20 Feb 2020 H Okshigrpoe Iemto 7 ,9.I H Tnadm See Standard Example, the (For in ﬁeld Electromagnetic T 9]

Home , Heisenberg picture

arXiv:2002.08996v1 [quant-ph] 20 Feb 2020 h okShigrpoe iemto 7 ,9.I h tnadm see standard example, the (for In field electromagnetic T 9]. external 8, 11]. the t Julia [7, 10, in proper and [7, described method to 5] form is time respect covariant 4, proper with the 3, Fock-Schwinger in equations 2, the equations differential [1, Heisenberg the Fock and case, A. Hamilton this Vladimir In us by by [6]. proposed appr described This be was also equation. approach can Dirac particles This the Dirac by The described picture. be Schrodinger can particles Dirac The Introduction 1 memo with method time process equat proper differential Schwinger dynamics; fractional environment; fractional with interaction particle; tems; particles Dirac and fields Keywords: of theory general in m topics statistical Other quantum 11.90.+t systems; PACS open methods Decoherence; calculus 03.65.Yz PACS fractional and Perturbation 45.10.Hj PACS hsc etr .22.AtceI:160.DI 10.1016/j.phys DOI: 126303. ID: Article 2020. A. Letters Physics ocwAito nttt Ntoa eerhUniversity) Research (National Institute Aviation Moscow r ugse.Teaypoi eairo h rpsdsolu proposed the expr of fract the behavior The and asymptotic memory, The with time. suggested. particle proper are Dirac to the respect describes c time which with a proper orders Fock-Schwinger is non-integer the maps use of dynamic We of memory. property with time d semigroup with dynamics finite system the can quantum on of time open changes violation an of proper The considered history be at the can particle particle on Dirac also the but of function time, behavior memory present account the the into by that take described account we is generalization which the environment, In with proposed. is field ia atcewt eoy rprTm Non-Locality Time Proper Memory: with Particle Dirac eeaiaino h tnadmdlo ia atcei e in particle Dirac of model standard the of generalization A 2 oooo ocwSaeUiest,Mso 191 Russia 119991, Moscow University, State Moscow Lomonosov aut IfrainTcnlge n ple Mathematic Applied and Technologies ”Information Faculty 1 kblsnIsiueo ula Physics, Nuclear of Institute Skobeltsyn -al [email protected] E-mail: aiyE Tarasov E. Vasily Abstract 1 1 , 2 in sdescribed. is tions neatoso hsparticle this of interactions o;fatoa eiaie Fock- derivative; fractional ion; ocw159,Russia 125993, Moscow , oa ieeta equation, differential ional sino t xc solution exact its of ession hsfnto ae into takes function This . o-akva dynamics. non-Markovian nevl nti aethe case this In interval. aatrsi rpryof property haracteristic tra electromagnetic xternal ethods ehdadderivatives and method pn o nya the at only not epend n h esnegpicture. Heisenberg the ing 6,[] .0-0,ad[9], and p.100-104, [7], [6], m r sdt aethe have to used are ime leta.2020.126303 y pnqatmsys- quantum open ry; ahcrepnst the to corresponds oach dlteDrcparticle Dirac the odel i prahi called is approach his .Shigrin Schwinger S. n s”, p.728-732). The standard theory of the Dirac particles does not take into account an interaction between this particle and the environment. We proposed a generalization of the Heisenberg equations for the Dirac particle in the external electromagnetic field. In this generalization we take into account an interaction of this particle with environment. The interaction with environment can be described by the memory function [12], that is also called the response function. Therefore the proposed model can 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 will use the Fock-Schwinger proper time method to preserve the relativistically covariant form of Heisenberg equations. The proposed Heisenberg equations are fractional differential equation with derivatives of non-integer orders with respect to proper time. The exact expressions of the solution of the generalized Heisenberg equations are suggested for particular case of the external electromagnetic field (the constant uniform field). The standard solution for the absence of memory is the special case of the proposed solution. In this article also suggests the asymptotic 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 the assumption that the Dirac particle is not open system. In the standard approach the interaction of Dirac particle with environment is not considered. The first description of physical system with memory was suggested by Ludwig Boltzmann in 1874 and 1876. He took into account that the stress at time t can depend on the strains not only at the present time t, but also on the history of changes at t1 < t. Boltzmann also proposed the linear position principle for memory and the memory fading principle. The linear position principle states that total strain is a linear sum of strains, which have arisen in the media at t1 < t. The principle of memory fading states that the increasing of the time interval lead to a decrease in the corresponding contribution to the stress at time t. The Boltzmann theory has been significantly developed in the works of the Vito Volterra in 1928 and 1930 in the form of the heredity concept and its application to physics. Then the memory effects in natural and social sciences are considered in different works (for example, see [13]-[21]). As example, we can note the thermodynamics of materials with fading memory [13, 14], the hereditary solid mechanics [15], the theory of wave 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], the fractional 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 to describe 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 considered time. An effective and powerful tool for describing memory effects is the fractional calculus of derivatives and integrals of non-integer orders [22, 23, 24, 25, 26]. These operators have a long history of more than three hundred years [27, 28, 29, 30, 31]. We should note that the characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and 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 derivatives of non-integer orders. These non-standard mathematical properties allow us to describe non-

2 standard processes and phenomena associated with non-locality and memory [20, 21]. On the other hand, these non-standard properties lead to difficulties [37] in sequential constructing a fractional generalizations of standard models. In article [37], we show how problems arise when building fractional generalizations of standard models. In this paper, we proposed a generalization of the standard model of the Dirac particle in the external electromagnetic field by taking into account the interaction with an environment that is described by memory function. The fractional nonlinear differential equation, which describes the proposed model, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described.

2 Dirac particle in external electromagnetic ﬁeld

The kinematics of the Dirac particle can be described by the commutation relations

[xµ,pν]= igµν, (1) − where pµ = i∂µ with where µ =0, 1, 2, 3, and < x x′ >= δ4(x x′). (2) | − We use the notation of the book [7],

g00 = g11 = g22 = g33 =1, (3) − − − 0 k and A = A0, A = Ak. Let us consider the− standard function (x, p) in the coordinate representation H e =(p eA)2 m2 σ F µν , (4) H − − − 2 µν where F µν is the antisymmetric ﬁeld strength tensor

F µν = ∂µAν ∂ν Aµ, (5) − and 1 σµν = i[γµ,γν]. (6) 2 The electromagnetic field strength tensor F µν can be expressed through the components of the electric field (Ek) and magnetic field Bk, where k =1, 2, 3, in the form

0 E1 E2 E3 E1 −0 −B3 −B2 F µν =  −  . (7) E2 B3 0 B1 −  E3 B2 B1 0   −    The idea of the Fock-Schwinger proper time method is to consider as Hamiltonian that describes the proper time evolution of some system. H

3 Using the operator π = p eA , (8) µ µ − µ the standard Heisenberg equations for xµ(τ) and πµ(τ) of the Dirac particle in the external electromagnetic ﬁeld can be written in the form

dx (τ) µ = i[ , x (τ)], (9) dτ H µ dπ (τ) µ = i[ , π (τ)], (10) dτ H µ where τ is proper time of the Dirac particle. Using the Hamiltonian e = π2 m2 σ F µν , (11) H − − 2 µν and the commutation relations [πµ, πν]= ieF µ,ν , (12) − [xµ, πν]= igµν, (13) − [xµ, xν]=0, (14) the Heisenberg equations of the Dirac particle takes the form

dx (τ) µ = 2π , (15) dτ − µ dπ (τ) e µ = 2eF πν ie∂ν F ∂ F σνρ. (16) dτ − µν − µν − 2 µ νρ Note that the coeﬃcient ”-2” in equations (15) and (16) is due to the form of the operator (8) the Hamiltonian (11) that are standard for the theory of Dirac particle in external electromagnetic ﬁeld (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 xµ(τ) and πµ(τ) can be described by as unitary evolution in the form

∗ ∗ xµ(τ)= U (τ)xµ(0)U(τ), πµ(τ)= U (τ)πµ(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 the standard model. Dynamics with memory (with proper time non-locality) cannot be considered as the unitary evolution. The evolution of fractional dynamic systems with memory cannot be described by unitary operators. The dynamics with memory is non-Markovian in general.

4 3 Dirac particle with memory in external electromagnetic ﬁeld

The first description of physical system with memory was suggested by Ludwig Boltzmann in 1874 and 1876. He takes into account that the stress at time t can depend on the strains not only at the present time t, but also on the history of changes at t1 < t. The Boltzmann theory has been significantly developed in the works of the Vito Volterra in 1928 and 1930 in the form of the heredity concept and its application to physics. Then the memory effects in natural and social 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 τ x = i[ , x (τ)] = 2π , (18) E0;M,n{ µ} H µ − µ e τ π = i[ , π (τ)] = 2eF πν ie∂ν F ∂ F σνρ, (19) E0;M,n{ µ} H µ − µν − µν − 2 µ νρ where the symbol τ denotes a certain operator that allows us to find the values of (x (τ), π (τ)) E0;M,n µ µ for any proper time τ, if the values (xµ(τ1), πµ(τ1)) are known for τ1 [0, τ]. We can say that τ ∈ 0;M,n is an operator, which is a mapping from one space of functions to another. In this paper, E τ we consider linear operators 0;M,n of a special kind, which is called the Volterra operator that is defined by the expression E τ τ = M(τ τ ) (n)(τ )dτ , (20) E0;M,n{Aµ} − 1 Aµ 1 1 Z0 (n) n n where µ (τ)= d µ(τ)/dτ is the derivative of the positive integer order n N with respect to properA time. TheA function M(τ τ ), which characterizes an interaction∈ of the particle − 1 with environment, is called the memory function. In statistical mechanics it is also called the response function. It is obvious that not every operators τ of the form (20) can be used to describe a E0;M,n memory. Restrictions on the memory functions and consideration of the examples of memory function are discussed in paper [38]. In paper [38], some general restrictions that can be imposed on the properties of memory are described. In addition, to the causality principle, these restrictions include the following three 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 of memory recovery). For example, in the operator (20) we can consider the memory function of the power-law form 1 M(τ τ )= (τ τ )n−α−1, (21) − 1 Γ(n α) − 1 − τ where α > 0. In the case of integer values of α, the operator 0;M,n µ is the derivative of integer order E {A } dn (τ) τ = Aµ . (22) E0;M,n{Aµ} dτ n 5 In the case of positive real values of α R , the operator τ is the Caputo fractional ∈ + E0;M,n{Aµ} derivative of the order α> 0. The Caputo fractional derivative [25] is defined by the equation

1 τ α (τ)= (τ τ )n−α−1 (n)(τ )dτ , (23) Dτ;0+Aµ Γ(n α) − 1 A 1 1 − Z0 where n = [α] + 1 for non-integer values of α, and n = α for α N, (see equation 2.4.3 in [25], (n) ∈ p.91), Γ(α) is the Gamma function, and µ (τ) is the derivative of the integer order n. It is A assumed that the function µ(τ) has derivatives up to the (n 1)th order, which are absolutely continuous functions on aA ﬁnite time interval [25]. If α =−n N and the usual derivative (n) ∈ (n) µ (τ) of order n exists, then the Caputo fractional derivative (23) coincides with µ (τ) (for example,A see equation 2.4.14 in [25], p.92, proof in [24], p.79), i.e. the condition (22)A holds. For the power-law memory function (20), equations (18) and (19) take the form

α x (τ)= i[ , x (τ)] = 2π , (24) Dτ,0+ µ H µ − µ e α π (τ)= i[ , π (τ)] = 2eF πν ie∂ν F ∂ F σνρ, (25) Dτ,0+ µ H µ − µν − µν − 2 µ νρ where α is the Caputo fractional derivative of the order α> 0. In the general case, we can Dτ,0+ consider the diﬀerent orders in equations (24) and (25). Here we consider τ as dimensionless variable 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 as an approximation of the equations with generalized memory functions. In the paper [39], using the generalized Taylor series in the Trujillo-Rivero-Bonilla form for the memory function, we proved that the equations with memory functions can be represented through the Riemann- Liouville fractional integrals (and the Caputo fractional derivatives) of non-integer orders for wide class of the memory functions. Note that the proposed generalized Heisenberg equations (24) and (25) with α > 1 can be obtained from the integral equations

dx (τ) τ µ = 2 M (τ τ )π (τ )dτ , (26) dτ − I − 1 µ 1 1 Z0 τ dπµ(τ) ν ν e νρ = MI (τ τ1) 2eFµν π (τ1) ie∂ Fµν ∂µFνρσ dτ1, (27) dτ 0 − − − − 2 Z where 1 M (τ τ )= (τ τ )β−1. (28) I − 1 Γ(β) − 1 This possibility is based on the fact that the Caputo fractional derivative is the left inverse operator for the Riemann-Liouville fractional integral (see equation of the Lemma 2.21 of [25], p.95), the has the form Dβ Iβ (τ)= (τ), (29) τ,0+ τ,0+Aµ Aµ 6 if β > 0, and (τ) L∞(0, T ) or (τ) C[0, T ]. Aµ ∈ Aµ ∈ 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 time nonlocality) 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> 0. (c) The standard equations (15) and (16) of the Dirac particle without memory are special cases of equations (24) and (25) for α =1. Note that the proper time nonlocality and memory effects can be caused by the interaction of the particle with environment [12].

4 Violation of semi-group property: Non-Markovian dynamics

In this section, we discuss the characteristic property of dynamics with memory. Let us consider the fractional diﬀerential equations

Dα µ (τ)=Λµ ν(τ), (30) τ;0+A ν A µ µ where 0 < α 1. For example, we can consider (τ) = π (τ) and Λµν = 2eF ν. Using ≤ µ µ A { } − the matrix notation Λ = Λ ν and (τ)= A (τ) , equation (30) can be represented as the matrix equation { } A { } Dα (τ)=Λ (τ). (31) τ;0+A A The Cauchy-type problem for equation (31), in which the initial condition is given at the time τ = 0 by (0), has solution that can be represented in the form A (τ) = Φ (α) (0), (τ 0), (32) A τ A ≥ where α Φτ (α)= Eα [Λτ ] . (33) α Here Eα [Λτ ] is the Mittag-Leﬄer function [40] with the matrix argument ∞ τ αk E [Λτ α]= Λk. (34) α Γ(αk + 1) Xk=0 The operator Φτ (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 Φτ (α) on operator algebra of quantum observables

α Dτ,0+Φτ (α) (τ)=ΛΦτ (α). (35) 7 The set Φ (α) τ > 0 , can be called a quantum dynamical groupoid [44]. Note that the { τ | } following properties are realized

Φ (α)I = I, (Φ (α) )∗ = Φ (α) (36) τ τ A τ A for self-adjoint operators ( ∗ = ), and A A A

lim Φτ (α)= I, (37) τ→0+ where I is an identity operator (I = ). As a result, the operators Φτ (α), τ > 0, are real and unit preserving maps on operatorA algebraA of quantum observables. For α = 1, we have Φ (1) = E [Λτ] = exp Λτ . (38) τ 1 { } The operators Φτ = Φτ (1) form a semigroup such that

Φτ Φs = Φτ+s, (τ,s> 0), Φ0 = I. (39)

This property holds since

exp Λτ exp Λs = exp Λ(τ + s) . (40) { } { } { } For α N we have 6∈ E [Λτ α] E [Λsα] = E [Λ(τ + s)α]. (41) α α 6 α Therefore the semigroup property [41, 42, 43] is not satisﬁed for non-integer values of α:

Φ (α)Φ (α) = Φ (α), (τ,s> 0). (42) τ s 6 τ+s As a result, we can formulate the following statement. Statement 2. The operators Φτ (α), which describe quantum dynamics of Dirac particle with memory, cannot form 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 with6∈ memory describe non-Markovian evolution. Note that the violation of this semigroup property is a characteristic property of dynamics with memory.

5 Dynamics with memory in constant electromagnetic ﬁeld

5.1 Equations for constant electromagnetic field Let us consider a constant electromagnetic field. The equations, which described Dirac particle with power-law memory in a constant field, can be written as

α xµ(τ) (t)= 2πµ(τ), (43) Dτ,0+ − 8 α πµ(τ) (t)= 2eF µ πν(τ). (44) Dτ,0+ − ν For α = 1 equations (43) and (44) are reduced to dxµ(τ) = 2πµ(τ), (45) dτ − dπµ(τ) = 2eF µ πν. (46) dτ − ν Equations (45) and (46) describe the Dirac particle without memory in constant electromagnetic ﬁeld. µ µ µ Using matrix notations x = x , π = π , F = F ν , equations (45) and (46) can be integrated, and we get the well-known{ } solutions{ } [7] in the{ form}

π(τ)= e−2eF τ π(0), (47)

e−2eF τ 1 x(τ)= x(0) + − π(0). (48) eF Let us obtain solutions of the fractional diﬀerential equations (43) and (44).

5.2 Solution of fractional diﬀerential equation for πµ(τ) Equation (44) can be rewritten in the form

α πµ(τ) (t)= 2eF µ πν, (49) Dτ,0+ − ν where n 1 < α n. Using µ(τ) = πµ(τ), and Λµ = 2eF µ , equation (49) is written in − ≤ A ν − ν the form Dα µ (τ)=Λµ ν(τ). (50) τ;0+A ν A µ µ Using the matrix notation Λ = Λ ν and (τ) = A (τ) , equation (50) can be represented as the matrix equation { } A { } Dα (τ)=Λ (τ). (51) τ;0+A A To solve equation (51), we can use Theorem 5.15 of book [25], p.323. The solution of equation (51) has the form

n−1 (τ)= τ k E [Λ τ α] (k)(0), (52) A α,k+1 A Xk=0 where n 1 <α n, and E [z] is the two-parameter Mittag-Leﬄer function [25, 40] that is − ≤ α,β deﬁned by the expression ∞ τ kα E [Λ τ α]= Λk, (53) α,β Γ(αk + β) Xk=0

9 where Γ(α) is the gamma function, α > 0, and β is an arbitrary real number. For 0 <α< 1, solution (52) takes the form (τ)= E [Λτ α] (0). (54) A α,1 A z For α = 1, we can use E1,1[z]= e . Then solution (52) gives the equation

(τ) = exp Λτ (0). (55) A { }A 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 πµ(τ) has the form

n−1 π(τ)= τ k E [ 2eF τ α] π(k)(0), (56) α,k+1 − Xk=0 µ where F = F ν and n 1 <α n. Expression{ (56)} describes− dynamics≤ of Dirac particle with memory in constant electromagnetic ﬁeld. For 0 <α 1, equation (56) takes the form ≤ π(τ)= E [ 2eF τ α] π(0). (57) α,1 − For α = 1, solution (56) takes the standard form (47), i.e., π(τ)= e−2eF τ π(0).

5.3 Eigenvalues of electromagnetic tensor for constant ﬁelds Let us consider the equation (56) that has the form

n−1 k α (k) π(τ)= τ Eα,k+1 [Λ τ ] π (0), (58) Xk=0 where Λ is the matrix 0 E1 E2 E3 1 3 2 µ E 0 B B Λ= 2eF ν = 2e  −  . (59) {− } − E2 B3 0 B1 −  E3 B2 B1 0   −    µ µρ To get expression (59), we use that F ν = F gρν and

10 0 0 µν 0 1 0 0 g = gµν =  −  , (60) { } { } 0 0 1 0 −  0 0 0 1   −    10 that gives the equations F µ = F µ0, F µ = F µk (k =1, 2, 3; µ =0, 1, 2, 3). (61) 0 k − µ µ µ We can use the diagonalization of the real skew-symmetric matrix Λν =2eFν , where Fν = gµρF , if all eigenvalues of this matrix are nonzero. Note that the 4 4 matrix Λ = Λµ is ρν × { ν } not skew-symmetric matrix. An n n matrix Λ over a ﬁeld R is diagonalizable if and only if there exists a basis of Rn 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 called −1 a modal matrix for Λ, such that L = NΛN is a diagonal matrix. The diagonal elements λk, k = 1, 2, ..., n, of this matrix L = diagonal λ0,λ1,λ2,λ3 are the eigenvalues of Λ. An n n matrix Λ is diagonalizable if it has n distinct{ eigenvalues,} i.e. if its characteristic polynomial× has n distinct roots. µ µρ It is known that the eigenvalues of the matrix F = F ν = F gρν, are the roots of the fourth- ∗ µν µν 2 2 degree polynomial with invariants (1/2)Fµν F = (E,~ B~ ) and (1/2)FµνF = (B E ) in the coeﬃcients (for example see [56, 57]). Therefore, the eigenvalues of the matrix Λ− = 2eF − are the roots of the equation 1 λ4 + Sλ2 P 2 =0, (62) − 4 where S and P are the invariants P =2e2F ∗F µν =2e2(E,~ B~ ), S =2e2F F µν =2e2 B2 E2 , (63) µν µν − The set of eigenvalues λs, where s = 0, 1, 2, 3 consists of two pairs, one real and the other imaginary λ +b, b;+ia, ia , (64) s ∈{ − − } where a and b are the scalar functions of invariants (63) in the form

1 a = S + √S2 + P 2 , (65) 2 r 1 b = S + √S2 + P 2 . (66) r2 − For details see [56, 57]. It should be noted that the matrix Λ can be represented in the form Λ= N −1 LN, (67) where N = columns ξ ,ξ ,ξ ,ξ , (68) { 0 1 2 3} λ0 0 0 0 0 λ1 0 0 L = diagonal λ0,λ1,λ2,λ3 =   , (69) { } 0 0 λ2 0  0 0 0 λ   3    11 and λµ are the complex coeﬃcients such that

Λξs = λsξs. (70)

Here ξs, s =0, 1, 2, 3, are eigenvectors and λs are eigenvalues of the matrix Λ, respectively. In equation (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 with proper time, the solution (58) of fractional differential equation (44) (or (49)) of the order n 1 <α n for the operator πµ(τ) can be written in the form − ≤ n−1 k −1 α (k) π(τ)= τ N Eα,k+1 [L τ ] Nπ (0). (71) Xk=0 where Eα,β[z] is the two-parametric Mittag-Leffler function (34). Equation (71) describes the evolution of the operator πµ(τ) of the Dirac particle with power- law memory that is considered as an open quantum system.

5.4 Asymptotic behavior of πµ(τ) Let us describe the asymptotic behavior of solution (56) at τ . To describe the behavior of the two-parameter Mittag-Leffler function E [λ τ α] at τ →∞, we can use equations 1.8.27 α,k+1 →∞ and 1.8.28 of book [25], p.43. For arg(λ) m and 0 <α< 2, where | | ≤ 0 πα < m < min π,πα , 2 0 { } we have the asymptotic behavior of the two-parameter Mittag-Leffler function in the form λ−β/α m λ−j 1 1 E [λ τ α]= τ −β exp λ1/α τ + O . (72) α,β+1 α − Γ(β +1 α j) τ αj τ α (m+1) j=1 − X For m < arg(λ) π and 0 <α< 2, we have the asymptotic behavior in the form 0 | | ≤ m λ−j 1 1 E [λ τ α]= + O . (73) α,β+1 − Γ(β +1 α j) τ αj τ α (m+1) j=1 X − 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 λs = b> 0, where b is defined by (66) (i.e. the case λ = Re[λ] > 0 with Im[λ]=0), and 0 <α< 2, we should use equation

λ−β/α m λ−j 1 1 E [λ τ α]= τ −β exp λ1/α τ + O (74) α,β+1 α s − Γ(β +1 α j) τ αj τ α (m+1) j=1 − X 12 for τ , where λ is a positive real number. →∞ For the eigenvalues λs = ia and λs = 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] < 0 with Im[λ] = 0), and 0 <α< 2, we should use equation 6

m λ−j 1 1 E [λ τ α]= s + O (75) α,β+1 − Γ(β +1 α j) τ αj τ α (m+1) j=1 X − for τ , where λ is a negative real number or λ is an imaginary number. The diﬀerence between→ 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 model with power-law memory fading parameter 0 <α< 1and1 <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

s µ Πs(τ)=(Nπ)s(τ)= N µπ (τ), (76) which characterizes the Dirac particle with memory in a constant ﬁeld, is described by the following equations. For the eigenvalues λ = ia and λ = b< 0, where a and b are deﬁned s ± s − by (65), (66) and 0 < n 1 <α

For the eigenvalue λs = b> 0, where b is deﬁned by (66) and 0 < n 1 <α

− n 1 λ−k/α m τ k−αjλ−j 1 Π (τ)= exp λ1/α τ s Π(k)(0) + O . (78) s α s − Γ(k +1 α j) s τ α (m+1)−k k=0 j=1 − ! X X Expressions (77) and (78) describe the behavior of π(τ) at τ for the Dirac particle with power-law memory in constant external electromagnetic ﬁeld.→ ∞

5.5 Solution of equation for xµ(τ) To solve equation (43), we can use Theorem 5.15 of book [25], p.323. Fractional diﬀerential equation (43) can be written in the form

Dα µ (τ)= Bµ(τ), (79) τ,0A

13 where µ(τ)= xµ(τ), Bµ(τ)= 2πµ(τ) and n 1 <α n. If Bµ(τ) are continuous functions A − − ≤ defined on the positive semiaxis (τ > 0), then equations (79) has the solution (for details see Lemma 2.22 and equation 2.4.42 in [25], p.96) in the form − n 1 k t µ τ µ(k) 1 α−1 µ (τ)= (0) + (τ τ1) B (τ1)dτ1. (80) A k! A Γ(α) 0 − Xk=0 Z As a result, we get the equation − n 1 k t µ τ µ(k) 2e α−1 µ x (τ)= x (0) (τ τ1) π (τ1)dτ1. (81) k! − Γ(α) 0 − Xk=0 Z For 0 <α< 1, we have τ 2e − xµ(τ) xµ(0) = − (τ τ )α 1 πµ(τ )dτ . (82) − Γ(α) − 1 1 1 Z0 Note that equation (82) can be written through the Riemann-Liouville fractional integral (the definition of this operator is given in [25], p.69-70) Using the matrix notation x(τ)= xµ(τ) , and π(τ)= πµ(τ) equation takes the form { } { } − n 1 k t τ (k) 2e α−1 x(τ)= x (0) (τ τ1) π(τ1)dτ1. (83) k! − Γ(α) 0 − Xk=0 Z Substitution of the expression (56) of the solution for π(τ) into expression (83) of x(τ) gives the equation − − n 1 k n 1 τ τ (k) 2e α−1 k α (k) x(τ)= x (0) (τ τ1) τ1 Eα,k+1 [Λ τ1 ] π (0)dτ1. (84) k! − Γ(α) 0 − Xk=0 Xk=0 Z µ where Λ = 2eF = 2eF ν . To calculate− integral{− of (84),} we can use equation 4.10.8 from [40], p.86, or equation 4.4.5 of [40], p.61, where Γ(µ) should be used instead of Γ(α), in the form τ − − τ β 1 E [λ τ α] (τ τ )µ 1 dτ = Γ(µ) τ β+µ−1 E [λ τ α] , (85) 1 α,β 1 − 1 1 α,β+µ Z0 where β > 0 and 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 differential equation (43) of the order n 1 <α n for the operator xµ(τ) has the form − ≤ − − n 1 τ k n 1 x(τ)= x(k)(0) 2e τ α+kE [Λ τ α] π(k)(0), (86) k! − α,α+k+1 Xk=0 Xk=0 14 where Λ= 2eF . − Note that using equation (85), we can easily generalize solutions to the case of different memory fading parameters for variables xµ(τ) and πν(τ) (for example, when the order of the Caputo 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) 2eτ αE [Λ τ α] π(0). (87) − α,α+1 Using that Λ = 2eF , equation (87) can be rewritten as − x(τ)= x(0) 2eτ αE [ 2eF τ α] π(0), (88) − α,α+1 − where 0 <α 1. Using equations 1.8.2 in [25], p.40, and 1.8.19 in [25], p.42, (see also equation 4.2.1 in [40],≤ p.57) in the form ez 1 E [z]= E [z]= ez, E [z]= − (89) 1,1 1 1,2 z 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 open quantum system.

5.6 Asymptotic behavior of xµ(τ)

The asymptotic behavior of xµ(τ), which is described by expression (86), can be obtained by using equations (72) and (73) for β = α + k. Using the notations

s µ s µ Xs(τ)=(Nx)s(τ)= N µx (τ), Πs(τ)=(Nπ)s(τ)= N µπ (τ), (90) we can formulate the following statement. Statement 7. For the eigenvalue λs = b> 0, where b is deﬁned by (66) and 0 <α< 2, equations (72) gives the asymptotic behavior of the operator xµ(τ) in the form

− − − n 1 τ k n 1 λ (α+k)/α X (τ)= X(k)(0) 2e s exp λ1/α τ s k! s − α s − k=0 k=0 X X (91) m λ−jτ k−α (j−1) 1 s Π(k)(0) + O Γ(k +1 α (j 1)) s τ α m−k j=1 X − − for τ , where λ is a positive real number. →∞ s For the eigenvalues λs = ia and λs = b < 0, where a and b are deﬁned by (65), (66), and 0 <α< 2, equations (73)±gives the asymptotic− behavior of xµ(τ) in the form

− − n 1 τ k n 1 m λ−jτ k−α (j−1) 1 X (τ)= X(k)(0) 2e s Π(k)(0) + O (92) s k! s − − Γ(k +1 α (j 1)) s τ α m−k j=1 ! Xk=0 Xk=0 X − − 15 for τ , where λ is a negative real number or λ is an imaginary number. →∞ s The diﬀerence between the formulas (91) and (92) is the absence of the term with the 1/α exponent exp λs τ . 6 Conclusion

In this article, a generalization of the theory of Dirac particle in external electromagnetic field is suggested. In this generalization we take into account an interaction of the Dirac particle with environment. In the standard approach the interaction of the Dirac particle with environment is not considered. The interaction with environment (an openness of the system) is described as proper time non-locality by the memory function. We use the Fock-Schwinger proper time method to have covariant description of dynamics and derivatives of non-integer orders to have non-locality in proper time. In the proposed theory we take into account that the behavior of the Dirac particle at proper time τ can depend not only at the present proper time τ, but also on the history of changes at τ1 < τ. In this case the Dirac particle is considered an open quantum system, dynamics of which can be characterized as non-Markovian evolution. We demonstrate the violation of the semigroup property of dynamic maps that is a characteristic property of dynamics with memory. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of 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 electromagnetic field strength tensor (or the matrix Λ) on the proper time, it is necessary to use the generalization of the time-ordered product and the ordered exponential proposed in [18, 45] for the processes with memory. The proposed model of the Dirac particle with proper time nonlocality (memory) can be generalized to other type of fading memory by using other type fractional derivatives and integrals. 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] and fractional generalizations of quantum Markov equations proposed in [49, 50, 51, 52, 53, 54, 55].

16 References [1] Fock, V.E. ”Proper time in classical and quantum mechanics”. Proceedings of the USSR Academy of Sciences. Physics Series. [Izvestiya AN SSSR. Seriya ﬁzika]. 1937. No.4-5. p.551-568. [in Russian] [2] Fock, W.A. ”Die Eigenzeit in der klassischen und in der Quantenmechanik”. Physikalische Zeitschrift der Sowjetunion. 1937. Bd.12. No.4. S.404-425. [in German] [3] Fock, V.E. ”Proper time in classical and quantum mechanics”. in Fock, V.E. Works on Quantum Field Theory. Leningrad. University of Leningrad Publishing House, 1957. 160 p. pp.141-158. [in Russian] [4] Fock, V.E. ”Proper time in classical and quantum mechanics”. Translated by D.H.Delphenich URL: http : //www.neo classical physics.info/uploads/3/4/3/6/34363841/fock wkb nd irac.pdf − − − a d [in English] [5] ”V.A. Fock - Selected works Quantum Mechanics and Quantum Field Theory”. Edited by L.D. Faddeev, L.A. Khalﬁn, I.V. Komarov. Boca Raton, London, New York: Chapman and Hall/CRC, 2004. 562 pages. ISBN: 0-415-30002-9 [6] Schwinger, J. ”On gauge invariance and vacuum polarization”. Physical Review. 1951. Vol.82. No.5. P.664-679. DOI: 10.1103/PhysRev.82.664 [7] Itzykson, C.; Zuber, J.B. Quantum Field Theory. New York: McGraw-Hill, 1980. 705 p. [8] Itzykson, C.; Zuber, J.B. Quantum Field Theory. New York: Dover Publications, 2006. 752 p. ISBN: 978-0486445687 [9] Schwartz, M.D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. 850 p. ISBN: 978-1-107-03473-0 [10] Tarasov, V.E. ”Relativistic non-Hamiltonian mechanics”. Annals of Physics. 2010. Vol.325. No.10. P.2103-2119. DOI: 10.1016/j.aop.2010.06.011 [11] Tarasov, V.E. ”Fractional dynamics of relativistic particle”. International Journal of Theoretical Physics. 2010. Vol.49. No.2. P.293-303. DOI: 10.1007/s10773-009-0202-z (arXiv:1107.5749) [12] Tarasov, V.E. ”The fractional oscillator as an open system”. Central European Journal of Physics. 2012. Vol.10. No.2. P.382-389. DOI: 10.2478/s11534-012-0008-0 [13] Day, W.A. The Thermodynamics of Simple Materials with Fading Memory. Berlin: Springer-Verlag, 1972. 134 p. [14] Amendola, G.; Fabrizio, M.; Golden, J.M. Thermodynamics of Materials with Mem- ory: Theory and Applications. Springer Science and Business Media, 2011. 576 p. DOI: 10.1007/978-1-4614-1692-0 [15] Rabotnov, Yu.N. Elements of Hereditary Solid Mechanics. Moscow: Mir Publishers, 1980. 387 p. [16] Lokshin, A.A.; Suvorova, Yu.V. Mathematical Theory of Wave Propagation in Media with Memory. Moscow: Moscow State University. 1982. 152 p. [in Russian] [17] Mainardi, F. Fractional Calculus and Waves Linear Viscoelasticity: An Introduction to Mathematical Models. London: Imperial College Press, 2010. 368 p.

17 [18] Tarasov, V.E.; Tarasova, V.V. ”Time-dependent fractional dynamics with memory in quantum and economic physics”. Annals of Physics. 2017. Vol.383. P.579–599. DOI: 10.1016/j.aop.2017.05.017 [19] Memory Function Approaches to Stochastic Problems in Condensed Matter. Edited by M.W. Evans, P. Grigolini, G.P. Parravicini. New York: Intersicence, De Gruyter, 1985. 556 Pages. [20] Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. New York: Springer, 2010. 505 p. DOI: 10.1007/978-3-642- 14003-7 [21] Handbook of Fractional Calculus with Applications. Edited by J.A. Tenreiro Machado. Volumes 1-8. Berlin, Boston: De Gruyter, 2019. [22] Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives Theory and Applications. New York: Gordon and Breach, 1993. 1006 p. ISBN: 9782881248641 [23] Kiryakova, V. Generalized Fractional Calculus and Applications. New York: Longman and J. Wiley, 1994. 360 p. ISBN 9780582219779 [24] Podlubny, I. Fractional Differential Equations. San Diego: Academic Press, 1998. 340 p. [25] Kilbas, A.A.; Srivastava, H.M.; Trujillo J.J. Theory and Applications of Fractional Differ- ential Equations. Amsterdam: Elsevier, 2006. 540 p. [26] Diethelm, K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Berlin: Springer-Verlag, 2010. 247 p. DOI: 10.1007/978-3-642-14574-2 [27] Letnikov, A.V. ”On the historical development of the theory of differentiation with arbitrary index”. Mathematical Collection [Matematicheskii Sbornik]. 1868. Vol.3. No.2. P.85– 112. [in Russian]. URL: http://mi.mathnet.ru/eng/msb8048 [28] Tenreiro Machado, J.; Kiryakova, V.; Mainardi, F. ”Recent history of fractional calculus”. Communications in Nonlinear Science and Numerical Simulation. 2011. Vol.16. No.3. P.1140–1153. DOI: 10.1016/j.cnsns.2010.05.027 [29] Tenreiro Machado, J.A.; Galhano, A.M.; Trujillo, J.J. ”Science metrics on fractional calculus development since 1966”. Fractional Calculus and Applied Analysis. 2013. Vol.16. No.2. P.479–500. DOI: 10.2478/s13540-013-0030-y [30] Tenreiro Machado, J.A.; Galhano, A.M.; Trujillo, J.J. ”On development of fractional calculus during the last fifty years”. Scientometrics. 2014. Vol.98. No.1. P.577–582. DOI: 10.1007/s11192-013-1032-6 [31] Tenreiro Machado, J.A.; Kiryakova, V. ”The chronicles of fractional calculus”. Fractional Calculus and Applied Analysis. 2017. Vol.20. No.2. P.307–336. DOI: 10.1515/fca-2017-0017 [32] Tarasov, V.E. ”No violation of the Leibniz rule. No fractional derivative”. Communications in Nonlinear Science and Numerical Simulation. 2013. Vol.18. No.11. P.2945–2948. DOI: 10.1016/j.cnsns.2013.04.001 [33] Ortigueira, M.D.; Tenreiro Machado, J.A. ”What is a fractional derivative?” Journal of Computational Physics. 2015. Vol.293. P.4–13. DOI: 10.1016/j.jcp.2014.07.019

18 [34] Tarasov, V.E. ”On chain rule for fractional derivatives”. Communications in Nonlinear Science and Numerical Simulation. 2016. Vol.30. No.1-3. P.1–4. DOI: 10.1016/j.cnsns.2015.06.007 [35] Tarasov, V.E. ”Leibniz rule and fractional derivatives of power functions”. Journal of Computational and Nonlinear Dynamics. 2016. Vol.11. No.3. Article ID 031014. DOI: 10.1115/1.4031364 [36] Tarasov, V.E. ”No nonlocality. No fractional derivative”. Communications in Nonlinear Science and Numerical Simulation. 2018. Vol.62. P.157–163. DOI: 10.1016/j.cnsns.2018.02.019 (arXiv:1803.00750) [37] Tarasov, V.E. ”Rules for fractional-dynamic generalizations: difficulties of constructing fractional dynamic models”. Mathematics. 2019. Vol.7. No.6. Article ID: 554. 50 pages. DOI: 10.3390/math7060554 [38] Tarasova, V.V.; Tarasov, V.E. ”Concept of dynamic memory in economics”. Communi- cations in Nonlinear Science and Numerical Simulation. 2018. Vol.55. P.127-145. DOI: 10.1016/j.cnsns.2017.06.032 [39] Tarasov, V.E. ”Generalized memory: fractional calculus approach”. Fractal and Fractional. 2018. Vol.2. No.4. Article ID: 23. DOI: 10.3390/fractalfract2040023 [40] Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S.V. Mittag-Leffler Functions, Related Topics and Applications. Berlin: Springer-Verlag, 2014. 443 p. [41] Peng, J.; Li, K. ”A note on property of the Mittag-Leffler function”. Journal of Mathematical Analysis and Applications. 2010. Vol.370. No.2. P.635-638. DOI: 10.1016/j.jmaa.2010.04.031 [42] Elagan, S.K. ”On the invalidity of semigroup property for the MittagLeffler function with two parameters”. Journal of the Egyptian Mathematical Society. 2016. Vol.24. No.2. P.200- 203. DOI: 10.1016/j.joems.2015.05.003 [43] Sadeghi, A.; Cardoso, J.R. ”Some notes on properties of the matrix Mittag-Leffler function”. Applied Mathematics and Computation. 2018. Vol.338. P.733-738. DOI: 10.1016/j.amc.2018.06.037 [44] Tarasov, V.E. ”Quantum dissipation from power-law memory” Annals of Physics. 2012. Vol.327. No.6. P.1719-1729. DOI: 10.1016/j.aop.2012.02.011 [45] Tarasov, V.E. ”Fractional quantum mechanics of open quantum systems”. Chapter 11 in Handbook of Fractional Calculus with Applications. Volume 5: Applications in Physics, Part B. Berlin, Germany: Walter de Gruyter, 2019. pp.257-277. DOI: 10.1515/9783110571721-011 [46] Davis, E.B. Quantum Theory of Open Systems. London, New York: Academic Press, 1976 179 p. ISBN: 0-12-206150-0 [47] Breuer, H.-P.; Petruccione, F. Theory of Open Quantum Systems. Oxford: Oxford Univer- sity Press, 2002. 625 p. ISBN: 0-19-852063-8 [48] Tarasov, V.E. Quantum Mechanics of Non-Hamiltonian and Dissipative Systems Amster- dam, London: Elsevier, 2008. 540 p. ISBN-13: 978-0-444-53091-2 [49] Tarasov, V.E. ”Fractional Heisenberg equation” Physics Letters A. 2008. Vol.372. No.17. P.2984-2988. DOI: 10.1016/j.physleta.2008.01.037. (arXiv:0804.0586)

19 [50] Tarasov, V.E. ”Path integral for quantum operations” Journal of Physics A. 2004. Vol.37. No.9. P.3241-3257. DOI: 10.1088/0305-4470/37/9/013 (arXiv:0706.2142) [51] Tarasov, V.E. ”Stationary states of dissipative quantum systems” Physics Letters A. 2002. Vol.299. No.2/3. P.173-178. DOI: 10.1016/S0375-9601(02)00678-3 (arXiv:1107.5907) [52] Tarasov, V.E. ”Uncertainty relation for non-Hamiltonian quantum systems”. Jour- nal of Mathematical Physics. 2013. Vol.54. No.1. Article ID 012112. 13 p. DOI: 10.1063/1.4776653 [53] Tarasov, V.E. ”Quantization of non-Hamiltonian and dissipative systems” Physics Letters A. 2001. Vol.288. No.3/4. P.173-182. DOI: 10.1016/S0375-9601(01)00548-5 (quant-ph/0311159) [54] Tarasov, V.E. ”Fractional quantum ﬁeld theory: From lattice to continuum”, Ad- vances in High Energy Physics. 2014. Vol.2014. Article ID 957863. 14 pages. DOI: 10.1155/2014/957863 [55] Tarasov, V.E. ”Fractional derivative regularization in QFT”, Advances in High Energy Physics. 2018. Vol.2018. Article ID 7612490. 8 pages. DOI: 10.1155/2018/7612490 [56] Fradkin, D.M. ”Covariant electromagnetic projection operators and covariant description of charged particle guiding centre motion”. Journal of Physics A: Mathematical and Gen- eral. 1978. Vol.11. No.6. P.1069-1086. DOI: 10.1088/0305-4470/11/6/010 [57] Yaremko, Yu. ”Exact solution to the Landau-Lifshitz equation in a constant electromagnetic ﬁeld”. Journal of Mathematical Physics. 2013. Vol.54. No.9. Article ID: 092901. 19 pages. DOI:: 10.1063/1.4820131 (arXiv:1412.1661)