aa r X i v : . [ qu a n t - ph ] F e b Physics Letters A. 2020. Article ID: 126303. DOI: 10.1016/j.physleta.2020.126303
Dirac Particle with Memory: Proper Time Non-LocalityVasily E. Tarasov , Skobeltsyn Institute of Nuclear Physics,Lomonosov Moscow State University, Moscow 119991, Russia
E-mail: [email protected] Faculty ”Information Technologies and Applied Mathematics”,Moscow Aviation Institute (National Research University), Moscow 125993, Russia
Abstract
A generalization of the standard model of Dirac particle in external electromagneticfield is proposed. In the generalization we take into account interactions of this particlewith environment, which is described by the memory function. This function takes intoaccount that the behavior of the particle at proper time can depend not only at thepresent time, but also on the history of changes on finite time interval. In this case theDirac particle can be considered an open quantum system with non-Markovian dynamics.The violation of the semigroup property of dynamic maps is a characteristic property ofdynamics with memory. We use the Fock-Schwinger proper time method and derivativesof non-integer orders with respect to proper time. The fractional differential equation,which describes the Dirac particle with memory, and the expression of its exact solutionare suggested. The asymptotic behavior of the proposed solutions is described.
PACS 45.10.Hj Perturbation and fractional calculus methodsPACS 03.65.Yz Decoherence; open systems; quantum statistical methodsPACS 11.90.+t Other topics in general theory of fields and particlesKeywords: Dirac particle; fractional dynamics; process with memory; open quantum sys-tems; interaction with environment; fractional differential equation; fractional derivative; Fock-Schwinger proper time method
The Dirac particles can be described by the Dirac equation. This approach corresponds to theSchrodinger picture. The Dirac particles can also be described by using the Heisenberg picture.This approach was proposed by Vladimir A. Fock [1, 2, 3, 4, 5] and Julian S. Schwinger in[6]. In this case, the differential equations with respect to proper time are used to have theHamilton and Heisenberg equations in the covariant form [7, 10, 11]. This approach is calledthe Fock-Schwinger proper time method [7, 8, 9]. In the standard model the Dirac particleis described in the external electromagnetic field (for example, see [6], [7], p.100-104, and [9],1.728-732). The standard theory of the Dirac particles does not take into account an interactionbetween this particle and the environment.We proposed a generalization of the Heisenberg equations for the Dirac particle in theexternal electromagnetic field. In this generalization we take into account an interaction ofthis particle with environment. The interaction with environment can be described by thememory function [12], that is also called the response function. Therefore the proposed modelcan be considered an open quantum system, and the dynamics of this Dirac particle is non-Markovian. The Dirac particle is considered in the Heisenberg picture. In this paper we willuse the Fock-Schwinger proper time method to preserve the relativistically covariant form ofHeisenberg equations. The proposed Heisenberg equations are fractional differential equationwith derivatives of non-integer orders with respect to proper time. The exact expressionsof the solution of the generalized Heisenberg equations are suggested for particular case ofthe external electromagnetic field (the constant uniform field). The standard solution for theabsence of memory is the special case of the proposed solution. In this article also suggests theasymptotic behavior of the solution as the proper time tends to infinity.From a physical point of view, the neglect of memory in the standard model is based on theassumption that the Dirac particle is not open system. In the standard approach the interactionof Dirac particle with environment is not considered. The first description of physical systemwith memory was suggested by Ludwig Boltzmann in 1874 and 1876. He took into accountthat the stress at time t can depend on the strains not only at the present time t , but alsoon the history of changes at t < t . Boltzmann also proposed the linear position principle formemory and the memory fading principle. The linear position principle states that total strainis a linear sum of strains, which have arisen in the media at t < t . The principle of memoryfading states that the increasing of the time interval lead to a decrease in the correspondingcontribution to the stress at time t . The Boltzmann theory has been significantly developedin the works of the Vito Volterra in 1928 and 1930 in the form of the heredity concept and itsapplication to physics. Then the memory effects in natural and social sciences are consideredin different works (for example, see [13]-[21]). As example, we can note the thermodynamicsof materials with fading memory [13, 14], the hereditary solid mechanics [15], the theory ofwave propagation in media with memory [16], the dynamics of media with viscoelasticity [17],the econophysics of processes with memory [18], the condensed matter with memory [19], thefractional dynamics of systems and processes with memory [20, 21].From a mathematical point of view, the neglect of memory effects is due to the fact that todescribe the system we use only equations with derivatives of an integer order, which are deter-mined by the properties of the function in an infinitely small neighborhood of the consideredtime. An effective and powerful tool for describing memory effects is the fractional calculus ofderivatives and integrals of non-integer orders [22, 23, 24, 25, 26]. These operators have a longhistory of more than three hundred years [27, 28, 29, 30, 31]. We should note that the character-istic properties of fractional derivatives of non-integer order are the violation of standard rulesand properties that are fulfilled for derivatives of integer order [32, 33, 34, 35, 36]. For example,the standard product (Leibniz) rule and the standard chain rule are violated for derivativesof non-integer orders. These non-standard mathematical properties allow us to describe non-2tandard processes and phenomena associated with non-locality and memory [20, 21]. On theother hand, these non-standard properties lead to difficulties [37] in sequential constructing afractional generalizations of standard models. In article [37], we show how problems arise whenbuilding fractional generalizations of standard models.In this paper, we proposed a generalization of the standard model of the Dirac particle inthe external electromagnetic field by taking into account the interaction with an environmentthat is described by memory function. The fractional nonlinear differential equation, whichdescribes the proposed model, and the expression of its exact solution are suggested. Theasymptotic behavior of the proposed solutions is described. The kinematics of the Dirac particle can be described by the commutation relations[ x µ , p ν ] = − ig µν , (1)where p µ = i∂ µ with where µ = 0 , , ,
3, and < x | x ′ > = δ ( x − x ′ ) . (2)We use the notation of the book [7], g = − g = − g = − g = 1 , (3)and A = A , A k = − A k .Let us consider the standard function H ( x, p ) in the coordinate representation H = ( p − eA ) − m − e σ µν F µν , (4)where F µν is the antisymmetric field strength tensor F µν = ∂ µ A ν − ∂ ν A µ , (5)and σ µν = 12 i [ γ µ , γ ν ] . (6)The electromagnetic field strength tensor F µν can be expressed through the components of theelectric field ( E k ) and magnetic field B k , where k = 1 , ,
3, in the form F µν = − E − E − E E − B B E B − B E − B B . (7)The idea of the Fock-Schwinger proper time method is to consider H as Hamiltonian thatdescribes the proper time evolution of some system.3sing the operator π µ = p µ − eA µ , (8)the standard Heisenberg equations for x µ ( τ ) and π µ ( τ ) of the Dirac particle in the externalelectromagnetic field can be written in the form dx µ ( τ ) dτ = i [ H , x µ ( τ )] , (9) dπ µ ( τ ) dτ = i [ H , π µ ( τ )] , (10)where τ is proper time of the Dirac particle. Using the Hamiltonian H = π − m − e σ µν F µν , (11)and the commutation relations [ π µ , π ν ] = − ieF µ,ν , (12)[ x µ , π ν ] = − ig µν , (13)[ x µ , x ν ] = 0 , (14)the Heisenberg equations of the Dirac particle takes the form dx µ ( τ ) dτ = − π µ , (15) dπ µ ( τ ) dτ = − eF µν π ν − ie∂ ν F µν − e ∂ µ F νρ σ νρ . (16)Note that the coefficient ”-2” in equations (15) and (16) is due to the form of the operator (8) theHamiltonian (11) that are standard for the theory of Dirac particle in external electromagneticfield (see original Schwinger’s article [6], Section 2.5.4 in [7], p.100-104, and Section 33 in [9],p.728-732).The dynamics of quantum observables x µ ( τ ) and π µ ( τ ) can be described by as unitaryevolution in the form x µ ( τ ) = U ∗ ( τ ) x µ (0) U ( τ ) , π µ ( τ ) = U ∗ ( τ ) π µ (0) U ( τ ) , (17)where U ( τ ) the unitary operator [7].Note that in general the evolution cannot be considered as the unitary evolution as in thestandard model. Dynamics with memory (with proper time non-locality) cannot be consideredas the unitary evolution. The evolution of fractional dynamic systems with memory cannot bedescribed by unitary operators. The dynamics with memory is non-Markovian in general.4 Dirac particle with memory in external electromag-netic field
The first description of physical system with memory was suggested by Ludwig Boltzmann in1874 and 1876. He takes into account that the stress at time t can depend on the strains notonly at the present time t , but also on the history of changes at t < t . The Boltzmann theoryhas been significantly developed in the works of the Vito Volterra in 1928 and 1930 in the formof the heredity concept and its application to physics. Then the memory effects in natural andsocial sciences are considered in different works (for example, see [13]-[21]).To take into account the memory effects for Dirac particle in external electromagnetic field,we can consider the equations E τ M,n { x µ } = i [ H , x µ ( τ )] = − π µ , (18) E τ M,n { π µ } = i [ H , π µ ( τ )] = − eF µν π ν − ie∂ ν F µν − e ∂ µ F νρ σ νρ , (19)where the symbol E τ M,n denotes a certain operator that allows us to find the values of ( x µ ( τ ) , π µ ( τ ))for any proper time τ , if the values ( x µ ( τ ) , π µ ( τ )) are known for τ ∈ [0 , τ ]. We can say that E τ M,n is an operator, which is a mapping from one space of functions to another. In this paper,we consider linear operators E τ M,n of a special kind, which is called the Volterra operator thatis defined by the expression E τ M,n {A µ } = Z τ M ( τ − τ ) A ( n ) µ ( τ ) dτ , (20)where A ( n ) µ ( τ ) = d n A µ ( τ ) /dτ n is the derivative of the positive integer order n ∈ N with respectto proper time. The function M ( τ − τ ), which characterizes an interaction of the particlewith environment, is called the memory function. In statistical mechanics it is also called theresponse function.It is obvious that not every operators E τ M,n of the form (20) can be used to describe amemory. Restrictions on the memory functions and consideration of the examples of memoryfunction are discussed in paper [38].In paper [38], some general restrictions that can be imposed on the properties of memoryare described. In addition, to the causality principle, these restrictions include the followingthree principles: the principle of memory fading; the principle of memory homogeneity on time(the principle of non-aging memory); the principle of memory reversibility (the principle ofmemory recovery).For example, in the operator (20) we can consider the memory function of the power-lawform M ( τ − τ ) = 1Γ( n − α ) ( τ − τ ) n − α − , (21)where α >
0. In the case of integer values of α , the operator E τ M,n {A µ } is the derivative ofinteger order E τ M,n {A µ } = d n A µ ( τ ) dτ n . (22)5n the case of positive real values of α ∈ R + , the operator E τ M,n {A µ } is the Caputo fractionalderivative of the order α >
0. The Caputo fractional derivative [25] is defined by the equation (cid:0) D ατ ;0+ A µ (cid:1) ( τ ) = 1Γ( n − α ) Z τ ( τ − τ ) n − α − A ( n ) ( τ ) dτ , (23)where n = [ α ] + 1 for non-integer values of α , and n = α for α ∈ N , (see equation 2.4.3 in [25],p.91), Γ( α ) is the Gamma function, and A ( n ) µ ( τ ) is the derivative of the integer order n . It isassumed that the function A µ ( τ ) has derivatives up to the ( n − α = n ∈ N and the usual derivative A ( n ) µ ( τ ) of order n exists, then the Caputo fractional derivative (23) coincides with A ( n ) µ ( τ ) (forexample, see equation 2.4.14 in [25], p.92, proof in [24], p.79), i.e. the condition (22) holds.For the power-law memory function (20), equations (18) and (19) take the form (cid:0) D ατ, x µ (cid:1) ( τ ) = i [ H , x µ ( τ )] = − π µ , (24) (cid:0) D ατ, π µ (cid:1) ( τ ) = i [ H , π µ ( τ )] = − eF µν π ν − ie∂ ν F µν − e ∂ µ F νρ σ νρ , (25)where D ατ, is the Caputo fractional derivative of the order α >
0. In the general case, we canconsider the different orders in equations (24) and (25). Here we consider τ as dimensionlessvariable in order to have standard physical dimensions of physical quantities. Equations (24)and (25) are fractional Heisenberg equations [20] for the Dirac particle.We should note that we use the power-law memory function since it can be considered asan approximation of the equations with generalized memory functions. In the paper [39], usingthe generalized Taylor series in the Trujillo-Rivero-Bonilla form for the memory function, weproved that the equations with memory functions can be represented through the Riemann-Liouville fractional integrals (and the Caputo fractional derivatives) of non-integer orders forwide class of the memory functions.Note that the proposed generalized Heisenberg equations (24) and (25) with α > dx µ ( τ ) dτ = − Z τ M I ( τ − τ ) π µ ( τ ) dτ , (26) dπ µ ( τ ) dτ = Z τ M I ( τ − τ ) (cid:16) − eF µν π ν ( τ ) − ie∂ ν F µν − e ∂ µ F νρ σ νρ (cid:17) dτ , (27)where M I ( τ − τ ) = 1Γ( β ) ( τ − τ ) β − . (28)This possibility is based on the fact that the Caputo fractional derivative is the left inverseoperator for the Riemann-Liouville fractional integral (see equation of the Lemma 2.21 of [25],p.95), the has the form (cid:16) D βτ, (cid:16) I βτ, A µ (cid:17)(cid:17) ( τ ) = A µ ( τ ) , (29)6f β >
0, and A µ ( τ ) ∈ L ∞ (0 , T ) or A µ ( τ ) ∈ C [0 , T ].The action of the Caputo derivative on equations (26) and (27) gives the suggested equations(24) and (25) with the order α = β + 1.As a result, we can formulate the following statement. Statement 1. (a) For Dirac particle in external electromagnetic field, the memory effects (the proper timenonlocality) can be described by equations (18) and (19) , or equations (26) and (27) .(b) The Dirac particle with memory, which is described by the power-law memory function (21) ,is described by the fractional differential equations (24) and (25) of the order α > .(c) The standard equations (15) and (16) of the Dirac particle without memory are special casesof equations (24) and (25) for α = 1 . Note that the proper time nonlocality and memory effects can be caused by the interactionof the particle with environment [12].
In this section, we discuss the characteristic property of dynamics with memory.Let us consider the fractional differential equations (cid:0) D ατ ;0+ A µ (cid:1) ( τ ) = Λ µν A ν ( τ ) , (30)where 0 < α ≤
1. For example, we can consider A ( τ ) = { π µ ( τ ) } and Λ µν = − eF µν . Usingthe matrix notation Λ = { Λ µν } and A ( τ ) = { A µ ( τ ) } , equation (30) can be represented as thematrix equation (cid:0) D ατ ;0+ A (cid:1) ( τ ) = Λ A ( τ ) . (31)The Cauchy-type problem for equation (31), in which the initial condition is given at thetime τ = 0 by A (0), has solution that can be represented in the form A ( τ ) = Φ τ ( α ) A (0) , ( τ ≥ , (32)where Φ τ ( α ) = E α [Λ τ α ] . (33)Here E α [Λ τ α ] is the Mittag-Leffler function [40] with the matrix argument E α [Λ τ α ] = ∞ X k =0 τ αk Γ( αk + 1) Λ k . (34)The operator Φ τ ( α ) describes dynamics of open quantum systems with power-law memory.The matrix Λ can be considered as a generator of the one-parameter groupoid Φ τ ( α ) on operatoralgebra of quantum observables (cid:0) D ατ, Φ τ ( α ) (cid:1) ( τ ) = ΛΦ τ ( α ) . (35)7he set { Φ τ ( α ) | τ > } , can be called a quantum dynamical groupoid [44]. Note that thefollowing properties are realizedΦ τ ( α ) I = I, (Φ τ ( α ) A ) ∗ = Φ τ ( α ) A (36)for self-adjoint operators A ( A ∗ = A ), andlim τ → Φ τ ( α ) = I, (37)where I is an identity operator ( I A = A ). As a result, the operators Φ τ ( α ), τ >
0, are realand unit preserving maps on operator algebra of quantum observables.For α = 1, we have Φ τ (1) = E [Λ τ ] = exp { Λ τ } . (38)The operators Φ τ = Φ τ (1) form a semigroup such thatΦ τ Φ s = Φ τ + s , ( τ, s > , Φ = I. (39)This property holds since exp { Λ τ } exp { Λ s } = exp { Λ( τ + s ) } . (40)For α N we have E α [Λ τ α ] E α [Λ s α ] = E α [Λ( τ + s ) α ] . (41)Therefore the semigroup property [41, 42, 43] is not satisfied for non-integer values of α :Φ τ ( α )Φ s ( α ) = Φ τ + s ( α ) , ( τ, s > . (42)As a result, we can formulate the following statement. Statement 2.
The operators Φ τ ( α ) , which describe quantum dynamics of Dirac particle with memory, cannotform a semi-group in general, since the semigroup property is violated for α N . This property means that equations (24) and (25) for Dirac particle with memory describenon-Markovian evolution. Note that the violation of this semigroup property is a characteristicproperty of dynamics with memory.
Let us consider a constant electromagnetic field. The equations, which described Dirac particlewith power-law memory in a constant field, can be written as (cid:0) D ατ, x µ ( τ ) (cid:1) ( t ) = − π µ ( τ ) , (43)8 D ατ, π µ ( τ ) (cid:1) ( t ) = − eF µν π ν ( τ ) . (44)For α = 1 equations (43) and (44) are reduced to dx µ ( τ ) dτ = − π µ ( τ ) , (45) dπ µ ( τ ) dτ = − eF µν π ν . (46)Equations (45) and (46) describe the Dirac particle without memory in constant electromagneticfield.Using matrix notations x = { x µ } , π = { π µ } , F = { F µν } , equations (45) and (46) can beintegrated, and we get the well-known solutions [7] in the form π ( τ ) = e − eF τ π (0) , (47) x ( τ ) = x (0) + (cid:18) e − eF τ − eF (cid:19) π (0) . (48)Let us obtain solutions of the fractional differential equations (43) and (44). π µ ( τ ) Equation (44) can be rewritten in the form (cid:0) D ατ, π µ ( τ ) (cid:1) ( t ) = − eF µν π ν , (49)where n − < α ≤ n . Using A µ ( τ ) = π µ ( τ ), and Λ µν = − eF µν , equation (49) is written inthe form (cid:0) D ατ ;0+ A µ (cid:1) ( τ ) = Λ µν A ν ( τ ) . (50)Using the matrix notation Λ = { Λ µν } and A ( τ ) = { A µ ( τ ) } , equation (50) can be representedas the matrix equation (cid:0) D ατ ;0+ A (cid:1) ( τ ) = Λ A ( τ ) . (51)To solve equation (51), we can use Theorem 5.15 of book [25], p.323. The solution ofequation (51) has the form A ( τ ) = n − X k =0 τ k E α,k +1 [Λ τ α ] A ( k ) (0) , (52)where n − < α ≤ n , and E α,β [ z ] is the two-parameter Mittag-Leffler function [25, 40] that isdefined by the expression E α,β [Λ τ α ] = ∞ X k =0 τ kα Γ( αk + β ) Λ k , (53)9here Γ( α ) is the gamma function, α >
0, and β is an arbitrary real number. For 0 < α < A ( τ ) = E α, [Λ τ α ] A (0) . (54)For α = 1, we can use E , [ z ] = e z . Then solution (52) gives the equation A ( τ ) = exp { Λ τ }A (0) . (55)As a result, we can formulate the following statement. Statement 3.
In the case of constant external electromagnetic field ( F = const ), solution of fractional differ-ential equation (44) (or (49) ) for the operator π µ ( τ ) has the form π ( τ ) = n − X k =0 τ k E α,k +1 [ − eF τ α ] π ( k ) (0) , (56) where F = { F µν } and n − < α ≤ n . Expression (56) describes dynamics of Dirac particle with memory in constant electromag-netic field.For 0 < α ≤
1, equation (56) takes the form π ( τ ) = E α, [ − eF τ α ] π (0) . (57)For α = 1, solution (56) takes the standard form (47), i.e., π ( τ ) = e − eF τ π (0). Let us consider the equation (56) that has the form π ( τ ) = n − X k =0 τ k E α,k +1 [Λ τ α ] π ( k ) (0) , (58)where Λ is the matrixΛ = {− eF µν } = − e E E E E B − B E − B B E B − B . (59)To get expression (59), we use that F µν = F µρ g ρν and { g µν } = { g µν } = − − − , (60)10hat gives the equations F µ = F µ , F µk = − F µk ( k = 1 , , µ = 0 , , , . (61)We can use the diagonalization of the real skew-symmetric matrix Λ µν = 2 eF µν , where F µν = g µρ F ρν , if all eigenvalues of this matrix are nonzero. Note that the 4 × { Λ µν } isnot skew-symmetric matrix.An n × n matrix Λ over a field R is diagonalizable if and only if there exists a basis of R n consisting of eigenvectors of Λ. If such a basis has been found, we can form the matrix N having these basis vectors as columns. Then there exists an invertible matrix N , which is calleda modal matrix for Λ, such that L = N Λ N − is a diagonal matrix. The diagonal elements λ k , k = 1 , , ..., n , of this matrix L = diagonal { λ , λ , λ , λ } are the eigenvalues of Λ. An n × n matrix Λ is diagonalizable if it has n distinct eigenvalues, i.e. if its characteristic polynomialhas n distinct roots.It is known that the eigenvalues of the matrix F = F µν = F µρ g ρν , are the roots of the fourth-degree polynomial with invariants (1 / F µν ∗ F µν = ( ~E, ~B ) and (1 / F µν F µν = ( B − E ) inthe coefficients (for example see [56, 57]). Therefore, the eigenvalues of the matrix Λ = − eF are the roots of the equation λ + Sλ − P = 0 , (62)where S and P are the invariants P = 2 e F µν ∗ F µν = 2 e ( ~E, ~B ) , S = 2 e F µν F µν = 2 e (cid:0) B − E (cid:1) , (63)The set of eigenvalues λ s , where s = 0 , , , λ s ∈ { + b, − b ; + ia, − ia } , (64)where a and b are the scalar functions of invariants (63) in the form a = r (cid:16) S + √ S + P (cid:17) , (65) b = r (cid:16) − S + √ S + P (cid:17) . (66)For details see [56, 57].It should be noted that the matrix Λ can be represented in the formΛ = N − L N, (67)where N = columns { ξ , ξ , ξ , ξ } , (68) L = diagonal { λ , λ , λ , λ } = λ λ λ
00 0 0 λ , (69)11nd λ µ are the complex coefficients such thatΛ ξ s = λ s ξ s . (70)Here ξ s , s = 0 , , ,
3, are eigenvectors and λ s are eigenvalues of the matrix Λ, respectively. Inequation (70) there is no sum on the index ρ .As a result, we can formulate the following statement. Statement 4.
In the case of constant external electromagnetic field ( F = const ), which does not change withproper time, the solution (58) of fractional differential equation (44) (or (49) ) of the order n − < α ≤ n for the operator π µ ( τ ) can be written in the form π ( τ ) = n − X k =0 τ k N − E α,k +1 [ L τ α ] N π ( k ) (0) . (71) where E α,β [ z ] is the two-parametric Mittag-Leffler function (34) . Equation (71) describes the evolution of the operator π µ ( τ ) of the Dirac particle with power-law memory that is considered as an open quantum system. π µ ( τ ) Let us describe the asymptotic behavior of solution (56) at τ → ∞ . To describe the behavior ofthe two-parameter Mittag-Leffler function E α,k +1 [ λ τ α ] at τ → ∞ , we can use equations 1.8.27and 1.8.28 of book [25], p.43.For | arg ( λ ) | ≤ m and 0 < α <
2, where πα < m < min { π, πα } , we have the asymptotic behavior of the two-parameter Mittag-Leffler function in the form E α,β +1 [ λ τ α ] = λ − β/α α τ − β exp (cid:0) λ /α τ (cid:1) − m X j =1 λ − j Γ( β + 1 − α j ) 1 τ α j + O (cid:18) τ α ( m +1) (cid:19) . (72)For m < | arg ( λ ) | ≤ π and 0 < α <
2, we have the asymptotic behavior in the form E α,β +1 [ λ τ α ] = − m X j =1 λ − j Γ( β + 1 − α j ) 1 τ α j + O (cid:18) τ α ( m +1) (cid:19) . (73)Therefore we can use the following equations for the asymptotic behavior of the two-parameter Mittag-Leffler functions, which are part of the solutions.For the eigenvalue λ s = b >
0, where b is defined by (66) (i.e. the case λ = Re [ λ ] > Im [ λ ] = 0), and 0 < α <
2, we should use equation E α,β +1 [ λ τ α ] = λ − β/α α τ − β exp (cid:0) λ /αs τ (cid:1) − m X j =1 λ − j Γ( β + 1 − α j ) 1 τ α j + O (cid:18) τ α ( m +1) (cid:19) (74)12or τ → ∞ , where λ is a positive real number.For the eigenvalues λ s = ± ia and λ s = − b <
0, where a and b are defined by (65), (66)(i.e. the case λ = Re [ λ ] = 0 with Im [ λ ] = 0, and the case λ = Re [ λ ] < Im [ λ ] = 0), and0 < α <
2, we should use equation E α,β +1 [ λ τ α ] = − m X j =1 λ − js Γ( β + 1 − α j ) 1 τ α j + O (cid:18) τ α ( m +1) (cid:19) (75)for τ → ∞ , where λ is a negative real number or λ is an imaginary number. The differencebetween the formulas (73) and (72) is the absence of the first term with the exponent.Asymptotic equations (74) and (75) allow us to describe the behavior at τ → ∞ in modelwith power-law memory fading parameter 0 < α < < α <
2. Substitution of expressions(74) and (75) with β = k into (58) can give the asymptotic expressions of solution.As a result, we can formulate the following statement. Statement 5.
The asymptotic behavior of the variable Π s ( τ ) = ( N π ) s ( τ ) = N sµ π µ ( τ ) , (76) which characterizes the Dirac particle with memory in a constant field, is described by thefollowing equations. For the eigenvalues λ s = ± ia and λ s = − b < , where a and b are definedby (65) , (66) and < n − < α < n , we have the equation Π s ( τ ) = − n − X k =0 m X j =1 τ k − α j λ − js Γ( k + 1 − α j ) Π ( k ) s (0) + O (cid:18) τ α ( m +1) − k (cid:19)! . (77) For the eigenvalue λ s = b > , where b is defined by (66) and < n − < α < n , the asymptoticbehavior of the variable (76) is described by the equation Π s ( τ ) = n − X k =0 λ − k/α α exp (cid:0) λ /αs τ (cid:1) − m X j =1 τ k − α j λ − js Γ( k + 1 − α j ) Π ( k ) s (0) + O (cid:18) τ α ( m +1) − k (cid:19)! . (78)Expressions (77) and (78) describe the behavior of π ( τ ) at τ → ∞ for the Dirac particlewith power-law memory in constant external electromagnetic field. x µ ( τ ) To solve equation (43), we can use Theorem 5.15 of book [25], p.323. Fractional differentialequation (43) can be written in the form (cid:0) D ατ, A µ (cid:1) ( τ ) = B µ ( τ ) , (79)13here A µ ( τ ) = x µ ( τ ), B µ ( τ ) = − π µ ( τ ) and n − < α ≤ n . If B µ ( τ ) are continuous functionsdefined on the positive semiaxis ( τ > A µ ( τ ) = n − X k =0 τ k k ! A µ ( k ) (0) + 1Γ( α ) Z t ( τ − τ ) α − B µ ( τ ) dτ . (80)As a result, we get the equation x µ ( τ ) = n − X k =0 τ k k ! x µ ( k ) (0) − e Γ( α ) Z t ( τ − τ ) α − π µ ( τ ) dτ . (81)For 0 < α <
1, we have x µ ( τ ) − x µ (0) = − e Γ( α ) Z τ ( τ − τ ) α − π µ ( τ ) dτ . (82)Note that equation (82) can be written through the Riemann-Liouville fractional integral (thedefinition of this operator is given in [25], p.69-70)Using the matrix notation x ( τ ) = { x µ ( τ ) } , and π ( τ ) = { π µ ( τ ) } equation takes the form x ( τ ) = n − X k =0 τ k k ! x ( k ) (0) − e Γ( α ) Z t ( τ − τ ) α − π ( τ ) dτ . (83)Substitution of the expression (56) of the solution for π ( τ ) into expression (83) of x ( τ ) givesthe equation x ( τ ) = n − X k =0 τ k k ! x ( k ) (0) − n − X k =0 e Γ( α ) Z τ ( τ − τ ) α − τ k E α,k +1 [Λ τ α ] π ( k ) (0) dτ . (84)where Λ = − eF = {− eF µν } .To calculate integral of (84), we can use equation 4.10.8 from [40], p.86, or equation 4.4.5of [40], p.61, where Γ( µ ) should be used instead of Γ( α ), in the form Z τ τ β − E α,β [ λ τ α ] ( τ − τ ) µ − dτ = Γ( µ ) τ β + µ − E α,β + µ [ λ τ α ] , (85)where β > µ >
0. In equation (85) we have µ = α , β = k + 1, where the parameter µ can be considered as a positive integer, or as a positive real number.As a result, we can formulate the following statement by using equation (85) with µ = α and β = k + 1 for solution (84). Statement 6.
In the case of constant external electromagnetic field ( F = const ), solution of fractional differ-ential equation (43) of the order n − < α ≤ n for the operator x µ ( τ ) has the form x ( τ ) = n − X k =0 τ k k ! x ( k ) (0) − e n − X k =0 τ α + k E α,α + k +1 [Λ τ α ] π ( k ) (0) , (86)14 here Λ = − eF . Note that using equation (85), we can easily generalize solutions to the case of differentmemory fading parameters for variables x µ ( τ ) and π ν ( τ ) (for example, when the order of theCaputo fractional derivative in equation (79) is equal to µ instead of α ).For 0 < α ≤
1, expression (86) takes the form x ( τ ) = x (0) − eτ α E α,α +1 [Λ τ α ] π (0) . (87)Using that Λ = − eF , equation (87) can be rewritten as x ( τ ) = x (0) − eτ α E α,α +1 [ − eF τ α ] π (0) , (88)where 0 < α ≤
1. Using equations 1.8.2 in [25], p.40, and 1.8.19 in [25], p.42, (see also equation4.2.1 in [40], p.57) in the form E , [ z ] = E [ z ] = e z , E , [ z ] = e z − z (89)the proposed solutions (87) give the standard solutions (48) for the case α = 1.Equations (71) and (86) describe the Dirac particle with power-law memory as an openquantum system. x µ ( τ ) The asymptotic behavior of x µ ( τ ), which is described by expression (86), can be obtained byusing equations (72) and (73) for β = α + k .Using the notations X s ( τ ) = ( N x ) s ( τ ) = N sµ x µ ( τ ) , Π s ( τ ) = ( N π ) s ( τ ) = N sµ π µ ( τ ) , (90)we can formulate the following statement. Statement 7.
For the eigenvalue λ s = b > , where b is defined by (66) and < α < , equations (72) givesthe asymptotic behavior of the operator x µ ( τ ) in the form X s ( τ ) = n − X k =0 τ k k ! X ( k ) s (0) − e n − X k =0 (cid:16) λ − ( α + k ) /αs α exp (cid:0) λ /αs τ (cid:1) − m X j =1 λ − js τ k − α ( j − Γ( k + 1 − α ( j − ( k ) s (0) + O (cid:18) τ α m − k (cid:19)(cid:17) (91) for τ → ∞ , where λ s is a positive real number.For the eigenvalues λ s = ± ia and λ s = − b < , where a and b are defined by (65) , (66) ,and < α < , equations (73) gives the asymptotic behavior of x µ ( τ ) in the form X s ( τ ) = n − X k =0 τ k k ! X ( k ) s (0) − e n − X k =0 − m X j =1 λ − js τ k − α ( j − Γ( k + 1 − α ( j − ( k ) s (0) + O (cid:18) τ α m − k (cid:19)! (92)15 or τ → ∞ , where λ s is a negative real number or λ is an imaginary number. The difference between the formulas (91) and (92) is the absence of the term with theexponent exp (cid:16) λ /αs τ (cid:17) . In this article, a generalization of the theory of Dirac particle in external electromagnetic field issuggested. In this generalization we take into account an interaction of the Dirac particle withenvironment. In the standard approach the interaction of the Dirac particle with environmentis not considered. The interaction with environment (an openness of the system) is describedas proper time non-locality by the memory function. We use the Fock-Schwinger proper timemethod to have covariant description of dynamics and derivatives of non-integer orders to havenon-locality in proper time. In the proposed theory we take into account that the behavior ofthe Dirac particle at proper time τ can depend not only at the present proper time τ , but also onthe history of changes at τ < τ . In this case the Dirac particle is considered an open quantumsystem, dynamics of which can be characterized as non-Markovian evolution. We demonstratethe violation of the semigroup property of dynamic maps that is a characteristic property ofdynamics with memory. The fractional differential equation, which describes the Dirac particlewith memory, and the expression of its exact solution are suggested. The asymptotic behaviorof the proposed solutions is described.Let us briefly describe three possible generalizations of the proposed model.To describe the Dirac particle in the case of an explicit dependence of the electromagneticfield strength tensor (or the matrix Λ) on the proper time, it is necessary to use the general-ization of the time-ordered product and the ordered exponential proposed in [18, 45] for theprocesses with memory.The proposed model of the Dirac particle with proper time nonlocality (memory) can begeneralized to other type of fading memory by using other type fractional derivatives andintegrals.In order to have a consistent description of the interaction of a particle with its environment,it is necessary to take into account the quantum theory of open quantum systems [46, 47, 48] andfractional generalizations of quantum Markov equations proposed in [49, 50, 51, 52, 53, 54, 55].16 eferences [1] Fock, V.E. ”Proper time in classical and quantum mechanics”. Proceedings of the USSRAcademy of Sciences. Physics Series. [Izvestiya AN SSSR. Seriya fizika]. . No.4-5.p.551-568. [in Russian][2] Fock, W.A. ”Die Eigenzeit in der klassischen und in der Quantenmechanik”.
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