A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian–non-Markovian mix

Physica A 391 (2012) 2167–2181

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A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian–non-Markovian mix G. Costanza ∗ Departamento de Física, Universidad Nacional de San Luis, Chacabuco 917, 5700 San Luis, Argentina

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Article history: Received 13 August 2011 Received in revised form 2 November 2011 Available online 4 December 2011 Keywords: Evolution equations Stochastic processes

abstract The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Markovian and non-Markovian evolution equations have been obtained in a very large number of works and by a wide variety of authors, who have applied these general concepts to almost everything. Illustrative lists of books and papers that contain works on Markovian evolution equations are given in Refs. [1–12]. On the other hand, non-Markovian evolution equations are studied in Refs. [13–22]. Recently, Markovian evolution equations were obtained from stochastic evolution equations in Ref. [23]. An extension that includes Markovian evolution equations with dynamical variables and weights that can be complex numbers, with a real as well as an imaginary part, was considered in Ref. [24], where the main result was to prove a theorem that allows one to find deterministic evolution equations from stochastic evolution equations after an average over realizations. The non-Markovian extension was proved in Ref. [25], allowing one to derive deterministic evolution equations that contain second-order time derivatives. A new extension of the theorem will be proved in this paper, allowing one to study evolution equations described by a mix of Markovian and non-Markovian dynamical variables. As an illustrative example, the deterministic evolution equations of quantum electrodynamics will be derived. As is well known, there are two field equations: the Dirac equation for the fermionic field and the bosonic field equation that describe the evolution of the electromagnetic four-potential. In this paper, the example is a little less simple than the previous examples studied in Refs. [23–25], in two aspects. First, a three-dimensional (3D) example is studied with some detail in order to provide the most general equations that describe quantum electrodynamics. Second, this is the first time that a mix of two

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G. Costanza / Physica A 391 (2012) 2167–2181

different dynamical variables is considered. Even though the calculations seem at first glance to be lengthy and tedious, the work can easily be implemented in commercial software that provides a symbolic language such as MAPLE [26], allowing one to obtain the required results quickly and efficiently. The paper is organized as follows. In Section 2, an introduction of the stochastic evolution rules corresponding to models with an updating of the dynamical variables that depends on the values of the dynamical variables at an arbitrary number of previous time steps and with subsets that are of different type is considered. In Section 3, a theorem allowing one to connect two sets of stochastic evolution equations with another two sets that contain deterministic weights is proved. This connection is proved for both the Markovian case and the non-Markovian case studied in Refs. [24,25], but for sets of different types of dynamical variable. In Section 4, the general procedure will be applied to quantum electrodynamics. The first set of equations, studied in Section 4.1, allows us to obtain the evolution equations with updating of the dynamical variables that depends on one previous time step, allowing us to derive the 3D Dirac equation. The second example, analyzed in Section 4.2, is an extension of the first example to updating of the dynamical variables that depends on two previous time steps, allowing us to derive the evolution equation for the four-potential. These two examples allow us to show some interesting features that possess a linear transformation that connects the connection parameters of the stochastic evolution of the dynamical variables with the coefficients of a deterministic differential equation. As is usual, the differential equations are obtained after a Taylor series expansion of the deterministic dynamical variables derived after an average over realizations of the stochastic dynamical variables. The conclusions and other possible generalizations are considered in Section 5. 2. Markovian and non-Markovian stochastic evolution updating for a set of complex dynamical variables and weights: basic definitions A 3D lattice Λ consisting of a set of points {x}, with periodic boundary conditions in an interval [−L0i /2, +L0i /2] for (r ) i = 1, . . . , 3 (L0i being finite or infinite), will be considered, and a set of complex dynamical variables {qs (x0 , x)} will be used for describing the value of each dynamical variable in a realization r, in a state s, at coordinate x = x1 , x2 , x3 and at time x0 . s designates the generic value of the set {1, . . . , S }, where S is the number of elements of the set. The separations between sites, or lattice constants, are a1 , a2 , a3 , and the time between two successive updates is a0 . In order to save space, and without loss of generality, both or one of the two constants will be set equal to one when this is suitable. The length of the lattice corresponding to each coordinate is Li = ai L0i and the number of lattice sites is M = (2L01 + 1)(2L02 + 1)(2L03 + 1). The evolution equation for the set of dynamical variables can be expressed, as in Ref. [24], in the following general form: q(sr ) (x0 + a0 , x) = q(sr ) (x0 , x) + G(sr ) (x0 , . . . , x0 − lk a0 , X0 , . . . , Xl0k , Xj , Xξ ),

∀s ∈ {1, . . . , S }, x0 ≥ 0, x ∈ Λ,

(1)

where G denotes the set of rules that define a given model and Xl0k , . . . ,Xl0k denotes the set of complex dynamical variables

{q(sr ) (x0 , x)}, . . . , {q(sr ) (x0 − l0k a0 , x)}, respectively. The sets of both discrete and continuous stochastic variables that confer stochasticity to the evolution equations are Xj = { j} and Xξ = {ξ }, respectively. Note that both j = j(r ) (x0 ) and ξ = ξ (r ) (x0 )

depend on the particular realization r and at time x0 . Below, the dependence on x0 is usually neglected and in j also the dependence on r, in order to save space. The sets of dynamical variables depend on the particular realization r and previous time x0 , . . . , x0 − l0k a0 . The number of previous time is k + 1 and the set is {l0α } = {0, . . . , k}, for any 0 ≥ α ≥ k. The stochastic variables are chosen in such a way that all of them are statistically independent and a factorization of each product that contains stochastic variables is then possible. Let us assume that the set of S dynamical variables is separated into subsets of S1 and S2 dynamical variables such that S = S1 + S2 . The first subset correspond to dynamical variables that evolve Markovianly and the second to those that evolve non-Markovianly. The stochastic evolution equations are of the form (r )

(r )

qA,s (x0 + a0 , x) = qA,s (x0 , x) +

+

 {s2 ,l2 }

 {s1 ,l1 }

wA(r,)s,s1 ,l1 q(Ar,)s1 (x0 , x11 + ∆x11 )

wA(r,)s,s1 ,s2 ,l2 q(Ar,)s1

(x0 , x11 + ∆x11 ) q(Ar,)s2 (x0 , x12 + ∆x12 )

+ · · · + wA(r,)s,source , ∀s, s1 , . . . ∈ {1, . . . , S1 }, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,  (r ) (r ) (r ) qB,s (x0 + a0 , x) = qB,s (x0 , x) + wB,s,s1 ,l1 q(Br,)s1 (x0 + l01 a0 , x11 + ∆x11 ) {s1 ,l1 }

+

 {s2 ,l2 }

wB(r,s),s1 ,s2 ,l2 q(Br,)s1

(x0 + l01 a0 , x11 + ∆x11 )

× q(Br,)s2 (x0 + l02 a0 , x12 + ∆x12 ) + · · · + wB(r,s),source , ∀s, s1 , . . . ∈ {S1 + 1, . . . , S1 + S2 }, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,

(2)

where x1α + ∆x1α = x1 + l1α a1 , x2 + l2α a2 , x3 + l3α a3 , for any α , was used. In order to derive Markovian as well as nonMarkovian deterministic equations, Eq. (2) will be used as the starting set of stochastic evolution equations. The shorthand notation l1 = l01 , l11 , l21 , l31 and l2 = l01 , l11 , l21 , l31 , l02 , l12 , l22 , l32 is used in order to save space. Likewise, s1 = s1 and

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s2 = s1 , s2 . The number of equations is M = SM. The stochastic weights and the dynamical variables in Eq. (2) are labeled with an index r, emphasizing that the value depends on a specific realization. The stochastic weights can, in general, be a ′ (r )

′′ (r )

complex number with a real part ws,lk and an imaginary part ws,lk , for any k, with lk = l01 , l11 , l12 , l13 , . . . , l0k , l1k , l2k , l3k . (r )

(r )

The last terms, wA,s,source and wB,s,source , are sums of products of dynamical variables of different types, ψl (x0 , x) and Ak (x0 , x), (r )

(r )

like the one given in Eq. (9) below, with coefficients or weights denoted by Im,k,l and Jn,u,v that are products of Kronecker deltas and Heaviside functions, as shown in Eq. (39). These circumstance require no additional consideration in the following demonstration of the theorem because these crossed terms are of the ‘‘same form’’ as those containing products of dynamical variables of the same type, and consequently the factorization is also valid. In order to be more formal, an arbitrary weight (r ) can be denoted by wu , where u is some set of indices u1 , . . . , uα not necessarily of the same type as in Eq. (2). A general expression of a weight as a product of Kronecker deltas and Heaviside functions can be written as

 (r )

wu =

 {k′ }

δik′ ,jk′

              ′ (r ) ′′ (r ) (r ) ′ θ Pv ′ − ξ v ′ δik′′ ,jk′′ θ Pv′′ − ξv′′ θ Pc ′ − ξc ′ + i θ Pc′′′′ − ξc ′′(r ) {v ′ }

{k′′ }

(3)

{v ′′ }

where {k′ } and {v ′ } are sets of indices that are used to label discrete and continuous factors, respectively. These indices (r ) correspond to the real part of the complex weight wu . c ′ denotes the index that connects the real part of the stochastic weight with the real part of the deterministic weight of some other approach. In the same way, {k′′ }, {v ′′ }, and c ′′ denote the (r ) indices corresponding to the imaginary part of wu . The imaginary unit is i. There are some key questions that allow the construction of deterministic evolution equations from an average over realizations of a stochastic evolution equation. First, the stochastic weights must be a product of conditionals expressed as products of some delta functions and theta functions whose arguments contain discrete as well as continuous stochastic variables, respectively. The definitions of these functions are as follows: δx,y is equal to 1 if x = y and 0 otherwise, and θ (x − y) is equal to 1 if x − y ≥ 0 and 0 if x − y < 0, for any x and y. Second, all these stochastic variables (discrete and continuous) are statistically independent, allowing the factorization of the averages. Third, two of the theta functions, corresponding to the real and imaginary parts of the stochastic weights, contain in their argument the functions Pc′ ′ and Pc′′′′ that allow one to connect the average over realizations of all the stochastic weights with the deterministic weights of any other deterministic approach (e.g. master equation, etc.). For the interpretation of these functions that define the weights see the first example in Section 4 of Ref. [24]. The above general definition of a generic stochastic weight allows us to demonstrate the following theorem. 3. A theorem connecting the average over realizations of the stochastic weights with the deterministic weights In the general case of an updating that depends on more than one previous time step, the theorem and the proof can be made in an almost verbatim way, with the appropriate changes in the notation, as that done in Ref. [25]. For the sake of completeness, the theorem and the proof are reproduced below. Theorem. A set of deterministic evolution equations is obtained after an average over realizations of a set of stochastic evolution equations like those given in Eq. (2) with stochastic coefficients of the general form of those given in Eq. (3). The connection with a set of deterministic evolution equations, obtained with other approach, is made after an appropriate election of the functions Pc′ ′ and Pc′′′′ . Proof. The proof is obtained in two steps in a very simple way. First, using standard results of statistical mechanics (see the Appendix of Ref. [24]), the general deterministic equations are obtained after an average over realizations on both sides of Eq. (2), in the following general form: (r )

(r )

qA,s (x0 + a0 , x) = qA,s (x0 , x) +

+

 {s2 ,l2 }

 {s1 ,l1 }

wA,s,s1 ,l1 q(Ar,)s1 (x0 , x11 + ∆x11 )

wA,s,s1 ,s2 ,l2 q(Ar,)s1 (x0 , x11 + ∆x11 ) q(Ar,)s2 (x0 , x12 + ∆x12 )

+ · · · + wA,s,source , ∀s, s1 , . . . ∈ {1, . . . , S1 }, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,  (r ) (r ) qB,s (x0 + a0 , x) = qB,s (x0 , x) + wB,s,s1 ,l1 q(Br,)s1 (x0 + l01 a0 , x11 + ∆x11 ) {s1 ,l1 }

+

 {s2 ,l2 }

wB,s,s1 ,s2 ,l2 q(Br,)s1 (x0 + l01 a0 , x11 + ∆x11 )

× q(Br,)s2 (x0 + l02 a0 , x12 + ∆x12 ) + · · · + wB,s,source , ∀s, s1 , . . . ∈ {S1 + 1, . . . , S1 + S2 }, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,

(4)

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G. Costanza / Physica A 391 (2012) 2167–2181

(r )

(r )

where wA,s,s1 ,l1 = wA,s,s1 ,l1 , wB,s,s1 ,l1 = wB,s,s1 ,l1 , . . ., are the weights corresponding to the product of one, two, . . ., dynamical variables. Note that a (1,0)-closure was used for the product of two dynamical variables, which is the simplest closure that can be used in the infinite hierarchy of evolution equations. The deterministic weights can be written in the usual form, wA,s,s1 ,...,lk = wA′ ,s,s1 ,...,lk + iwA′′,s,s1 ,...,lk and wB,s,s1 ,...,lk = wB′ ,s,s1 ,...,lk + iwB′′,s,s1 ,...,lk . Note that the factorization of the averages over realization was used because it was assumed that the discrete and continuous stochastic variables in all the weights w are statistically independent and also are independent of all the dynamical variables. For a demonstration that the product of two functions of complex stochastic variables factorizes, see the Appendix of Ref. [24]. Second, the last step needed to obtain the connection between the approaches is to make an average over realizations on both two sides of Eq. (3). The result is (r )

wu =



δik′ ,jk′

{k′ }

=

{v ′ }

{k′′ }



 1 {k′ }

               ′ (r ) ′′ (r ) (r ) ′ θ Pv ′ − ξ v ′ θ Pc ′ − ξ c ′ + i δik′′ ,jk′′ θ Pv′′ − ξv′′ θ Pc′′′′ − ξc ′′(r )  

Mk′

Pv ′

 1

Pc′ ′ + i

{v ′ }

{k′′ }

Mk′′



{v ′′ }

 

Pv ′′

Pc′′′′ ,

(5)

{v ′′ }

where Mk′ and Mk′′ are the numbers of elements of the k-th discrete set. Note that it was assumed that all the intervals of variation of all the continuous stochastic variables is [0, 1]. If some of the intervals are different, the result of Eq. (56) in the Appendix of Ref. [24] must be used. The connection with another approach is easily obtained. Equating the coefficients of   (r )

the expressions of the weights wu = wc , Pc′ ′ and Pc′′′′ can be found as Pc′ ′ =

 {k′ }

1 Mk′

wc′  

Pc′′′′ =

 {k′′ }

1 Mk′′

{v ′ }

,

(6)

Pv ′

wc′′   {v ′′ }

,

(7)

Pv ′′

where wc′ and wc′′ are the real part and the imaginary part of wc , respectively. If the deterministic evolution equation is expressed as a partial differential equation like those given in the example in Section 5, Pc′ ′ and Pc′′′′ , in Eqs. (6) and (7), must be multiplied by a0 in order to recover the correct deterministic weights. These expressions allow one to establish the complete equivalence with the deterministic weights corresponding to some other approach.  4. Illustrative example In this section, a simple illustrative example will be given in some detail in order to show the basic steps necessary to obtain a set of deterministic evolution equations from a set of stochastic evolution equations after an average over realizations. In this case we will analyze the quantum electrodynamics evolution equations corresponding to the fermionic field ψ and to the bosonic field A; the first is described by a partial differential equation that contains only a first-order derivative with respect to time x0 , and the second is described by a partial differential equation that contains a secondorder time derivative. As is well known, both fields are 4-vectors, and in Appendix A the partial differential equations of both fields are obtained using the Lagrangian approach. As was shown in Ref. [24], the Markovian evolution equations are obtained using a set of dynamical variables like those (r ) labeled qA in Eq. (2), and in this example we will use ψ (r ) . Likewise, for the non-Markovian evolution equations, those (r )

labeled qB , we will use A(r ) . Note that, in the Markovian evolution equations, A is a label and, in the non-Markovian evolution equations, A designates a 4-vector, which is the four-potential. In order to obtain the evolution equation corresponding to the quantum electrodynamics, we will use the following set of equations equivalent to those of Eq. (2):

ψm(r ) (x0 + a0 , x) = ψm(r ) (x0 , x) +

 {m1 ,l1 }

wm(r,)m1 ,l1 ψm(r1) (x0 , x11 + ∆x11 )

+ wm(r,)source1 ,

m, m1 , . . . = 1, . . . , 4, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,  (r ) An (x0 + a0 , x) = An (x0 , x) + wn,l1 A(nr1) (x0 + l01 a0 , x11 (r )

(r )

{n1 ,l1 }

) + ∆x11 ) + wn(r,source2 ,

n, n1 , . . . = 0, . . . , 3, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,

(8)

G. Costanza / Physica A 391 (2012) 2167–2181

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where (r ) (r ) (r ) wm(r,)source1 = Im ,k,l Ak (x0 , x)ψl (x0 , x), ) wn(r,source2

=

m, l = 1, . . . , 4 and k = 0, . . . , 3,

Jn(r,u) ,v ψu∗(r ) (x0 , x)ψv(r ) (x0 , x),

n = 0, . . . , 3 and u, v = 1, . . . , 4.

(9)

As in Appendix B, sums over repeated indices are assumed. Note that, in Eqs. (8) and (9), we have used m and n instead of s, with the corresponding set of values {1, . . . 4} and {0, . . . 3}, respectively, in order to use a notation more closely related to that usually used in the literature. The last step, in order to complete the explicit evolution equations, is to provide the explicit form of the sums in the right-hand sides of Eq. (8). The full expressions needed to obtain the equations for the fermionic field are given by (r )

(r )

Σm(r ) = wm,1,0,0,0,0 ψ1 (x0 , x1 , x2 , x3 )

+ wm(r,)1,0,−1,0,0 ψ1(r ) (x0 , x1 − a1 , x2 , x3 ) − wm(r,)1,0,+1,0,0 ψ1(r ) (x0 , x1 + a1 , x2 , x3 ) + wm(r,)1,0,0,−1,0 ψ1(r ) (x0 , x1 , x2 − a2 , x3 ) − wm(r,)1,0,0,+1,0 ψ1(r ) (x0 , x1 , x2 + a2 , x3 ) + wm(r,)1,0,0,0,−1 ψ1(r ) (x0 , x1 , x2 , x3 − a3 ) − wm(r,)1,0,0,0,+1 ψ1(r ) (x0 , x1 , x2 , x3 + a3 ) + wm(r,)2,0,0,0,0 ψ2(r ) (x0 , x1 , x2 , x3 ) + wm(r,)2,0,−1,0,0 ψ2(r ) (x0 , x1 − a1 , x2 , x3 ) − wm(r,)2,0,+1,0,0 ψ2(r ) (x0 , x1 + a1 , x2 , x3 ) + wm(r,)2,0,0,−1,0 ψ2(r ) (x0 , x1 , x2 − a2 , x3 ) − wm(r,)2,0,0,+1,0 ψ2(r ) (x0 , x1 , x2 + a2 , x3 ) + wm(r,)2,0,0,0,−1 ψ2(r ) (x0 , x1 , x2 , x3 − a3 ) − wm(r,)2,0,0,0,+1 ψ2(r ) (x0 , x1 , x2 , x3 + a3 ) + wm(r,)3,0,0,0,0 ψ3(r ) (x0 , x1 , x2 , x3 ) + wm(r,)3,0,−1,0,0 ψ3(r ) (x0 , x1 − a1 , x2 , x3 ) − wm(r,)3,0,+1,0,0 ψ3(r ) (x0 , x1 + a1 , x2 , x3 ) + wm(r,)3,0,0,−1,0 ψ3(r ) (x0 , x1 , x2 − a2 , x3 ) − wm(r,)3,0,0,+1,0 ψ3(r ) (x0 , x1 , x2 + a2 , x3 ) + wm(r,)3,0,0,0,−1 ψ3(r ) (x0 , x1 , x2 , x3 − a3 ) − wm(r,)3,0,0,0,+1 ψ3(r ) (x0 , x1 , x2 , x3 + a3 ) + wm(r,)4,0,0,0,0 ψ4(r ) (x0 , x1 , x2 , x3 ) + wm(r,)4,0,−1,0,0 ψ4(r ) (x0 , x1 − a1 , x2 , x3 ) − wm(r,)4,0,+1,0,0 ψ4(r ) (x0 , x1 + a1 , x2 , x3 ) + wm(r,)4,0,0,−1,0 ψ4(r ) (x0 , x1 , x2 − a2 , x3 ) − wm(r,)4,0,0,+1,0 ψ4(r ) (x0 , x1 , x2 + a2 , x3 ) + wm(r,)4,0,0,0,−1 ψ4(r ) (x0 , x1 , x2 , x3 − a3 ) − wm(r,)4,0,0,0,+1 ψ4(r ) (x0 , x1 , x2 , x3 + a3 ),

(10)

and for the bosonic field (r )

(r )

Σn(r ) = wn,0,0,0,0 A(nr ) (x0 , x1 , x2 , x3 ) + wn,−1,0,0,0 A(nr ) (x0 − a0 , x1 , x2 , x3 )

+ wn(r,0) ,−1,0,0 A(nr ) (x0 , x1 − a1 , x2 , x3 ) + wn(r,0) ,+1,0,0 A(nr ) (x0 , x1 + a1 , x2 , x3 ) + wn(r,0) ,0,−1,0 A(nr ) (x0 , x1 , x2 − a2 , x3 ) + wn(r,0) ,0,+1,0 A(nr ) (x0 , x1 , x2 + a2 , x3 ) + wn(r,0) ,0,0,−1 A(nr ) (x0 , x1 , x2 , x3 − a3 ) + wn(r,0) ,0,0,+1 A(nr ) (x0 , x1 , x2 , x3 + a3 ) ) (r ) (r ) (r ) + wn(r,− 1,−1,0,0 An (x0 − a0 , x1 − a1 , x2 , x3 ) + wn,−1,+1,0,0 An (x0 − a0 , x1 + a1 , x2 , x3 ) ) (r ) (r ) (r ) + wn(r,− 1,0,−1,0 An (x0 − a0 , x1 , x2 − a2 , x3 ) + wn,−1,0,+1,0 An (x0 − a0 , x1 , x2 + a2 , x3 ) ) (r ) (r ) (r ) + wn(r,− 1,0,0,−1 An (x0 − a0 , x1 , x2 , x3 − a3 ) + wn,−1,0,0,+1 An (x0 − a0 , x1 , x2 , x3 + a3 )

(11)

where, in order to save space, the following notation has been used:

Σm(r ) =

 {m1 ,l1 }

(r )

Σn =

 {n1 ,l1 }

wm(r,)m1 ,l1 ψm(r1) (x0 , x11 + ∆x11 ),

wn(r,l)1 A(nr1) (x0 + l01 a0 , x11 ).

(12)

Explicit expressions for the weights (w ) and the corresponding averages over realizations are given in Appendix B. After an average over realizations on both sides of the two equations given as Eq. (8), the following discrete deterministic evolution equations are obtained:

ψm (x0 + a0 , x1 , x2 , x3 ) = ψm (x0 , x1 , x2 , x3 ) + Σm + wm,source1 , An (x0 + a0 , x1 , x2 , x3 ) = An (x0 , x1 , x2 , x3 ) + Σm + wn,source2 ,

m = 1, . . . , 4, x0 ≥ 0, x1 , . . . , x3 ∈ Λ, n = 0, . . . , 3, x0 ≥ 0, x1 , . . . , x3 ∈ Λ,

(13)

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G. Costanza / Physica A 391 (2012) 2167–2181

(r )

(r )

(r )

(r )

where ψm = ψm , An = An , Σm = Σm , Σn Eqs. (9)–(11) become

) = Σn , wm(r,)source1 = wm,source1 , and wn(r,source2 = wn,source2 were used. Note that

wm,source1 = Im,k,l Ak (x0 , x)ψl (x0 , x) = Im,k,l Ak ψl ,

m, l = 1, . . . , 4 and k = 0, . . . , 3,

wn,source2 = Jn,u,v ψu (x0 , x)ψv (x0 , x) = Jn,u,v ψu ψv , Σm = wm,1,0,0,0,0 ψ1 (x0 , x1 , x2 , x3 ) .. . + wm,4,0,0,0,−1 ψ4 (x0 , x1 , x2 , x3 − a3 ) ∗

Σn

∗

n = 0, . . . , 3 and u, v = 1, . . . , 4,

(14)

+ wm,4,0,0,0,+1 ψ4 (x0 , x1 , x2 , x3 + a3 ), = wn,0,0,0,0 An (x0 , x1 , x2 , x3 ) + wn,−1,0,0,0 An (x0 − a0 , x1 , x2 , x3 ) .. . + wn,−1,0,0,−1 An (x0 − a0 , x1 , x2 , x3 − a3 )

(15)

+ wn,−1,0,0,1 A,n (x0 − a0 , x1 , x2 , x3 + a3 ),

(16) (r )

(r )

(r )

(r )

∗(r )

ψv(r )

∗(r )

ψv(r )

where, in Eq. (14), the (1, 0)-closure was used, allowing us to write Ak ψl = Ak ψl = Ak ψl and ψu = ψu = ψu∗ ψv . In Eqs. (15) and (16), the vertical dots indicate that the other summands are obtained in an obvious way from Eqs. (10) and (11), respectively. 4.1. The fermionic field equations In order to obtain the set of partial differential equations that describe the evolution of the four components of the fermionic field, we must expanded all terms in the first equation of Eq. (13), up to O(aα ) for α = 0, . . . , 3, given Wm,0,m ∂0 ψm + Wm,1,1 ∂1 ψ1 + Wm,1,2 ∂1 ψ2 + Wm,1,3 ∂1 ψ3 + Wm,1,4 ∂1 ψ4

+ Wm,2,1 ∂2 ψ1 + Wm,2,2 ∂2 ψ2 + Wm,2,3 ∂2 ψ3 + Wm,2,4 ∂2 ψ4 + Wm,3,1 ∂3 ψ1 + Wm,3,2 ∂3 ψ2 + Wm,3,3 ∂3 ψ3 + Wm,3,4 ∂3 ψ4 + Wm,1 ψ1 + Wm,2 ψ2 + Wm,3 ψ+ Wm,4 ψ4 + Im,k,l Ak ψl + O(a20 ) + O(a21 ) + O(a22 ) + O(a23 ) = 0 for m = 1, . . . , 4,

(17)

where Wm,0,m = −a0 , Wm,1,1 = −(wm,1,0,−1,0,0 + wm,1,0,1,0,0 )a1 , Wm,1,2 = −(wm,2,0,−1,0,0 + wm,2,0,1,0,0 )a1 , Wm,1,3 = −(wm,3,0,−1,0,0 + wm,3,0,1,0,0 )a1 , Wm,1,4 = −(wm,4,0,−1,0,0 + wm,4,0,1,0,0 )a1 , Wm,2,1 = −(wm,1,0,0,−1,0 + wm,1,0,0,1,0 )a2 , Wm,2,2 = −(wm,2,0,0,−1,0 + wm,2,0,0,1,0 )a2 , Wm,2,3 = −(wm,3,0,0,−1,0 + wm,3,0,0,1,0 )a2 , Wm,2,4 = −(wm,4,0,0,−1,0 + wm,4,0,0,1,0 )a2 , Wm,3,1 = −(wm,1,0,0,0,−1 + wm,1,0,0,0,1 )a3 , Wm,3,2 = −(wm,2,0,0,0,−1 + wm,2,0,0,0,1 )a3 , Wm,3,3 = −(wm,3,0,0,0,−1 + wm,3,0,0,0,1 )a3 , Wm,3,4 = −(wm,4,0,0,0,−1 + wm,4,0,0,0,1 )a3 , Wm,1 = wm,1,0,0,0,0 + wm,1,0,−1,0,0 − wm,1,0,1,0,0

+ wm,1,0,0,0,−1 − wm,1,0,0,0,1 + wm,1,0,0,−1,0 − wm,1,0,0,1,0 ,

Wm,2 = wm,2,0,0,0,0 + wm,2,0,0,0,−1 − wm,2,0,0,0,1

+ wm,2,0,−1,0,0 − wm,2,0,1,0,0 + wm,2,0,0,−1,0 − wm,2,0,0,1,0 ,

Wm,3 = wm,3,0,0,0,0 + wm,3,0,0,0,−1 − wm,3,0,0,0,1

+ wm,3,0,−1,0,0 − wm,3,0,1,0,0 + wm,3,0,0,−1,0 − wm,3,0,0,1,0 ,

Wm,4 = wm,4,0,0,0,0 + wm,4,0,0,0,−1 − wm,4,0,0,0,1

+ wm,4,0,−1,0,0 − wm,4,0,1,0,0 + wm,4,0,0,−1,0 − wm,4,0,0,1,0 .

(18)

G. Costanza / Physica A 391 (2012) 2167–2181

2173

Since the number of w s is greater than the number of W s, some subsidiary condition must be imposed in order to obtain the number of equations being equal to the number of unknowns. The only physical condition that can be imposed is space homogeneity, meaning that the following equalities must be imposed:

wm,1,0,−1,0,0 wm,2,0,−1,0,0 wm,3,0,−1,0,0 wm,4,0,−1,0,0 wm,1,0,0,−1,0 wm,2,0,0,−1,0 wm,3,0,0,−1,0 wm,4,0,0,−1,0 wm,1,0,0,0,−1 wm,2,0,0,0,−1 wm,3,0,0,0,−1 wm,4,0,0,0,−1

= wm,1,0,1,0,0 , = wm,2,0,1,0,0 , = wm,3,0,1,0,0 , = wm,4,0,1,0,0 , = wm,1,0,0,1,0 , = wm,2,0,0,1,0 , = wm,3,0,0,1,0 , = wm,4,0,0,1,0 , = wm,1,0,0,0,1 , = wm,2,0,0,0,1 , = wm,3,0,0,0,1 , = wm,4,0,0,0,1 ,

and the resulting set of equations is easily obtained: Wm,1,1 = −2wm,1,0,1,0,0 a1 , Wm,1,2 = −2wm,2,0,1,0,0 a1 , Wm,1,3 = −2wm,3,0,1,0,0 a1 , Wm,1,4 = −2wm,4,0,1,0,0 a1 , Wm,2,1 = −2wm,1,0,0,1,0 a2 , Wm,2,2 = −2wm,2,0,0,1,0 a2 , Wm,2,3 = −2wm,3,0,0,1,0 a2 , Wm,2,4 = −2wm,4,0,0,1,0 a2 , Wm,3,1 = −2wm,1,0,0,0,1 a3 , Wm,3,2 = −2wm,2,0,0,0,1 a3 , Wm,3,3 = −2wm,3,0,0,0,1 a3 , Wm,3,4 = −2wm,4,0,0,0,1 a3 , Wm,1 = wm,1,0,0,0,0 , Wm,2 = wm,2,0,0,0,0 , Wm,3 = wm,3,0,0,0,0 , Wm,4 = wm,4,0,0,0,0 ,

(19)

allowing us to determine all the w s as a function of the W s straightforwardly. It is not difficult to write the set of equations, given in Eq. (17), after defining the following set of vectors and matrices:

 ψ1 ψ  ψ =  2 , ψ3 ψ4  

W1,0,1  0 0 W = − 0 0

0

0 0

0 0 0



W2,0,2 0 0

W3,0,3 0

W4,0,4



W1,1,2 W2,1,2 W3,1,2 W4,1,2

W1,1,3 W2,1,3 W3,1,3 W4,1,3

W1,1,4 W2,1,4  , W3,1,4  W4,1,4



W1,2,2 W2,2,2 W3,2,2 W4,2,2

W1,2,3 W2,2,3 W3,2,3 W4,2,3

W1,2,4 W2,2,4  , W3,2,4  W4,2,4

W1,1,1 W2,1,1 1 W = − W3,1,1 W4,1,1 W1,2,1 W2,2,1 2 W = − W3,2,1 W4,2,1

 , 



2174

G. Costanza / Physica A 391 (2012) 2167–2181



W1,3,1 W2,3,1 3 W = − W3,3,1 W4,3,1



W1,3,2 W2,3,2 W3,3,2 W4,3,2



W1,3,3 W2,3,3 W3,3,3 W4,3,3

W1,3,4 W2,3,4  , W3,3,4  W4,3,4



W1,1 W2,1 4 W = W3,1 W4,1

W1,2 W2,2 W3,2 W4,2

W1,3 W2,3 W3,3 W4,3



I1,0,1 I2,0,1 I0 =  I3,0,1 I4,0,1

I1,0,2 I2,0,2 I3,0,2 I4,0,2

I1,0,3 I2,0,3 I3,0,3 I4,0,3

I1,0,4 I2,0,4  , I3,0,4  I4,0,4



I1,1,1 I2,1,1 I1 =  I3,1,1 I4,1,1

I1,1,2 I2,1,2 I3,1,2 I4,1,2

I1,1,3 I2,1,3 I3,1,3 I4,1,3

I1,1,4 I2,1,4  , I3,1,4  I4,1,4



I1,2,1 I2,2,1 I2 =  I3,2,1 I4,2,1

I1,2,2 I2,2,2 I3,2,2 I4,2,2

I1,2,3 I2,2,3 I3,2,3 I4,2,3

I1,2,4 I2,2,4  , I3,2,4  I4,2,4



I1,3,2 I2,3,2 I3,3,2 I4,3,2

I1,3,3 I2,3,3 I3,3,3 I4,3,3

I1,3,4 I2,3,4  , I3,3,4  I4,3,4

I1,3,1 I2,3,1 I3 =  I3,3,1 I4,3,1

W1,4 W2,4  , W3,4  W4,4









in the following matrix form: W µ ∂µ ψ − W 4 ψ = Iµ Aµ ψ.

(20)

The easiest way to check that Eq. (20) is the same as Eq. (17) is by expanding Eq. (20) and comparing each component to each equation in Eq. (17), corresponding to the four possible values of m. It is not difficult to see that the Dirac equation is nothing but a special case of Eq. (20), namely when W µ = iγ µ , W 4 = mI ,

Iµ = eγµ ,

(21)

where I, in Eq. (21), is the 4 × 4 identity matrix. 4.2. The bosonic field equations In order to obtain the set of partial differential equations that describe the evolution of the four components of the bosonic field, all terms in the second equation in Eq. (13) must be expanded up to O(a2α ) for α = 0, . . . , 3, obtaining Wn,1,1,4,4,0 ∂32 ∂02 An + Wn,1,1,3,3,0 ∂22 ∂02 An + Wn,1,1,2,2,0 ∂12 ∂02 An + Wn,1,4,4,0 ∂32 ∂0 An

+ Wn,1,3,3,0 ∂22 ∂0 An + Wn,1,2,2,0 ∂12 ∂0 An + Wn,1,1,4,0 ∂3 ∂02 An + Wn,1,1,3,0 ∂2 ∂02 An + Wn,1,1,2,0 ∂1 ∂02 An + Wn,4,4,0 ∂32 An + Wn,3,3,0 ∂22 An + Wn,2,2,0 ∂12 An + Wn,1,1,0 ∂02 An + Wn,1,4,0 ∂3 ∂0 An + Wn,1,3,0 ∂2 ∂0 An + Wn,1,2,0 ∂1 ∂0 An + Wn,4,0 ∂3 An + Wn,3,0 ∂2 An + Wn,2,0 ∂1 An + Wn,1,0 ∂0 An + Wn,0 An + Jn,u,v ψu∗ ψv + O(a33 ) + O(a32 ) + O(a31 ) + O(a30 ) = 0 for n = 0, . . . , 3, where Wn,1,1,4,4,0 = wn,−1,0,0,−1 + wn,−1,0,0,1



Wn,1,1,3,3,0 = wn,−1,0,−1,0 + wn,−1,0,1,0



1 4 1 4

a23 a20 , a22 a20 ,

(22)

G. Costanza / Physica A 391 (2012) 2167–2181

Wn,1,1,2,2,0 = wn,−1,1,0,0 + wn,−1,−1,0,0

Wn,1,3,3,0 = Wn,1,2,2,0 = Wn,1,1,4,0 = Wn,1,1,3,0 = Wn,1,1,2,0 =

1

a21 a20 , 4  1 − wn,−1,0,0,−1 + wn,−1,0,0,1 a23 a0 , 2  1 2 − wn,−1,0,1,0 + wn,−1,0,−1,0 a2 a0 , 2  1 2 − wn,−1,1,0,0 + wn,−1,−1,0,0 a1 a0 , 2 1 2  wn,−1,0,0,1 − wn,−1,0,0,−1 a3 a0 , 2 1 2  wn,−1,0,1,0 − wn,−1,0,−1,0 a2 a0 , 2 1 2  wn,−1,1,0,0 − wn,−1,−1,0,0 a1 a0 , 2



Wn,1,4,4,0 =

2175

Wn,4,4,0 = wn,0,0,0,1 + wn,0,0,0,−1 + wn,−1,0,0,−1 + wn,−1,0,0,1



Wn,3,3,0 = wn,−1,0,−1,0 + wn,0,0,−1,0 + wn,0,0,1,0 + wn,−1,0,1,0



Wn,2,2,0 = wn,−1,1,0,0 + wn,−1,−1,0,0 + wn,0,−1,0,0 + wn,0,1,0,0



1 2 1 2 1 2

a23 , a22 , a21 ,

1 −1 + wn,−1,0,0,0 + wn,−1,1,0,0 + wn,−1,−1,0,0 + wn,−1,0,0,1 + wn,−1,0,0,−1 a20 2  1 2 + wn,−1,0,1,0 + wn,−1,0,−1,0 a0 , 2   Wn,1,4,0 = wn,−1,0,0,−1 − wn,−1,0,0,1 a3 a0 ,   Wn,1,3,0 = wn,−1,0,−1,0 − wn,−1,0,1,0 a2 a0 ,   Wn,1,2,0 = wn,−1,−1,0,0 − wn,−1,1,0,0 a1 a0 ,   Wn,4,0 = wn,−1,0,0,1 − wn,−1,0,0,−1 + wn,0,0,0,1 − wn,0,0,0,−1 a3 ,   Wn,3,0 = wn,0,0,1,0 − wn,0,0,−1,0 + wn,−1,0,1,0 − wn,−1,0,−1,0 a2 ,   Wn,2,0 = wn,−1,1,0,0 − wn,−1,−1,0,0 + wn,0,1,0,0 − wn,0,−1,0,0 a1 ,   Wn,1,0 = − 1 + wn,−1,0,0,0 + wn,−1,1,0,0 + wn,−1,−1,0,0 + wn,−1,0,1,0 + wn,−1,0,−1,0 a0   − wn,−1,0,0,1 + wn,−1,0,0,−1 a0 , Wn,1,1,0 =



Wn,0 = wn,0,0,0,1 + wn,−1,0,0,1 + wn,0,0,0,−1 + wn,−1,0,0,−1 + wn,0,0,0,0 + wn,−1,0,0,0

+ wn,−1,1,0,0 + wn,0,−1,0,0 + wn,0,1,0,0 + wn,−1,−1,0,0 + wn,−1,0,−1,0 + wn,−1,0,1,0 + wn,0,0,1,0 + wn,0,0,−1,0 .

(23)

Note that, in Eq. (23), there are 21 equations and 14 unknowns; some of the equations are linearly dependent. The linearly dependent equations are the 4th, 5th, 6th, 13th, 14th, 15th, and 20th. The remaining 14 equations must be used to obtain the 14 unknowns. Solving the linear system of equations we obtain the following:

wn,0,0,0,1 =

1

wn,0,0,0,−1 = wn,0,0,1,0 =

1 1

a3 a20 Wn,4,0 − 2a20 Wn,4,4,0 + 4Wn,1,1,4,4,0 − 2a3 Wn,1,1,4,0 ,



2a20 Wn,3,3,0 − 4Wn,1,1,3,3,0 + a2 a20 Wn,3,0 − 2a2 Wn,1,1,3,0 ,

 1



a2 a20 Wn,3,0 − 2a20 Wn,3,3,0 + 4Wn,1,1,3,3,0 − 2a2 Wn,1,1,3,0 ,



2a22 a20 1 1 2a21 a20



2a20 Wn,2,2,0 − 4Wn,1,1,2,2,0 + a1 a20 Wn,2,0 − 2a1 Wn,1,1,2,0 ,



2a21 a20

wn,0,−1,0,0 =





2a23 a20

2a22 a20

wn,0,0,−1,0 = wn,0,1,0,0 =

2a20 Wn,4,4,0 − 4Wn,1,1,4,4,0 + a3 a20 Wn,4,0 − 2a3 Wn,1,1,4,0 ,



2a23 a20



a1 a20 Wn,2,0 − 2a20 Wn,2,2,0 + 4Wn,1,1,2,2,0 − 2a1 Wn,1,1,2,0 ,





2176

G. Costanza / Physica A 391 (2012) 2167–2181



2Wn,1,1,4,4,0 + a3 Wn,1,1,4,0

wn,−1,0,0,1 =

 ,

a23 a20

 wn,−1,0,0,−1 = − 

a3 Wn,1,1,4,0 − 2Wn,1,1,4,4,0



a22 a20



2Wn,1,1,3,3,0 − a2 Wn,1,1,3,0

wn,−1,0,−1,0 = 

,

a21 a20 Wn,1,1,2,2,0 − a1 Wn,1,1,2,0

wn,−1,−1,0,0 =



a21 a20 1

2a23 a22 a21 Wn,1,1,0 + a23 a22 a21 a20

+

a23 a22 a21 a20 Wn,0 + 2a22 a21 a20 Wn,4,4,0 − 2a23 a22 a21 Wn,1,1,0 − a23 a22 a21 a20



a23 a22 a21 a20

+



  −4a22 a21 Wn,1,1,4,4,0 − 4a23 a21 Wn,1,1,3,3,0 − 4a23 a22 Wn,1,1,2,2,0 ,

a23 a22 a21 a20 1

,



a23 a22 a21 a20 1

+ wn,0,0,0,0 =

,





wn,−1,0,0,0 =

, 

a22 a20

2Wn,1,1,2,2,0 + a1 Wn,1,1,2,0

wn,−1,1,0,0 =

,

a23 a20

2Wn,1,1,3,3,0 + a2 Wn,1,1,3,0

wn,−1,0,1,0 =



1

4a22 a21 Wn,1,1,4,4,0 + 4a23 a21 Wn,1,1,3,3,0 + 4a23 a22 Wn,1,1,2,2,0



a23 a22 a21 a20 1

a23 a22 a21 a20

  −2a23 a21 a20 Wn,3,3,0 − 2a23 a22 a20 Wn,2,2,0 .



 (24)

The last step, needed to find the evolution equation for the bosonic field, is to choose the coefficients in Eq. (22) appropriately. Choosing Wn,4,4,0 = Wn,3,3,0 = Wn,2,2,0 = −1, Wn,1,1,0 = 1,

for n = 0, . . . , 3,

(25)

and zero otherwise, the evolution equation becomes

 2  −∂0 + ∂12 + ∂22 + ∂32 An = Jn,u,v ψu∗ ψv ,

for n = 0, . . . , 3.

(26)

In order to have a version of the evolution equation that is the same as that obtained in Appendix A, it is necessary to rewrite the right-hand side of Eq. (26) in the following matrix form:

Jn,u,v ψu∗ ψv = ψ ∗ J µ ψ,

(27)

where the row 4-vector ψ ∗ is defined as

ψ ∗ = (ψ1∗ , ψ2∗ , ψ3∗ , ψ4∗ ), and



J0,1,1 J0,2,1 0 J = J0,3,1 J0,4,1

J0,1,2 J0,2,2 J0,3,2 J0,4,2

J0,1,3 J0,2,3 J0,3,3 J0,4,3

J0,1,4 J0,2,4  , J0,3,4  J0,4,4



J1,1,1 J1,2,1 1 J = J1,3,1 J1,4,1

J1,1,2 J1,2,2 J1,3,2 J1,4,2

J1,1,3 J1,2,3 J1,3,3 J1,4,3

J1,1,4 J1,2,4  , J1,3,4  J1,4,4



J2,1,1 J2,2,1 2 J = J2,3,1 J2,4,1

J2,1,2 J2,2,2 J2,3,2 J2,4,2

J2,1,3 J2,2,3 J2,3,3 J2,4,3

J2,1,4 J2,2,4  , J2,3,4  J2,4,4



J3,1,2 J3,2,2 J3,3,2 J3,4,2

J3,1,3 J3,2,3 J3,3,3 J3,4,3

J3,1,4 J3,2,4  , J3,3,4  J3,4,4

J3,1,1 J3,2,1 3 J = J3,3,1 J3,4,1









G. Costanza / Physica A 391 (2012) 2167–2181

2177

allowing us to rewrite Eq. (26) in the following standard form:



 −∂02 + ∂12 + ∂22 + ∂32 Aµ = ψ ∗ J µ ψ,

for µ = 0, . . . , 3.

(28)

Note that we also used Am = Aµ . Complete equivalence is achieved after making J µ = eγ 0 γ µ , obtaining the following usual form:

 2  −∂0 + ∂12 + ∂22 + ∂32 Aµ = eψ ∗ γ 0 γ µ ψ = eψγ µ ψ.

(29)

Note that an overline in ψ is used because this is the standard notation for the Dirac adjoint, used in the literature, and this must not be confused with the average over realizations used in this article. 5. Conclusions and other possible generalizations The extension of the general approach to the case where dynamical variables are of different type was analyzed and a theorem, previously proved for Markovian as well as non-Markovian evolution equations, was extended and used to obtain continuum evolution equations from sets of discrete stochastic evolution equations. It was used for obtaining quantum electrodynamics evolution equations for both the fermionic and the bosonic field equations. The example was chosen because it is, perhaps, the best-known paradigmatic example of a field theory. Of course other cases can be implemented along the general lines shown in this paper. Extension of models that contain more than two types of dynamical variable can also be implemented. Formally, extensions for models described by a k-vector, for k > 4, could be obtained. Relativistic field equations, such as the Einstein field equations for geodesics, can also be obtained after replacing the evolution parameter x0 by the arc usually denoted by s, etc. It must be emphasized that, even though the notation used could at first glance seem a little cumbersome, all the Taylor series expansions and the solutions of the sets of linear equations are easily implemented in a symbolic language such as MAPLE [26]. Acknowledgment The author thanks Dr. V.D. Pereyra for reading the manuscript. Appendix A One of the usual ways to derive evolution equations is the Lagrangian approach, consisting of giving a Lagrangian function and plugging it in the Euler–Lagrange equation; then the evolution equations for the dynamical variables are obtained. As an example, the evolution equations for quantum electrodynamics will be given, and they will be compared to the derivation obtained by using the stochastic evolution equations. As is well known, the Lagrangian is

LQED = i ψγ µ ∂µ ψ − eψγ µ Aµ ψ − mψψ −

1 4

Fµν F µν ,

(30)

where the γ µ are Dirac matrices, and ψ and ψ = ψ ∗ γ 0 are the fermionic field and the Dirac adjoint, respectively. The charge and the mass are e and m, respectively, and the fermionic field is Aµ . The covariant and contravariant versions of the strength tensor are Fµν and F µν , respectively. The Euler–Lagrange equation for the fermionic field is

 ∂µ



LQED

∂ ∂µ ψ 

−



∂ LQED = 0, ∂ψ

(31)

and the evolution equation for the fermionic field is iγ µ ∂µ ψ − mψ = eγµ Aµ ψ.

(32)

On the other hand, the Euler–Lagrange equation to obtain the bosonic field equation is

 ∂ν



LQED

∂ ∂ν Aµ 



−

∂ LQED = 0, ∂ Aµ

(33)

and the evolution equation for the bosonic field, after imposing the Lorentz–Gauge condition (the divergence of the fourpotential vanishes), is



 −∂02 + ∂12 + ∂22 + ∂32 Aµ = eψγ µ ψ.

(34)

Note that h¯ = c = 1 was used. Appendix B The connection of the deterministic with the stochastic coefficients can be made in the following way. For example,

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G. Costanza / Physica A 391 (2012) 2167–2181

wm(r,)1,0,0,0,0 = δi1 ,j1 δi2 ,j2 δi3 ,j3 δ1,jm θ (Pm′ ,1,0,0,0,0 − ξm(,r1),0,0,0,0 ) ′

+ iδi1 ,j1 δi2 ,j2 δi3 ,j3 δ1,jm θ (Pm′′ ,1,0,0,0,0 − ξm(,r1),0,0,0,0 ). ′′

(35)

After an average over realizations is performed, the deterministic weight can be easily obtained using the factorization as prescribed by the theorem, given

wm,1,0,0,0,0 = δi1 ,j1 δi2 ,j2 δi3 ,j3 δ1,jm θ (Pm′ ,1,0,0,0,0 − ξm(,r1),0,0,0,0 ) + iδi1 ,j1 δi2 ,j2 δi3 ,j3 δ1,jm θ (Pm′′ ,1,0,0,0,0 − ξm(,r1),0,0,0,0 ) ′

= =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4 1 4M

′′

Pm,1,0,0,0,0

Pm,1,0,0,0,0 ,

(36)

′ ′′ where Pm,1,0,0,0,0 = Pm ,1,0,0,0,0 + iPm,1,0,0,0,0 . The explicit form for all the weights is

  ′ ′′ wm,1,0,0,0,0 = δi1 ,j1 δi2 ,j2 δi3 ,j3 δ1,jm × θ (Pm′ ,1,0,0,0,0 − ξm(,r1),0,0,0,0 ) + i θ (Pm′′ ,1,0,0,0,0 − ξm(,r1),0,0,0,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

Pm,1,0,0,0,0 =

1 4M

Pm,1,0,0,0,0 ,

wm,1,0,−1,0,0 = δi1 −1,j1 δi2 ,j2 δi3 ,j3 δ1,jm θ (P − ξm(r,)1 )   ′ ′′ × θ (Pm′ ,1,0,−1,0,0 − ξm(,r1),0,−1,0,0 ) + i θ (Pm′′ ,1,0,−1,0,0 − ξm(,r1),0,−1,0,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,1,0,−1,0,0 =

1 4M

PPm,1,0,−1,0,0

wm,1,0,0,−1,0 = δi1 ,j1 δi2 −1,j2 δi3 ,j3 δ1,jm θ (P − ξm(r,)2 )   ′ ′′ × θ (Pm′ ,1,0,0,−1,0 − ξm(,r1),0,0,−1,0 ) + i θ (Pm′′ ,1,0,0,−1,0 − ξm(,r1),0,0,−1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,1,0,0,−1,0 =

1 4M

PPm,1,0,0,−1,0

wm,1,0,0,0,−1 = δi1 ,j1 δi2 ,j2 δi3 −1,j3 δ1,jm θ (P − ξm(r,)3 )   ′ ′′ × θ (Pm′ ,1,0,0,0,−1 − ξm(,r1),0,0,0,−1 ) + i θ (Pm′′ ,1,0,0,0,−1 − ξm(,r1),0,0,0,−1 )

wm,2,0,0,0,0

1

1

1

1

1

1

1

1

1

PPm,1,0,0,0,−1 = PPm,1,0,0,0,−1 (2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M   ′ ′′ = δi1 ,j1 δi2 ,j2 δi3 ,j3 δ2,jm × θ (Pm′ ,2,0,0,0,0 − ξm(,r2),0,0,0,0 ) + i θ (Pm′′ ,2,0,0,0,0 − ξm(,r2),0,0,0,0 )

=

=

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

Pm,2,0,0,0,0 =

1 4M

PPm,1,0,0,0,−1

wm,2,0,−1,0,0 = δi1 −1,j1 δi2 ,j2 δi3 ,j1 3 δ2,jm θ (P − ξm(r,)4 )   ′ ′′ × θ (Pm′ ,2,0,−1,0,0 − ξm(,r2),0,−1,0,0 ) + i θ (Pm′′ ,2,0,−1,0,0 − ξm(,r2),0,−1,0,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,2,0,−1,0,0 =

1 4M

PPm,2,0,−1,0,0

wm,2,0,0,−1,0 = δi1 ,j1 δi2 −1,j2 δi3 ,j1 3 δ2,jm θ (P − ξm(r,)5 )   ′ ′′ × θ (Pm′ ,2,0,0,−1,0 − ξm(,r2),0,0,−1,0 ) + i θ (Pm′′ ,2,0,0,−1,0 − ξm(,r2),0,0,−1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,2,0,0,−1,0 =

1 4M

PPm,2,0,0,−1,0

wm,2,0,0,0,−1 = δi1 ,j1 δi2 ,j2 δi3 −1,j3 δ2,jm θ (P − ξm(r,)6 )   ′ ′′ × θ (Pm′ ,2,0,0,0,−1 − ξm(,r2),0,0,0,−1 ) + i θ (Pm′′ ,2,0,0,0,−1 − ξm(,r2),0,0,0,−1 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,2,0,0,0,−1 =

1 4M

PPm,2,0,0,0,−1 ,

G. Costanza / Physica A 391 (2012) 2167–2181

2179

wm,3,0,0,0,0 = δi1 ,j1 δi2 ,j2 δi3 ,j3 δ3,jm   ′ ′′ × θ (Pm′ ,3,0,0,0,0 − ξm(,r3),0,0,0,0 ) + i θ (Pm′′ ,3,0,0,0,0 − ξm(,r3),0,0,0,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

Pm,3,0,0,0,0 =

1 4M

Pm,3,0,0,0,0

wm,3,0,−1,0,0 = δi1 −1,j1 δi2 ,j2 δi3 ,j3 δ3,jm θ (P − ξm(r,)7 )   ′ ′′ × θ (Pm′ ,3,0,−1,0,0 − ξm(,r3),0,−1,0,0 ) + i θ (Pm′′ ,3,0,−1,0,0 − ξm(,r3),0,−1,0,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,3,0,−1,0,0 =

1 4M

PPm,3,0,−1,0,0

wm,3,0,0,−1,0 = δi1 ,j1 δi2 −1,j2 δi3 ,j1 3 δ3,jm θ (P − ξm(r,)8 )   ′ ′′ × θ (Pm′ ,3,0,0,−1,0 − ξm(,r3),0,0,−1,0 ) + i θ (Pm′′ ,3,0,0,−1,0 − ξm(,r3),0,0,−1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,3,0,0,−1,0 =

1 4M

PPm,3,0,0,−1,0

wm,3,0,0,0,−1 = δi1 ,j1 δi2 ,j2 δi3 −1,j3 δ3,jm θ (P − ξm(r,)9 )   ′ ′′ × θ (Pm′ ,3,0,0,0,−1 − ξm(,r3),0,0,0,−1 ) + i θ (Pm′′ ,3,0,0,0,−1 − ξm(,r3),0,0,0,−1 )

wm,4,0,0,0,0

1

1

1

1

1

1

1

1

1

PPm,3,0,0,0,−1 = PPm,3,0,0,0,−1 (2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M = δi1 ,j1 δi2 ,j2 δi3 ,j3 δ4,jm   ′ ′′ × θ (Pm′ ,4,0,0,0,0 − ξm(,r4),0,0,0,0 ) + i θ (Pm′′ ,4,0,0,0,0 − ξm(,r4),0,0,0,0 )

=

=

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

Pm,4,0,0,0,0 =

1 4M

Pm,4,0,0,0,0

wm,4,0,−1,0,0 = δi1 −1,j1 δi2 ,j2 δi3 ,j3 δ4,jm θ (P − ξm(r,)10 )   ′ ′′ × θ (Pm′ ,4,0,−1,0,0 − ξm(,r4),0,−1,0,0 ) + i θ (Pm′′ ,4,0,−1,0,0 − ξm(,r4),0,−1,0,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,4,0,−1,0,0 =

1 4M

PPm,4,0,−1,0,0

wm,4,0,0,−1,0 = δi1 ,j1 δi2 −1,j2 δi3 ,j3 δ4,jm θ (P − ξm(r,)11 )   ′ ′′ × θ (Pm′ ,4,0,0,−1,0 − ξm(,r4),0,0,−1,0 ) + i θ (Pm′′ ,4,0,0,−1,0 − ξm(,r4),0,0,−1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,4,0,0,−1,0 =

1 4M

PPm,4,0,0,−1,0

wm,4,0,0,0,−1 = δi1 −1,j1 δi2 ,j2 δi3 −1,j3 δ4,jm θ (P − ξm(r,)12 )   ′ ′′ × θ (Pm′ ,4,0,0,0,−1 − ξm(,r4),0,0,0,−1 ) + i θ (Pm′′ ,4,0,0,0,−1 − ξm(,r4),0,0,0,−1 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPm,4,0,0,0,−1 =

1 4M

PPm,4,0,0,0,−1 .

Likewise,

wn,0,0,0,0

= δi1 ,j1 δi2 ,j2 δi3 ,j3 δn,jn   ′ ′′ × θ (Pn′ ,0,0,0,0 − ξn(,0r ),0,0,0 ) + i θ (Pn′′,0,0,0,0 − ξn,(0r,)0,0,0 ) =

1 4M

Pn,0,0,0,0 ,

wn,−1,0,0,0 = δi1 ,j1 δi2 ,j2 δi3 ,j3 δn,jn   ′ r) ′′ (r ) ′′ × θ (Pn′ ,−1,0,0,0 − ξn(,− 1,0,0,0 ) + i θ (Pn,−1,0,0,0 − ξn,−1,0,0,0 )

(37)

2180

G. Costanza / Physica A 391 (2012) 2167–2181

=

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

Pn,−1,0,0,0 =

1 4M

Pn,−1,0,0,0 ,

wn,0,−1,0,0 = δi1 −1,j1 δi2 ,j2 δi3 ,j3 δn,jn θ (P − ξn(,r1) )   ′ ′′ ) × θ (Pn′ ,0,−1,0,0 − ξn(,0r ),−1,0,0 ) + i θ (Pn′′,0,−1,0,0 − ξn,(0r,− 1,0,0 )

wn,0,+1,0,0

1

1

1

1

1

1

1

1

1

PPn,0,−1,0,0 = PPn,0,−1,0,0 , (2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M   = δi1 +1,j1 δi2 ,j2 δi3 ,j3 δn,jn 1 − θ (P − ξn(,r2) )   ′ ′′ ) × θ (Pn′ ,0,+1,0,0 − ξn(,0r ),+1,0,0 ) + i θ (Pn′′,0,+1,0,0 − ξn,(0r,+ 1,0,0 )

=

=

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

(1 − P )Pn,0,+1,0,0 =

1 4M

(1 − P )Pn,0,+1,0,0 ,

wn,0,0,−1,0 = δi1 ,j1 δi2 −1,j2 δi3 ,j3 δn,jn θ (P − ξn(,r3) )   ′ ′′ × θ (Pn′ ,0,0,−1,0 − ξn(,0r ),0,−1,0 ) + i θ (Pn′′,0,0,−1,0 − ξn,(0r,)0,−1,0 ) =

wn,0,0,+1,0

1

1

1

1

PPn,0,0,−1,0 =

1

PPn,0,0,−1,0 ,

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M   (r ) = δi1 ,j1 δi2 +1,j2 δi3 ,j3 δn,jn 1 − θ (P − ξn,4 )   ′ ′′ × θ (Pn′ ,0,0,+1,0 − ξn(,0r ),0,+1,0 ) + i θ (Pn′′,0,0,+1,0 − ξn,(0r,)0,+1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

(1 − P )Pn,0,0,+1,0 =

1 4M

(1 − P )Pn,0,0,+1,0 ,

wn,0,0,0,−1 = δi1 ,j1 δi2 ,j2 δi3 −1,j3 δn,jn θ (P − ξn(,r5) )   ′ ′′ × θ (Pn′ ,0,0,0,−1 − ξn(,0r ),0,0,−1 ) + i θ (Pn′′,0,0,0,−1 − ξn,(0r,)0,0,−1 )

wn,0,0,0,+1

1

1

1

1

1

1

1

1

1

PPn,0,0,0,−1 = PPn,0,0,0,−1 , (2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M   = δi1 ,j1 δi2 ,j2 δi3 +1,j3 δn,jn 1 − θ (P − ξn(,r6) )   ′ ′′ × θ (Pn′ ,0,0,0,+1 − ξn(,0r ),0,0,+1 ) + i θ (Pn′′,0,0,0,+1 − ξn,(0r,)0,0,+1 )

=

=

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

(1 − P )Pn,0,0,0,+1 =

1 4M

(1 − P )Pn,0,0,0,+1 ,

wn,−1,−1,0,0 = δi1 −1,j1 δi2 ,j2 δi3 ,j3 δn,jn θ (P − ξn(,r7) )   ′ r) ′′ (r ) ′′ × θ (Pn′ ,−1,−1,0,0 − ξn(,− ) + i − ξ ) θ ( P 1,−1,0,0 n,−1,−1,0,0 n,−1,−1,0,0

wn,−1,+1,0,0

1

1

1

1

1

1

1

1

1

PPn,−1,−1,0,0 = PPn,−1,−1,0,0 , (2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M   = δi1 +1,j1 δi2 ,j2 δi3 ,j3 δn,jn 1 − θ (P − ξn(,r8) )   ′ r) ′′ (r ) ′′ × θ (Pn′ ,−1,+1,0,0 − ξn(,− 1,+1,0,0 ) + i θ (Pn,−1,+1,0,0 − ξn,−1,+1,0,0 )

=

=

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

(1 − P )Pn,−1,+1,0,0 =

1 4M

(1 − P )Pn,−1,+1,0,0 ,

wn,−1,0,−1,0 = δi1 ,j1 δi2 −1,j2 δi3 ,j3 δn,jn θ (P − ξn(,r9) )   ′ r) ′′ (r ) ′′ × θ (Pn′ ,−1,0,−1,0 − ξn(,− 1,0,−1,0 ) + i θ (Pn,−1,0,−1,0 − ξn,−1,0,−1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPn,−1,0,−1,0 =

1 4M

PPn,−1,0,−1,0 ,

G. Costanza / Physica A 391 (2012) 2167–2181

2181

 (r ) 1 − θ (P − ξn,10 )



wn,−1,0,+1,0 = δi1 ,j1 δi2 +1,j2 δi3 ,j3 δn,jn   ′ r) ′′ (r ) ′′ × θ (Pn′ ,−1,0,+1,0 − ξn(,− 1,0,+1,0 ) + i θ (Pn,−1,0,+1,0 − ξn,−1,0,+1,0 ) =

1

1

1

1

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

(1 − P )Pn,−1,0,+1,0 =

1 4M

(1 − P )Pn,−1,0,+1,0 ,

) wn,−1,0,0,−1 = δi1 ,j1 δi2 ,j2 δi3 −1,j3 δn,jn θ (P − ξn(,r11 )   ′ (r ) ′′ (r ) × θ (Pn′ ,−1,0,0,−1 − ξn,−1,0,0,−1 ) + i θ (Pn′′,−1,0,0,−1 − ξn,− ) 1,0,0,−1

wn,−1,0,0,+1

1

1

1

1

1

1

1

1

1

PPn,−1,0,0,−1 , (2L1 + 1) (2L2 + 1) (2L3 + 1) 4 4M   ) = δi1 ,j1 δi2 ,j2 δi3 +1,j3 δn,jn 1 − θ (P − ξn(,r12 )   ′ r) ′′ (r ) ′′ × θ (Pn′ ,−1,0,0,+1 − ξn(,− 1,0,0,+1 ) + i θ (Pn,−1,0,0,+1 − ξn,−1,0,0,+1 )

=

=

(2L1 + 1) (2L2 + 1) (2L3 + 1) 4

PPn,−1,0,0,−1 =

(1 − P )Pn,−1,0,0,+1 =

1 4M

(1 − P )Pn,−1,0,0,+1 ,

(38)

and 11 Pm,k,l , 44 11 (r ) Jn,u,v = δu,ju δv,jv θ (Pn,u,v − ξn,u,v ) = Pn,u,v . 44 (r )

Im,k,l = δk,jk δl,jl θ (Pm,k,l − ξm,k,l ) =

(39)

Note that M is the number of sites of the lattice and (2L1 + 1), (2L2 + 1), and (2L3 + 1) are the numbers of sites corresponding to the space components x1 , x2 , and x3 , respectively. Also, 4 is the number of components of a 4-vector. The set of equations above defines a linear transformation allowing one to express all the Ps as a function of the w s, and it provides the complete relation between the coefficients of the deterministic evolution equation (the W s) with the connecting functions of the discrete stochastic evolution equation (the Ps). One more comment about the equations in this 1 1 appendix is in order. Note that, if P = 1/2, in the above equations, all equations such as 4M PPα or 4M (1 − P )Pα take the general form

1 P , 8M α

for any set of indices α .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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