Stochastic mechanics for ponderomotive effect of a spin-less electron interacting with a high-intensity plane wave laser field

In recent years, plasma physics assisted by intense electromagnetic fields has been investigated by high-intensity laser facilities [1–6] in the world. Its typical idea is irradiation of high-intensity laser beams on several types of targets. When a target material is irradiated by such a high-intensity laser, it is expected that electrons are emitted from atoms in that target material due to a tunneling-ionization process [7]. Those free electrons may be accelerated up to the speed of light when a field intensity of a background laser field is larger than 10¹⁸W/cm². Let us consider a classical equation of motion of a free electron in a laser field briefly: ${m}_{0}d(\gamma {\boldsymbol{v}})/{dt}=-e{\boldsymbol{E}}\,-e{\boldsymbol{v}}\times {\boldsymbol{B}}$ with $\gamma := {(1-{{\boldsymbol{v}}}^{2}/{c}^{2})}^{-1/2}$, ${\boldsymbol{E}}:= -{\partial }_{t}{\boldsymbol{A}}$ and ${\boldsymbol{B}}:= {\rm{\nabla }}\times {\boldsymbol{A}}$ for ${\boldsymbol{A}}={\boldsymbol{A}}({\omega }_{{\rm{L}}}t-{{\boldsymbol{k}}}_{{\rm{L}}}\cdot {\boldsymbol{x}})$ as a laser field of an angular frequency ω_L and a wave vector ${{\boldsymbol{k}}}_{{\rm{L}}}$. Where, ${m}_{0},-e$, ${\boldsymbol{v}}$ and c are the rest mass, charge, velocity of an electron and the speed of light, respectively. For instance, ${\boldsymbol{v}}=0$ at t = 0. The following is derived from the above equation of motion (we will see its 4-vector form later):

$$\begin{eqnarray}&&{m}_{0}\displaystyle \frac{d\gamma {\boldsymbol{v}}}{{dt}}=-e{\boldsymbol{E}}+{{\boldsymbol{f}}}_{\mathrm{NL}},\quad {{\boldsymbol{f}}}_{\mathrm{NL}}:= -\displaystyle \frac{{e}^{2}}{2\gamma {m}_{0}}{\rm{\nabla }}{{\boldsymbol{A}}}^{2}.\end{eqnarray} \tag{ 1 }$$

Where, $\langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle := {(2\pi /{\omega }_{{\rm{L}}})}^{-1}\times {\int }_{0}^{2\pi /{\omega }_{{\rm{L}}}}{{\boldsymbol{f}}}_{\mathrm{NL}}({\boldsymbol{x}}(t),t){dt}$, the time average of the non-linear force ${{\boldsymbol{f}}}_{\mathrm{NL}}$ for a single laser circle 2π/ω_L, is the so-called ponderomotive force in plasma physics [8–10]. We regard $\langle { \mathcal X }\rangle$ as a slowly varying part of ${ \mathcal X }$ in this article, abstractly. When we consider ω_L is large enough, $\langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle =-{e}^{2}/4{m}_{0}\langle \gamma \rangle \times {\rm{\nabla }}{{\boldsymbol{A}}}_{0}^{2}$ for ${\boldsymbol{A}}={{\boldsymbol{A}}}_{0}({\omega }_{{\rm{L}}}t-{{\boldsymbol{k}}}_{{\rm{L}}}\cdot {\boldsymbol{x}})\times \cos ({\omega }_{{\rm{L}}}t-{{\boldsymbol{k}}}_{{\rm{L}}}\cdot {\boldsymbol{x}}+\delta )$. The non-linear force ${{\boldsymbol{f}}}_{\mathrm{NL}}$ corresponds to $-e{\boldsymbol{v}}\times {\boldsymbol{B}}$. Generally speaking, the factor ${\rm{\nabla }}{{\boldsymbol{A}}}^{2}$ is non-zero even in the rest frame of an electron. An appearance of large enough $\langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle$ is regarded as a signature of the relativistic regime since $-e{\boldsymbol{v}}\times {\boldsymbol{B}}$ becomes zero in the non-relativistic regime. Thus, the ponderomotive effect due to $\langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle$ is treated as one of the research topics in relativistic laser-electron interactions. The force $\langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle \propto {\rm{\nabla }}{{\boldsymbol{A}}}_{0}^{2}$ is a vector parallel to ${{\boldsymbol{k}}}_{{\rm{L}}}$ since ${{\boldsymbol{A}}}_{0}^{2}$ is a function of ${\omega }_{{\rm{L}}}t-{{\boldsymbol{k}}}_{{\rm{L}}}\cdot {\boldsymbol{x}}$. This is the reason why a high-intensity laser field accelerates an electron to the positive direction of ${{\boldsymbol{k}}}_{{\rm{L}}}$ relativistically [11]. The ponderomotive effect is applied to the plasma channeling and the ponderomotive self-focusing effect [12–14], the laser wake field acceleration method [15–17], the hole boring effect [18], etc.

An electron has to be expressed as a relativistic and quantum mechanical object in non-linear and quantum processes by high-intensity lasers [1–6, 19, 20]. It is necessary to describe the ponderomotive force acting on an electron in relativistic quantum mechanics, too. In fact, non-linear processes in quantum electrodynamics are expressed by an electron dressing a high-intensity laser field. We can see this via the ponderomotive effect in classical dynamics: the ponderomotive effect is regarded as a propagation of an electron dressing a laser field. We will see that detail in the mass correction

$$\begin{eqnarray}&&{m}_{\mathrm{eff}}^{2}:= {m}_{0}^{2}\times \left[1+\displaystyle \frac{{e}^{2}\langle {{\boldsymbol{A}}}^{2}\rangle }{{m}_{0}^{2}{c}^{2}}\right].\end{eqnarray} \tag{ 2 }$$

The existence of m_eff suggests that an electron dresses an additional mass $\delta m={m}_{\mathrm{eff}}-{m}_{0}$ due to a laser field. The contribution of δ m cannot be omitted in general, even in the case of the rest frame.

In this article, we investigate the relativistic ponderomotive effect in relativistic stochastic mechanics as quantum mechanics. Let us explain the importance on the relativistic covariance of equations. This is the reason why that covariance ensures to use the Lorentz transformation. Our idea consists of two problems. The first issue is to decide the definitive edition of the relativistically covariant ponderomotive force acting on a classical and spin-less electron. We will confirm that by solving equation (3) strictly with an external laser field of a plane wave in section 2, though there are several proposals of the relativistic models [21–23]. We extract a slowly varying component from the strict solution, as the ponderomotive effect. Kibble's form [21, 22] is derived as a result of that and this is the preparation for the later discussion in the quantum regime. The second problem is how we can express the ponderomotive effect in relativistic quantum mechanics. Let us formulate it by stochastic mechanics developed by E. Nelson [24, 25]. Stochastic mechanics is a formulation of quantum mechanics by stochastic analysis. We introduce its basic strategy in section 3. We hereby expand stochastic mechanics from Nelson's model in the non-relativistic regime to a relativistically covariant model: a quanta draws a 4D stochastic process $\hat{x}$ as its trajectory given by stochastic kinematics ${d}_{\pm }{\hat{x}}^{\mu }={{ \mathcal V }}_{\pm }^{\mu }(\hat{x})d\tau +(\mathrm{randomness})$ with dynamics ${m}_{0}{{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }(x)=-e{\hat{{ \mathcal V }}}_{\nu }(x){F}^{\mu \nu }(x)$. We define this dynamics as the Klein-Gordon (KG) equation of a spin-less electron.Stochastic mechanics is the unique method to illustrate a realistic trajectory of a particle in quantum mechanics. It is expected to employ the same method of classical mechanics in the quantum regime since stochastic mechanics has both kinematics and dynamics. The ponderomotive effect in stochastic mechanics is discussed in section 4. It becomes one of few applications of stochastic mechanics. The ponderomotive effect is described by a physical process satisfying the following KG equation: ${(i{\hslash }\partial )}^{2}\langle \phi \rangle -{m}_{\mathrm{eff}}^{2}{c}^{2}\langle \phi \rangle =0$. We will confirm this naive result by stochastic mechanics. The readers may find the fact that the ponderomotive effect in the quantum regime is not so different from one in the classical regime described by section 2, except randomness in the present model. Especially, the effective mass correction should be taken into account in both regimes. This is the signature that the ponderomotive effect acts on a charged particle in both classical and quantum mechanics universally. Once again let m₀ and $-e$ be the rest mass and charge of a spin-less electron.

Let us derive the relativistic ponderomotive effect in classical mechanics. Hence, we only deal with the basic equation of motion

$$\begin{eqnarray}&&{m}_{0}\displaystyle \frac{{{dv}}^{\mu }}{d\tau }=-{{eF}}^{\mu \nu }{v}_{\nu }\end{eqnarray} \tag{ 3 }$$

for a spin-less electron in this section. Equation (3) is equivalent to ${m}_{0}d(\gamma {\boldsymbol{v}})/{dt}=-e{\boldsymbol{E}}-e{\boldsymbol{v}}\times {\boldsymbol{B}}$ as its 3D expression. Where, $v=\gamma \times (c,{\boldsymbol{v}})$ is a 4-velocity of a spin-less electron and F is an electromagnetic field tensor characterized by ${F}^{\mu \nu }:= {\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$ with a 4-potential A of a laser field. We also introduce the metric $g=(+,\,-,\,-,\,-)$ for $A\cdot B:= {g}_{\mu \nu }{A}^{\mu }{B}^{\nu }={g}^{\mu \nu }{A}_{\mu }{B}_{\nu }$. Recall v satisfies v · v = c² due to this metric. Suppose the external field of a laser beam is a plane wave, i.e., A = A(k · x) for a 4-wave vector $k:= ({\omega }_{{\rm{L}}}/c,{{\boldsymbol{k}}}_{{\rm{L}}})$. Let the laser field be now $A:= (0,{\boldsymbol{A}})$ with ${A}^{2}=A\cdot A=-| {\boldsymbol{A}}{| }^{2}$. We set the Lorenz gauge condition ∂_μA^μ = 0 and the initial condition A(θ(0)) = 0 for $\theta (\tau ):= k\cdot x(\tau )$, too. Under such a laser field of a plane wave, equation (3) has its exact solution below [26]:

$$\begin{eqnarray}\begin{array}{rcl}{v}^{\mu }(\theta (\tau )) & = & {v}^{\mu }(\theta (0))-\displaystyle \frac{e}{{m}_{0}}\displaystyle \frac{v(\theta (0))\cdot A(\theta (\tau ))}{k\cdot v}{k}^{\mu }\\ & & +\displaystyle \frac{e}{{m}_{0}}{A}^{\mu }(\theta (\tau ))-\displaystyle \frac{{e}^{2}}{2{m}_{0}^{2}}\displaystyle \frac{{A}^{2}(\theta (\tau ))}{k\cdot v}{k}^{\mu }\end{array}\end{eqnarray} \tag{ 4 }$$

We wrote v(θ(τ)) = v(τ) for our readability. Since v = dx/dτ, an electron draws

$$\begin{eqnarray}\begin{array}{rcl}{x}^{\mu }(\theta (\tau )) & = & {x}^{\mu }(\theta (0))+{v}^{\mu }(\theta (0))\times \tau \\ & & -\displaystyle \frac{e}{{m}_{0}}{\displaystyle \int }_{0}^{\tau }d\tau ^{\prime} \displaystyle \frac{v(\theta (0))\cdot A(\theta (\tau ^{\prime} ))}{k\cdot v}{k}^{\mu }\\ & & +\displaystyle \frac{e}{{m}_{0}}{\displaystyle \int }_{0}^{\tau }d\tau ^{\prime} {A}^{\mu }(\theta (\tau ^{\prime} ))\\ & & -\displaystyle \frac{{e}^{2}}{2{m}_{0}^{2}}{\displaystyle \int }_{0}^{\tau }d\tau ^{\prime} \displaystyle \frac{{A}^{2}(\theta (\tau ^{\prime} ))}{k\cdot v}{k}^{\mu }.\end{array}\end{eqnarray} \tag{ 5 }$$

Where, ${k}_{\mu }{v}^{\mu }$ is a Lorentz invariant since $d({k}_{\mu }{v}^{\mu })d\tau =-e/{m}_{0}\times {k}_{\mu }{F}^{\mu \nu }{v}_{\nu }=0$. The readers may find

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dv}}^{\mu }}{d\theta (\tau )} & = & -\displaystyle \frac{e}{{m}_{0}}v(\theta (0))\cdot \displaystyle \frac{{dA}}{d\theta (\tau )}\displaystyle \frac{{k}^{\mu }}{k\cdot v}+\displaystyle \frac{e}{{m}_{0}}\displaystyle \frac{{{dA}}^{\mu }}{d\theta (\tau )}\\ & & -\displaystyle \frac{{e}^{2}}{2{m}_{0}^{2}}\displaystyle \frac{{{dA}}^{2}(\theta (\tau ))}{d\theta (\tau )}\displaystyle \frac{{k}^{\mu }}{k\cdot v}\end{array}\end{eqnarray} \tag{ 6 }$$

by using equation (4). Since d/dτ = k · v × d/dθ(τ),

$$\begin{eqnarray}&&{m}_{0}\displaystyle \frac{{{dv}}^{\mu }}{d\tau }=-{{eF}}^{\mu \nu }(\theta (\tau ))\cdot {v}_{\nu }(0)+{f}_{\mathrm{NL}}^{\mu }(\theta (\tau )),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{NL}}^{\mu }(\theta (\tau )):= -\displaystyle \frac{{e}^{2}}{2{m}_{0}}{\partial }^{\mu }{A}^{2}(\theta (\tau )),\end{eqnarray} \tag{ 8 }$$

is derived strictly as the one equivalent to equation (3). The non-linear force ${f}_{\mathrm{NL}}$ is extracted from the Lorentz force $-{{eF}}^{\mu \nu }{v}_{\nu }$ in equation (3). We remark equations (7)–(8) do not require any assumption except an external laser field of a plane wave. We just transformed the exact solution of equation (3). Let $\langle { \mathcal X }\rangle$ be a slowly varying part of ${ \mathcal X }$, i.e. $\langle { \mathcal X }\rangle$ omits rapidly oscillating components like ω_L and its harmonics in ${ \mathcal X }$. Consider the operation of $\langle \circ \rangle$ wrt Equations (7)–(8):

$$\begin{eqnarray}&&{m}_{0}\displaystyle \frac{d\langle {v}^{\mu }\rangle }{d\tau }=\langle {f}_{\mathrm{NL}}(\langle x\rangle )\rangle ,\quad \langle {f}_{\mathrm{NL}}\rangle =-\displaystyle \frac{{e}^{2}}{2{m}_{0}}{\partial }^{\mu }\langle {A}^{2}\rangle .\end{eqnarray} \tag{ 9 }$$

Equation (9) gives the slow components of equations (4)–(5):

$$\begin{eqnarray}&&\langle {v}^{\mu }(\tau )\rangle =\langle {v}^{\mu }(0)\rangle -\displaystyle \frac{{e}^{2}}{2{m}_{0}^{2}}\displaystyle \frac{\langle {A}^{2}(\langle \theta (\tau )\rangle )\rangle }{k\cdot \langle v(0)\rangle }{k}^{\mu }\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\langle {x}^{\mu }(\tau )\rangle =\langle {x}^{\mu }(0)\rangle +\langle {v}^{\mu }(0)\rangle \times \tau -\displaystyle \frac{{e}^{2}}{2{m}_{0}^{2}}{\int }_{0}^{\tau }d\tau ^{\prime} \displaystyle \frac{\langle {A}^{2}(\langle \theta (\tau )\rangle )\rangle }{k\cdot \langle v(0)\rangle }{k}^{\mu }\end{eqnarray} \tag{ 11 }$$

With $\langle v(0)\rangle =v(0)$ and $\langle x(0)\rangle =x(0)$ as the initial conditions. $\langle \theta (\tau )\rangle =k\cdot \langle x(\tau )\rangle$ as the index of $\langle {A}^{2}\rangle$ is also corrected from θ(τ) here. Equation (11) represents a trajectory of a slowly varying motion of a spin-less electron. We can confirm the 3D form of equation (9): since ${\partial }^{i=\mathrm{1,2,3}}=-\partial /\partial {x}^{i}$ and $\langle {A}^{2}\rangle =-\langle {{\boldsymbol{A}}}^{2}\rangle$, equation (9) becomes

$$\begin{eqnarray}&&{m}_{0}\displaystyle \frac{d\langle \gamma \rangle \langle {\boldsymbol{v}}\rangle }{{dt}}=\langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle ,\quad \langle {{\boldsymbol{f}}}_{\mathrm{NL}}\rangle =-\displaystyle \frac{{e}^{2}}{4{m}_{0}\langle \gamma \rangle }{\rm{\nabla }}{{\boldsymbol{A}}}_{0}^{2},\end{eqnarray} \tag{ 12 }$$

with ${\boldsymbol{A}}(\theta (\tau ))={{\boldsymbol{A}}}_{0}(\theta (\tau ))\times \cos \theta (\tau )$. This is an equation of a slowly varying motion w.r.t. Equation (1).

The force $\langle {f}_{\mathrm{NL}}\rangle$ in equation (9) may be considered as a relativistically covariant ponderomotive force. However, we still cannot suggest that since its time component $\langle {f}_{\mathrm{NL}}^{0}\rangle$ is unclear and $\langle v{\rangle }^{2}={c}^{2}-{e}^{2}\langle {A}^{2}\rangle /{m}_{0}^{2}$ due to equation (10). The operation $\langle \circ \rangle$ acting on equation (3) implies $\langle v{\rangle }^{2}\ne {c}^{2}$. We can recover that squared velocity rule by the method of T. W. B. Kibble [22]. Suppose $\langle p\rangle := {m}_{0}\langle v\rangle ={m}_{0}d\langle x\rangle /d\tau$, $\langle p{\rangle }^{2}={m}_{0}^{2}[1-{e}^{2}\langle {A}^{2}\rangle /{m}_{0}^{2}{c}^{2}]{c}^{2}$. Then, we define the effective mass

$$\begin{eqnarray}&&{m}_{\mathrm{eff}}:= {m}_{0}\times \sqrt{1-\displaystyle \frac{{e}^{2}\langle {A}^{2}\rangle }{{m}_{0}^{2}{c}^{2}}}\end{eqnarray} \tag{ 13 }$$

and introduce a new proper time τ_eff such as $d{\tau }_{\mathrm{eff}}/d\tau =\sqrt{1-{e}^{2}\langle {A}^{2}\rangle /{m}_{0}^{2}{c}^{2}}\geqslant 1$. Let $\langle X({\tau }_{\mathrm{eff}})\rangle =\langle x(\tau )\rangle$, we obtain ${[d\langle X\rangle /d{\tau }_{\mathrm{eff}}]}^{2}={c}^{2}$ and

$$\begin{eqnarray}&&\langle {p}^{\mu }\rangle ={m}_{0}\displaystyle \frac{d\langle {x}^{\mu }\rangle }{d\tau }={m}_{\mathrm{eff}}\displaystyle \frac{d\langle {X}^{\mu }\rangle }{d{\tau }_{\mathrm{eff}}}\end{eqnarray} \tag{ 14 }$$

which gives $\langle p{\rangle }^{2}={m}_{\mathrm{eff}}^{2}{c}^{2}$. Then, the following is the relativistically covariant equation of a slowly varying motion of a spin-less electron in a high-intensity field:

$$\begin{eqnarray}&&\displaystyle \frac{d}{d{\tau }_{\mathrm{eff}}}\left[{m}_{\mathrm{eff}}\displaystyle \frac{d\langle {X}^{\mu }\rangle }{d{\tau }_{\mathrm{eff}}}\right]={\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}\end{eqnarray} \tag{ 15 }$$

Consider ${f}^{\mu }={\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}$ as a conservative force, then,

$$\begin{eqnarray}&&U(\langle X\rangle ):= {m}_{\mathrm{eff}}{c}^{2}\end{eqnarray} \tag{ 16 }$$

is the so-called ponderomotive potential in the relativistic covariant form. Equations (13), (15) for $\langle X\rangle$ are equivalent to equation (9) for $\langle x\rangle$ because both expressions derive the same solution of equation (11). We hereby conclude that the relativistically covariant equation of motion for the ponderomotive effect is expressed by equation (15) with the effective mass of equation (13).

Stochastic mechanics is one of formulation of quantum mechanics by using stochastic analysis. Its non-relativistic model was completed by E. Nelson [24, 25] with the following achievements: (A) it is equivalent to the Schrödinger equation, (B) it has stochastic kinematics to draw a trajectory of a quanta and (C) Born's rule is naturally satisfied in this model. Especially, the ability-(B) is useful to confirm the classical-quantum correspondence of the ponderomotive effect. We investigate its relativistic version for that purpose since a high-intensity laser accelerates a spin-less electron to nearly the speed of light. Though the several authors had proposed each relativistic models [27–30], however, its definitive edition has not been found. There is no practice to draw a trajectory by them, too. Here we propose a new model of relativistic stochastic mechanics including quantum dynamics of the Klein-Gordon (KG) equation for a spin-less electron expressed by ϕ interacting with an external laser field A:

$$\begin{eqnarray}&&(i{\hslash }{\partial }_{\nu }+{{eA}}_{\nu })(i{\hslash }{\partial }^{\nu }+{eA}{}^{\nu })\phi -{m}_{0}^{2}{c}^{2}\phi =0\end{eqnarray} \tag{ 17 }$$

Our new model can draw a trajectory of a spin-less electron in a laser field of a plane wave as we see it later. We give the overview of relativistic stochastic mechanics in section 3.1 and its minimum remarks in section 3.2.

We hereby recall the fact in classical mechanics before discussing relativistic stochastic mechanics. Classical kinematics dx = vdτ is coupled with dynamics of equation (3). Consider v² = constant given by equation (3) in general. To fix this constant on c² is not derived by only equation (3). The equation v² = c² is introduced as a holonomic constraint of classical dynamics via the definition $v=\gamma \times (c,{\boldsymbol{v}})$. The three of classical kinematics, dynamics and its holonomic constraint are required to draw a classical trajectory of a particle. Relativistic stochastic mechanics takes over this situation perfectly due to the similarity to classical mechanics.

3.1. Overview

Suppose the axiom such as stochastic kinematics imposes an existence probability of a particle in quantum mechanics [24, 25]. Here we reformulate relativistic quantum mechanics with this axiom. Namely, relativistic quantum mechanics consists of relativistic stochastic kinematics, relativistic quantum dynamics of equation (17) and its holonomic constraint as an analogy of relativistic classical mechanics. The readers should recall the fact that kinematics is not included in the standard construction of quantum dynamics: matrix mechanics, wave mechanics and the path integral method do not provide the ability to illustrate a realistic trajectory of a particle in quantum mechanics.

In order to stochastic kinematics, a path (a sample) labeled by ω happens with a probability ${\mathscr{P}}(\omega )$. Hence, the probability which one of independent ω₁ and ω₂ happens is ${\mathscr{P}}(\{{\omega }_{1},{\omega }_{2}\})={\mathscr{P}}({\omega }_{1})+{\mathscr{P}}({\omega }_{2})\leqslant 1$ (σ-additivity). Let Ω be the set of all samples and $\hat{x}(\omega ):= \{\hat{x}(\tau ,\omega )\}{}_{\tau \in {\mathbb{R}}}$ a 4D trajectory of a spin-less electron for ω. Therefore, ${\mathscr{P}}(\omega )$ is an existence probability of $\hat{x}(\omega )$. A stochastic process is non-differentiable, but continuous. Then, the following two vectors are different for a positive dτ: ${d}_{+}\hat{x}(\tau ,\omega ):= \hat{x}(\tau +d\tau ,\omega )-\hat{x}(\tau ,\omega )$ and ${d}_{-}\hat{x}(\tau ,\omega ):= \hat{x}(\tau ,\omega )-\hat{x}(\tau -d\tau ,\omega )$. We remark an inequality ${d}_{+}\hat{x}(\tau ,\omega )\ne {d}_{-}\hat{x}(\tau ,\omega )$. Where, $\hat{x}({\tau }_{b},\omega )-\hat{x}({\tau }_{a},\omega )$ is decomposed by the following two methods with $d\tau := ({\tau }_{b}-{\tau }_{a})/n$ the resolution of them:

$$\begin{eqnarray}&&\hat{x}({\tau }_{b},\omega )-\hat{x}({\tau }_{a},\omega )=\displaystyle \sum _{k=0}^{n-1}{d}_{+}\hat{x}({\tau }_{a}+k\times d\tau ,\omega )\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\,=\,\displaystyle \sum _{k=1}^{n}{d}_{-}\hat{x}({\tau }_{a}+k\times d\tau ,\omega )\,\end{eqnarray} \tag{ 19 }$$

It reminds us of idea that $\hat{x}({\tau }_{b},\omega )-\hat{x}({\tau }_{a},\omega )$ is calculated by the integrals of ${d}_{+}\hat{x}(\tau ,\omega )$ and ${d}_{-}\hat{x}(\tau ,\omega )$ w.r.t. large enough n. Let us express it by

$$\begin{eqnarray}&&\hat{x}({\tau }_{b},\omega )-\hat{x}({\tau }_{a},\omega )={\int }_{{\tau }_{a}}^{{\tau }_{b}}{d}_{\pm }\hat{x}(\tau ,\omega ).\end{eqnarray} \tag{ 20 }$$

Recall dx(τ) = v(x(τ))dτ as classical kinematics to draw x. As an analogy of this, we suppose stochastic kinematics as follows:

$$\begin{eqnarray}&&{d}_{\pm }\hat{x}(\tau ,\omega )={{ \mathcal V }}_{\pm }(\hat{x}(\tau ,\omega ))d\tau +\sqrt{\displaystyle \frac{{\hslash }}{{m}_{0}}}\times {{dW}}_{\pm }(\tau ,\omega )\end{eqnarray} \tag{ 21 }$$

The first term in RHS of equation (21) shows its drift like classical kinematics. ${W}_{+}(\omega ):= {\{{W}_{+}(\tau ,\omega )\}}_{\tau \in {\mathbb{R}}}$ and ${W}_{-}(\omega ):= {\{{W}_{-}(\tau ,\omega )\}}_{\tau \in {\mathbb{R}}}$ are 4D random processes. We set each averages of ${\{{{dW}}_{+}(\tau ,\omega )\}}_{\omega \in {\rm{\Omega }}}$ and ${\{{{dW}}_{-}(\tau ,\omega )\}}_{\omega \in {\rm{\Omega }}}$ are zero for all τ: ${\mathbb{E}}[\kern-2pt[ {{dW}}_{\pm }(\tau ,\bullet )]\kern-2pt] =0$ with ${\mathbb{E}}[\kern-2pt[ f(\bullet )]\kern-2pt] := {\int }_{{\rm{\Omega }}}f(\omega )d{\mathscr{P}}(\omega )$ an expectation of $\{f(\omega )\}{}_{\omega \in {\rm{\Omega }}}$ wrt ${\mathscr{P}}$. Both W_± (ω) provide non-differentiability of $\hat{x}(\omega )$. The randomness $\sqrt{{\hslash }/{m}_{0}}\times {{dW}}_{\pm }(\tau ,\omega )$ represents quantum uncertainty of a position of a quanta.

Quantum dynamics is required for the definition of ${{ \mathcal V }}_{\pm }$ in equation (21). Recall equation (3) for dx = vdτ. Quantum dynamics has to be the KG equation (17) in the present case. Let us express quantum dynamics of equation (17) by

$$\begin{eqnarray}&&{m}_{0}{{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }(x)=-e{\hat{{ \mathcal V }}}_{\nu }(x){F}^{\mu \nu }(x)\end{eqnarray} \tag{ 22 }$$

with operators

$$\begin{eqnarray}&&{\hat{{ \mathcal V }}}^{\mu }(x):= {{ \mathcal V }}^{\mu }(x)+\displaystyle \frac{i{\hslash }}{2{m}_{0}}\times {\partial }^{\mu },\quad {{\mathfrak{D}}}_{\tau }:= {\hat{{ \mathcal V }}}^{\mu }(x)\cdot {\partial }_{\mu }\end{eqnarray} \tag{ 23 }$$

as [31] suggests. The form of equation (22) is similar to equation (3). Suppose ${ \mathcal V }:= ({{ \mathcal V }}_{+}+{{ \mathcal V }}_{-})/2-i({{ \mathcal V }}_{+}-{{ \mathcal V }}_{-})/2$ and

$$\begin{eqnarray}&&{{ \mathcal V }}^{\mu }(x):= \displaystyle \frac{1}{{m}_{0}}\times \left[i{\hslash }\partial {}^{\mu }\mathrm{ln}\phi (x)+{{eA}}^{\mu }(x)\right],\end{eqnarray} \tag{ 24 }$$

too. Therefore, the drift velocities in equation (21) are given by ${{ \mathcal V }}_{\pm }=\mathrm{Re}\{{ \mathcal V }\}\mp \mathrm{Im}\{{ \mathcal V }\}$. The equivalency between equation (22) and the KG equation (17) is demonstrated by

$$\begin{eqnarray}\begin{array}{rcl}{{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }+\displaystyle \frac{e}{{m}_{0}}{\hat{{ \mathcal V }}}_{\nu }{F}^{\mu \nu } & = & \displaystyle \frac{1}{2}{\partial }^{\mu }\left[\displaystyle \frac{\left(i{\rm{\hslash }}{\partial }_{\nu }+{{eA}}_{\nu }\right)\left(i{\rm{\hslash }}{\partial }^{\nu }+{{eA}}^{\nu }\right)\phi }{{m}_{0}^{2}\phi }\right]\\ & = & 0\end{array}\end{eqnarray} \tag{ 25 }$$

with an assumption that ϕ is a solution of equation (17) [31].

However when we don't know that ϕ is a solution of equation (17), equation (25) leads $(i{\hslash }{\partial }_{\nu }+{{eA}}_{\nu })(i{\hslash }{\partial }^{\nu }+{eA}{}^{\nu })\phi -{m}_{0}^{2}C\phi =0$ with a free constant C. We need to put an extra rule, C = c², for the equivalency between equation (17) and equation (22). That is the holonomic constraint in relativistic stochastic mechanics. As [27] proposed, this constraint expressed by

$$\begin{eqnarray}&&{\mathbb{E}}\unicode{x0301A}{{ \mathcal V }}_{\mu }^{* }(\hat{x}(\tau ,\bullet )){{ \mathcal V }}^{\mu }(\hat{x}(\tau ,\bullet ))\unicode{x0301B}={c}^{2}\end{eqnarray} \tag{ 26 }$$

is useful as an analogy of the classical one: v² = c². Where, A^* denotes the complex conjugate of A. The compatibility between equation (22) and equation (26) will be shown in equation (36) mathematically. We regard the minimum construction of relativistic stochastic mechanics is the triplet of equation (21), equation (22) and equation (26) as quantum mechanics.

3.2. Mathematical remarks

We recall mathematical remarks for equation (21) and equation (26) briefly. A probability measure ${\mathscr{P}}$ for $\hat{x}$ provides $p(\circ ,\tau )$ a distribution of sample trajectories at τ wrt Equation (21). Suppose a probability density

$$\begin{eqnarray}\begin{array}{rcl}p(x,\tau ) & := & {\int }_{{\rm{\Omega }}}{\delta }^{4}(x-\hat{x}(\tau ,\omega ))d{\mathscr{P}}(\omega )\\ & = & {\mathbb{E}}[\kern-2pt[ {\delta }^{4}(x-\hat{x}(\tau ,\bullet ))]\kern-2pt] \end{array}\end{eqnarray} \tag{ 27 }$$

which follows the Fokker-Planck (FP) equation below:

$$\begin{eqnarray}&&{\partial }_{\tau }p(x,\tau )+{\partial }_{\mu }[{{ \mathcal V }}_{\pm }^{\mu }(x)p(x,\tau )]=\pm \displaystyle \frac{{\hslash }}{2{m}_{0}}(-{g}^{\mu \nu }){\partial }_{\mu }{\partial }_{\nu }p(x,\tau )\end{eqnarray} \tag{ 28 }$$

In the classical limit ${\hslash }\to 0$, equation (28) expresses a classical advection by ${\partial }_{\tau }p(x,\tau )+{\partial }_{\mu }[{v}^{\mu }(x)p(x,\tau )]=0$ with ${{ \mathcal V }}_{\pm }\to v$, i.e., $p(x,\tau )={\delta }^{4}(x-x(\tau ))$ with classical kinematics $x(\tau )=x(0)+{\int }_{0}^{\tau }v(\tau ^{\prime} )d\tau ^{\prime}$. When we accept the equivalency between stochastic kinematics of equation (21) and the FP equation (28), the conditions ${{ \mathcal V }}_{\pm }=0$ give the rules of $\sqrt{{\hslash }/{m}_{0}}\times {W}_{\pm }(\omega )$ in equation (21), respectively¹ :

$$\begin{eqnarray}&&{\partial }_{\tau }{p}_{\pm }(x,\tau )=\pm \displaystyle \frac{{\hslash }}{2{m}_{0}}(-{g}^{\mu \nu }){\partial }_{\mu }{\partial }_{\nu }{p}_{\pm }(x,\tau )\end{eqnarray} \tag{ 29 }$$

Hence, p_± in equation (29) describe the distributions of the random processes $\sqrt{{\hslash }/{m}_{0}}\times {W}_{\pm }(\omega )$ at τ. Stochastic kinematics of equation (21) is now characterized mathematically.

Let us introduce the following operator:

$$\begin{eqnarray}&&{{\mathfrak{D}}}_{\tau }^{\pm }:= {{ \mathcal V }}_{\pm }^{\mu }{\partial }_{\mu }\pm \displaystyle \frac{{\hslash }}{2{m}_{0}}(-{g}^{\mu \nu }){\partial }_{\mu }{\partial }_{\nu }\end{eqnarray} \tag{ 30 }$$

We can find ${{\mathfrak{D}}}_{\tau }^{\pm }\hat{x}={{ \mathcal V }}_{\pm }^{\mu }(\hat{x})$ from the above. Since p follows the FP equation (28),

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d}{d\tau }{\mathbb{E}}[\kern-2pt[ f(\hat{x}(\tau ,\bullet ))]\kern-2pt] & = & {\displaystyle \int }_{{{\mathbb{R}}}^{4}}f(x){\partial }_{\tau }p(x,\tau ){d}^{4}x\\ & = & {\mathbb{E}}[\kern-2pt[ {{\mathfrak{D}}}_{\tau }^{\pm }f(\hat{x}(\tau ,\bullet ))]\kern-2pt] .\end{array}\end{eqnarray} \tag{ 31 }$$

This formulae of ± are applied to many transformations in stochastic mechanics. For example when f(x) = x,

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathbb{E}}[\kern-2pt[ {\hat{x}}^{\mu }(\tau ,\bullet )]\kern-2pt] }{d\tau }={\mathbb{E}}[\kern-2pt[ {{ \mathcal V }}_{\pm }^{\mu }(\hat{x}(\tau ,\bullet ))]\kern-2pt] .\end{eqnarray} \tag{ 32 }$$

The readers may find the expansion of equation (31) bellow:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d}{d\tau }{\mathbb{E}}[\kern-2pt[ {\alpha }_{\mu }(\hat{x}(\tau ,\bullet )){\beta }^{\mu }(\hat{x}(\tau ,\bullet ))]\kern-2pt] \\ \quad =\,{\mathbb{E}}\unicode{x0301A}{{\mathfrak{D}}}_{\tau }^{\pm }{\alpha }_{\mu }(\hat{x}(\tau ,\bullet ))\cdot {\beta }^{\mu }(\hat{x}(\tau ,\bullet ))+{\alpha }_{\mu }(\hat{x}(\tau ,\bullet ))\cdot {{\mathfrak{D}}}_{\tau }^{\mp }{\beta }^{\mu }(\hat{x}(\tau ,\bullet ))\unicode{x0301B}\end{array}\end{eqnarray} \tag{ 33 }$$

This is the so-called differential form of Nelson's partial integral formula [25]. The complex version of equation (33) is much more useful. Recall the definitions of ${{\mathfrak{D}}}_{\tau }$ and ${ \mathcal V }$,

$$\begin{eqnarray}&&{{\mathfrak{D}}}_{\tau }=\displaystyle \frac{{{\mathfrak{D}}}_{\tau }^{+}+{{\mathfrak{D}}}_{\tau }^{-}}{2}-i\displaystyle \frac{{{\mathfrak{D}}}_{\tau }^{+}-{{\mathfrak{D}}}_{\tau }^{-}}{2}.\end{eqnarray} \tag{ 34 }$$

The superposition of equation (33) of '+' and '−' signs gives

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d}{d\tau }{\mathbb{E}}[\kern-2pt[ {\alpha }_{\mu }(\hat{x}(\tau ,\bullet )){\beta }^{\mu }(\hat{x}(\tau ,\bullet ))]\kern-2pt] \\ \quad =\,{\mathbb{E}}\unicode{x0301A}{{\mathfrak{D}}}_{\tau }^{* }{\alpha }_{\mu }(\hat{x}(\tau ,\bullet ))\cdot {\beta }^{\mu }(\hat{x}(\tau ,\bullet ))+{\alpha }_{\mu }(\hat{x}(\tau ,\bullet ))\cdot {{\mathfrak{D}}}_{\tau }{\beta }^{\mu }(\hat{x}(\tau ,\bullet ))\unicode{x0301B}\end{array}\end{eqnarray} \tag{ 35 }$$

When ${\alpha }_{\mu }={{ \mathcal V }}_{\mu }^{* }$ and ${\beta }^{\mu }={{ \mathcal V }}^{\mu }$ in equation (35),

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d}{d\tau }{\mathbb{E}}\unicode{x0301A}{{ \mathcal V }}_{\mu }^{* }(\hat{x}(\tau ,\bullet )){{ \mathcal V }}^{\mu }(\hat{x}(\tau ,\bullet ))\unicode{x0301B}\\ \quad =\,{\mathbb{E}}\unicode{x0301A}{{ \mathcal V }}_{\mu }^{* }(\hat{x}(\tau ,\bullet ))\cdot {{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }(\hat{x}(\tau ,\bullet ))+{{\mathfrak{D}}}_{\tau }^{* }{{ \mathcal V }}_{\mu }^{* }(\hat{x}(\tau ,\bullet ))\cdot {{ \mathcal V }}^{\mu }(\hat{x}(\tau ,\bullet ))\unicode{x0301B}\\ \quad =\,\displaystyle \frac{{{\rm{\hslash }}}^{2}e}{2{m}_{0}^{3}}\,{\int }_{{{\mathbb{R}}}^{4}}{d}^{4}x\,p(x,\tau )\cdot {\partial }_{\mu }{\partial }_{\nu }{F}^{\mu \nu }(x)\\ \quad =\,0\end{array}\end{eqnarray} \tag{ 36 }$$

is found by using equation (22). This result bridges between equation (22) and equation (26). This situation is same as the classical one: dv²/dτ = 0 by v² = c² and equation (3).

3.3. Ehrenfest's theorem

One of the concerns in quantum mechanics is an expectation of that. Let us consider it in relativistic stochastic mechanics. In order to equation (35), we get

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathbb{E}}[\kern-2pt[ {\hat{x}}^{\mu }(\tau ,\bullet )]\kern-2pt] }{d\tau }={\mathbb{E}}[\kern-2pt[ {{ \mathcal V }}^{\mu }(\hat{x}(\tau ,\bullet ))]\kern-2pt] ,\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{\mathbb{E}}[\kern-2pt[ {\hat{x}}^{\mu }(\tau ,\bullet )]\kern-2pt] }{d{\tau }^{2}}={\mathbb{E}}[\kern-2pt[ {{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }(\hat{x}(\tau ,\bullet ))]\kern-2pt] .\end{eqnarray} \tag{ 38 }$$

To substitute equation (22) for RHS of equation (38),

$$\begin{eqnarray}&&{m}_{0}\displaystyle \frac{{d}^{2}{\mathbb{E}}[\kern-2pt[ {\hat{x}}^{\mu }(\tau ,\bullet )]\kern-2pt] }{d{\tau }^{2}}={\mathbb{E}}[\kern-2pt[ -e{\hat{{ \mathcal V }}}_{\nu }(\hat{x}(\tau ,\bullet )){F}^{\mu \nu }(\hat{x}(\tau ,\bullet ))]\kern-2pt] .\end{eqnarray} \tag{ 39 }$$

This is so-called Ehrenfest's theorem [34] which is an average mechanics of equation (22) as the KG equation (17).

Let us discuss the ponderomotive effect in relativistic stochastic mechanics of a spin-less electron. The KG equation (17) for a slowly varying motion is below: ${(i{\hslash }\partial )}^{2}\langle \phi \rangle -[{m}_{0}^{2}{c}^{2}-{e}^{2}\langle {A}^{2}\rangle ]\langle \phi \rangle =0$. This is an equation which has eliminated the fast oscillation of A in equation (17). This suggests that an effective mass ${m}_{\mathrm{eff}}={m}_{0}\times \sqrt{1-{e}^{2}\langle {A}^{2}\rangle /{m}_{0}^{2}{c}^{2}}$ appears in a propagation of a spin-less electron in quantum mechanics. Against this fact, here we confirm the above in stochastic mechanics with the same strategy of one in classical mechanics discussed in section 2. Namely, we will give the strict solution of stochastic mechanics at first, then find an equation of a slowly varying motion. Let us consider dW_± (ω) in equation (21) are another types of oscillations from one of A.

In this section, we consider the '+' sign of equation (21). Therefore, the determination of ${{ \mathcal V }}_{+}$ is our first problem. This is found by ${{ \mathcal V }}_{+}=\mathrm{Re}\{{ \mathcal V }\}-\mathrm{Im}\{{ \mathcal V }\}$ with equation (24). To give ${ \mathcal V }$ by equation (24), ϕ a wave function of the KG equation is required. Again suppose a laser field is a plane wave, i.e., A = A(k · x) and let p be an initial 4-momentum of a spin-less electron ($p\cdot p={m}_{0}^{2}{c}^{2}$). Under this situation, we know the Volkov solution [35] which is an analytic solution of the KG equation (17).

$$\begin{eqnarray}&&\phi (x)=\mathrm{Const}.\times \exp \displaystyle \frac{i}{{\hslash }}\left[-p\cdot x+{\int }_{0}^{k\cdot x}d\xi \displaystyle \frac{2{eA}(\xi )\cdot p+{e}^{2}{A}^{2}(\xi )}{2k\cdot p}\right]\end{eqnarray} \tag{ 40 }$$

${ \mathcal V }$ of equation (24) is calculated strictly as follows:

$$\begin{eqnarray}&&{{ \mathcal V }}^{\mu }(x)=\displaystyle \frac{{p}^{\mu }}{{m}_{0}}-\displaystyle \frac{e}{{m}_{0}}\displaystyle \frac{A(k\cdot x)\cdot p}{k\cdot p}{k}^{\mu }+\displaystyle \frac{e}{{m}_{0}}{A}^{\mu }(k\cdot x)-\displaystyle \frac{{e}^{2}}{2{m}_{0}}\displaystyle \frac{{A}^{2}(k\cdot x)}{k\cdot p}{k}^{\mu }\end{eqnarray} \tag{ 41 }$$

The readers should compare this with equation (4). ${{ \mathcal V }}_{+}={{ \mathcal V }}_{-}={ \mathcal V }$ is satisfied since $\mathrm{Im}\{{ \mathcal V }\}=0$ in our present case. Therefore, equation (41) is a drift velocity of a spin-less electron. We hereby express a trajectory of a spin-less electron in a background laser field of a plane wave by the integral of equation (21) corresponding to equation (5):

$$\begin{eqnarray}\begin{array}{rcl}{\hat{x}}^{\mu }(\tau ,\omega ) & = & {\hat{x}}^{\mu }(0,\omega )+\displaystyle \frac{{p}^{\mu }}{{m}_{0}}\times \tau \\ & & -\displaystyle \frac{e}{{m}_{0}}{\displaystyle \int }_{0}^{\tau }d\tau ^{\prime} \displaystyle \frac{p\cdot A(k\cdot \hat{x}(\tau ^{\prime} ,\omega ))}{k\cdot p}{k}^{\mu }\\ & & +\displaystyle \frac{e}{{m}_{0}}{\displaystyle \int }_{0}^{\tau }d\tau ^{\prime} {A}^{\mu }(k\cdot \hat{x}(\tau ^{\prime} ,\omega ))\\ & & -\displaystyle \frac{{e}^{2}}{2{m}_{0}}{\displaystyle \int }_{0}^{\tau }d\tau ^{\prime} \displaystyle \frac{{A}^{2}(k\cdot \hat{x}(\tau ^{\prime} ,\omega ))}{k\cdot p}{k}^{\mu }\\ & & +\sqrt{\displaystyle \frac{{\hslash }}{{m}_{0}}}{\displaystyle \int }_{0}^{\tau }{{dW}}_{+}^{\mu }(\tau ^{\prime} ,\omega ),{\mathscr{P}} \mbox{-} {\rm{a}}.{\rm{s}}.\end{array}\end{eqnarray} \tag{ 42 }$$

Recall the fact that we have to give the three of a holonomic constraint, stochastic kinematics and quantum dynamics of a spin-less electron dressing a laser field in relativistic stochastic mechanics. Let us find that triplet for the ponderomotive effect from the holonomic constraint. Consider the Itô formula of ${{ \mathcal V }}_{+}(\hat{x}(\tau ,\omega ))$ for the '+' sign of equation (21)² :

$$\begin{eqnarray}\begin{array}{rcl}{d}_{+}{{ \mathcal V }}_{+}^{\mu }(\hat{x}(\tau ,\omega )) & = & {{\mathfrak{D}}}_{\tau }^{+}{{ \mathcal V }}_{+}^{\mu }(\hat{x}(\tau ,\omega ))d\tau \\ & & +\sqrt{\displaystyle \frac{{\hslash }}{{m}_{0}}}{\partial }_{\alpha }{{ \mathcal V }}_{+}^{\mu }(\hat{x}(\tau ,\omega ))\cdot {{dW}}_{+}^{\alpha }(\tau ,\omega ),{\mathscr{P}} \mbox{-} {\rm{a}}.{\rm{s}}.\end{array}\end{eqnarray} \tag{ 43 }$$

For ${{\mathfrak{D}}}_{\tau }^{+}{{ \mathcal V }}_{+}^{\mu }={{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }=-e/{m}_{0}{F}^{\mu \nu }{{ \mathcal V }}_{\nu }$ since ${{ \mathcal V }}_{+}={{ \mathcal V }}_{-}={ \mathcal V }$ and ${\partial }_{\nu }{\partial }^{\nu }{{ \mathcal V }}^{\mu }(x)=0$ in the present situation,

$$\begin{eqnarray}\begin{array}{rcl}\langle {m}_{0}{d}_{+}{{ \mathcal V }}_{+}^{\mu }(\langle \hat{x}(\tau ,\omega )\rangle )\rangle & = & -\displaystyle \frac{{e}^{2}}{2{m}_{0}}{\partial }^{\mu }\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle d\tau \\ & & -\sqrt{\displaystyle \frac{{\hslash }}{{m}_{0}}}\displaystyle \frac{{e}^{2}{\partial }_{\alpha }\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle {k}^{\mu }}{2k\cdot p}\cdot {{dW}}_{+}^{\alpha }(\tau ,\omega )\end{array}\end{eqnarray} \tag{ 44 }$$

is the quantum version of equation (9). Where, $\hat{\theta }(\tau ,\omega ):= k\cdot \langle \hat{x}(\tau ,\omega )\rangle$. The operation $\langle \circ \rangle$ to extract a slowly varying part acts on only A of an external laser field. Conversely, let $\langle {{dW}}_{+}(\tau ,\omega )\rangle ={{dW}}_{+}(\tau ,\omega )$ since the random oscillation of W₊(ω) is independent from A. The second term of randomness in RHS of equation (44) is the difference from classical mechanics after the operation $\langle \circ \rangle$. We can find that $\langle {m}_{0}{d}_{+}{{ \mathcal V }}_{+}\rangle$ is parallel to the 4-wave vector k even if there is that random term. A 4-velocity in a slowly varying motion

$$\begin{eqnarray}&&\langle {{ \mathcal V }}_{+}^{\mu }(\langle \hat{x}(\tau ,\omega )\rangle )\rangle =\displaystyle \frac{{p}^{\mu }}{{m}_{0}}-\displaystyle \frac{{e}^{2}}{2{m}_{0}}\displaystyle \frac{\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle }{k\cdot p}{k}^{\mu }\end{eqnarray} \tag{ 45 }$$

(use equation (41)) provides the following relation:

$$\begin{eqnarray}&&\langle {{ \mathcal V }}_{+}(\langle \hat{x}(\tau ,\omega )\rangle )\rangle \cdot \langle {{ \mathcal V }}_{+}(\langle \hat{x}(\tau ,\omega )\rangle )\rangle ={c}^{2}\times \left[1-\displaystyle \frac{{e}^{2}\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle }{{m}_{0}^{2}{c}^{2}}\right]\end{eqnarray} \tag{ 46 }$$

Instead of this relation, the readers may find its another expression such as

$$\begin{eqnarray}&&\langle {{ \mathcal V }}_{+}(\langle \hat{x}(\tau ,\omega )\rangle )\rangle \cdot \langle {d}_{+}{{ \mathcal V }}_{+}(\langle \hat{x}(\tau ,\omega )\rangle )\rangle ={c}^{2}\times {d}_{\mathrm{+}}\,\left[1-\displaystyle \frac{{e}^{2}\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle }{2{m}_{0}^{2}}\right]\end{eqnarray} \tag{ 47 }$$

for a stochastic process $\langle \hat{x}\rangle$. Again, let us introduce the effective mass

$$\begin{eqnarray}&&{m}_{\mathrm{eff}}:= {m}_{0}\times \sqrt{1-\displaystyle \frac{{e}^{2}\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle }{{m}_{0}^{2}{c}^{2}}}\end{eqnarray} \tag{ 48 }$$

and the effective proper time τ_eff by

$$\begin{eqnarray}&&\displaystyle \frac{d{\tau }_{\mathrm{eff}}}{d\tau }:= \sqrt{1-\displaystyle \frac{{e}^{2}\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle }{{m}_{0}^{2}{c}^{2}}}.\end{eqnarray} \tag{ 49 }$$

Consider the replacements of variables: $\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle := \langle \hat{x}(\tau ,\omega )\rangle$, $\hat{{\rm{\Theta }}}({\tau }_{\mathrm{eff}},\omega ):= \hat{\theta }(\tau ,\omega )$ and

$$\begin{eqnarray}&&\langle {{ \mathcal V }}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )\rangle := \displaystyle \frac{\langle {{ \mathcal V }}_{+}(\langle \hat{x}(\tau ,\omega )\rangle )\rangle }{\sqrt{1-\displaystyle \frac{{e}^{2}\langle {A}^{2}(\hat{\theta }(\tau ,\omega ))\rangle }{2{m}_{0}^{2}{c}^{2}}}}.\end{eqnarray} \tag{ 50 }$$

Now the rule of the 4-velocity squared in our model is recovered for all ω:

$$\begin{eqnarray}&&\langle {{ \mathcal V }}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )\rangle \cdot \langle {{ \mathcal V }}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )\rangle ={c}^{2}\end{eqnarray} \tag{ 51 }$$

This is the holonomic constraint of a spin-less electron dressing a plane wave laser field described by the effective variables. This is a special case of equation (26): ${\mathbb{E}}[\kern-2pt[ \mathrm{LHS}\,\mathrm{of}\,\mathrm{Eq}.(51)]\kern-2pt] ={c}^{2}$, too. The plus sign of stochastic kinematics of equation (21) as the second item in stochastic mechanics is replaced by

$$\begin{eqnarray}&&{d}_{+}^{\mathrm{eff}}\langle {\hat{X}}^{\mu }({\tau }_{\mathrm{eff}},\omega )\rangle =\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )\rangle d{\tau }_{\mathrm{eff}}+\sqrt{\displaystyle \frac{{\hslash }}{{m}_{\mathrm{eff}}}}\times {{dW}}_{+}^{\mu }({\tau }_{\mathrm{eff}},\omega )\end{eqnarray} \tag{ 52 }$$

with ${{dW}}_{+}^{\alpha }({\tau }_{\mathrm{eff}},\omega )=\sqrt{d{\tau }_{\mathrm{eff}}/d\tau }\times {{dW}}_{+}^{\alpha }(\tau ,\omega )$ due to equation (29): ${\partial }_{{\tau }_{\mathrm{eff}}}{P}_{+}=-{\hslash }/2{m}_{\mathrm{eff}}\times {\partial }_{\mu }{\partial }^{\mu }{P}_{+}$ for ${P}_{+}(x,{\tau }_{\mathrm{eff}})={p}_{+}(x,\tau )$ and the fact that m_eff is a function of $k\cdot \langle \hat{X}\rangle$. Namely, a stochastic trajectory is,

$$\begin{eqnarray}\begin{array}{rcl}\langle {\hat{X}}^{\mu }({\tau }_{\mathrm{eff}}^{f},\omega )\rangle & = & \langle {\hat{X}}^{\mu }({\tau }_{\mathrm{eff}}^{i},\omega )\rangle +{\displaystyle \int }_{{\tau }_{\mathrm{eff}}^{i}}^{{\tau }_{\mathrm{eff}}^{f}}\displaystyle \frac{{p}^{\mu }d\tau }{{m}_{\mathrm{eff}}}\\ & & -{\displaystyle \int }_{{\tau }_{\mathrm{eff}}^{i}}^{{\tau }_{\mathrm{eff}}^{f}}\displaystyle \frac{{e}^{2}}{2{m}_{\mathrm{eff}}}\displaystyle \frac{\langle {A}^{2}(\hat{{\rm{\Theta }}}({\tau }_{\mathrm{eff}},\omega ))\rangle }{k\cdot p}{k}^{\mu }d{\tau }_{\mathrm{eff}}\\ & & +{\displaystyle \int }_{{\tau }_{\mathrm{eff}}^{i}}^{{\tau }_{\mathrm{eff}}^{f}}\sqrt{\displaystyle \frac{{\hslash }}{{m}_{\mathrm{eff}}}}{{dW}}_{+}^{\mu }({\tau }_{\mathrm{eff}},\omega )\end{array}\end{eqnarray} \tag{ 53 }$$

when ${\tau }_{\mathrm{eff}}^{i}\leqslant {\tau }_{\mathrm{eff}}^{f}$. The readers have to confirm the relation below: ${d}_{+}^{\mathrm{eff}}\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle ={d}_{+}\langle \hat{x}(\tau ,\omega )\rangle$. Define the operator

$$\begin{eqnarray}&&{{\mathfrak{D}}}_{\tau }^{\mathrm{eff}}:= \langle {{ \mathcal V }}_{\mathrm{eff}}^{\alpha }\rangle {\partial }_{\alpha }-\displaystyle \frac{{\hslash }}{2{m}_{\mathrm{eff}}}{\partial }^{\alpha }{\partial }_{\alpha }\end{eqnarray} \tag{ 54 }$$

appearing in the following Itô formula for $\langle \hat{X}(\omega )\rangle$ by equation (52) with its probability ${{\mathscr{P}}}_{\mathrm{eff}}(\omega )$:

$$\begin{eqnarray}\begin{array}{rcl}{d}_{+}^{\mathrm{eff}}f(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle ) & = & {{\mathfrak{D}}}_{\tau }^{\mathrm{eff}}f(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )d{\tau }_{\mathrm{eff}}\\ & & +\sqrt{\displaystyle \frac{{\hslash }}{{m}_{\mathrm{eff}}}}{\partial }_{\alpha }f(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )\cdot {{dW}}_{+}^{\alpha }({\tau }_{\mathrm{eff}},\omega ),{{\mathscr{P}}}_{\mathrm{eff}} \mbox{-} {\rm{a}}.{\rm{s}}.\end{array}\end{eqnarray} \tag{ 55 }$$

Then, quantum dynamics as the third important element in our model for the ponderomotive effect is

$$\begin{eqnarray}&&{{\mathfrak{D}}}_{\tau }^{\mathrm{eff}}({m}_{\mathrm{eff}}\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }\rangle )={\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}.\end{eqnarray} \tag{ 56 }$$

This is derived from $\langle {m}_{0}{{\mathfrak{D}}}_{\tau }^{+}{{ \mathcal V }}_{+}^{\mu }\rangle =\langle -{{eF}}^{\mu \nu }{{ \mathcal V }}_{\nu }\rangle$. Equation (56) is equivalent to the KG equation

$$\begin{eqnarray}&&{\left(i{\hslash }\partial \right)}^{2}\langle \phi \rangle -{m}_{\mathrm{eff}}^{2}{c}^{2}\langle \phi \rangle =0\end{eqnarray} \tag{ 57 }$$

as we discussed in the first part of this section with

$$\begin{eqnarray}&&\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }(x)\rangle := \displaystyle \frac{i{\hslash }\partial {}^{\mu }\mathrm{ln}\langle \phi (x)\rangle }{{m}_{\mathrm{eff}}},\end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray}&&\langle \phi (x)\rangle =\mathrm{Const}.\times \exp \displaystyle \frac{i}{{\hslash }}\left[-p\cdot x+{\int }_{0}^{k\cdot x}\displaystyle \frac{{e}^{2}\langle {A}^{2}(\xi )\rangle }{2k\cdot p}d\xi \right].\end{eqnarray} \tag{ 59 }$$

Equation (57) shows that the ponderomotive effect is a free propagation of a spin-less electron dressing an external field. We could formulate quantum dynamics of equation (56) similar to equation (15) in the classical regime. When f in equation (55) is replaced by ${m}_{\mathrm{eff}}\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }\rangle$,

$$\begin{eqnarray}\begin{array}{rcl}{d}_{+}^{\mathrm{eff}}[{m}_{\mathrm{eff}}\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )\rangle ] & = & [{\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}]d{\tau }_{\mathrm{eff}}\\ & & +\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{{\rm{\hslash }}}{{m}_{\mathrm{eff}}}}\displaystyle \frac{{\partial }_{\alpha }{m}_{\mathrm{eff}}^{2}{c}^{2}}{k\cdot p}{k}^{\mu }\cdot {{dW}}_{+}^{\alpha }({\tau }_{\mathrm{eff}},\omega ),{{\mathscr{P}}}_{\mathrm{eff}} \mbox{-} {\rm{a.s.}}\end{array}\end{eqnarray} \tag{ 60 }$$

expresses the time evolution of a momentum ${m}_{\mathrm{eff}}\langle {{ \mathcal V }}_{\mathrm{eff}}\rangle ={m}_{0}\langle {{ \mathcal V }}_{+}\rangle$ along $\langle \hat{X}(\omega )\rangle$ instead of equation (44). The triplet of equation (51), equation (52) and equation (56) are relativistic stochastic mechanics for the ponderomotive effect acting on a spin-less electron at $\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle$.

Consider the FP equation for equation (52):

$$\begin{eqnarray}&&{\partial }_{{\tau }_{\mathrm{eff}}}{p}_{\mathrm{eff}}(x,{\tau }_{\mathrm{eff}})+{\partial }_{\mu }[\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }(x)\rangle {p}_{\mathrm{eff}}(x,{\tau }_{\mathrm{eff}})]=-\displaystyle \frac{{\hslash }}{2{m}_{0}}{\partial }_{\mu }{\partial }^{\mu }\left[\displaystyle \frac{{m}_{0}}{{m}_{\mathrm{eff}}}{p}_{\mathrm{eff}}(x,{\tau }_{\mathrm{eff}})\right]\end{eqnarray} \tag{ 61 }$$

With ${p}_{\mathrm{eff}}(x,{\tau }_{\mathrm{eff}}):= d{{\mathscr{P}}}_{\mathrm{eff}}({{\rm{\Omega }}}_{x})/{d}^{4}x$ the probability density at τ_eff for ${{\rm{\Omega }}}_{x}:= \{\omega | \hat{X}({\tau }_{\mathrm{eff}},\omega )=x\}$. We can expect p_eff is a function of k · x and τ_eff since $\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }\rangle$ is a vector function of k · x. Then, the RHS of the above equation becomes zero since k_μk^μ = 0. Therefore, stochastic kinematics of equation (52) imposes the following advection equation of its probability distribution p_eff:

$$\begin{eqnarray}&&{\partial }_{{\tau }_{\mathrm{eff}}}{p}_{\mathrm{eff}}(x,{\tau }_{\mathrm{eff}})+{\partial }_{\mu }[\langle {{ \mathcal V }}_{\mathrm{eff}}^{\mu }(x)\rangle {p}_{\mathrm{eff}}(x,{\tau }_{\mathrm{eff}})]=0\end{eqnarray} \tag{ 62 }$$

So, a profile of p_eff free propagates in the spacetime with the 4-velocity of $\langle {{ \mathcal V }}_{\mathrm{eff}}\rangle$. In order to the rule of randomness ${\mathbb{E}}[\kern-2pt[ {{dW}}_{+}^{\alpha }({\tau }_{\mathrm{eff}},\bullet )]\kern-2pt] =0$, the expectation of equations (51), (52), (56) gives its Ehrenfest's theorem

$$\begin{eqnarray}&&\displaystyle \frac{d}{d{\tau }_{\mathrm{eff}}}\left[{m}_{\mathrm{eff}}\displaystyle \frac{d{\mathbb{E}}[\kern-2pt[ \langle {X}^{\mu }({\tau }_{\mathrm{eff}},\bullet )\rangle ]\kern-2pt] }{d{\tau }_{\mathrm{eff}}}\right]={\mathbb{E}}[\kern-2pt[ {\partial }^{\mu }{m}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle ){c}^{2}]\kern-2pt] \end{eqnarray} \tag{ 63 }$$

and this is comparable with equation (15), too, where we dare to write the index of m_eff in equation (63) since

$$\begin{eqnarray}&&{\mathbb{E}}[\kern-2pt[ {\partial }^{\mu }{m}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle )]\kern-2pt] \ne {\partial }^{\mu }{m}_{\mathrm{eff}}({\mathbb{E}}[\kern-2pt[ \langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle ]\kern-2pt] )\end{eqnarray} \tag{ 64 }$$

in general. It shows a difference between the present model and equation (15) of classical mechanics. A spin-less electron stays in ${\mathbb{V}}({\tau }_{\mathrm{eff}}):= \mathrm{supp}({p}_{\mathrm{eff}}(\circ ,{\tau }_{\mathrm{eff}}))$ at τ_eff. For

$$\begin{eqnarray}&&{r}_{e}:= {\left[{\int }_{{\mathbb{V}}({\tau }_{\mathrm{eff}})}\displaystyle \sum _{\mu =1}^{3}{\left({X}^{\mu }-{\mathbb{E}}[\kern-2pt[ \langle {\hat{X}}^{\mu }({\tau }_{\mathrm{eff}},\bullet )\rangle ]\kern-2pt] \right)}^{2}{p}_{\mathrm{eff}}(X,{\tau }_{\mathrm{eff}}){d}^{4}X\right]}^{1/2}\end{eqnarray} \tag{ 65 }$$

a typical radius of a spin-less electron, let r_e be the Compton wavelength, i.e., r_e = 6.5 × 10⁻¹²m. A typical pulse duration of a high-intensity laser is the order of 30fsec equivalent to 10μ m and its wavelength is order of 1 μ m [1–6] against the order of r_e. Hence, the domain ${\mathbb{V}}({\tau }_{\mathrm{eff}})$ is point-like for $\langle {A}^{2}\rangle$ at τ_eff. Therefore,

$$\begin{eqnarray}&&{\mathbb{E}}[\kern-2pt[ {\partial }^{\mu }{m}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle )]\kern-2pt] \approx {\partial }^{\mu }{m}_{\mathrm{eff}}({\mathbb{E}}[\kern-2pt[ \langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle ]\kern-2pt] )\end{eqnarray} \tag{ 66 }$$

is reasonable for the interactions of a spin-less electron with a high-intensity femtosecond laser field. Then, equation (63) with equation (66) becomes equivalent to equation (15) of classical mechanics. The similarity of formulae between equation (15) and equations (51), (52), (56) shows a universality of a ponderomotive effect acting on a spin-less electron interacting with a high-intensity femtosecond laser field of a plane wave in both classical and quantum regimes.

We discussed relativistically covariant formalism of the ponderomotive effect of a spin-less electron interacting with a high-intensity plane wave laser field in classical and in quantum mechanics. The key words in our discussion were, relativistic covariance and a slowly varying component of variables. In section 2, we recalled the relativistically covariant form of the ponderomotive effect in classical mechanics. We derived this by solving equation (3) strictly with a plane wave laser field. Where, the effective mass m_eff of equation (13) was defined to express a spin-less electron dressing a laser field. Its gradient force ${\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}$ was identified as the relativistically covariant ponderomotive force. To discuss the above in the quantum regime, we introduced relativistic stochastic mechanics in section 3. It was constructed by stochastic kinematics of equation (21), quantum dynamics of equation (22) equivalent to the KG equation (17) and the holonomic constraint of equation (26). By using this in section 4, we expressed the ponderomotive effect in the quantum regime. Namely, the triplet of equations (51), (52), (56) was derived as relativistic stochastic mechanics for the ponderomotive effect. It related with $\langle \phi \rangle$ the wave function of a slowly varying spin-less electron of its mass m_eff. Thus, relativistic stochastic mechanics also imposed m_eff in the ponderomotive effect. Where, $\langle \hat{X}(\omega )\rangle$ of equation (53) is a realistic trajectory of a scalar electron. The ability to draw $\langle \hat{X}(\omega )\rangle$ by the present model is the unique achievement in quantum mechanics since stochastic kinematics has been absent in usual quantum mechanics so far. For $\langle \hat{X}(\omega )\rangle$,

$$\begin{eqnarray}&&{f}^{\mu }(\langle \hat{X}({\tau }_{\mathrm{eff}},\omega )\rangle )d{\tau }_{\mathrm{eff}}=[{\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}]d{\tau }_{\mathrm{eff}}+\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{{\rm{\hslash }}}{{m}_{\mathrm{eff}}}}\displaystyle \frac{{\partial }_{\alpha }{m}_{\mathrm{eff}}^{2}{c}^{2}}{k\cdot p}{k}^{\mu }\cdot {{dW}}_{+}^{\alpha }({\tau }_{\mathrm{eff}},\omega )\end{eqnarray} \tag{ 67 }$$

in equation (60) provided the momentum transfer of a spin-less electron due to the ponderomotive force f. Both classical and quantum dynamics have the term of ${\partial }^{\mu }{m}_{\mathrm{eff}}{c}^{2}$. This imposed the universality of the ponderomotive effect bridging between both regimes with a typical high-intensity femtosecond laser of a plane wave. Hence, we concluded relativistic classical mechanics is enough to express the purely ponderomotive effect in laser-plasmas produced by a high-intensity femtosecond laser via equation (66). Let us consider the inequality of equation (64). Equation (66) was derived by the treatment such that ${\mathbb{E}}[\kern-2pt[ {\partial }^{\mu }{m}_{\mathrm{eff}}(\langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle )]\kern-2pt]$ was approximated by ${\partial }^{\mu }{m}_{\mathrm{eff}}({\mathbb{E}}[\kern-2pt[ \langle \hat{X}({\tau }_{\mathrm{eff}},\bullet )\rangle ]\kern-2pt] )$ since ${\mathbb{V}}({\tau }_{\mathrm{eff}})$ the domain of the probability density of a spin-less electron was point-like for $\langle {A}^{2}\rangle$ of a high-intensity femtosecond laser in equation (63). Therefore, we can find the inequality of equation (64) when ${\mathbb{V}}({\tau }_{\mathrm{eff}})$ is not point-like for $\langle {A}^{2}\rangle$. This inequality expresses quantumness of the a scalar electron dressing a laser field. A possibility to see this may be examined by extremely short pulse lasers like recent attosecond lasers (for example [2]) or forthcoming high-intensity zeptosecond lasers like IZEST project [36].

Finally, KS was impressed by Kibble's work of [22]. He also argued the classical-quantum correspondence in that paper. He explained that by Ehrenfest's theorem like equation (63) in the present article. Then, ${m}_{0}[{\partial }_{t}{\boldsymbol{v}}+{\boldsymbol{v}}\cdot {\rm{\nabla }}{\boldsymbol{v}}]=-e({\boldsymbol{E}}+{\boldsymbol{v}}\times {\boldsymbol{B}})$ as equation (50) in [22] was suggested. This is almost same as following Nelson's stochastic mechanics

$$\begin{eqnarray}&&{m}_{0}[{\partial }_{t}{\boldsymbol{v}}+{\boldsymbol{v}}\cdot {\rm{\nabla }}{\boldsymbol{v}}]=-e({\boldsymbol{E}}+{\boldsymbol{v}}\times {\boldsymbol{B}})+\displaystyle \frac{{{\hslash }}^{2}}{2{m}_{0}}{{\rm{\nabla }}}^{2}\displaystyle \frac{{{\rm{\nabla }}}^{2}\sqrt{p}}{\sqrt{p}}\end{eqnarray} \tag{ 68 }$$

with $p=| \psi {| }^{2}$ assisted by the Schrödinger equation $i{\hslash }{\partial }_{t}\psi =1/2{m}_{0}\times {[-i{\hslash }{\rm{\nabla }}+{eA}]}^{2}\psi$. This Nelson's model is coupled with a 3D stochastic process given by its kinematics [24, 25]. Our discussion in section 4 for relativistic dynamics ${m}_{0}{{\mathfrak{D}}}_{\tau }{{ \mathcal V }}^{\mu }=-e{\hat{{ \mathcal V }}}_{\nu }{F}^{\mu \nu }$ is regarded as the natural expansion of the above non-relativistic dynamics, with stochastic kinematics to illustrate a realistic trajectory of a quanta.

KS acknowledges Prof K A Tanaka (ELI-NP), Dr J F Ong (ELI-NP), Dr Y Nakamiya (ELI-NP), Mr J Tamlyn (ELI-NP) and Dr J K Koga (QST, Japan) for useful discussion. This work was supported by the Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase II, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund—the Competitiveness Operational Programme (1/07.07.2016, COP, ID 1334).