Radiation Reaction on Brownian Motions

Radiation Reaction on Brownian Motions Keita Seto

arXiv:1603.03379v2 [math-ph] 13 Mar 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.∗ (Dated: March 15, 2016) Tracking the “real” trajectory of a quantum particle is one of the interpretation problem and it is expressed by the Brownian (stochastic) motion suggested by E. Nelson. Especially the dynamics of a radiating electron, namely, “radiation reaction” which requires us to track its trajectory becomes important in the high-intensity field physics by PW-class lasers at present. It has been normally treated by the Furry picture in non-linear QED, but it is difficult to draw the real trajectory of a quantum particle. For the improvement of this, I propose the representation of a stochastic particle interacting with fields and show the way to describe radiation reaction on its Brownian motion. PACS numbers: 41.60.-m, 02.50.-r, 05.40.Jc , 42.50.Lc, 45.50.-j

When we draw “real” trajectory of a particle under the quantum fluctuation, the stochastic quantization proposed by E. Nelson is the candidate of the method for it. He assumed the Brownian motion as quantum fluctuation and he also demonstrated not only his approach is essentially equivalent to the Schr¨odinger equation, but answered why the square of the wave function is regarded as the probability density [1, 2]. However when we consider the coupled system between a stochastic particle and fields, this treatment is not obviously. Therefore, the expression of the field generated from the current of a stochastic particle is interesting. Anyway on the recent research programs by 10PWclass lasers [3–5], it is predicted the non-linear QED effects under these high-intensity laser fields [6]. As a basic interest of them, “radiation reaction” acting on a radiating electron is regarded as the feedback effect of nonlinear QED radiation [7, 8] via the Furry picture [6, 9]. Though the Furry picture can describe the propagator of the Dirac particle absorbing photons non-linearly in a high-intensity field, but QED radiation in this method is treated as the lowest order of its photon emission [10–13]. Since we may find more higher order interactions in the real experiments, the possibility of the non-linear and non-perturbative regime which the Furry picture can’t address to should not be disregarded. On the other hand, the model of radiation reaction derived by P. A. M. Dirac is named the Lorentz-Abraham-Dirac (LAD) equation in classical dynamics [14], but non-linear and non-perturbative equation.   dv µ m0 τ0 d2 v µ ν d2 v ν µ µν m0 v − v = −eFex vν + 2 vν (1) dτ c dτ 2 dτ 2 Here, τ0 = e2 /6πε0 m0 c3 = O(10−25 sec) [15] . When we discover the direct way to the quantization from this LAD equation, there is the possibility which improves the non-linear and non-pertubative methods. For this purpose, we discuss a possibility of stochastic quantization of radiation reaction under the meaning of Nelson. In

this paper, we discuss a single Klein-Gordon particle interacting with fields. The contents are the kinematics of the Klein-Gordon particle including the definition of its proper time, the Lagrangian as the full dynamics of this system and the derivation of radiation reaction in this framework. A similar dynamics of the LAD equation (1) on its Brownian trajectory can be found. The first part is the kinematics of a Klein-Gordon particle. For describing it, we can’t avoid considering the probability space (Ω , A , P) for the quantum fluctuation. Here, A is the σ-algebra generated from a certain non-empty set Ω which is the collection of the label of stochastic (Brownian) paths [16]. P is the probability measure defined on Ω . Then, we also need to define the trajectory itself. The trajectory what we want to analyze is, namely, x ˆ : T × Ω → A4 (V4M , g). Here T = R ∪ {±∞} represents the parameter of time [17] and A4 (V4M , g) is the 4-dimensional metric affine space as the spacetime itself with the 4-dimensional standard vector space V4M and the Minkowski metric g [18]. I also use the symbol A4 as the affine space excluding the structure of metric g. The evolution of stochastic trajectories should be assumed as the continuous semi-martingales [1, 2, 19, 20]. Under this construction of a certain stochastic process x ˆ(τ ), we consider how to analyze this time evolution. Due to the quantum fluctuation, the both categories of the time-forward and the time-backward evolution have to be considered. In other words, one is the (time) increasing family ”Past” = {Pτ |τ ∈ T } for the timeforward evolution and the another is the decreasing family ”Future” = {Fτ |τ ∈ T } for the time-backwarding [1, 2, 19]. Let’s consider Pτ is the condition from the past to the present time τ and Fτ for one from the future to the present. Under these structures, we can consider the kinematics of the Brownian particle by using Pτ measurable and Fτ -measurable function xˆ(τ ) under the definition of the Itˆ o (stochastic) integral [21].  µ V+ (ˆ x(τ ))dτ + λ × dW+µ (τ ), xˆ(τ ) ∈ Pτ dˆ xµ (τ ) = (2) µ V− (ˆ x(τ ))dτ + λ × dW−µ (τ ), xˆ(τ ) ∈ Fτ

2 p Here, λ = ~/m0 represents the amplitude of the Brownian motion [22]. The Brownian fluctuation is characterized by the the standard Wiener process W± (τ ) satisfying E[dW+µ (τ )|Pτ ] = 0 and E[dW−µ (τ )|Fτ ] = 0 [23]. Especially in the case of standard Wiener process, the following Itˆ o rules are satisfied [24]; dτ · dτ = 0, dτ · dW±µ (τ ) = 0, dW±µ (τ ) · dW±ν (τ ) = ∓g µν dτ.

(3)

µ Moreover we also define the white noise ξˆ± (τ ) as the time derivatives of W± (τ ) [25]. Via this introduction and putting the new symbols d± x ˆµ (τ ) as the RHS in Eq.(2), Eq.(2) can be recognized as the summation of the drift µ velocity and the randomness ξˆ± (τ ) = dW±µ /dτ (τ ),

d± x ˆµ µ µ (τ ) = V± (ˆ x(τ )) + λ × ξˆ± (τ ). dτ

(4)

Corresponding to Eq.(3), the modified Itˆ o rule for this white noise can be derived as µ ν ξˆ± (τ )ξˆ± (τ ′ ) = ∓g µν δ(τ − τ ′ ).

(5)

The conditional expectation of Eq.(4) at time τ implies µ the drift velocities , V+ (ˆ x(τ )) = E[d+ x ˆµ /dτ (τ )|Pτ ] and µ µ V− (ˆ x(τ )) = E[d− xˆ /dτ (τ )|Fτ ] [26]. This kinematics induces the Itˆ o lemma [21] as the method of the directional derivatives of f (ˆ x(τ )) along the stochastic path xˆ(τ ), λ2 µ ∂ ∂µ f (ˆ x(τ ))dτ. 2 (6) µ Especially by limiting the path x ˆ (τ ) on γτ ⊆ Pτ ∩ Fτ as the “present” τ , the superposition of the above “±” evolution is introduced. Nottale suggested the following complex differential and the velocity are essential manners for the dynamics [22]. d± f (ˆ x(τ )) = ∂µ f (ˆ x(τ ))d± x ˆµ (τ ) ∓

1−i 1+i dˆ ≡ d+ + d− 2 2

(7)

2 ˆ (ˆ ˆxµ (τ ) − iλ ∂ µ ∂µ f (ˆ x(τ ))dτ (8) df x(τ )) = ∂µ f (ˆ x(τ ))dˆ 2

1−i µ 1+i µ V+ (ˆ x(τ )) + V− (ˆ x(τ )) 2 2 (9) ˆ Here, Dτ ≡ E[d/dτ |γτ ]. Choosing the wave function φ ∈ L2 (A4 , µA4 ) like Ref.[22], V µ (ˆ x(τ )) ≡ Dτ x ˆ(τ ) =

V α (x) = iλ2 × ∂ α ln φ(x) +

e α A (x) m0

(10)

on the measure space (A4 , AA4 , µA4 ). We can find φ satisfies with the Klein-Gordon equation in the later study. The following identity is also derived [22]. Vα∗ (x)V α (x) = c2

(11)

Here the metric is chosen as g = diag(+1, −1, −1, −1). So this is the new invariant parameter for the definition of “the proper time” of a stochastic particle; q 1 ˆxν (τ )|γτ ]. (12) dτ ≡ × E[dˆ∗ xˆµ (τ )|γτ ] · E[dˆ c It requires us to consider the stochastic paths should be the patch-works under the path-class ΓT = {γτ |τ ∈ T } for the definition of dτ . In order to realize this kinematics Eq.(2) or Eq.(4), we need to investigate the behavior of the complex vector V µ (ˆ x(τ )). The action integral for classical dynamics in Minkowski spacetime (A4 , AA4 , µA4 , g) is ˆ ∞ ˆ ∞ m0 vα v α − e dτ Aα v α S= dτ 2 −∞ −∞ ˆ 1 d4 x Fαβ F αβ . (13) + 4µ0 c A4

Here, the measure µA4 on A4 denotes dµA4 = d4 x. We construct the quantized system from Eq.(13). At first we don’t touch the part of the field propagation, ´ 1/4µ0 c × d4 x Fαβ F αβ . Though Nottale suggested a particle Lagrangian interacting with the field(s) satisfies L0 (τ, x, V+ , V− ) = L(τ, x, V) [22], however, let’s consider L0 (τ, x, V+ , V− ) = L(τ, x, V, V ∗ ) as the expansion of it. V ∗ is the complex conjugate of V. Corresponding to Eq.(13), I propose the following new action integral for the Klein-Gordon particle system. ˆ ∞  m0 ∗ S=E Vα (ˆ x(τ ))V α (ˆ x(τ )) dτ 2 −∞   ˆ ∞ dτ Aα (ˆ x(τ ))Re{V α (ˆ x(τ ))} + E −e ˆ −∞ 1 d4 x Fαβ (x)F αβ (x) (14) + 4µ0 c A4 Resulting the variational calculus, the dynamics of the particle and the field is induced (Vˆ µ = V µ − iλ2 /2 × ∂ µ ν and Vreal (x) = Re{V ν (x)}). m0 Dτ V µ (ˆ x(τ )) = −eVˆ ν (ˆ x(τ ))Fνµ (ˆ x(τ )) ∂µ F

µν



(x) = µ0 E −ec

ˆ

∞

−∞

dτ

′

ν Vreal (x)δ 4 (x

(15)

 −x ˆ(τ )) ′

(16) Here, I recall the Euler-Lagrange equation (the Yasue equation [19, 20]) for the particle-Lagrangian Lparticle = α E[m0 /2 × Vα∗ V α − eAα (ˆ x(τ ))Vreal (ˆ x(τ ))], ∂Lparticle ∂Lparticle ∂Lparticle − Dτ − D∗τ =0 ∗ ∂xµ ∂Vµ ∂Vµ

(17)

which derives Eq.(15). The Maxwell equation comes from the variational of Eq.(14) by Aν . Applying the formula of Eq.(10), we can demonstrate Eq.(15) is equal to the

3 Klein-Gordon equation [22] and Eqs.(15-16) describe the scalar QED with the assumption, ˆ ∞  ′ 4 ′ φ(x)φ(x) = E dτ δ (x − xˆ(τ )) (18) −∞

for the realization of the ´current term of Klein-Gordon µ ν particle, jK-G = E[−ec dτ ′ Vreal (x)δ 4 (x − x ˆ(τ ′ ))] = 2 −ieµ0 λ /2 × g µν (φ∗ Dν φ − φDν∗ φ∗ ). Instead of the action integral Eq.(14), the same dynamics of Eqs.(15-16) are also derived by using following form;   ˆ ∞ p x(τ ))V α (ˆ x(τ )) S ′ = E m0 c dτ Vα∗ (ˆ −∞   ˆ ∞ dτ Aα (ˆ x(τ ))Re{V α (ˆ x(τ ))} + E −e ˆ −∞ 1 + d4 x Fαβ (x)F αβ (x). (19) 4µ0 c A4 p ´ x(τ ))V α (ˆ x(τ ))] expresses The term E[m0 c dτ Vα∗ (ˆ the ´quantization from the classical action integral p m0 c dτ vα (τ )v α (τ ) on the Brownian trajectories. The final topics in this paper is the “radiation reaction” effects as the solution of the Maxwell equation (16) at the point of a stochastic particle. From here, we consider how to derive the field strength tensor as the mimic of classical dynamics like Ref.[14, 27]. Considering the Green function ∂µ ∂ µ G(±) (x − x′ ) = δ 4 (x − x′ ) under the Lorentz condition ∂ν Aν(±) = 0, the solution of Eq.(16) is Aν(±) (x)

= −ecµ0

ˆ

∞

dτ

−∞

′

ν E[Vreal (ˆ x(τ ′ ))G(±) (x

′

−x ˆ(τ ))].

(20) Where, G(±) (∆x) = θ(±∆x0 )/2π × δ(∆xµ ∆xµ ) with ∆xµ = xµ − x′µ and “+/−” represents the retarded/advanced Green function. By following Ref.[27], the field strength becomes, ˆ ∞ µν F(±) (x) = −ecµ0 dτ ′ −∞

µ ν E[(Vreal (ˆ x(τ ′ ))∂ µ − Vreal (ˆ x(τ ′ ))∂ ν )G(±) (δx)]. (21)

Here δxµ = x − x ˆµ (τ ′ ) and the treatment of ∂ α G(±) (δx) with the stochastic valued index δx is not trivial. We want to assume s2 ≡ δxµ δxµ |x=ˆxµ (τ ) ∼ c2 × (τ − τ ′ )2 for it, under the small time duration τ − τ ′ . In the fact, that condition can be found under the limited path-class. The Itˆ o integral is introduced at Eq.(2). By observing a certain stochastic path x ˆ, the evolution d+ x ˆµ (τ ) µ and d− x ˆ (τ ) should be automatically defined at time τ . When we considered the path-class Γ[τ,τ ′ ] = {γτ˜ |γτ˜ ⊂ Pτ˜ ∩ Fτ˜ , τ˜ ∈ [τ ′ , τ ]} ⊂ ΓT ,

(22)

then the proper time was realized on the stochastic particle in the domain of [τ ′ , τ ]. Since the Itˆ o integral on it

is xˆµ (τ ) − x ˆµ (τ ′ ) =

ˆ

τ

d± x ˆµ (τ ′′ ),

(23)

τ′

x ˆµ (τ )− x ˆµ (τ ′ ) is described by using both of the evolution, µ d+ x ˆ /dτ and d− x ˆµ /dτ under the Γ[τ,τ ′ ] -class. We have ˆxν (τ )|γτ ] as known the fact c2 dτ 2 = E[dˆ∗ x ˆµ (τ )|γτ ] · E[dˆ the definition of the proper time at Eq.(12). The length of the world line cdτ √ is equal to the small distance between the two point d s2 . Therefore expanding this naive idea, s2 = δxµ δxµ |x=ˆxµ (τ ) satisfies with s2 = E

ˆ

τ

τ′

2

 ˆ dˆ∗ x ˆµ (τ ′′ ) · E ′ 2

τ τ′

 ˆxµ (τ ′′ ) dˆ

= c × (τ − τ ) + O((τ − τ ′ )3 )

(24)

under the condition of the small time duration |τ − τ ′ |. Here our hope s2 ∼ c2 × (τ − τ ′ )2 was realized on the Γ[τ,τ ′ ] -class path. Therefore, s2 = E[δxµ ] · E[δxµ ] is defined dauntlessly reflecting Eq.(24) when |τ − τ ′ | is too small. This expansion leads the useful relation via the following Nelson’s partial integration formula [1, 2, 19, 20]; ˆ τ  E[δxµ ]x=ˆxµ (τ ) = E dτ ′′ V µ (τ ′′ ) (25) τ′ ˆ τ  µ =E dτ ′′ Vreal (τ ′′ ) . (26) τ′

Many difficulties can be removed via this treatment. Considering ds2 /dτ ′ = −2 × E[δxµ ] · E[dδxµ /dτ ′ ] = −2c2 × (τ − τ ′ ) + O((τ − τ ′ )2 ), dG(±) E[δxα ] . (27) ∂ α G(±) (δx) = − 2 × c (τ − τ ′ ) dτ ′ x=ˆxµ (τ )

Substituting Eq.(27) for Eq.(21), the following expression is derived. ˆ eµ0 ∞ dτ ′ µν F(±) (ˆ xµ (τ )) = c −∞ τ − τ ′    ν dG(±) 2 E[δxµ ]Vreal (ˆ x(τ ′ )) E (s ) µ −E[δxν ]Vreal (ˆ x(τ ′ )) dτ ′ (28) ´τ µ Here E[δxµ ]x=ˆxµ (τ ) = E[ τ ′ dτ ′′ Vreal (ˆ x(τ ′′ ))] should be chosen. Then considering the expansion of an arbitrary function E[F (ˆ x(τ ′ ))] with the signatures of the conditional expectations of derivatives D+ τ = E[d+ /dτ |Pτ ] and D− τ = E[d− /dτ |Fτ ], "ˆ ′ # τ

E[F (ˆ x(τ ′ ))] = E[F (ˆ x(τ ))] + E

dτ ′′ D± x(τ ′′ )) τ F (ˆ

τ

∞ X  (τ ′ − τ )n  ± n E (Dτ ) F (ˆ x(τ )) = n! n=0

(29)

4 is satisfied strictly. The rest treatments are completely same as the method provided by F. V. Hartemann [27] excluding the directions of the stochastic evolution. Again, there are two Itˆ o integrals. The each of evolution can be applied to the method by Hartemann. Operating this manner, the following two radiation fields at the point of a particle, namely the LAD-radiation fields are derived.   ± µ E[D± x(τ ))] τ Dτ Vreal (ˆ ν m0 τ0  ×E[Vreal (ˆ x(τ ))]  µν  FLAD(±) (ˆ x(τ )) = − 2  ± ± ν x(τ ))]  ec  −E[Dτ Dτ Vreal (ˆ µ ×E[Vreal (ˆ x(τ ))] (30) We need to consider the both of fields act on a certain stochastic trajectory in the ΓT -class. Proposing the field µν µν Fµν = [FLAD(+) + FLAD(−) ]/2, the radiation field by a stochastic Klein-Gordon particle is formulated as   (2) µ E[Dτ Vreal (ˆ x(τ ))] ν m0 τ0  ×E[Vreal (ˆ x(τ ))]   . (31) Fµν (ˆ x(τ )) = − 2  (2) ν  ec −E[Dτ Vreal (ˆ x(τ ))]  µ ×E[Vreal (ˆ x(τ ))] (2)

Here, Dτ ≡ (D∗τ Dτ + Dτ D∗τ )/2. This field satisfies with the source-free condition, i.e., ∂µ Fµν = 0. Finally, the dynamics of the particle is µν m0 Dτ V µ (ˆ x(τ )) = −eVˆν Fex (ˆ x(τ )) − eFµν Vν (ˆ x(τ )). (32)

This is the dynamics of a radiating stochastic particle corresponding to the LAD equation what we want to derive. However it includes many higher order derivatives. By reducing terms like the Ford-O’Connell [28]/LandauLifshitz [29] schemes, e Vα Vβ∗ + Vα∗ Vβ α µβ ∂ Fex + O(τ0 , ~, ∂ 2 ), m0 2 (33) the estimations become a bit easier. We discussed the formulation of radiation reaction on a Brownian trajectory. For the realization of it, the kinematics was limited under the path-class ΓT = {γτ |γτ ⊆ Pτ ∩ Fτ , τ ∈ T } at first. The proper time [Eq.(12)] and the new Lagrangians [Eq.(14, 19)] were defined on the ΓT -class path. Then, these defined Lagrangians implied the dynamics of Scalar QED [Eq.(15, 16)]. Finally, the Maxwell equation with the current of a stochastic particle was solved as the radiation field [Eq.(31)] and the expression of the equation of a radiating stochastic particle’s motion was derived [Eq.(32)]. The expected future works should be the check of this idea with ordinal QED form [6]. For example, the probability space (Ω , A , P) was defined with the probability measure P, however I couldn’t touch the detail of it in this article. From the mathematical research, it is expressed by the OnsagerMachlup functional [30–32]. It is expected this choice of the probability measure is equivalent to the method by G. µ D(2) τ Vreal ∼ −

Parisi and Y. S. Wu [33]. On the other hand being more realistic for the experiments, comparing Eq.(32) and the orders of Feynman diagrams must be the concrete interests of high-intensity field physics driven by PW-class lasers. The numerical simulations are required to investigate these researches, too. Moreover, it can describe the time evolution of a radiating quantum particle. When we can apply it to material science, we may realize the new simulation method. This work is supported by Extreme Light Infrastructure – Nuclear Physics (ELI-NP) – Phase I, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund.

∗

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