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Research Article Fractional Quantum Field Theory: from Lattice to Continuum

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Research Article Fractional Quantum Field Theory: from Lattice to Continuum 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 mod- els 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 frac- tional 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, (◻+ )() =() , (1) we are dealing with the case of an innumerable number of ◻ () degrees of freedom, labeled by the space-time coordinates at where is the d’Alembert operator, is a real field, and ∈R4 least. To give the path integrals a precise meaning, we can dis- 1,3 is the space-time vector with components ,where cretize space and time; that is, we can introduce a space-time = 0, 1, 2,. 3 This field equation follows from stationary lattice. The introduction of a lattice space-time corresponds action principle, [],wheretheaction =0 [] 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 [] =− ∫ 4 () (◻+2) () . 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 () ̇ lization procedure can be carried out for path integrals in and momenta ()̇ become the operators, Φ() and Φ(), 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 ( ,..., ) = ⟨Ω {Φ ( ) ⋅⋅⋅Φ( )} Ω⟩ , the path integral since the lattice has the denumerable num- 1 1 (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 { } denotes the time-ordered product of the theories. In the expression of the Fourier integral for lattice operators Φ(). These Green functions have a path integral fields, the momentum integration with respect to wave-vector representation [28]intheform components ( = 1, 2, 3,) 4 is restricted by the Brillouin zone ∈[−/,/],where arethelatticeconstants. ∫(( )⋅⋅⋅( )) [] ( ,..., )= 1 , The second step of the renormalization is a continualization 1 [] (4) procedure that removes the lattice structure by a continuum ∫ limit, where the lattice constants tend to zero. In this step of the renormalization process the momentum cut-off where ∫ is the sum over all possible configurations of is removed by the continuum limit. the field (). 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 → − . (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 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 [] 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 →−4,where 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 ()=(,⃗ 4 ) fractional field theory for lattice space-time is proposed. In instead of (,⃗ ),where(,⃗ 4 ) is not obtained from Section 4, the lattice-continuum transformation of the lattice (, )⃗ by substituting 4 for ,notingthat()= fractional theories is discussed. A short conclusion is given in (,⃗ 4 ) is a real field which is a function of the Section 5. Euclidean variable , − 2. Fractional Field Theory
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