>0) , { 2 2 (26) where sgn(𝑘𝜇) is the sign function that extracts the sign of a − 𝛼 with 𝑛∈N and 𝛼∈R+. real number (𝑘𝜇).For0<𝛼<1the operator D𝐶 [ 𝜇 ] can be considered as the conjugate Riesz derivative [30]withrespect Note that the distinction between the continuum frac- to 𝑥𝜇. Therefore, the operator21 ( ) will be called a generalized − 𝛼 tional derivatives D𝐶 [ 𝜇 ] and the Riesz 4-dimensional frac- conjugate derivative of the Riesz type. |𝑘 |𝛼 |k|𝛼 D− [ 𝛼 ] tional derivative consists [2]intheuseof 𝜇 instead of . The fractional operator 𝐶 𝜇 will be defined separately For integer odd values of 𝛼,wehave for the following three cases: (a) 𝛼>1;(b)𝛼=1;(c)0<𝛼< 1 2𝑚+1 . − 2𝑚 + 1 𝑚 𝜕 𝜑𝐶 (x) D𝐶 [ ]𝜑𝐶 (x) = (−1) , + 𝛼 𝜇 𝜕𝑥2𝑚+1 Definition 2. Continuum fractional derivatives D𝐶 [ 𝜇 ] of the 𝜇 (27) Riesz type are defined by the following equations. (𝑚∈N) . + 𝛼 (a) For 𝛼>1, the fractional operator D𝐶 [ 𝜇 ] is defined Equation (27) means that the fractional derivatives by the equation − 𝛼 D𝐶 [ 𝜇 ] of the odd orders 𝛼 are local operators represented 𝛼 by the usual derivatives of integer orders. Note that the D− [ ]𝜑 (x) − 2𝑚 𝐶 𝑗 𝐶 continuum derivative D𝐶 [ 𝜇 ] with 𝑚∈N cannot be considered as a local derivative of the order 2𝑚 with respect 1 𝜕 1 𝑚 to 𝑥𝜇.For𝛼=2the generalized conjugate Riesz derivative is = ∫ 󵄨 󵄨𝛼 (Δ 𝑧 𝜑) (x) 𝑑𝑧𝜇, (22) 2 2 − 𝛼 𝑑 (𝑚, 𝛼) −1 𝜕𝑥 R1 󵄨 󵄨 𝜇 𝜕 /𝜕 𝑥 D [ 𝜇 ] 1 𝜇 󵄨𝑧𝜇󵄨 not the local derivative 𝜇. The derivatives 𝐶 for even orders 𝛼=2𝑚,where𝑚∈N,arenonlocaloperators 2𝑚 2𝑚 (1<𝛼<𝑚+1) , that cannot be considered as usual derivatives 𝜕 /𝜕𝑥𝜇 . It is important to note that the usual Leibniz rule for the 𝑚 derivative of products of two or more functions does not hold where (Δ 𝑧 𝜑)(x) isafinitedifferencethatisdefinedin(15). 𝜇 for derivatives of noninteger orders and for integer orders different from one [31]. This violation of the usual Leibniz (b) For integer values 𝛼=1,wehave rule is a characteristic property of all types of fractional 1 𝜕𝜑 (x) derivatives. D− [ ] 𝜑 (x) = 𝐶 . 𝐶 𝜇 𝐶 𝜕𝑥 (23) Equations (18)and(27) allow us to state that the partial 𝜇 derivatives of integer orders are obtained from the fractional Advances in High Energy Physics 5

± 𝛼 4 derivatives of the Riesz type D𝐶 [ 𝜇 ] for odd values 𝛼=2𝑚𝑗+ 2𝛼,± ± 2𝛼 − 𝛼 ◻𝐸,𝐶 𝜑𝐿 (x) = ∑D𝐶 [ ] 𝜑𝐶 (x) , (31) 1>0by D𝐶 [ 𝜇 ] only, and for even values 𝛼=2𝑚>0(𝑚∈ 𝜇 + 𝛼 𝜇=1 N), by D𝐶 [ 𝜇 ]. The continuum derivatives of the Riesz type D− [ 2𝑚 ] D+ [ 2𝑚+1 ] ± 𝐶 𝜇 and 𝐶 𝜇 are nonlocal differential operators of where D𝐶 are defined in Definitions 1 and 2. integer orders. In formulation of fractional analogs of classical field theo- The violation of the semigroup property (29)leadstothe ries, we need to generalize some field equations with partial fact that the operators (30)and(31) do not coincide in general. 𝛼,𝛼,− 2𝛼,+ differential equations of integer order. It is obvious that we It should be noted that the operators ◻𝐸,𝐶 and ◻𝐸,𝐶 for would like to have a fractional generalization of these integer- integer 𝛼=1gives the usual (local) 4-dimensional Laplacian order differential equations so as to obtain the original ◻𝐸 that is defined by (7); that is, equations in the limit case, when the orders of generalized ◻1,1,− =◻2,+ =◻ . derivatives become equal to initial integer values. In order 𝐸,𝐶 𝐸,𝐶 𝐸 (32) for this requirement to hold we can use the following rules ◻𝛼,𝛼,+ ◻2𝛼,− 𝛼=1 of generalization: The operators 𝐸,𝐶 and 𝐸,𝐶 for integer are non- localoperatorsofthesecondordersthatcannotbeconsidered 2𝑚 as ◻𝐸: 𝜕 2𝑚 𝛼 = (−1)𝑚D+ [ ]󳨀→(−1)𝑚D+ [ ], 2𝑚 𝐶 𝜇 𝐶 𝜇 1,1,+ 2,− 𝜕𝑥𝜇 ◻𝐸,𝐶 =◻̸ 𝐸,◻𝐸,𝐶 =◻̸ 𝐸. (33) (𝑚∈N,2𝑚−1<𝛼<2𝑚+1) , Therefore we should use only the continuum fractional 4- ◻𝛼,𝛼,− ◻2𝛼,+ (28) dimensional Laplace operators 𝐸,𝐶 or 𝐸,𝐶 in the fractional 𝜕2𝑚+1 2𝑗 + 1 𝛼 ◻𝛼,𝛼,+ ◻2𝛼,− = (−1)𝑚D− [ ] 󳨀→ (−1)𝑚D− [ ], field theory, since the operators 𝐸,𝐶 or 𝐸,𝐶 do not satisfy 2𝑚+1 𝐶 𝜇 𝐶 𝜇 𝜕𝑥𝜇 the correspondence principle for 𝛼=1. Fractional Laplace operators have been suggested by (𝑚∈N,2𝑚<𝛼<2𝑚+2) . Riesz in [14] for the first time. The fractional Laplacian 𝛼/2 (−Δ)𝐶 in the Riesz form for 4-dimensional Euclidean space- 4 In order to derive a fractional generalization of differential time R can be considered as an inverse Fourier’s integral −1 𝛼 equation with partial derivatives of integer orders, we should transform F of |k| by replace the usual derivatives of odd orders with respect to 𝑥𝜇 − 𝛼 ((−Δ)𝛼/2𝜑) (x) = F−1 (|k|𝛼 (F𝜑) (k)) , by the continuum fractional derivatives D𝐶 [ 𝜇 ] and the usual 𝐶 (34) 𝑥 derivatives of even orders with respect to 𝜇 by the continuum 4 + 𝛼 where 𝛼>0and x ∈ R . fractional derivatives of the Riesz type D𝐶 [ 𝜇 ]. Definition 4. For 𝛼>0, the fractional Laplacian of the Riesz 2.4. Continuum Fractional 4-Dimensional Laplacian. The 4- form is defined as the hypersingular integral dimensional Laplacian ◻𝐸,𝐶 is defined by (10)asanoperator 1 1 of second order for Euclidean space-time. ((−Δ)𝛼/2𝜑 ) (x) = ∫ (Δ𝑚𝜑 ) (z) 𝑑4z, 𝐶 𝐶 𝛼+4 z 𝐶 Fractional-order generalizations of the d’Alembert oper- 𝑑4 (𝑚, 𝛼) R4 |z| ator ◻ and the 𝑁-dimensional Laplacian ◻𝐸 are considered in (35) [14] and in Section 28 of [1]. 𝑚>𝛼 (Δ𝑚𝜑)(z) 𝑚 It is important to note that an action of two repeated where and z is a finite difference of order 4 fractional derivatives of order 𝛼 is not equivalent to the action of a field 𝜑𝐶(x) with a vector step z ∈ R and centered at the 4 of the fractional derivative of the double order 2𝛼, point x ∈ R : 𝑚 𝑚! ± 𝛼 ± 𝛼 ± 2𝛼 (Δ𝑚𝜑) (z) = ∑(−1)𝑗 𝜑(x −𝑗z). D𝐶 [ ] D𝐶 [ ] ≠ D𝐶 [ ], (𝛼>0) . (29) z (36) 𝜇 𝜇 𝜇 𝑗=0 𝑗! (𝑚 − 𝑗)! 𝑑 (𝑚, 𝛼) The continuum 4-dimensional Laplacian of noninteger The constant 4 is defined by order for the scalar field 𝜑𝐶(x) can be defined by two different 𝜋3𝐴 (𝛼) 𝑑 (𝑚, 𝛼) = 𝑚 , (37) equations, where the first expression contains the two lattice 4 2𝛼Γ (1+𝛼/2) Γ (2+𝛼/2) (𝜋𝛼/2) operators of order 𝛼, and the second expression contains the sin fractional derivatives of the doubled order 2𝛼. where 𝑚 𝑚! Definition 3. The continuum 4-dimensional Laplace opera- 𝐴 (𝛼) = ∑(−1)𝑗−1 𝑗𝛼. 𝛼,𝛼,± 2𝛼,± 𝑚 𝑗! (𝑚 − 𝑗)! (38) tors ◻𝐸,𝐶 and ◻𝐸,𝐶 of noninteger order 2𝛼 for the scalar field 𝑗=0 𝜑𝐶(x) are defined by the different equations: Note that the hypersingular integral (35) does not depend 4 2 onthechoiceof𝑚>𝛼.TheFouriertransformF of 𝛼,𝛼,± ± 𝛼 𝛼/2 ◻ 𝜑 (x) = ∑ (D [ ]) 𝜑 (x) , the fractional Laplacian is given by F{(−Δ)𝐶 𝜑}(k)= 𝐸,𝐶 𝐶 𝐶 𝜇 𝐶 (30) 𝛼 𝜇=1 |k| (F𝜑)(k). This equation is valid for the Lizorkin space [1] 6 Advances in High Energy Physics

∞ 4 and the space 𝐶 (R ) of infinitely differentiable functions on 3. Fractional Field Theory on 4 R with compact support. Lattice Space-Time 3.1. Lattice Space-Time. In quantum field theory, a lattice 2.5. Fractional Field Equations. The Euclidean action 𝑆𝐸[𝜑𝐶] for fractional scalar fields can be defined by the expression approach is based on lattice space-time instead of the continuum of space-time. Lattice models originally occurred 𝑆(𝛼) [𝜑 ,𝐽 ] in the condensed matter physics, where the atoms of a crystal 𝐸 𝐶 𝐶 formalattice.Theunitcellisrepresentedintermsofthe 1 lattice parameters, which are the lengths of the cell edges (a𝜇, = ∫ 𝑑4x𝜑 (x) (◻2𝛼,+ +𝑀2 )𝜑 (x) + ∫ 𝑑4x𝐽 (x) 𝜑 (x) , 2 𝐶 𝐸,𝐶 𝐶 𝐶 𝐶 𝐶 where 𝜇 = 1, 2, 3, 4) and the angles between them. (39) Let us consider an unbounded space-time lattice characterized by the noncoplanar vectors a𝜇, 𝜇 = 1, 2, 3,,thatare 4 2𝛼,+ the shortest vectors by which a lattice can be displaced and where ◻𝐸,𝐶 denotes the fractional 4-dimensional Laplacian (31) for dimensionless variables x of continuum space-time. be brought back into itself. For simplification, we assume that 2,+ a𝜇, 𝜇 = 1, 2, 3,, 4 are mutually perpendicular primitive lattice Here we take into account (18)intheform◻𝐸,𝐶 =−◻𝐸,𝐶. (𝛼) vectors. We choose directions of the axes of the Cartesian Using the stationary action principle, 𝛿𝑆𝐸 [𝜑𝐶,𝐽𝐶]=0, coordinate system coinciding with the vector a𝜇.Thena𝜇 = we derive the fractional field equation 𝑎𝜇e𝜇,where𝑎𝜇 =|a𝜇|,ande𝜇,(𝜇 = 1, 2, 3,), 4 are the 2𝛼,+ 2 basis vectors of the Cartesian coordinate system for Euclidean (◻ +𝑀 )𝜑𝐶 (x) =𝐽𝐶 (x) . (40) 4 𝐸,𝐶 𝐶 space-time R . This simplification means that the lattice is a primitive 4-dimensional orthorhombic Bravais lattice. Similarly, we can consider the fractional field theories that are The position vector of an arbitrary lattice site is written described by the fractional field equations as 𝛼,𝛼,− 2 (◻𝐸,𝐶 +𝑀𝐶)𝜑𝐶 (x) =𝐽𝐶 (x) , 4 (41) x n = ∑𝑛 a , 𝛼/2 2 ( ) 𝜇 𝜇 (45) ((−Δ)𝐶 +𝑀𝐶)𝜑𝐶 (x) =𝐽𝐶 (x) , 𝜇=1 where the fractional 4-dimensional Laplacians (30)and(35) where 𝑛𝜇 are integer. In a lattice the sites are numbered by n, are used. (𝛼) so that the vector n =(𝑛1,𝑛2,𝑛4,𝑛4) can be considered as a The Green functions 𝐺 (x1,...,x𝑠)=⟨𝜑𝐶(x1)⋅⋅⋅ 𝑠,𝐶,𝐸 number vector of the corresponding lattice site. 𝜑 (x )⟩(𝛼) 𝐶 𝑠 𝐸 for Euclidean space-time and dimensionless vari- As the lattice fields we consider real-valued functions for ables have the following path integral representation: n-sites. For simplification, we consider the scalar field 𝜑𝐿(n) for lattice sites that is defined by n =(𝑛1,𝑛2,𝑛3,𝑛4).Inmany −𝑆(𝛼)[𝜑 ,𝐽 ] 𝐸 𝐶 𝐶 𝜑 (n) 𝑙 (𝛼) ∫𝐷𝜑𝐶 (𝜑𝐶 (x1)⋅⋅⋅𝜑𝐶 (x𝑠)) 𝑒 cases, we can assume that 𝐿 belongs to the Hilbert space 2 𝐺 (x1,...,x𝑠)= , 𝑠,𝐶,𝐸 −𝑆(𝛼)[𝜑 ,𝐽 ] of square-summable sequences to apply the discrete Fourier ∫𝐷𝜑 𝑒 𝐸 𝐶 𝐶 𝐶 transform. For simplification, we will consider operators (42) for the lattice scalar fields 𝜑𝐿(n)=𝜑(𝑛1,𝑛2,𝑛3,𝑛4).All consideration can be easily generalized to the case of the ∫𝐷𝜑 where 𝐶 is the sum over all possible configurations of vector fields and other types of fields. 𝜑 (x) the field 𝐶 for continuum space-time. Note that the path- For continuum fractional field theory, we use the dimen- integral approach for space-fractional quantum mechanics is sionless quantities (8). In the lattice fractional theory, we also considered in [3, 4, 32]. will be using the physically dimensionless quantities such as The Euclidean Green functions (42) of fractional field 𝑎𝜇, 𝑛𝜇, x(n), e𝜇,and𝜑𝐿(n). theory can be derived from the generating functional 3.2. Lattice Fractional Derivative. Let us give a definition (𝛼) 𝛼 (𝛼) −𝑆𝐸 [𝜑𝐶,𝐽𝐶] ± 𝑍0,𝐶 [𝐽𝐶]=∫ 𝐷𝜑𝐶𝑒 . (43) of lattice partial derivative D𝐿 [ 𝜇 ] of arbitrary positive real order 𝛼 in the direction e𝜇 = a𝜇/|a𝜇| in the lattice space- Using the integer-order differentiation of (43)withrespectto time. the sources 𝐽𝑛, we can obtain the correlation functions. The Definition 5. Lattice fractional partial derivatives are the 𝑠-point fractional correlation function is ± 𝛼 operators D𝐿 [ 𝜇 ] such that 𝑠 (𝛼) (𝛼) 𝛿 𝑍0,𝐶 [𝐽𝐶] ⟨𝜑 (x )⋅⋅⋅𝜑 (x )⟩ = . (44) +∞ 𝐶 1 𝐶 𝑠 𝐸 ± 𝛼 1 ± 𝛿𝐽𝐶 (x1)⋅⋅⋅𝛿𝐽𝐶 (x𝑠) (D [ ]𝜑 ) (n) = ∑ 𝐾 (𝑛 −𝑚 )𝜑 (m) , 𝐿 𝜇 𝐿 𝑎𝛼 𝛼 𝜇 𝜇 𝐿 𝜇 𝑚𝜇=−∞ Quantum fluctuations correspond to the contributions to (𝜇=1,2,3,4), the integral (43) coming from field configurations which are not solutions to the classical field equations40 ( )and(41). (46) Advances in High Energy Physics 7

± ̂− 𝛼 where 𝛼∈R, 𝛼>0, 𝑛𝜇, 𝑚𝜇 ∈ Z, and the kernels 𝐾𝛼 (𝑛 − 𝑚) the inverse relation of (51)with𝐾𝛼 (𝑘) = 𝑖 sgn(𝑘)|𝑘| in the are defined by the equations form 1 𝜋 𝜋𝛼 𝛼+1 1 𝛼+3 𝜋2(𝑛−𝑚)2 𝐾− (𝑛) =− ∫ 𝑘𝛼 (𝑛𝑘) 𝑑𝑘 (𝛼∈R,𝛼>0) , 𝐾+ (𝑛−𝑚) = 𝐹 ( ; , ;− ), 𝛼 𝜋 sin (54) 𝛼 𝛼+11 2 2 2 2 4 0 − − we get (48)forthekernel𝐾 (𝑛 − 𝑚).Notethat𝐾 (0) = 0. 𝛼>0, 𝛼 𝛼 The lattice operators46 ( )with(47)and(48)forinteger and noninteger orders 𝛼 can be interpreted as a long-range (47) interactions of the lattice site defined by 𝑛 with all other sites − with 𝑚 =𝑛̸ . 𝐾𝛼 (𝑛−𝑚) 𝜋𝛼+1 (𝑛−𝑚) 𝛼+2 3 𝛼+4 𝜋2(𝑛−𝑚)2 3.3. Lattice Operators of Integer Orders. Let us give exact =− 𝐹 ( ; , ;− ), 𝐾̂±(𝑘) 𝛼∈N 𝛼+2 1 2 2 2 2 4 forms of the kernels 𝛼 for integer positive .Equa- tions (47)and(48)forthecase𝛼∈N can be simplified. ̂± 𝛼>0, To obtain the simplified expressions for kernels 𝐾𝛼 (𝑘) with positive integer 𝛼=𝑚,weusetheintegralsofSec.2.5.3.5 in (48) ± [35]. The kernels 𝐾𝛼 (𝑛) for integer positive 𝛼=𝑚are defined by the equations where 1𝐹2 is the Gauss hypergeometric function [33, 34]. The parameter 𝛼>0will be called the order of the lattice [(𝛼−1)/2] (−1)𝑛+𝑘𝑠!𝜋𝛼−2𝑘−2 1 derivatives (46). 𝐾+ (𝑛) = ∑ 𝛼 (𝛼−2𝑛−1)! 𝑛2𝑘+2 ± 𝑘=0 The kernels 𝐾𝛼 (𝑛) are real-valued functions of integer (55) + + [(𝛼+1)/2] variable 𝑛∈Z.Thekernel𝐾𝛼 (𝑛) is even function 𝐾𝛼 (−𝑛) = (−1) 𝑠! (2 [(𝛼+1) /2] −𝛼) + − − − + , +𝐾𝛼 (𝑛),and𝐾𝛼 (𝑛) is odd function 𝐾𝛼 (−𝑛) = −𝐾𝛼 (𝑛) for all 𝜋𝑛𝛼+1 𝑛∈Z. 𝐾±(𝑛 − 𝑚) [𝛼/2] (−1)𝑛+𝑘+1𝑠!𝜋𝛼−2𝑘−1 1 The reasons to define the kernels 𝛼 in the forms 𝐾− (𝑛) =−∑ 𝛼 (𝛼−2𝑛)! 2𝑘+2 (47)and(48) are based on the expressions of their Fourier 𝑘=0 𝑛 series transforms. The Fourier series transform (56) (−1)[𝛼/2]𝑠! (2 [𝛼/2] −𝛼+1) +∞ ∞ − , ̂+ −𝑖𝑘𝑛 + + + 𝛼+1 𝐾𝛼 (𝑘) = ∑ 𝑒 𝐾𝛼 (𝑛) =2∑𝐾𝛼 (𝑛) cos (𝑘𝑛) +𝐾𝛼 (0) 𝜋𝑛 𝑛=−∞ 𝑛=1 [𝑥] 𝑥 𝑛∈N (49) where is the integer part of the value ,and .Here 2[(𝑚 + 1)/2] − 𝑚 =1 for odd 𝑚,and2[(𝑚 + 1)/2] − 𝑚 =0 + for even 𝑚. for the kernel 𝐾𝛼 (𝑛) defined by (47) satisfies the condition Using (55) or direct integration (53)forintegervalues𝛼= + ̂+ 𝛼 1 and 𝛼=2, we get the simplest examples of 𝐾𝛼 (𝑛) in the form 𝐾𝛼 (𝑘) = |𝑘| , (𝛼>0) . (50) 1−(−1)𝑛 2(−1)𝑛 𝐾+ (𝑛) =− ,𝐾+ (𝑛) = , (57) The Fourier series transforms 1 𝜋𝑛2 2 𝑛2 +∞ ∞ − −𝑖𝑘𝑛 − − + 𝑚 ̂ where 𝑛 =0̸ , 𝑛∈Z,and𝐾𝑚(0) = 𝜋 /(𝑚 + 1) for all 𝑚∈N. 𝐾𝛼 (𝑘) = ∑ 𝑒 𝐾𝛼 (𝑛) =−2𝑖∑𝐾𝛼 (𝑛) sin (𝑘𝑛) (51) 𝑛=−∞ 𝑛=1 Using (56) or direct integration (54)for𝛼=1and 𝛼=2,we − get examples of 𝐾𝛼 (𝑛) in the form − for the kernels 𝐾 (𝑛) defined by (48) satisfies the condition 𝛼 (−1)𝑛 (−1)𝑛𝜋 2(1−(−1)𝑛) 𝐾− (𝑛) = ,𝐾− (𝑛) = + , ̂− 𝛼 1 𝑛 2 𝑛 𝜋𝑛3 𝐾𝛼 (𝑘) =𝑖sgn (𝑘) |𝑘| , (𝛼>0) . (52) (58) Note that we use the minus sign in the exponents of (49)and − where 𝑛 =0̸ , 𝑛∈Z,and𝐾𝑚(0) = 0 for all 𝑚∈N.Notethat (51) instead of plus in order to have the plus sign for plane (1 − (−1)𝑛)=2 𝑛 (1 − (−1)𝑛)=0 𝑛 wavesandfortheFourierseries. for odd ,and for even . 𝐾+(𝑛 − 𝑚) In the definition of lattice fractional derivatives46 ( )the The form (47)ofthekernel 𝛼 is completely 𝜇 = 1, 2, 3, 4 𝑛 determined by the requirement (50). If we use an inverse value characterizes the component 𝜇 of the ̂+ 𝛼 lattice vector n with respect to which this derivative is taken. relation of (49)with𝐾𝛼 (𝑘) = |𝑘| that has the form It is similar to the variable 𝑥𝜇 in the usual partial derivatives 𝜋 4 ± 𝛼 + 1 𝛼 for the space-time R . The lattice operators D𝐿 [ 𝜇 ] are 𝐾𝛼 (𝑛) = ∫ 𝑘 cos (𝑛𝑘) 𝑑𝑘, (𝛼∈R,𝛼>0) , (53) 𝛼 𝜋 0 analogous to the partial derivatives of order with respect to coordinates 𝑥𝜇 for continuum field theory. The lattice + ± 𝛼 then we get (47)forthekernel𝐾𝛼 (𝑛 − 𝑚).Theform(48)of derivative D𝐿 [ 𝜇 ] is an operator along the vector e𝜇 = a𝜇/|a𝜇| − the term 𝐾𝛼 (𝑛 − 𝑚) is completely determined by (52). Using in the lattice space-time. 8 Advances in High Energy Physics

3.4. Lattice Operators with Other Kernels. In general, we can 3.5. Lattice Fractional 4-Dimensional Laplacian. An action of weaken the conditions (50)and(52) to determine a more two repeated lattice operators of order 𝛼 isnotequivalentto wider class of the lattice fractional derivatives. For this aim, the action of the lattice operator of double order 2𝛼: we replace the exact conditions (50)and(52)bytheasympto- 𝛼 𝛼 2𝛼 D± [ ] D± [ ] ≠ D± [ ], (𝛼>0) . tical requirements 𝐿 𝜇 𝐿 𝜇 𝐿 𝜇 (64) ̂+ 𝛼 𝛼 𝐾𝛼 (𝑘) = |𝑘| +𝑜(|𝑘| ), (𝑘󳨀→0) , (59) Note that these properties are similar to noninteger order ̂− 𝛼 𝛼 𝐾𝛼 (𝑘) =𝑖sgn (𝑘) |𝑘| +𝑜(|𝑘| ), (𝑘󳨀→0) , (60) derivatives [2]. 𝛼 where the little-o notation 𝑜(|𝑘| ) meansthetermsthat Definition 6. The lattice 4-dimensional fractional Laplacian |𝑘| |𝑘|𝛼 𝛼,𝛼,± 2𝛼,± include higher powers of than . The conditions (59) operators ◻𝐸,𝐿 and ◻𝐸,𝐿 for a scalar lattice field 𝜑𝐿(m) and (60) mean that we can consider arbitrary functions are defined by the following two equations, where the first ± ̂± 𝐾𝛼 (𝑛 − 𝑚) for which 𝐾𝛼 (𝑘) are asymptotically equivalent to 𝛼 𝛼 𝛼 expression contains the two lattice operators of order , |𝑘| and 𝑖 sgn(𝑘)|𝑘| as |𝑘|→0,respectively. 𝐾+(𝑛 − 𝑚) 4 𝛼 2 As an example of the kernel 𝛼 ,whichcangive ◻𝛼,𝛼,±𝜑 (m) = ∑ (D± [ ]) 𝜑 (m) , 𝐸,𝐿 𝐿 𝐿 𝜇 𝐿 (65) the lattice fractional derivatives (46)with(59), has been sug- 𝜇=1 gested in [18–20]intheform 𝑛 and the second expression contains the lattice operator of the + (−1) Γ (𝛼+1) 𝐾 (𝑛) = , order 2𝛼 in the form 𝛼 Γ (𝛼/2 + 1 + 𝑛) Γ (𝛼/2+1−𝑛) (61) 4 2𝛼 ◻2𝛼,±𝜑 (m) = ∑ D± [ ]𝜑 (m) . whereweuserelation5.4.8.12from[35]. This kernel has been 𝐸,𝐿 𝐿 𝐿 𝜇 𝐿 (66) suggested in [18, 19] to describe long-range interactions of the 𝜇=1 lattice particles for noninteger values of 𝛼.Forintegervalues + The violation of the semigroup property (64)leadstothe of 𝛼∈N,thekernel𝐾𝛼 (𝑛 − 𝑚) =0 for |𝑛−𝑚|≥ 𝛼/2+ + fact that operators (65)and(66) do not coincide in general. 1.For𝛼=2𝑗,wehave𝐾𝛼 (𝑛 − 𝑚) =0 for all |𝑛 − 𝑚| ≥ + Using (46), expression (66) can be represented by 𝑗+1.Thefunction𝐾𝛼 (𝑛 − 𝑚) with even value of 𝛼=2𝑗 can be interpreted as an interaction of the 𝑛-particle with 2𝑗 4 1 +∞ 𝑛±1⋅⋅⋅𝑛±𝑗 (◻2𝛼,±𝜑 ) (n) = ∑ ∑ 𝐾± (𝑛 −𝑚 )𝜑 (m) . particles with numbers .Notethatthelong-range 𝐸,𝐿 𝐿 2𝛼 2𝛼 𝜇 𝜇 𝐿 (67) 𝑎 𝑚 =−∞ interaction with the kernel (61)ispartiallyconnectedwiththe 𝜇=1 𝜇 𝜇 long-range interaction of the Grunwald-Letnikov-Riesz¨ type The correspondent continuum fractional Laplace opera- [24]. It is easy to see that expression (47) is more complicated tors are defined by (30)and(31). The continuum operators than (61). ◻𝛼,𝛼,− ◻2𝛼,+ 𝛼=1 𝐾−(𝑛 − 𝑚) 𝐸,𝐶 and 𝐸,𝐶 for integer give the usual (local) 4- As an example of the kernel 𝛼 ,whichcangive ◻ the lattice fractional derivatives (46)with(60), has been dimensional Laplacian 𝐸 that is defined by (7). The operators ◻𝛼,𝛼,+ ◻2𝛼,− 𝛼=1 suggested in [20]intheform 𝐸,𝐶 and 𝐸,𝐶 for integer are nonlocal operators and cannot get a correspondence with the usual (nonfractional) (−1)(𝑛+1)/2 (2 [(𝑛+1) /2] −𝑛) Γ (𝛼+1) 𝐾− (𝑛) = , field theories. Therefore we should use the lattice fractional 𝛼 𝛼 (62) 𝛼,𝛼,− 2𝛼,+ 2 Γ ((𝛼+𝑛) /2 + 1) Γ ((𝛼−𝑛) /2 + 1) Laplace operators ◻𝐸,𝐿 or ◻𝐸,𝐿 in the lattice fractional field theories. where the brackets [] mean the integral part, that is, the floor function that maps a real number to the largest previous 3.6. Lattice Riesz 4-Dimensional Laplacian. Let us define a (2[(𝑛+1)/2]−𝑛) integer number. The expression is equal to lattice analog of the fractional Laplace operator of the Riesz 𝑛=2𝑚 𝑛=2𝑚−1 zero for even ,anditisequalto1forodd . type [2, 14] which is an operator for scalar fields on the lattice To get the expression, we use relation 5.4.8.13 from [35]. Note space-time. that the kernel (62) is real valued function since we have zero, (𝑛+1)/2 when the expression (−1) becomes a complex number. Definition 7. The lattice fractional Laplace operator of the For 0<𝛼≤2, we can give other examples of the kernels 𝛼/2 Riesz type (−Δ)𝐿 for 4-dimensional Euclidean space-time with the property (59) which are given in Section 8 of the is defined by the equation book [36]. For example, the most frequently used kernel is +∞ 𝛼/2 1 + + 𝐴 (𝛼) ((−Δ) 𝜑 ) (n) = ∑ K (n − m) 𝜑 (m) , 𝐾 (𝑛) = , 𝐿 𝐿 𝛼 𝛼 𝐿 (68) 𝛼 𝛼+1 (63) 𝑎 |𝑛| 𝑚1⋅⋅⋅𝑚4=−∞ 𝐴(𝛼) = (2Γ(−𝛼) (𝜋𝛼/2))−1 4 2 𝛼/2 where we use the multiplier cos , where the constant 𝑎 is 𝑎=(∑ 𝑎 ) and the kernel ̂+ ̂+ 𝛼 𝜇=1 𝜇 which has the asymptotic behavior 𝐾𝛼 (𝑘) = 𝐾𝛼 (0) + |𝑘| + + 𝛼 K𝛼(n − m) is defined by the equation 𝑜(|𝑘| ), (𝑘 → 0),forthecases0<𝛼<2and 𝛼 =1̸ , 𝐾̂+(0) 𝜁(𝑧) 𝛼/2 with nonzero term 𝛼 ,where is the Riemann zeta- 1 𝜋 𝜋 4 4 K+ (n) = ∫ 𝑑𝑘 ⋅⋅⋅∫ 𝑑𝑘 (∑𝑘2 ) ∏ (𝑛 𝑘 ), function.Totakeintoaccountthisexpression,weusethe 𝛼 4 1 4 𝜇 cos 𝜇 𝜇 ̂+ 𝜋 0 0 𝜇 𝜇=1 asymptotic condition for 𝐾𝛼 (𝑘) in the form (50) that includes ̂+ (69) 𝐾𝛼 (0). For details see Section 8.11-8.12 in [36]. Advances in High Energy Physics 9

4 where n = ∑𝜇=1 𝑛𝜇e𝜇 and the parameter 𝛼>0is the order of The lattice action 𝑆𝐸[𝜑𝐿,𝐽𝐿] is not unique, and we can the lattice operator (68). choose the simplest one. We have only the requirement that any lattice action should reproduce the correct continuum Note that the kernel (69)isconnectedwith(47)bythe expression in the continuum limit 𝑎𝜇 →+0. equation The action used in the path integral74 ( )canbeconsid- ered in the forms 1 𝜋 𝜋 𝛼/2 ∫ 𝑑𝑘 ⋅⋅⋅∫ 𝑑𝑘 (𝑘2 ) (𝑛 𝑘 ) 4 1 4 𝜇 cos 𝜇 𝜇 1 2𝛼,± 2 𝜋 0 0 𝑆𝐸 [𝜑𝐿,𝐽𝐿]= ∑𝜑𝐿 (n) (◻𝐸,𝐿 +𝑀𝐿)𝜑𝐿 (m) 2 n,m 2 2 (70) 𝜋𝛼 𝛼+1 1 𝛼+3 𝜋 (𝑛 ) (75) 𝜇 + ∑𝜑 m 𝐽 m . = 1𝐹2 ( ; , ;− ), 𝐿 ( ) 𝐿 ( ) 𝛼+1 2 2 2 4 m For lattice theory with the lattice Riesz fractional Laplacian where n𝜇 =𝑛𝜇e𝜇 without the sum over 𝜇. ̂+ + the action is The Fourier series transform K𝛼(k) of the kernels K𝛼(n) 1 𝛼/2 2 in the form 𝑆𝐸 [𝜑𝐿,𝐽𝐿]= ∑𝜑𝐿 (n) ((−Δ)𝐿 +𝑀𝐿)𝜑𝐿 (m) 2 n,m +∞ + −𝑖 ∑4 𝑘 𝑛 + (76) K̂ (k) = ∑ 𝑒 𝜇=1 𝜇 𝜇 K (n) 𝛼 𝛼 (71) + ∑𝜑 (m) 𝐽 (m) . 𝑛 ⋅⋅⋅𝑛 =−∞ 𝐿 𝐿 1 4 m satisfies the condition Using (67), we rewrite expressions (75)intheform 𝛼/2 4 1 4 +∞ K̂+ (k) = |k|𝛼 = (∑𝑘2 ) , (𝛼>0) . (72) 𝑆 [𝜑 ,𝐽 ]= ∑ ∑ 𝜑 (n) 𝑃 (2𝛼) 𝜑 (m) 𝛼 𝜇 𝐸 𝐿 𝐿 𝐿 𝑛𝜇𝑚𝜇 𝐿 𝜇 2 𝜇=1𝑛 ,𝑚 =−∞ 𝜇 𝜇 (77) K+(n) The form (69)ofthekernel 𝛼 is completely determined + ∑𝜑𝐿 (m) 𝐽𝐿 (m) , by the requirement (72). The inverse relation to(71)with(72) m has the form (69). 𝑃 (2𝛼) where the kernel 𝑛𝜇𝑚𝜇 is given by If the lattice field 𝜑𝐿(m) depends only on one variable 𝑚𝜇 𝜇∈{1,2,3,4} m = m =𝑚 e with fixed ,thatis, 𝜇 𝜇 𝜇 without the 𝑃𝑛 𝑚 (2𝛼) sum over 𝜇,thenwehave 𝜇 𝜇 2 2 𝛼 1 𝜋2𝛼 2𝛼 + 1 1 2𝛼 + 3 𝜋 (𝑛 −𝑚 ) (−Δ)𝛼/2𝜑 (m )=D+ [ ]𝜑 (m) . = 𝐹 ( ; , ;− 𝜇 𝜇 ) 𝐿 𝐿 𝜇 𝐿 𝜇 𝐿 (73) 2𝛼 1 2 𝑎𝜇 2𝛼 + 1 2 2 2 4 𝛼/2 The lattice fractional Laplacian (−Δ)𝐿 in the Riesz 2 +𝑀 𝛿𝑛 ,𝑚 , form for 4-dimensional lattice space-time can be considered 𝐿 𝜇 𝜇 𝛼/2 (78) as a lattice analog of the fractional Laplacian (−Δ)𝐶 for R4 continuum Euclidean space-time that is defined by (35). where 1𝐹2 is the Gauss hypergeometric function [33, 34]. Expression (78)canbeusedforallpositiverealvalues𝛼 3.7.Lattice Fractional Field Theory. The path integral11 ( )does including positive integer values. This kernel describes the not have a precise mathematical definition. To give a defi- space-time lattice with long-range properties that can be nition of the path integrals, we can introduce a space-time interpreted as a lattice space-time with power-law nonlocal- lattice with “lattice constants” a𝜇. Every point on the lattice ity. For the lattice with the nearest-neighbor interactions, the 𝑃 (𝛼) is then specified by four integers which are denoted by the kernel 𝑛𝜇𝑚𝜇 can defined by n =(𝑛,𝑛 ,𝑛 ,𝑛 ) vector 1 2 3 4 ,wherethelastcomponentwill 1 denotealatticeanalogoftheEuclideantime. 𝑃𝑛 𝑚 (2) =− ∑ (𝛿𝑛 +𝑠 ,𝑚 +𝛿𝑛 −𝑠 ,𝑚 −2𝛿𝑛 ,𝑚 ) 𝜇 𝜇 𝑎2 𝜇 𝜇 𝜇 𝜇 𝜇 𝜇 𝜇 𝜇 In the path integral expression for lattice fields, we should 𝜇 𝑠 >0 𝜇 (79) use dimensionless variables only. Note that by convention all +𝑀2𝛿 . variables of the lattice theory are dimensionless variables. 𝐿 𝑛𝜇,𝑚𝜇 For lattice fractional fied theory the path-integral expression of the Green functions is Note that the kernel (78)with𝛼=2reproduces the same continuum fractional field theory as79 ( ). ⟨𝜑 (n )⋅⋅⋅𝜑 (n )⟩ 𝐿 1 𝐿 𝑠 Using (68), we rewrite expression (76)intheform

𝑠 −𝑆𝐸[𝜑𝐿,𝐽𝐿] +∞ ∫ ∏ 𝑑𝜑𝐿 (n𝑗)(𝜑𝐿 (n1)⋅⋅⋅𝜑𝐿 (n𝑠)) 𝑒 (74) 1 = 𝑗=1 . 𝑆 [𝜑 ,𝐽 ]= ∑ 𝜑 (n) 𝑃 (𝛼) 𝜑 (m) 𝑠 −𝑆 [𝜑 ,𝐽 ] 𝐸 𝐿 𝐿 𝐿 nm 𝐿 ∫∏ 𝑑𝜑 (n )𝑒 𝐸 𝐿 𝐿 2 𝑛,𝑚=−∞ 𝑖=1 𝐿 𝑖 (80)

The structure of the path integral74 ( )isanalogoustothat + ∑𝜑𝐿 (m) 𝐽𝐿 (m) , used in the statistical mechanics of lattice system. m 10 Advances in High Energy Physics

𝑃 (2𝛼) where the kernel 𝑛𝜇𝑚𝜇 is given by For the action (80) the generating functional is defined by the equation 4 1 + 2 𝑃nm (𝛼) = K (n − m) + ∑𝑀 𝛿𝑛 ,𝑚 , 1 1 −1 𝛼 𝛼 L 𝜇 𝜇 (81) 𝑍 [𝐽 ]= ( 𝐽 n 𝑃 𝛼 𝐽 m ). 𝑎 𝜇=1 0,𝐿 𝐿 exp ∑ 𝐿 ( ) nm ( ) 𝐿 ( ) √det 𝑃 (𝛼) 2 n,m + and K𝛼(n − m) is defined by the expression (69). (89) For the lattice fractional field theory we can define the Using the integer-order differentiation of (89)withrespectto generating functional in the form the sources 𝐽𝐿, we can obtain the correlation functions for the

−𝑆𝐸[𝜑𝐿,𝐽𝐿] lattice fractional field theory. The 2-point correlation function 𝑍0,𝐿 [𝐽𝐿]=∫∏𝑑𝜑𝐿 (n) 𝑒 . (82) n is 2 𝛿 𝑍0,𝐿 [𝐽𝐿] −1 It can be easily calculated, since the multiple integral is of the ⟨𝜑𝐿 (n) 𝜑𝐿 (m)⟩= =𝑃nm (𝛼) . (90) Gaussian type. Apart from an overall constant, which we will 𝛿𝐽𝐿 (n) 𝛿𝐽𝐿 (m) always drop since it plays no role when computing ensemble Using the discrete Fourier representation, one finds that averages, we have that 𝑃nm(𝛼) is given by 𝑍0,𝐿 [𝐽𝐿] −1 ̂ 𝑃nm (𝛼) = FΔ {𝑃𝛼 (k)} 1 1 4 +∞ −1 4 1 = exp( ∑ ∑ 𝐽𝐿 (n) 𝑃𝑛 𝑚 (2𝛼) 𝐽𝐿 (m)), 2 𝜇 𝜇 =(∏ ) √det 𝑃 (2𝛼) 𝜇=1𝑛 ,𝑚 =−∞ 𝜇 𝜇 𝜇=1 𝑘0𝜇 (91) (83) +𝑘01/2 +𝑘04/2 𝑃−1 (2𝛼) × ∫ ⋅⋅⋅∫ 𝑑4k𝑃̂ (k) 𝑒𝑖(k,(x(n)−x(m))), where 𝑛𝜇𝑚𝜇 is the inverse of the matrix (78)and 𝛼 −1 −𝑘01/2 −𝑘04/2 det 𝑃(2𝛼) is the determinant of 𝑃𝑛 𝑚 (2𝛼).Theinversematrix 𝜇 𝜇 𝑘 =2𝜋/𝑎 𝑃−1 (2𝛼) where 0𝜇 𝜇 and 𝑛𝜇𝑚𝜇 is defined by the equation 4 𝛼/2 +∞ 𝑃̂ (k) = |k|𝛼 +𝑀2 =(∑𝑘2 ) +𝑀2. ∑ 𝑃 𝑃−1 =𝛿 , (𝜇 = 1, 2, 3,) 4 𝛼 𝐿 𝜇 𝐿 (92) 𝑛𝜇𝑠𝜇 𝑠𝜇𝑚𝜇 𝑛𝜇𝑚𝜇 (84) 𝜇=1 𝑠=−∞ and it can be easily derived by using the momentum space, Here we use the notations 𝛿 4 4 where 𝑛𝜇𝑚] is given by x (n) = ∑𝑛𝜇a𝜇, (k, x (n)) = ∑𝑘𝜇𝑛𝜇𝑎𝜇. (93) +𝑘 /2 1 0𝜇 𝜇=1 𝜇=1 𝑖𝑘𝜇(𝑛𝜇−𝑚𝜇)𝑎𝜇 𝛿𝑛 𝑚 = ∫ 𝑑𝑘𝜇𝑒 , (85) 𝜇 𝜇 𝑘 −1 0𝜇 −𝑘0𝜇/2 The inverse matrix 𝑃nm(𝛼) has the form where 𝑘0𝜇/2 = 𝜋/𝑎𝜇 and the integration is restricted by the −1 −1 ̂−1 𝑃nm (𝛼) = FΔ {𝑃𝛼 (k)} Brillouin zone, 𝑘𝜇 ∈[−𝑘0𝜇/2, 𝑘0𝜇/2]. Using the discrete Fourier representation, one finds that 4 1 𝑃 (2𝛼) =(∏ ) 𝑛𝜇𝑚𝜇 is given by 𝜇=1 𝑘0𝜇 𝑃 (2𝛼) = F−1 {𝑃̂ (𝑘 )} 𝑛𝜇𝑚𝜇 Δ 2𝛼 𝜇 +𝑘01/2 +𝑘04/2 −1 × ∫ ⋅⋅⋅∫ 𝑑4k(𝑃̂ (k)) 𝑒𝑖(k,(x(n)−x(m))). +𝑘0𝜇/2 (86) 𝛼 1 𝑖𝑘 (𝑛 −𝑚 )𝑎 −𝑘 /2 −𝑘 /2 = ∫ 𝑑𝑘 𝑃̂ (𝑘 )𝑒 𝜇 𝜇 𝜇 𝜇 , 01 04 𝑘 𝜇 2𝛼 𝜇 0𝜇 −𝑘0𝜇/2 (94) where The right-hand side of expression (94) depends on the n m 󵄨 󵄨2𝛼 lattice sites and and on the dimensionless mass parameter 𝑃̂ (𝑘 )=󵄨𝑘 󵄨 +𝑀2. 2𝛼 𝜇 󵄨 𝜇󵄨 𝐿 (87) 𝑀𝐿. Let us indicate this dependence explicitly, by using the −1 notation 𝐺𝑃(n, m,𝑀𝐿,𝛼) =nm 𝑃 (𝛼). Then substituting92 ( ) Note that the integration in (86)isrestrictedtotheBrillouin into (94), we have zone, 𝑘𝜇 ∈[−𝑘0𝜇/2, 𝑘0𝜇/2],where𝜇 = 1, 2, 3, 4 and 𝑘0𝜇/2 = 4 𝜋/𝑎𝜇. 1 𝐺 (n, m,𝑀 ,𝛼)=(∏ ) The inverse matrix is 𝑃 𝐿 𝜇=1 𝑘0𝜇 −1 −1 ̂−1 𝑃𝑛 𝑚 (2𝛼) = FΔ {𝑃2𝛼 (𝑘𝜇)} 𝜇 𝜇 +𝑘 /2 +𝑘 /2 𝑖(k,(x(n)−x(m))) 4 01 04 𝑒 𝑑 k × ∫ ⋅⋅⋅∫ . +𝑘 /2 𝑖𝑘𝜇(𝑛𝜇−𝑚𝜇)𝑎𝜇 𝛼/2 1 0𝜇 𝑒 (88) 4 2 −𝑘01/2 −𝑘04/2 (∑ 𝑘2 ) +𝑀 = ∫ 𝑑𝑘𝜇 . 𝜇=1 𝜇 𝐿 𝑘 󵄨 󵄨2𝛼 2 0𝜇 −𝑘0𝜇/2 󵄨𝑘 󵄨 +𝑀 󵄨 𝜇󵄨 𝐿 (95) Advances in High Energy Physics 11

We can study continuum limit of (95) in order to extract ⟨𝜑 (x)𝜑 (y)⟩ From lattice 𝜑 (x) the physical two-point correlation function, 𝐶 𝐶 .To 𝜑L(n) C 𝑎 →0 𝑥 → to continuum take the limit 𝜇 ,weshouldtakeintoaccountthat 𝜇 −1 ℱ ∘ Lim ∘ℱΔ 𝑛𝜇𝑎𝜇 and 𝑦𝜇 →𝑚𝜇𝑎𝜇. In our case, the continuum limit can give the correct continuum limit Fourier series Inverse Fourier integral −1 4 4 transform ℱΔ transform ℱ 𝑥𝜇 𝑦𝜇 ⟨𝜑𝐶 (x) 𝜑𝐶 (y)⟩ = 𝐺𝑃 (∑ e𝜇, ∑ e𝜇,𝑀𝐶,𝛼) 𝐸 𝑎lim→0 𝜇 𝜇=1 𝑎𝜇 𝜇=1 𝑎𝜇 (96) 𝜑(̂ k) Limit 𝜑(̃ k) a𝜇 →0 that reproduces the result for the scalar two-point function for fractional filed theory with continuum space-time. Figure 1: Diagram of sets of operations for scalar fields. 4. Continuum Fractional Field Theory from Lattice Theory (2)Thepassagetothelimit𝜑(̂ k)→Lim{𝜑(̂ k)} = 𝜑(̃ k), whereweuse𝑎𝜇 →0(or 𝑘0𝜇 →∞), allows us to In this section, we use the methods suggested in [18–20]to derive the function 𝜑(̃ k) from 𝜑(̂ k). By definition 𝜑(̃ k) 𝜑 (n) define the operation that transforms a lattice field 𝐿 and is the Fourier integral transform of the continuum 𝜑 (x) lattice operators into a field 𝐶 and operators for con- field 𝜑𝐶(x),andthefunction𝜑(𝑘)̂ is the Fourier series tinuum space-time. transform of the lattice field 𝜑𝐿(n),where The transformation of the field is following. We consider 4 the lattice scalar field 𝜑𝐶(n) as Fourier series coefficients of 2𝜋 𝜑(̂ k) 𝑘 ∈[−𝑘/2, 𝑘 /2] 𝜇= 𝜑𝐿 (n) = (∏ ) 𝜑𝐶 (x (n)) , (101) some function for 𝜇 0𝜇 0𝜇 ,where 𝜇=1 𝑘0𝜇 1, 2, 3, 4 and 𝑘0𝜇/2 = 𝜋/𝑎𝜇. As a next step we use the 𝑎 →0+(k →∞) 𝜑(̃ k) continuous limit 𝜇 0 to obtain .Finally and x(n)=𝑛𝜇𝑎𝜇 =2𝜋𝑛𝜇/𝑘0𝜇 → x.Notethat we apply the inverse Fourier integral transform to obtain the 2𝜋/𝑘0𝜇 =𝑎𝜇. continuum scalar field 𝜑𝐶(x). Let us give some details for (3) The inverse Fourier integral transform 𝜑(̃ k)→ these transformations of a lattice field into a continuum field F−1{𝜑(̃ k)} = 𝜑 (x) [18–20]. 𝐶 is defined by T 1 The lattice-continuum transform operation 𝐿→𝐶 is the 𝜑 (x) = ∫ 𝑑4k𝑒𝑖(k,x)𝜑̃ (k) = F−1 {𝜑̃ (k)} , F−1 F 𝐶 4 (102) combination of the operations ,Lim,and Δ in the form (2𝜋) R4 −1 T = F ∘ ∘ F (97) 4 𝐿→𝐶 Lim Δ where (k, x)=∑𝜇=1 𝑘𝜇𝑥𝜇,andtheFourierintegral transform of the continuum scalar field 𝜑𝐶(x) is that maps lattice field theory into the continuum field theory, where these operations are defined by the following. 4 −𝑖(k,x) 𝜑̃ (k) = ∫ 𝑑 x𝑒 𝜑𝐶 (x) = F {𝜑𝐶 (x)} . (103) R4 (1) The Fourier series transform 𝜑𝐿(n)→FΔ{𝜑𝐿(n)} = 𝜑(̂ k) of the lattice scalar field 𝜑𝐿(n) is defined by These transformations can be represented by the diagram in Figure 1. +∞ 𝜑̂ (k) = F {𝜑 (n)} = ∑ 𝜑 (n) 𝑒−𝑖(k,x(n)), Comparing (98)-(99)and(102)-(103), we see the existence Δ 𝐿 𝐿 (98) of a cut-off in the momentum in the lattice field theory. In the 𝑛1,...,𝑛4=−∞ theory of the lattice fields 𝜑𝐿(n), the momentum integration with respect to the wave-vector components 𝑘𝜇 is restricted where the inverse Fourier series transform is by the Brillouin zones 𝑘∈[−𝑘0𝜇/2, 𝑘0𝜇/2],where𝑘0𝜇 = −1 2𝜋/𝑎𝜇. 𝜑𝐿 (n) = FΔ {𝜑̂ (k)} In the lattice 4-dimensional space-time, all four com- 4 +𝑘01/2 +𝑘04/2 ponents of momenta 𝑘𝜇 are restricted by the interval 𝑘∈ 1 4 𝑖(k,x(n)) =(∏ ) ∫ ⋅⋅⋅∫ 𝑑 k𝜑̂ (k) 𝑒 . [−𝑘0𝜇/2, 𝑘0𝜇/2]. Therefore the introduction of a lattice space- 𝑘 −𝑘 /2 −𝑘 /2 𝜇=1 0𝜇 01 04 time provides a momentum cut-off of the order of the inverse (99) lattice constants, 𝑘0𝜇 =2𝜋/𝑎𝜇. Using the lattice-continuum transform operation T𝐿→𝐶, Here we use the notations (95)and(96) give the expression for the continuum fractional field theory: 4 4 x (n) = ∑𝑛 a , (k, x (n)) = ∑𝑘 𝑛 𝑎 1 𝑒𝑖(k,x−y) 𝜇 𝜇 𝜇 𝜇 𝜇 (100) ⟨𝜑 (x) 𝜑 (y)⟩ = ∫ 𝑑4k . 𝜇=1 𝜇=1 𝐶 𝐶 𝐸 4 𝛼/2 (2𝜋) R4 4 2 2 (∑𝜇=1 𝑘𝜇) +𝑀𝐶

and 𝑎𝜇 =2𝜋/𝑘0𝜇 is the lattice constants. (104) 12 Advances in High Energy Physics

Let us formulate and prove a proposition about the con- and the limit 𝑎𝜇 →0gives nection between the lattice fractional derivative and contin- ̃+ 1 ̂+ 󵄨 󵄨𝛼 uum fractional derivatives of noninteger orders with respect 𝐾 (𝑘𝜇)= lim 𝐾 (𝑘𝜇𝑎𝜇)=󵄨𝑘𝜇󵄨 , 𝛼 𝑎 →0𝑎𝛼 𝛼 󵄨 󵄨 to coordinates. 𝜇 𝜇 (111) Proposition 8. ̃− 1 ̂− 󵄨 󵄨𝛼−1 The lattice-continuum transform operation 𝐾 (𝑘𝜇)= lim 𝐾 (𝑘𝜇𝑎𝜇)=𝑖𝑘𝜇󵄨𝑘𝜇󵄨 . 𝛼 𝑎 →0𝑎𝛼 𝛼 󵄨 󵄨 T𝐿→𝐶 maps the lattice fractional derivatives 𝜇 𝜇

+∞ 𝛼 1 As a result, the limit 𝑎𝜇 →0for (109)gives (D± [ ]𝜑 ) (n) = ∑ 𝐾± (𝑛 −𝑚 )𝜑 (m) , 𝐿 𝜇 𝐿 𝛼 𝛼 𝜇 𝜇 𝐿 (105) 𝑎𝜇 𝑚 =−∞ 𝛼 𝜇 ∘ F (D± [ ]𝜑 (m))=𝐾̂± (𝑘 ) 𝜑̂ (k) , Lim Δ 𝐿 𝜇 𝐿 𝛼 𝜇 (112) ± where 𝐾𝛼 (𝑛 − 𝑚) are defined by (47), (48), into the continuum fractional derivatives of order 𝛼 with respect to coordinate 𝑥𝜇 where 󵄨 󵄨𝛼 󵄨 󵄨𝛼−1 by 𝐾̃+ (𝑘 )=󵄨𝑘 󵄨 , 𝐾̃− (𝑘 )=𝑖𝑘 󵄨𝑘 󵄨 , 𝛼 𝜇 󵄨 𝜇󵄨 𝛼 𝜇 𝜇󵄨 𝜇󵄨 𝛼 𝛼 (113) T (D± [ ]𝜑 (m))= D± [ ]𝜑 (x) . ̃ ̂ 𝐿󳨀→𝐶 𝐿 𝜇 𝐿 𝐶 𝜇 𝐶 (106) 𝜑 (k) = Lim𝜑 (k) . The inverse Fourier transforms of (112)havetheform Proof. Let us multiply (105) by the expression exp(−𝑖𝑘𝜇𝑛𝜇𝑎𝜇) 𝑛 −∞ +∞ 𝛼 𝛼 andthensumover 𝜇 from to .Then F−1 ∘ ∘ F (D+ [ ]𝜑 (m))= D+ [ ]𝜑 (x) Lim Δ 𝐿 𝜇 𝐿 𝐶 𝜇 𝐶 𝛼 F (D± [ ]𝜑 (m)) Δ 𝐿 𝜇 𝐿 (𝛼>0) ,

+∞ −1 − 𝛼 − 𝛼 −𝑖𝑘 𝑛 𝑎 ± 𝛼 F ∘ ∘ F (D [ ]𝜑 (m))= D [ ]𝜑 (x) = ∑ 𝑒 𝜇 𝜇 𝜇 D [ ]𝜑 (m) Lim Δ 𝐿 𝜇 𝐿 𝐶 𝜇 𝐶 𝐿 𝜇 𝐿 (107) 𝑛𝜇=−∞ (𝛼>0) , 1 +∞ +∞ −𝑖𝑘𝜇𝑛𝜇𝑎𝜇 ± (114) = ∑ ∑ 𝑒 𝐾𝛼 (𝑛𝜇 −𝑚𝜇)𝜑𝐿 (m) . 𝑎𝜇 𝑛 =−∞𝑚 =−∞ 𝜇 𝜇 where we use the connection between the continuum fractional derivatives of the order 𝛼 and the correspondent Using (98),theright-handsideof(107)gives Fourier integrals transforms +∞ +∞ −𝑖𝑘 𝑛 𝑎 ± 𝜇 𝜇 𝜇 + 𝛼 󵄨 󵄨𝛼 ∑ ∑ 𝑒 𝐾 (𝑛𝜇 −𝑚𝜇)𝜑𝐿 (m) F (D [ ]𝜑 (x))=󵄨𝑘 󵄨 𝜑̂ (k) , 𝛼 𝐶 𝜇 𝐶 󵄨 𝜇󵄨 𝑛𝜇=−∞𝑚𝜇=−∞ (115) +∞ +∞ 𝛼 󵄨 󵄨𝛼 − 󵄨 󵄨 −𝑖𝑘𝜇𝑛𝜇𝑎𝜇 ± F (D𝐶 [ ]𝜑𝐶 (x))=𝑖sgn (𝑘𝜇) 󵄨𝑘𝜇󵄨 𝜑̂ (k) . = ∑ 𝑒 𝐾𝛼 (𝑛𝜇 −𝑚𝜇) ∑ 𝜑𝐿 (m) 𝜇 󵄨 󵄨 𝑛𝜇=−∞ 𝑚𝜇=−∞ As a result, we obtain that lattice fractional derivatives are +∞ (108) −𝑖𝑘 𝑛󸀠 𝑎 ± 󸀠 transformed by the lattice-continuum transform operation = ∑ 𝑒 𝜇 𝜇 𝜇 𝐾 (𝑛 ) 𝛼 𝜇 T𝐿→𝐶 into continuum fractional derivatives of the Riesz 𝑛󸀠 =−∞ 𝜇 type. +∞ This ends the proof. −𝑖𝑘𝜇𝑚𝜇𝑎𝜇 ̂± × ∑ 𝜑𝐿 (m) 𝑒 = 𝐾𝛼 (𝑘𝜇𝑎𝜇) 𝜑̂ (k) , 𝑚 =−∞ We have similar relations for other lattice fractional 𝜇 differential operators. Using this Proposition, it is easy to 󸀠 prove that the lattice-continuum transform operation T𝐿→𝐶 where 𝑛 =𝑛𝜇 −𝑚𝜇. 𝜇 maps the lattice Laplace operators (65), (66), and (68)intothe As a result, (107)hastheform continuum 4-dimensional Laplacians of noninteger orders ± 𝛼 1 ̂± that are defined by (30), (31), and (35)suchthatwehave FΔ (D [ ]𝜑𝐿 (m))= 𝐾 (𝑘𝜇𝑎𝜇) 𝜑̂ (k) , 𝐿 𝜇 𝛼 𝛼 (109) 2𝛼,± 2𝛼,± 𝑎𝜇 T𝐿→𝐶((◻𝐸,𝐿 𝜑𝐿) (n))=(◻𝐸,𝐶 𝜑𝐶) (x) , where FΔ is an operator notation for the discrete Fourier 𝛼,𝛼,± 𝛼,𝛼,± T𝐿→𝐶((◻ 𝜑𝐿) (n))=(◻ 𝜑𝐶) (x) , (116) transform. 𝐸,𝐿 𝐸,𝐶 Thenweuse 𝛼/2 𝛼/2 T𝐿→𝐶(((−Δ)𝐿 𝜑𝐿) (n))=((−Δ)𝐶 𝜑𝐶) (x) . + 󵄨 󵄨𝛼 𝐾̂ (𝑎 𝑘 )=󵄨𝑎 𝑘 󵄨 , 𝛼 𝜇 𝜇 󵄨 𝜇 𝜇󵄨 As a result, the continuous limits of the lattice fractional (110) − 󵄨 󵄨𝛼 field equations give the continuum fractional-order field 𝐾̂ (𝑎 𝑘 )=𝑖 (𝑘 ) 󵄨𝑎 𝑘 󵄨 , 𝛼 𝜇 𝜇 sgn 𝜇 󵄨 𝜇 𝜇󵄨 equations for continuum space-time. Advances in High Energy Physics 13

5. Conclusion [12] S. C. Lim, “Fractional derivative quantum fields at positive temperature,” Physica A,vol.363,no.2,pp.269–281,2006. In this paper, an approach to formulate the fractional field [13] S. C. Lim and L. P.Teo, “Casimir effect associated with fractional theory on a lattice space-time has been suggested. Note that Klein-Gordon field,” in Fractional Dynamics,J.Klafter,S.C. lattice approaches to the fractional field theories were not Lim, and R. Metzler, Eds., pp. 483–506, World Science Pub- previously considered. A fractional-order generalization of lisher,Singapore,2012. the lattice field theories has not been proposed before. The [14] M. Riesz, “L’integrale´ de Riemann-Liouville et le probleme` suggested approach, which is suggested in this paper, can de Cauchy,” Acta Mathematica,vol.81,no.1,pp.1–222,1949 be considered from two following points of view. Firstly it (French). allows us to give lattice analogs of the fractional field theories. [15]C.G.BolliniandJ.J.Giambiagi,“Arbitrarypowersofd’Alem- Secondly, it allows us to formulate fractional-order analogs of bertians and the Huygens principle,” Journal of Mathematical the lattice quantum field theories. The lattice analogs of the Physics,vol.34,no.2,pp.610–621,1993. fractional-order derivatives for fields on the lattice space-time [16] D. G. Barci, C. G. Bollini, L. E. Oxman, and M. Rocca, “Lorentz- are suggested to formulate lattice fractional field theories. invariant pseudo-differential wave equations,” International The space-time lattices are characterized by the long-range Journal of Theoretical Physics,vol.37,no.12,pp.3015–3030,1998. properties of power-law type instead of the usual lattices [17] R. L. P.G. do Amaral and E. C. 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