A Brownian Particle and Fields I: Construction of Kinematics and ...

preprint:ELI-NP/RA5-TDR 0003

arXiv:1611.05861v1 [math-ph] 14 Nov 2016

A Brownian Particle and Fields I: Construction of Kinematics and Dynamics Keita Seto∗ November 18, 2016

Extreme Light Infrastructure – Nuclear Physics (ELI-NP) / Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), 30 Reactorului St., Bucharest-Magurele, jud. Ilfov, P.O.B. MG-6, RO-077125, Romania.

Abstract Tracking the “real” trajectory of a quantum particle still has been treated as the interpretation problem. It shall be expressed by the Brownian (stochastic) motion suggested by E. Nelson, but the well-defined mechanism of field generation from a stochastic particle hasn’t been proposed. For the improvement of this, I propose the extension of Nelson’s quantum dynamics for a relativistic scalar electron with its radiation, which is equivalent to the Klein-Gordon particle and field system.

Keyword: [Physics] Stochastic quantum dynamics, relativistic motion, field generation [mathematics] Applications of stochastic analysis This Volume I is reproduced from the parts of arXiv:1603.03379 .

∗ [email protected]

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Contents 1 Introduction

3

2 Kinematics of a scalar electron 2.1 Stochastic process . . . . . . 2.2 Complex velocity . . . . . . . 2.3 Fokker-Planck equations . . . 2.4 Proper time . . . . . . . . . .

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3 Dynamics of a scalar electron and 3.1 Euler-Lagrange (Yasue) equation 3.2 Action integral . . . . . . . . . . 3.3 Dynamics of a scalar electron . . 3.4 Dynamics of fields . . . . . . . .

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11 11 13 15 16

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4 Conclusion and discussion

18

Acknowledgement

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References

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Notation and Conventions Symbol

Description

c ~ m0 e V4M g A4 (V4M , g) B(I) (A4 (V4M , g), B(A4 (V4M , g)), µ) (Ω , D(P), P) ´ ˆ ˆ := Ω dP(ω) X(ω) EJX(•)K ˆ EJX(•)|C K Pτ ⊂ D(P) Fτ ⊂ D(P) x ˆ(◦, •) := {ˆ x(τ, ω)|τ ∈ R, ω ∈ Ω}

Speed of light Planck’s constant Rest mass of an electron Charge of an electron 4-dimentional standard vector space for the metric affine space Metric on V4M with its signature g := sign(+1, −1, −1, −1) 4-dimentional metric affine space with respect to V4M and g Borel σ-algebra of a topological space I. Measurable Minkowski spacetime Probability space with the probability measure P ˆ ˆ := {X(ω)|ω Expectation of X(•) ∈ Ω} ˆ Conditional expectation of X(•) on C ⊂ D(P). Sub-σ-algebra in the increasing family ”Past” = {Pτ |τ > −∞} Sub-σ-algebra in the decreasing family ”Future” = {Fτ |τ < ∞} Dual progressively measurable process (D-progressive, D-process) as the B(R) × D(P)/B(A4 (V4M , g)) measurable map [Collection of a scalar (spin-less) electron’s trajectory] e α A (x) Complex velocity; V α (x) := iλ2 × ∂ α ln φ(x) + m0

V ∈ V4M ⊕ iV4M

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Introduction

This series of the papers propose the quantum dynamics coupled with stochastic kinematics of a scalar (spinless) electron generating a light field as the extension from the model by E. Nelson [1, 2]. Especially, our main purpose by using Nelson’s stochastic quantization is the investigation of radiation reaction which is the kicked-back effect acting on an electron by radiation [3]. Many works of radiation reaction have been discussed in classical dynamics from early 1900’s, however, the corrections by non-linear QED becomes important in high-intensity field physics produced by the state-of-the-art O(10PW) laser experiments [4, 5, 6, 7, 8]. Comparing the radiation formula in these two regimes, the factor of q(χ) can be found in the following formula [9, 10, 11]: dWclassical dWQED = q(χ) × dt dt Where, dWclassical /dt = e2 /6πε0 c3 × dvα /dτ · dv α /dτ denotes Larmor’s formula as the energy loss by classical radiation [12] with respect to the Minkowski metric g := sign(+1, −1, −1, −1) and the 4-velocity v. The uniqueness of radiation reaction in high-intensity field physics is the dependence on the field strength via this factor q(χ), since q 3 ~ µα νβ := vα )(−eFex vβ ) = O(Fex × γ) . χ −gµν (−eFex 2 m20 c3 However as we will discuss it in Volume II [13], this is the applicable formula only in the case of an electron interacting with an external plane wave field [14, 15] in order to the fomulation of the non-linear Compton scattering [16, 17, 18]. Thus, the focused or superpositioned light must be out of this applicable range, it is very natural to consider how to generalize. Hence, we can expect to modify the factor of q(χ) in the general condition, however, what is the origin of this factor or this modification? Due to this reason, to identify the origin of q(χ) under the general fields by the stochastic quantization is the strong research motivation in high-intensity field physics. For this aspect, we choose quantum dynamics by using Brownian motions in this article series. Since Nelson’s scheme has very high-compatibility between classical and quantum dynamics of a particle, it is expected the bridge from well-known classical model to one in quantum dynamics. Here, let us summarize Nelson’s stochastic quantization. E. Nelson provided a quantization by the Brownian kinematics. A certain quanta draws a 3-dimentional ˆ ω) = Brownian motion as its trajectory in the non-relativistic regime [1, 2]. By employing the kinematics dx(t, p ˆ ω), t)dt + ~/2m0 × dW± (t, ω) and the Fokker-Planck equations derived from this kinematics, he V± (x(t, succeeded to demonstrate not only (A) his classical-style dynamics with the sub equations m0



 ~ 2 ∂t v(x, t) + v(x, t) · ∇v(x, t) − u(x, t) · ∇u(x, t) − ∇ u(x, t) = −∇V (x, t) 2m0  ~ ∇ ln ψ(x, t) m0   V+ (x, t) − V− (x, t) ~ u(x, t) = ∇ ln ψ(x, t) = Re 2 m0 v(x, t) =

V+ (x, t) + V− (x, t) = Im 2



is equivalent to the Schrödinger equation  ~2 2 ∇ + V (x, t) ψ(x, t) , i~∂t ψ(x, t) = − 2m0 

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but also (B) he answered why the square of the wave function should be regarded as the probability density ρ(x, t) = ψ ∗ (x, t)ψ(x, t) [1, 2]. The biggest advantage of this method is (C) the ability to draw the “real” trajectory of a quantum particle by its Brownian kinematics. It denotes (D) the transition between classical and quantum dynamics due to its semi-martingale, too. However, the feasibility of the coupled system between a stochastic particle and fields is not enough established. Hence, the realization of the field-generating mechanism by a stochastic scalar particle is the milestone in this Volume I for radiation reaction in Volume II [13]. This first volume dedicates the fundamental construction of the stochastic kinematics of a Klein-Gordon particle as a scalar electron and their dynamics including its light field generation. Then, Volume II [13] focuses on the topic of radiation reaction along its stochastic trajectory. In Section 2, we introduce the kinematics of a scalar electron before considering its dynamics. At first, we define the D-progressively measurable process xˆ(◦, •) as the trajectory class of the relativistic scalar electron (the modification of Nelson’s (S3)-process [2] for the Klein-Gordon equation), as the increasing and decreasing families of the σ-algebras. The basic kinematics dˆ x(τ, ω) = V± (ˆ x(τ, ω))dτ + λ× dW± (τ, ω) is found in the Minkowski spacetime at the same time. The complex velocity V(ˆ x(τ, ω)) := (1 − i)/2 × V+ (ˆ x(τ, ω)) + µ (1 + i)/2 × V− (ˆ x(τ, ω)) [19] is introduced by using this rule, and this V plays a role of the main cast in the present dynamics of a scalar electron in Section 3. Since its trajectory is considered as the Brownian diffusion process, we have to employ the Fokker-Planck equation. In the end of this section, we discuss the most delicate problem how should we define the proper time holding the transition between classical physics to quantum physics. For the calculation of dˆ x(τ, ω) = V± (ˆ x(τ, ω))dτ + λ × dW± (τ, ω), we need to define the evolutions of the velocities V± (ˆ x(◦, •)) and V(ˆ x(◦, •)). In Section 3, the dynamics of a quantum scalar electron interacting with fields is the main scope. Therefore, the mechanism of the field generation from a stochastic particle is also discussed here. Thus, our attention is devoted to the construction and demonstration of the KleinGordon equation and the Maxwell equation. Since we want to satisfy the transition between the classical and quantum regimes, the new action integral (the functional) like classical dynamics are examined. Hereby, the following dynamics of a scalar stochastic particle and fields is realized, as the transition from classical dynamics. m0 Dτ V µ (ˆ x(τ, ω)) = −eVˆν (ˆ x(τ, ω))F µν (ˆ x(τ, ω)) { s ˆ ν 4 µν µν ∂µ [F (x) + δf ] = µ0 × E −ec dτ Re {V (x)} δ (x − xˆ(τ, •)) R

Finally, we summarize this Volume I and propose the motivation for Volume II [13] in Section 4. Where, the transition between quantum and classical dynamics will be discussed by using Ehrenfest’s theorem [20] which is main tool to consider radiation reaction.

2

Kinematics of a scalar electron

The first part is the kinematics of a scalar electron. Let A4 (V4M , g) be the 4-dimensional metric affine space with respect to the 4-dimensional standard vector space V4M and the Minkowski metric g [21]. Defining the measurable space (A4 (V4M , g), B(A4 (V4M , g)), µ), we consider this as the measurable Minkowski spacetime. Where, B(I) denotes the Borel σ-algebra of a topological space I. For describing the stochastic 4

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processes, the probability space (Ω , D(P), P) has to be introduced, too [1, 2]. In these measurable spaces (A4 (V4M , g), B(A4 (V4M , g)), µ) and (Ω , D(P), P), the stochastic process of a scalar electron xˆ(◦, •), the complex velocity V(ˆ x(◦, •)), the Fokker-Planck equations and the proper time τ are discussed in this Section. Let us regard the coordinate map ϕ(x) := (x0 , x1 , x2 , x3 ) has been already introduced for x ∈ A4 (V4M , g) ˆ or x ∈ V4M even if we do not declare it explicitly. And let EJX(•)K be the expectation of the stochastic ´ ˆ ˆ ˆ ˆ := {X(ω)|ω := Ω dP(ω) X(ω). Introduce PA (B) as the conditional process X(•) ∈ Ω}, i.e., EJX(•)K probability of B given A. For {An }∞ n=1 of the countable decomposition of Ω, its minimum σ-algebra C = P∞ ´ ′ ˆ ′ ˆ σ({An }∞ ) is introduced. Then, EJ X(•)|C K(ω) := n=1 n=1 { Ω dPAn (ω )X(ω )}1An (ω) is defined as the ˆ / An ) = 0. conditional expectation of X(•) given C ⊂ D(P); 1An (ω) satisfies 1An (ω ∈ An ) = 1 and 1An (ω ∈

2.1

Stochastic process

Consider the the map x ˆ : R × Ω → A4 (V4M , g). By following Ref.[1, 2, 22, 23, 24], let us introduce the sub-σ-algebras Pτ , Fτ ⊂ D(P) and their filtration; the increasing family ”Past” = {Pτ |τ > −∞} and the decreasing family ”Future” = {Fτ |τ < ∞}. The stochastic process x ˆ(◦, •) := {ˆ x(τ, ω)|τ ∈ R, ω ∈ Ω} is defined as the {Pτ }-progressively measurable and the {Fτ }-progressively measurable process in (Ω , D(P), P). Definition 1 (D-progressive xˆ(◦, •)). Consider the {Pτ }-progressively measurable and the {Fτ }progressively measurable process xˆ(◦, •). [Nelson’s (S1)] For each (τ, ω) ∈ R × Ω , when the following B((−∞, τ ]) × Pτ measurable function µ µ V+ (ˆ x(◦, •)) and the B([τ, ∞)) × Fτ measurable function V− (ˆ x(◦, •)) exist as the limit in L1 , x ˆ(◦, •) is named “Nelson’s (S1)-process” [2]: µ V+ (ˆ x(τ, ω)) =

µ V− (ˆ x(τ, ω))

=

lim E

s

lim E

s

δt→0+

δt→0+

{ x ˆµ (τ + δτ, •) − x ˆµ (τ, •) P τ (ω) δτ { x ˆµ (τ, •) − x ˆµ (τ − δτ, •) Fτ (ω) δτ

(1)

(2)

[D-progressive] Let W+ (◦, •) and W− (◦, •) be the forward and backward standard Wiener processes. For a given set (τ, ω) ∈ R × Ω with respect to τa ≤ τ ≤ τb , consider the following {Pτ }-progressive and {Fτ }-progressive Itô process [25]. µ

x ˆ (τ, ω) = =

x ˆ (τa , ω) +

ˆ

x ˆµ (τb , ω) −

ˆ

µ

τ

τa τb

τ

dτ

′

µ V+ (ˆ x(τ ′ , ω))

+λ×

µ dτ ′ V− (ˆ x(τ ′ , ω)) − λ ×

ˆ

τ

τ

dW+µ (τ ′ , ω)

ˆ aτb τ

dW−µ (τ ′ , ω)

(3) (4)

p Where, λ := ~/m0 ∈ R [19]. This stochastic process includes Nelson’s (S1)-process obviously. Then, introduce the modified rule of Nelson’s (S2) and (S3)-processes [2] as the limit in L2 : g = − lim E

s

g = + lim E

s

δt→0+

δt→0+

{ [W+ (τ + δτ, •) − W+ (τ, •)] ⊗ [W+ (τ + δτ, •) − W+ν (τ, •)] Pτ (ω) δτ { [W− (τ, •) − W− (τ − δτ, •)] ⊗ [W− (τ, •) − W− (τ − δτ, •)] Fτ (ω) δτ 5

(5)

(6)

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We name “the dual-progressively measurable process”, or shortening “D-progressive” and also “the Dprocess”, such a {Pτ }-progressive and {Fτ }-progressive xˆ(◦, •) (3-4) instead of Nelson’s (S2) and (S3)process [2]. Of cause, g ∈ V4M ⊗V4M is the metric in the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ) with its signature g = diag(+1, −1, −1, −1). The differential form of (3-4) is also employed: µ dˆ xµ (τ, ω) = V± (ˆ x(τ, ω))dτ + λ × dW±µ (τ, ω)

(7)

By using this definition of the dual-progressively measurable process, let us assume the following idea: Conjecture 2. The D-progressive xˆ(◦, •) draws the trajectory of a scalar electron as the solution of the Klein-Gordon equation. The demonstration of Conjecture 2 is the first topic in this paper and its feasibility is found in Section 3. Under the construction of the D-progressive xˆ(◦, •), the standard Wiener processes W+ (◦, •) and W− (◦, •) satisfy the following conditional expectations: (

EJdW+µ (τ, •)|Pτ K(ω) = 0

EJdW−µ (τ, •)|Fτ K(ω) = 0

(8)

These conditional expectations induce the expectation formula EJdW±µ (τ, •)K = 0, too. These standard Wiener processes in the D-process impose the following Itô rules [26]; dτ · dτ = 0 ,

(9)

dτ · dW±µ (τ, ω) = 0 ,

(10)

dW±µ (τ, ω) · dW±ν (τ, ω) = ∓g µν dτ .

(11)

Moreover, let ξˆ± (τ, ω) be the white noise as the time derivatives of W± (τ, ω) as a means of the generalized´ ´ µ (τ, ω) with respect to a test function function satisfying R dτ dΦ/dτ (τ, ω) · W±µ (τ, ω) = − R dτ Φ(τ, ω) · ξˆ± Φ for each ω ∈ Ω. By introducing the new symbols d± xˆ(τ, ω) as the RHS in (7), (7) is recognized as the µ summation of the drift velocity V± (ˆ x(τ, ω)) and the randomness λ × ξˆ± (τ, ω) = λ × dW±µ /dτ (τ, ω), d± x ˆµ µ µ (τ, ω) = V± (ˆ x(τ, ω)) + λ × ξˆ± (τ, ω) . dτ

(12)

Corresponding to (9-11), the modified Itô rule for this white noise is derived as µ ν ξˆ± (τ, ω)ξˆ± (τ ′ , ω) = ∓g µν δ(τ − τ ′ ) .

(13)

µ µ Since EJξˆ+ (τ, •)|Pτ K = 0 and EJξˆ− (τ, •)|Fτ K = 0, it is repeatedly emphasized the drift velocities V± ∈ V4M as the conditional expectation of (12) at time τ (the mean-derivative): µ V+ (ˆ x(τ, ω))

:= =

s

{ d+ xˆµ (τ, •) Pτ (ω) dτ s µ { x ˆ (τ + δτ, •) − xˆµ (τ, •) lim E P τ (ω) δt→0+ δτ E

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µ V− (ˆ x(τ, ω))

s

{ d− x ˆµ (τ, •) Fτ dτ { s µ x ˆ (τ, •) − xˆµ (τ − δτ, •) := F lim E τ (ω) δt→0+ δτ =

E

(15)

In general, the D-progressive x ˆ(◦, •) is induce the following Itô formula [21, 25].

Lemma 3 (Itô formula). Consider the function f : A4 (V4M , g) → C as the C 2 -locally square integrate function f ∈ L2loc (A4 (V4M , g), µ). Let ∂µ f and ∂µ ∂ µ f also be the elements in L2loc (A4 (V4M , g), µ) under the meaning of the generalized-function, d± f along the D-progressive x ˆ(◦, •) satisfies the following Itô formula with respect to each (τ, ω); d± f (ˆ x(τ, ω)) = ∂µ f (ˆ x(τ, ω))d± xˆµ (τ, ω) ∓

λ2 ∂µ ∂ µ f (ˆ x(τ, ω))dτ . 2

(16)

This is also expressed by the form of its integral, f (ˆ x(τb , ω)) − f (ˆ x(τa , ω)) = =

ˆ

τb

d± x ˆµ (τ, ω) ∂µ f (ˆ x(τ, ω)) ∓

τa

2.2

(17)

d± f (ˆ x(τ, ω))

τa ˆ τb

λ2 2

ˆ

τb

dτ ∂µ ∂ µ f (ˆ x(τ, ω)) .

(18)

τa

Complex velocity

Especially by limiting the class in γτ = Pτ ∩ Fτ as the “present” τ , the superposition of d+ and d− is introduced. L. Not tale introduced that the following complex differential dˆ and the complex velocity V(ˆ x(◦, •)) as the essential manners in quantum dynamics [19]. Definition 4 (Complex differential and velocity). Consider the C 2 -locally square integrate function f ∈ L2loc (A4 (V4M , g), µ), the derivatives d+ f and d− f characterized by Lemma 3. Let dˆ be the complex differential defined at the point x ˆ(τ, ω) in γτ = Pτ ∩ Fτ with the Markov property: 1−i 1+i dˆ := d+ + d− 2 2 2 ˆ (ˆ ˆxµ (τ, ω) + iλ ∂ µ ∂µ f (ˆ x(τ, ω))dτ df x(τ, ω)) = ∂µ f (ˆ x(τ, ω))dˆ 2 Then, consider the conditional expectation of the derivative under the condition γτ is denoted by | t ˆ iλ2 µ df µ x(τ, ω))∂µ f (ˆ x(τ, ω)) + (ˆ x (τ, •)) γτ (ω) = V µ (ˆ ∂ ∂µ f (ˆ x(τ, ω)) , E dτ 2

(19) (20)

(21)

especially when f (ˆ x(τ, ω)) = x ˆ(τ, ω), it derives the complex velocity V ∈ V4M ⊕ iV4M , µ

V (ˆ x(τ, ω)) := E

t

| ˆxµ 1−i µ dˆ 1+i µ (τ, •) γτ (ω) = V+ (ˆ x(τ, ω)) + V− (ˆ x(τ, ω)) . dτ 2 2 7

(22)

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By choosing the wave function φ ∈ L2loc (A4 (V4M , g), µ) like Ref.[19], the following is defined: V α (x) := iλ2 × ∂ α ln φ(x) +

e α A (x) m0

(23)

This V α (x) behaves as the eigenvalue of the operator i~Dα /m0 := [i~∂ α + eAα (x)]/m0 with respect to φ(x). In addition, one of the interesting characteristics of V is the following the gauge invariance. Theorem 5 (Gauge invariance of V). For a given C 1 -function Λ : A4 (V4M , g) → R, the complex velocity V α (x) satisfies the local U (1) gauge symmetry in the transformation (φ, A) 7→ (φ′ , A′ ): A′α (x) = Aα (x) − ∂ α Λ(x)

φ′ (x) = e−ieΛ(x)/~ × φ(x) ,

(24)

This is the one of the basic rule in scalar QED. It is found the fact that φ satisfies the Klein-Gordon equation in the later study.

2.3

Fokker-Planck equations

In the probability space (Ω , D(P), P) and the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ), then let us consider a C 2 -locally square integrate function f ∈ L2loc (A4 (V4M , g), µ) and its expectation EJf (ˆ x(◦, •))K along the D-progressive x ˆ(◦, •) : R × Ω → A4 (V4M , g). Where, the measure in the Minkowski spacetime is µ : A4 (V4M , g) → [0, ∞). From the definition of the expectation, there are a certain set Θτ ⊂ Ω and the C 2,1 -probability density function p : A4 (V4M , g) × R → [0, ∞) characterized by the following relation with respect to each τ ∈ R: ˆ P(Θτ ) :=

dµ(x) p(x, τ ) = 1

(25)

x ˆ(τ,Θτ )⊂A4 (V4M ,g)

Here, Θτ is the set satisfying x ˆ(τ, Θτ ) := supp(p(◦, τ )) ⊂ A4 (V4M , g) where a scalar electron can exist in A4 (V4M , g). Since P(Ω\Θτ ) = 0, the domains of P and µ are expanded: P(Ω) =

ˆ

dµ(x) p(x, τ ) = 1

(26)

A4 (V4M ,g)

Hence, the expectation of f (ˆ x(τ, •)) is, EJf (ˆ x(τ, •))K

ˆ

:=

dP(ω) f (ˆ x(τ, ω))

ˆΩ

=

dµ(x) f (x)p(x, τ ) .

(27)

dµ(x) f (x)∂τ p(x, τ ) .

(28)

A4 (V4M ,g)

Consider the derivative of it with respect to τ , d EJf (ˆ x(τ, •))K = dτ

ˆ

A4 (V4M ,g)

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The LHS of this equation (28) along the evolution d± x ˆ(τ, ω) should be considered as follows; d EJf (ˆ x(τ, •))K dτ

= =

s { λ2 µ µ E V± (ˆ x(τ, •))∂µ f (ˆ x(τ, •)) ∓ ∂ ∂µ f (ˆ x(τ, •)) 2   ˆ λ2 µ µ dµ(x) f (x) −∂µ [V± (x)p(x, τ )] ∓ ∂ ∂µ p(x, τ ) . 2 A4 (V4M ,g)

(29)

For an arbitrary C 2 -function f ∈ L2loc (A4 (V4M , g), µ), the following Fokker-Planck equations of the Dprogressive x ˆ(◦, •) are derived from (28) and (29). Theorem 6 (Fokker-Planck equation). Consider the D-progressive x ˆ(◦, •) in the probability space (Ω , D(P), P) characterized by Definition 1. Let the C 2,1 -generalized function p : A4 (V4M , g)×R → [0, ∞) be the probability density satisfying the following Fokker-Planck equation with respect to x ∈ A4 (V4M , g): µ ∂τ p(x, τ ) + ∂µ [V± (x)p(x, τ )] ±

λ2 µ ∂ ∂µ p(x, τ ) = 0 2

(30)

By using the definition of the complex velocity V ∈ V4M ⊕ iV4M (see (22)), the superpositions of the “±”Fokker-Planck equations are imposed by using the real and imaginary part of V: ∂τ p(x, τ ) + ∂µ [Re{V µ (x)}p(x, τ )] = 0

Im{V µ (x)}

= =

λ2 × ∂ µ ln p(x, τ ) 2 ˆ λ2 µ × ∂ ln dτ p(x, τ ) , 2 R x ∈ supp(p(◦, τ ))

(31)

(32) (33)

Equation (31) represents the equation of continuity of the probability density p(x, τ ) in 4 + 1 dimensional space, (32-33) are a mimic of the osmotic pressure formula [1, 2]. The reason why there are the two expressions of (32) and (33), is the following should be derived from (30); Im{V µ (x)}p(x, τ ) −

λ2 µ ∂ p(x, τ ) = 0 . 2

(34)

However, since this equation doesn’t include ∂τ , λ2 Im{V (x)} dτ p(x, τ ) − ∂ µ 2 R µ

ˆ

ˆ

dτ p(x, τ ) = 0

(35)

R

´ is also satisfied and derives (33). Therefore, p(x, τ ) := ρ(x) × T (τ )/ R dτ ′ T (τ ′ ) is satisfied for the certain C 2 -function ρ : A4 (V4M , g) → [0, ∞) and the C 1 -function T : R → [0, ∞).

2.4

Proper time

One of the delicate problem in this paper is the definition of the proper time on the stochastic trajectory in the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ). Since we want to consider the D-process of a scalar 9

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electron as its quantization from classical dynamics, the limit ~ → 0 should imply the classical definition of the proper time. The proper time in classical dynamics is dτ |classical =

1 × c

q dxµ (τ )dxµ (τ ) ,

(36)

Here, the metric is selected as g = diag(+1, −1, −1, −1). Let us consider the following equation for quantum dynamics in advance; µ ˆxν (τ, ω) = V ∗ (ˆ dˆ∗ xˆµ (τ, ω)dˆ x(τ, ω))dτ 2 . (37) µ x(τ, ω))V (ˆ Of cause, A∗ represents the complex conjugate of A. Again, recall the definition of the complex velocity V µ (x) =

1 i~∂ µ φ(x) + eAµ (x)φ(x) i~Dµ φ(x) 1 × × = , m0 φ(x) m0 φ(x)

(38)

Vµ∗ (x)V µ (x) becomes Vµ∗ (x)V µ (x)

=

φ(x)(−i~D∗µ ) · (−i~D∗µ )φ∗ (x) + φ∗ (x)(i~Dµ ) · (i~Dµ )φ(x) 1 × 2m20 φ∗ (x)φ(x) 2 µ ∗ ~ ∂µ ∂ [φ(x) · φ (x)] + × . 2m20 φ∗ (x)φ(x)

(39)

Let φ(x) ∈ L2loc (A4 (V4M , g), µ) be the wave function of the complex Klein-Gordon equation, (i~Dµ )·(i~Dµ )φ(x)− m20 c2 φ(x) = 0. The validity of this equivalency is given in Section 3. Due to this assumption, the first term of RHS in (39) must be a constant of c2 . Then the issue is the behavior of ~2 /m20 × ∂µ ∂ µ [φ∗ (x)φ(x)]/φ∗ (x)φ(x). Where, we follow the proposal by T. Zastawniak in Ref.[27]. By defining the function φ(x) := exp[R(x)/~ + iS(x)/~] with respect to real valued functions of R and S, we obtain the relation φ∗ (x)φ(x) = exp[2R(x)/~]. From the definition of (38), ∂ µ R(x) = Im{m0 V µ (x)} = ~/2 × ∂ µ ln p(x, τ ) is fulfilled (see (32)); ∂µ ∂ µ [φ(x) · φ∗ (x)] ∂µ ∂ µ p(x, τ ) ~2 ~2 × × = . 2m20 φ∗ (x)φ(x) 2m20 p(x, τ )

(40)

Hence, the second term in RHS is non-zero in general. However, let us introduce the expectation of (40) after the substitution x = x ˆ(τ, ω), s

~2 ∂µ ∂ µ p(ˆ x(τ, •), τ ) E × 2 2m0 p(ˆ x(τ, •), τ )

{

=

~2 × 2m20

=

~2 2m20

 ∂µ ∂ µ p(x, τ ) p(x, τ ) dµ(x) p(x, τ ) A4 (V4M ,g) ˆ dµ(x) ∂µ ∂ µ p(x, τ ) = 0, × ˆ



(41)

A4 (V4M ,g)

under the acceptable condition of ∂ µ p(x, τ )|x∈∂A4 (V4M ,g) = 0 with respect to the boundary of A4 (V4M , g), ∂A4 (V4M , g). Therefore the following relation is realized. Lemma 7 (Lorentz invariant). Consider the D-progressive x ˆ(◦, •). A relativistic-stochastic scalar electron satisfies the following Lorentz invariant for each τ ∈ R [27]. q y E Vµ∗ (ˆ x(τ, •))V µ (ˆ x(τ, •)) = c2 10

(42)

ELI-NP/IFIN-HH

Keita Seto

This is the relation what we need to use. Due to this Lemma 7, the proper time is defined as the mimic of classical dynamics (36). Definition 8 (Proper time). For each τ ∈ R, the proper time of a stochastic scalar electron is defined as the following invariant parameter; dτ :=

1 × c

q ˆxν (τ, •)K . EJdˆ∗ xˆµ (τ, •) · dˆ

(43)

Here I want to mention the fact that Definition 8 realizes the compatibility from classical dynamics to quantum dynamics and cdτ is not the length of the world line along a Brownian motion.

3

Dynamics of a scalar electron and fields

In order to the realization this kinematics (7), i.e., dˆ x(τ, ω) = V± (ˆ x(τ, ω))dτ + λ × dW± (τ, ω), we need to µ 4 investigate the behavior of the complex velocity V (ˆ x(τ, ω)) ∈ VM ⊕ iV4M . For the derivation of V(ˆ x(τ, ω)), the action integral (the functional) on a stochastic trajectory is required. Before entering the main body, we consider the variational calculus for obtaining the Euler-Lagrange equation of a stochastic particle. After this explanation, let us proceed the concrete style of the action integral of the system between a stochastic scalar electron and fields corresponding to the styles in classical dynamics.

3.1

Euler-Lagrange (Yasue) equation

In this small section, we focus the action integral of a Brownian particle interacting with a field. Concerning the method by the complex velocity V ∈ V4M ⊕ iV4M, L. Not tale suggests the following Lagrangian due to the forward and the backward evolution; L0 (τ, x ˆ, V+ , V− ) = L(τ, x ˆ, V) [19]. However, I propose the extension of ∗ ∗ 4 it, namely, L0 (τ, xˆ, V+ , V− ) = L(τ, x ˆ, V, V ). Here V ∈ VM ⊕ iV4M is the complex conjugate of V. Recalling the definition of V in (22),   1−i 1+i 1+i 1−i L0 (τ, x ˆ, V+ , V− ) = L τ, x ˆ, V+ + V− , V+ + V− 2 2 2 2

(44)

along a sample path in the D-progressive xˆ(◦, ω) ⊂ A4 (V4M , g). For the simplification, the following signatures are introduced (γτ = Pτ ∩ Fτ as the “present” τ ): D± τ

Dτ := E

t

:= E

{ d± γτ dτ

s

(45)

| dˆ 1−i + 1+i − Dτ + Dτ γτ = dτ 2 2

(46)

µ By using these expressions, D± ˆµ (τ, ω) = V± (ˆ x(τ, ω)) is obviously satisfied. The variation of the functional τ x ´ τ2 dτ E JL (τ, x ˆ , V , V )K with respect to x ˆ is 0 + − τ1

δ

ˆ

τ2

τ1

dτ E JL0 (τ, x ˆ, V+ , V− )K

=

ˆ

τ2

τ1

dτ E

s

11

∂L0 µ ∂L0 µ ∂L0 µ δˆ x + µ δV+ + µ δV− ∂x ˆµ ∂V+ ∂V−

{

ELI-NP/IFIN-HH

Keita Seto

=

ˆ

τ2

dτ E

τ1

s

∂L µ ∂L ∂L ∗ µ δˆ x + Dτ δˆ xµ + D δˆ x ∂x ˆµ ∂V µ ∂V ∗µ τ

{

,

(47)

where, the following relations are introduced. ∂L ∂L0 = µ ∂x ˆ ∂x ˆµ

(48)

∂L0 1 − i ∂L 1 + i ∂L + µ = ∂V+ 2 ∂V µ 2 ∂V ∗µ

(49)

∂L0 1 − i ∂L 1 + i ∂L + µ = ∂V− 2 ∂V µ 2 ∂V ∗µ

(50)

Then, we need to recall the following Nelson’s partial integral [1, 2, 23]. Lemma 9 (Nelson’s partial integral). Let α, β ∈ L2loc (A4 (V4M , g), µ) be the C 2 -local square integrate functions defined along the D-progressive x ˆ(◦, •), the following partial integral formula is fulfilled; ˆ

τ2

τ1

q y µ dτ E D± x(τ, •)) · β µ (ˆ x(τ, •)) + αµ (ˆ x(τ, •)) · D∓ x(τ, •)) τ αµ (ˆ τ β (ˆ

or its differential form,

= E Jαµ (ˆ x(τ2 , •))β µ (ˆ x(τ2 , •)) − αµ (ˆ x(τ1 , •))β µ (ˆ x(τ1 , •))K ,

d E Jαµ (ˆ x(τ, •))β µ (ˆ x(τ, •))K = E dτ

t

D± x(τ, •)) · β µ (ˆ x(τ, •)) τ αµ (ˆ

|

µ +αµ (ˆ x(τ, •)) · D∓ x(τ, •)) τ β (ˆ

.

(51)

(52)

By using the superposition of the above “±”-formulas, it can be switched to the formula for the complex derivatives. t | µ D α (ˆ x (τ, •)) · β (ˆ x (τ, •)) τ µ d E Jαµ (ˆ x(τ, •))β µ (ˆ x(τ, •))K = E dτ +αµ (ˆ x(τ, •)) · D∗τ β µ (ˆ x(τ, •)) t ∗ | µ Dτ αµ (ˆ x(τ, •)) · β (ˆ x(τ, •)) = E (53) +αµ (ˆ x(τ, •)) · Dτ β µ (ˆ x(τ, •))

Proof. Consider the following relation at first;

since

q y µ E D+ x(τ, •)) · β µ (ˆ x(τ, •)) + αµ (ˆ x(τ, •)) · D− x(τ, •)) τ αµ (ˆ τ β (ˆ y q µ x(τ, •)) · β µ (ˆ x(τ, •)) + αµ (ˆ x(τ, •)) · D+ x(τ, •)) , = E D− τ αµ (ˆ τ β (ˆ y q µ E D+ x(τ, •)) · β µ (ˆ x(τ, •)) + αµ (ˆ x(τ, •)) · D− x(τ, •)) τ β (ˆ τ αµ (ˆ q y µ −E D− x(τ, •)) · β µ (ˆ x(τ, •)) + αµ (ˆ x(τ, •)) · D+ x(τ, •)) τ αµ (ˆ τ β (ˆ ( " #) ˆ ∂ν αµ (x) · β µ (x) 4 ν = −λ × dµ(x) ∂ p(x, τ ) A4 (V4M ,g) −αµ (x) · ∂ν β µ (x) 12

= 0.

(54)

(55)

ELI-NP/IFIN-HH

Keita Seto

Then using the Fokker-Planck equation (30), d E Jαµ (ˆ x(τ, •))β µ (ˆ x(τ, •))K dτ ˆ

dµ(x) αµ (x)β µ (x)∂τ p(x, τ )

=

A4 (V4M ,g)

1 = ×E 2

t

− (D+ x(τ, •)) · β µ (ˆ x(τ, •)) τ + Dτ )αµ (ˆ

|

− µ +αµ (ˆ x(τ, •)) · (D+ x(τ, •)) τ + Dτ )β (ˆ

.

(56)

By combining it with (54), (52) is demonstrated. And also considering the superposition of “±” of (52), (53) is also imposed. By considering (53), (47) becomes δ

ˆ

τ2

τ1

dτ E JL0 (τ, xˆ, V+ , V− )K

=

s

{  ∂L ∂L µ ∗ ∂L δˆ x − D − D τ τ ∂x ˆµ ∂V µ ∂V ∗µ τ1 { s ˆ τ2 ∂L ∂L µ d µ , δˆ x + δˆ x E dτ + dτ ∂V µ ∂V ∗µ τ1

ˆ

τ2

dτ E

(57)

the following Theorem 10 is derived with the boundary conditions δˆ xµ (τ1,2 , •) = 0. Theorem 10 (Euler-Lagrange (Yasue) equation). Let the functional S[ˆ x, V, V ∗ ] =

ˆ

τ2

τ1

dτ E JL (τ, x ˆ(τ, •), V(ˆ x(τ, •)), V ∗ (ˆ x(τ, •)))K

(58)

be the action integral of a particle along the D-progressive xˆ(◦, •). By the variation of this action integral with respect to x ˆ, the following Euler-Lagrange (Yasue) equation is induced: ∂L ∂L ∂L − D∗τ − Dτ =0 ∂x ˆµ ∂V µ ∂V ∗µ

(59)

This is the version of the extended Euler-Lagrange equation for a stochastic particle, namely, this equation (59) corresponds to the Yasue equation in Nelson’s framework [23, 24].

3.2

Action integral

Let us consider the action integral of “classical” dynamics in the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ); Sclassical

=

ˆ

R

dτ

m0 vα (τ )v α (τ ) − 2 ˆ +

ˆ

dτ eAα (x(τ ))v α (τ )

R

A4 (V4M ,g)

dµ(x)

1 Fαβ (x)F αβ (x) . 4µ0 c

(60)

Corresponding to (60), I propose the new action integral for Klein-Gordon particle-field system via the introduction of the mass measure and the charge measure.

13

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Keita Seto

Definition 11 (Mass and charge measures). Let M and E be the mass measure and the charge measure of a stochastic scalar electron defined in the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ). For the positive constants m0 and e with respect to τ ∈ R, M and E are characterized by ˆ ˆ q y dµ(x) E δ 4 (x − xˆ(τ, •)) , (61) dM(x, τ ) := m0 × A4 (V4M ,g)

A4 (V4M ,g)

ˆ

dE(x, τ ) := e ×

ˆ

A4 (V4M ,g)

A4 (V4M ,g)

The following is also introduced for simply writing,

q y dµ(x) E δ 4 (x − x ˆ(τ, •)) .

(62)

q y dM(x, τ ) = m0 × E δ 4 (x − xˆ(τ, •)) dµ(x),

(63)

q y dE(x, τ ) = e × E δ 4 (x − x ˆ(τ, •)) dµ(x).

(64)

q y The key of this definition is the appearance of the smeared distribution E δ 4 (x − x ˆ(τ, •)) dµ(x) from δ 4 (x − x(τ ))dµ(x) in classical dynamics. By this new idea with the complex velocity V(x) ∈ V4M ⊕ iV4M (22-23), let us introduce the functional S as the action integral of the dynamics. Theorem 12 (Action integral). The following functional S is the action integral deriving the dynamics of a “stochastic” scalar electron and fields characterized by V ∈ V4M ⊕ iV4M , A ∈ V4M with the help by F ∈ V4M ⊗ V4M and a given tensor δf ∈ V4M ⊗ V4M : S[ˆ x, V, V ∗ , A] =

1 dM(x, τ ) Vα∗ (x)V α (x) 2 A4 (V4M ,g) R ˆ ˆ − dτ dE(x, τ ) Aα (x)Re {V α (x)} ˆ

dτ

ˆ

A4 (V4M ,g)

R

+

ˆ

dµ(x)

A4 (V4M ,g)

1 [Fαβ (x) + δfαβ (x)] · [F αβ (x) + δf αβ (x)] 4µ0 c

(65)

Writing the detail of the measures explicitly, sˆ

{ m0 ∗ α S[ˆ x, V, V , A] = E V (ˆ x(τ, •))V (ˆ x(τ, •)) dτ 2 α { sR ˆ α x(τ, •))Re{V (ˆ x(τ, •))} +E − dτ e Aα (ˆ ˆR 1 + dµ(x) [Fαβ (x) + δfαβ (x)] · [F αβ (x) + δf αβ (x)] . 4µ0 c A4 (V4M ,g) ∗

(66)

Where, F αβ (x) := ∂ µ Aν − ∂ ν Aµ . Hence, the Lagrangian density L is also introduced; L(x, xˆ, V, V ∗ , A) =

ˆ

dτ

R

+



dE(x, τ ) 1 dM (x, τ )Vα∗ (x)V α (x) − Aα (x)Re {V α (x)} 2 dµ dµ

1 [Fαβ (x) + δfαβ (x)] · [F αβ (x) + δf αβ (x)] 4µ0 c 14

 (67)

ELI-NP/IFIN-HH

Keita Seto

such that S[ˆ x, V, V ∗ , A] =

3.3

´

A4 (V4M ,g)

dµ(x) L(x, xˆ, V, V ∗ , A)

Dynamics of a scalar electron

Here, the Lagrangan of a stochastic scalar electron with the interaction is ˆ 1 Lparticle [ˆ x, V, V ∗ ] := dM(x, τ ) Vα∗ (x)V α (x) 2 A4 (V4M ,g) ˆ dE(x, τ ) Aα (x)Re {V α (x)} . −

(68)

A4 (V4M ,g)

Substituting this for (59), n o Re m0 Dτ V µ (ˆ x(τ, ω)) + eVˆν (ˆ x(τ, ω))F µν (ˆ x(τ, ω)) = 0 .

(69)

Here, the following new signature Vˆ µ (x) and the relations are introduced with the Lorenz gauge ∂µ Aµ = 0 [19]: iλ2 µ Vˆ µ (x) := V µ (x) + ∂ (70) 2 Dτ = Vˆ µ (x)∂µ

(71)

Dτ Aµ (ˆ x(τ, ω)) = Vˆ ν (ˆ x(τ, ω))∂ν Aµ (ˆ x(τ, ω))

(72)

D∗τ Aµ (ˆ x(τ, ω)) = Vˆ ∗ν (ˆ x(τ, ω))∂ν Aµ (ˆ x(τ, ω))

(73)

Theorem 13 (Equation of a stochastic particle’s motion). The equation of a “stochastic” motion of a scalar electron interacting with fields is dM(x, τ ) Dτ V µ (x) = −dE(x, τ ) Vˆν (x)F µν (x)

(74)

derived from the action integral (65-66). Considering its integral with respect to x ∈ A4 (V4M , g), m0 Dτ V µ (ˆ x(τ, ω)) = −eVˆν (ˆ x(τ, ω))F µν (ˆ x(τ, ω)) .

(75)

These equations are equivalent to the Klein-Gordon equation. Proof. Let an arbitrary smooth C 1,0 -function f : A4 (V4M , g) × R → R be a degree of freedom of the imaginary part of (69), namely, m0 Dτ V µ (ˆ x(τ, ω)) = −eVˆν (ˆ x(τ, ω))F µν (ˆ x(τ, ω)) +

i µ ∂ f (ˆ x(τ, ω), τ ) . 2m0

(76)

This equation is the general solution of (69). Transforming Dτ V µ + e/m0 × Vˆν F µν with (71),   e µν e ˆ µν ν µ ˆ = Vˆν ∂ µ V ν , Vν F = Vν ∂ V + F Dτ V + m0 m0 µ

since the identity ∂αV β − ∂β V α = 15

e αβ F m0

(77)

(78)

ELI-NP/IFIN-HH

Keita Seto

derived from (23) (also see Ref[19]). Substituting (23) and (70) for this (76), Vˆν ∂ µ V ν −

i µ ∂ f 2m20

= =

   e ν i µ e iλ2 2 ν µ iλ × ∂ν ln φ + Aν + A − ∂ f ∂ν × ∂ iλ × ∂ ln φ + m0 2 m0 2m20   1 µ (i~∂ν + eAν )(i~∂ ν + eAν )φ − if φ = 0. (79) ∂ 2 m20 φ



2

Therefore, the quasi-Klein-Gordon equation is found by putting an arbitrary constant c2 , (i~∂ν + eAν )(i~∂ ν + eAν )φ − (m20 c2 + if )φ = 0 .

(80)

Thus, f implies the non-electromagnetic interaction of φ. This interaction should be removed or be rounded into the free-propagation term of a scalar electron as the mass. Hence f ≡ 0 can be selected. Then, the normal Klein-Gordon equation is derived; (i~∂ν + eAν )(i~∂ ν + eAν )φ − m20 c2 φ = 0 .

(81)

Therefore, the equation of motion (74 or 75) is equivalent to the Klein-Gordon equation. ˆ Dτ and F Concerning Theorem 5, the following theorem is essential and obviously fulfilled since V, V, themselves are U (1) gauge invariant which the Klein-Gordon equation satisfies, too. Theorem 14 (Gauge symmetry). The equation of motion (74 or 75) satisfies the U (1) gauge symmetry characterized by Theorem 5. This equation (74 or 75) is very similar style of classical dynamics. Eherenfest’s theorem [20] of it implies the average behavior of this stochastic scalar electron. It is discussed in Section 4. In the fact, the nonelectromagnetic interaction i/2m0 × ∂ µ f (ˆ x(τ, ω), τ ) is hidden under observation by Theorem 18.

3.4

Dynamics of fields

Let us proceed the dynamics of the field interacting with a stochastic scalar electron. The Maxwell equation is derived by the variation of (66) with respect to A ∈ V4M , namely, the Euler-Lagrange equation ∂µ [∂Lfield /∂(∂µ Aν )] − ∂Lfield /∂Aν = 0. Where, the Lagrangian density for the field dynamics is, Lfield [ˆ x, A] =

−

ˆ

R

+

dτ

dE(x, τ ) Aα (x)Re {V α (x)} dµ

1 [Fαβ (x) + δfαβ (x)] · [F αβ (x) + δf αβ (x)] . 4µ0 c

(82)

Theorem 15 (Maxwell equation). Let the D-progressive x ˆ(◦, •) be the trajectory of a stochastic scalar electron. The variation of the action integral (66) with respect to the field A ∈ V4M derives the following Maxwell equation. { s ˆ ∂µ [F µν (x) + δf µν (x)] = µ0 × E −ec dτ Re {V ν (x)} δ 4 (x − xˆ(τ, •)) R

16

(83)

ELI-NP/IFIN-HH

Keita Seto

Where, δf ∈ V4M ⊗ V4M is the given field and the current of a stochastic scalar electron s { ˆ µ jstochastic (x) := E −ec dτ Re {V µ (x)} δ 4 (x − x ˆ(τ, •))

(84)

R

is equivalent to the current of Klein-Gordon particle µ jK-G (x) = −

ieλ2 × g µν [φ∗ (x)Dν φ(x) − φ(x)D∗ν φ∗ (x)] . 2

(85)

Where, Dµ := ∂ µ − ie/~ × Aµ (x). Remark 16. The tensor δf µν (x) is introduced to remove the field singularity at the point of an electron. For the discussion of radiation reaction in Volume II [13], the generated field has to be separated in the homogeneous solution F ∈ V4M ⊗ V4M such that ∂µ F µν = 0, and the singularity part defined by the equation µ ∂µ δf µν = µ0 × jstochastic . We regards (83) as the superposition of these two equations. The detail of these concrete solutions is in Volume II [13]. µ Proof. The derivation of the Maxwell equation (83) is obvious. The current jstochastic (x) is calculated as follows by employing (23): µ jstochastic (x)

ˆ = −ec dτ Re {V α (x)} p(x, τ ) R ´ dτ p(x, τ ) µ R = × jK−G (x) φ∗ (x)φ(x)

(86)

µ Hereby, jstochastic (x) ∈ V4M satisfies ∂µ jstochastic (x) = 0 due to the equation of continuity (31) with the natural µ boundary condition p(x, τ = ∂R) = 0. Of cause, ∂µ jK-G (x) = 0 is held, too. Due to the divergences of these currents, ´ dτ p(x, τ ) R = Constant (87) ∗ φ (x)φ(x) ν is imposed, the Maxwell equation with the current by a Klein-Gordon particle ∂µ [F µν (x)+δf µν (x)] = µ0 jK-G ´ is realized by the condition R dτ p(x, τ )/φ∗ (x)φ(x) = 1.

The following rule is implied via the above discussion. Assumption 17. For the realization of the Klein-Gordon equation and the Maxwell equation from the action integral (65-66), the following relation should be satisfied with respect to x ∈ supp(p(◦, τ )): φ∗ (x)φ(x)

:= =

ˆ

ˆR

q y dτ E δ 4 (x − x ˆ(τ, •)) dτ p(x, τ )

R

= Where, p(x, τ ) := ρ(x) × T (τ )/

´

R

ρ(x)

(88)

dτ ′ T (τ ′ ) is ensured by the Fokker-Planck equation.

´ The plot of ρ(x) = R dτ EJδ 4 (x − xˆ(τ, •))K denotes the distribution of a scalar electron in the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ). Here, the coupling system between a stochastic particle and fields 17

ELI-NP/IFIN-HH

Keita Seto

is exactly regarded as the system between a Klein-Gordon particle and fields.

4

Conclusion and discussion

In this Volume I, we discussed the relativistic stochastic kinematics of a scalar electron with its dynamics and a field. For the kinematics of a particle, the dual-progressively measurable process (D-progressive) process x ˆ(◦, •) in the Minkowski spacetime (A4 (V4M , g), B(A4 (V4M , g)), µ) was defined as the extended stochastic process from Nelson’s (S3)-process [2] at Definition 1 in Section 2. It introduced the two types of the µ velocities V± (ˆ x(◦, •)) and the metric defined by the modified Itô rule dW±µ (τ, ω) · dW±ν (τ, ω) = ∓g µν dτ . We needed to consider the probability density p : A4 (V4M , g) × R → [0, ∞) characterized by Theorem 6 and Assumption 17, and the definition of the proper time dτ (43) as the mimic of the kinematics in classical dynamics. The complex differential dˆ (19) and the complex velocity Vˆ (23) [19] which are the main casts of the present model were also introduced. In Section 3, the dynamics of the stochastic particle was proposed. We introduced the new action integral (65-66) corresponding to the form in classical dynamics. Hence, we could obtain the dynamics of a stochastic particle and fields by the variational calculus of this action integral. The special remarks at here are, this method can derive the Maxwell equation coupled with the current of a stochastic particle (see (83)). Let us summarize the results of this article. Conclusion 18 (System of a scalar electron and a field). Consider the probability space (Ω , D(P), P) and the Minkowski space (A4 (V4M , g), B(A4 (V4M , g)), µ). When the sub-σ-algebras of Pτ ∈R and Fτ ∈R with their filtration are included in D(P), the D-progressive x ˆ(◦, •) := {ˆ x(τ, ω) ∈ A4 (V4M , g)|τ ∈ R, ω ∈ Ω} characterized by µ dˆ xµ (τ, ω) = V± (ˆ x(τ, ω))dτ + λ × dW±µ (τ, ω)

(89)

is defined as the kinematics of a stochastic scalar electron [Definition 1]. The following action integral [Theorem 12] ∗

S[ˆ x, V, V , A] =

ˆ

dτ

A4 (V4M ,g)

R

−

ˆ

R

+

ˆ

ˆ

dτ

1 dM(x, τ ) Vα∗ (x)V α (x) 2

ˆ

dE(x, τ ) Aα (x)Re {V α (x)}

A4 (V4M ,g)

A4 (V4M ,g)

dµ(x)

1 [Fαβ (x) + δfαβ (x)] · [F αβ (x) + δf αβ (x)] 4µ0 c

(90)

provides the following dynamics of a stochastic scalar electron [Theorem 13] and a field [Theorem 15] characterized by V := (1 − i)/2 × V+ + (1 + i)/2 × V− ∈ V4M ⊕ iV4M and F ∈ V4M ⊗ V4M : m0 Dτ V µ (ˆ x(τ, ω)) = −eVˆν (ˆ x(τ, ω))F µν (ˆ x(τ, ω)) { s ˆ ˆ(τ ′ , ω)) ∂µ [F µν (x) + δf µν (x)] = µ0 × E −ec dτ ′ Re {V ν (x)} δ 4 (x − x R

18

(91) (92)

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Keita Seto

Here, the dynamics of (91) is equivalent to the Klein-Gordon equation. These dynamics fulfill the U (1) gauge symmetry such that A′α (x) = Aα (x) − ∂ α Λ(x) .

φ′ (x) = e−ieΛ(x)/~ × φ(x) ,

(93)

We could conclude the quantum dynamics of a scalar particle, however, how does this equation correspond to classical behavior? It can be described by following Ehrenfest’s theory and it becomes one of the key idea for investigations of radiation reaction in Volume II. Theorem 19 (Ehrenfest’s theorem). The expectation of the equation of motion (75) derives Ehrenfest’s theorem of the Klein-Gordon equation. µ µ Proof. Due to the identity EJdW±µ (τ, •)K = 0, then EJV+ (ˆ x(τ, •))K = EJV− (ˆ x(τ, •))K is satisfied. Considering the expectation of the equation of motion (91),

m0

d E JV µ (ˆ x(τ, •))K dτ

=

m0

Eq.(53)

=

d E JRe{V µ (ˆ x(τ, •))}K dτ

Re {E Jm0 Dτ V µ (ˆ x(τ, •))K} E JRe {f µ (ˆ x(τ, •))}K .

=

(94)

xµ (τ, •)K = E JV µ (ˆ x(τ, •))K, Where, f µ (ˆ x(τ, ω)) := −eVˆν (ˆ x(τ, ω))F µν (ˆ x(τ, ω)) ∈ V4M ⊕ iV4M . Since d/dτ E Jˆ m0

d2 E Jˆ xµ (τ, •)K = E JRe {f µ (ˆ x(τ, •))}K dτ 2

(95)

is satisfied and it is Ehrenfest’s theorem which is the average motion of a scalar electron. Due to Theorem 19, the correspondence of the velocities between classical and quantum dynamics is v µ (τ ) ↔

d E Jˆ x(τ, •)K = E JRe{V(ˆ x(τ, •))}K . dτ

(96)

Furthermore, d/dτ (vµ v µ ) = 2 × vµ dv µ /dτ = 0 must be satisfied in classical dynamics. The present dynamics of a stochastic particle provides the similar relation. y d q ∗ E Vµ (ˆ x(τ, •))V µ (ˆ x(τ, •)) dτ

= = = =

E

t

Vµ∗ (ˆ x(τ, •)) · Dτ V µ (ˆ x(τ, •))

|

x(τ, •)) · V µ (ˆ x(τ, •)) +Dτ ∗ Vµ∗ (ˆ r z 2 λ e × E Im{Vµ (ˆ − x(τ, •))} · ∂ν F µν (ˆ x(τ, •)) m0 ˆ λ4 e dµ(x) ∂µ p(x, τ ) · ∂ν F µν (x) × − 2m0 A4 (V4M ,g) ˆ λ4 e dµ(x) p(x, τ ) · ∂µ ∂ν F µν (x) = 0 × 2m0 A4 (V4M ,g)

(97)

Where, the natural boundary condition p(x, τ )|x∈∂A4 (V4M ,g) = 0 is selected. This calculation also supports the Lorentz invariant EJVµ∗ (ˆ x(τ, •))V µ (ˆ x(τ, •))K = c2 (constant). 19

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Keita Seto

Lemma 20. The trajectory of a stochastic scalar electron satisfies the following relation; y d q ∗ E Vµ (ˆ x(τ, •))V µ (ˆ x(τ, •)) = 0 . dτ

(98)

The non-linear QED effects and moreover non-perturbative effects are derived from the averaged valued interaction E JRe {f µ (ˆ x(τ, •))}K. We will demonstrate the existance a certain function Ξ : R → R corresponding to q(χ) in the formula dWQED /dt = q(χ) × dWclassical /dt, introduced by a certain integral transformulation ˆ K,

dWStochastic (τ ) dt

ˆ dWclassical (τ ) = K dt dWclassical (EJˆ x(τ, •)K) = Ξ(τ ) × dt

(99)

in Volume II for radiation reaction acting on a stochastic scalar electron.

Acknowledgement This work is supported by Extreme Light Infrastructure – Nuclear Physics (ELI-NP) – Phase I, and also Phase II, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund.

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[10] K. Seto, Prog. Theor. Exp. Phys., 2015, 103A01 (2015). [11] I. V. Sokolov, N. M. Naumova, and J. A. Nees, Phys. Plasmas 18, 093109 (2011). [12] J. Larmor, Phil. Trans. Roy. Soc. London A 190 205 (1897). [13] K. Seto, "A Brownian Particle and Fields II: Radiation Reaction as an Application" (2016). [14] S. Zakowicz, J. Math. Phys. 46, 032304 (2005). [15] M. Boca, and V. Florescu, Rom. J. Phys. 55, 511 (2010). [16] L.L. Brown, and T. W. B. Kibble, Phys Rev. 133, A705 (1964). [17] A. I. Nikishov, and V. I. Ritus, Zh. Eksp. Teor. Fiz. 46, 776 (1963) [Sov. Phys. JETP 19, 529 (1964)]. [18] A. I. Nikishov, and V. I. Ritus, Zh. Eksp. Teor. Fiz. 46, 1768 (1964) [Sov. Phys. JETP 19, 1191 (1964)]. [19] L. Not tale, "Scale Relativity and Fractal Space-time" (Imperial College Press, 2011). [20] P. Ehrenfest, Z. Phy. 45, 455 (1927). [21] A. Arai, "Gendai Butsuri-Sugaku Handbook [Handbook of Mordern Physical Mathematics]" (Asakura Publishing Co. Ltd., 2005). [22] K. Yasue, Int. J. Theor. Phys. 18, 861 (1979). [23] K. Yasue, J. Func. Ana. 41, 327 (1981). [24] K. Yasue, J. Math. Phys. 22, 1010 (1981). [25] K. Itô, Imp. Acad. 20, 519 (1944). [26] C. Gradiner, "Stochastic Methods: A Handbook for the Natural and Social Sciences" (Springer, 2009). [27] T. Zastawniak, Europhys. Lett., 13, 13 (1990).

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