Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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376 15 The Newton-Nelson Equation

t,0

0,t

= α

; Γ

t,0

0,t

= α

; Γ

t,0

0,t

Ric = Ric;

t,0

0,t

tr∇α

(t, ξ(t))(α

)=tr∇α

(t, ξ(t))(α

);

t,0

0,t

tr∇Ric(t, ξ(t))(Ric) = tr∇Ric(t, ξ(t))(Ric)

and so on. Thus the fact that ξ(t) satisﬁes (15.29) follows by direct calcula-

tion. 

To obtain a solution with initial conditions ξ(0) = η and Dξ(0) = a(η), in

the above construction one should take a

in the form b(0)

−1

(a(η)). 

15.3 Relativistic Stochastic Mechanics

15.3.1 Stochastic mechanics in Minkowski space

In this section we present a modiﬁcation of stochastic mechanics which is

well-posed for describing relativistic particles in Minkowski space (see Exam-

ple 13.4). In particular we obtain generalizations of the existence theorems

of Section 15.1 to the relativistic case. In our constructions, following [103],

we apply the relativistic deﬁnition of mean derivatives suggested by Guerra

and Ruggiero [55, 142] and Zastawniak’s idea of transition from stochastic

processes in Minkowski space to those in the underlying Euclidean space (see,

e.g., [237, 238]). The material of this section forms the basis for the construc-

tion of stochastic mechanics in the general relativistic case in Section 15.3.2

below (see [130]).

For the sake of simplicity, in this section we work in a system of units in

which the speed of light c = 1. We also consider particles with rest mass 1.

In this subsection we denote by M

the Minkowski space with inner prod-

uct (·, ·)(seeExample13.4). Choose a certain orthonormal frame (with re-

spect to (·, ·)) in M

.LetR

be the (underlying) Euclidean space in which

the above frame is orthonormal in the Euclidean sense. The (positive-deﬁnite)

inner product in R

will be denoted by a dot ·. The main idea here is to con-

sider Itˆo processes ξ(τ )oftheform(8.15) as processes in M

while Wiener

processes are deﬁned with respect to R

. This idea originated in the work

of Zastawniak (see [237]). Here τ is an invariant parameter which plays the

role of proper time. For such ξ(τ),accordingtoanideaofGuerraandRug-

giero [55, 142], we deﬁne the relativistic forward mean derivative D

ξ(τ)and

relativistic backward mean derivative D

−

ξ(τ) as follows:

ξ(τ)

= lim

Δτ↓0



ξ(τ + Δτ) − ξ(τ )

Δτ



, (ξ(τ + Δτ ) − ξ(τ))

≤ 0



(15.36)

+ lim

Δτ↓0



ξ(τ) −ξ(τ − Δτ)

Δτ



, (ξ(τ ) − ξ(τ − Δτ))

≥ 0



;

15.3 Relativistic Stochastic Mechanics 377

−

ξ(τ)

= lim

Δτ↓0



ξ(τ) −ξ(τ − Δτ)

Δτ



, (ξ(τ ) − ξ(τ − Δτ))

≤ 0



(15.37)

+ lim

Δτ↓0



ξ(τ + Δτ) − ξ(τ )

Δτ



, (ξ(τ + Δτ ) − ξ(τ))

≥ 0



It should be noted that (15.36)and(15.37) are covariant under the Lorentz

transformations. Below we will deal with both D

and D

−

on the one hand

and D, D

∗

and D

on the other hand, applied to a process ξ(τ )whereD,

∗

and D

will be calculated in R

by formulae (8.1), (8.2)and(8.13),

respectively.

Of course, D

ξ(τ)=Y

(τ,ξ(τ)) for the vector ﬁeld

(τ,x)

= lim

Δτ↓0



ξ(τ + Δτ) − ξ(τ )

Δτ



ξ(τ)=x, (ξ(τ + Δτ ) −ξ(τ))

≤ 0



+ lim

Δτ↓0



ξ(τ) −ξ(τ − Δτ)

Δτ



ξ(τ)=x, (ξ(τ ) −ξ(τ − Δτ))

≥ 0



and D

−

ξ(τ)=Y

−

(τ,ξ(τ)) for the vector ﬁeld

−

(τ,x)

= lim

Δτ↓0



ξ(τ) − ξ(τ − Δτ)

Δτ



ξ(τ)=x, (ξ(τ ) −ξ(τ − Δτ))

≤ 0



+ lim

Δτ↓0



ξ(τ + Δτ) − ξ(τ )

Δτ



ξ(τ)=x, (ξ(τ + Δτ ) −ξ(τ))

≥ 0



Consider the coordinate decomposition of some process ζ(τ ) with re-

spect to the frame mentioned above: ζ(τ)=(ζ

(τ),

ζ(τ )) where

ζ(τ )=

(ζ

(τ),ζ

(τ)). Then clearly

ζ(τ )=(D

(τ),D

∗

ζ(τ )) (15.38)

and

−

ζ(τ )=(D

∗

(τ),D

ζ(τ )), (15.39)

where Dζ and D

∗

ζ are deﬁned in R

by (8.7)and(8.8), respectively.

Let us introduce the relativistic current velocity ¯v

(τ)=

ζ(τ )+

−

ζ(τ )) and the relativistic osmotic velocity ¯u

(τ)=

ζ(τ ) −D

−

ζ(τ )).

We have ¯v

(τ)=¯v

(τ,ζ(τ)) where ¯v

(τ,x)=

(τ,x)+Y

−

(τ,x)) and

¯u

(τ)=¯u

(τ,ζ(τ)) where ¯u

(τ,x)=

(τ,x) − Y

−

(τ,x)).

Lemma 15.33 The relativistic current velocity ¯v(τ) and the usual current

velocity v(τ), calculated in R

by Deﬁnition 8.16,coincide.

To prove that ¯v

(τ)=v

(τ), compare formulae (15.38)and(15.39).

378 15 The Newton-Nelson Equation

Lemma 15.34 The relativistic osmotic velocity takes the form

¯u

(τ)=



(τ), −D

ζ(τ )



while the usual osmotic velocity, calculated in R

by Deﬁnition 8.16, takes

the form

(τ)=



(τ),D

ζ(τ )



and so they do not coincide.

Lemma 15.34 follows immediately from (15.38)and(15.39).

Lemma 15.35

¯u

(τ,x)=−

Grad log ρ

(τ,x) (15.40)

where ρ

(τ,x) is the density of ζ(τ) with respect to Lebesgue measure, deﬁned

in Section 8.1,andGrad is the gradient calculated with respect to the inner

product (·, ·) in M

Lemma 15.35 follows from the above coordinate decomposition for ¯u

(τ),

formula (8.18) and the deﬁnition of the inner product (·, ·)inM

(note in

particular, that Grad f =(−

∂f

∂x

∂f

∂x

∂f

∂x

∂f

∂x

) while grad f =(

∂f

∂x

∂f

∂x

∂f

∂x

∂f

∂x

) for any function f on M

Lemma 15.36 Formula (8.19) remains true for ¯v

and ρ

Indeed, this follows from the fact that ¯v

(τ)=v

(τ)(seeabove).

For a vector ﬁeld Z(τ,x)onM

deﬁne the relativistic forward D

Z(τ,ζ(τ))

and backward D

−

Z(τ,ζ(τ)) mean derivatives along ζ(τ) by the formulae

Z(τ,ζ(τ)) = (15.41)

lim

Δτ↓0



Z(τ + Δτ, ζ(τ + Δτ)) − Z(τ,ζ(τ))

Δτ



, (ζ(τ + Δτ ) − ζ(τ))

≤ 0



lim

Δτ↓0



Z(τ,ζ(τ)) − Z(τ −Δτ, ζ(τ − Δτ))

Δτ



, (ζ(τ ) − ζ(τ − Δτ))

≥ 0



and

−

Z(τ,ζ(τ)) = (15.42)

lim

Δτ↓0



Z(τ,ζ(τ)) − Z(τ −Δτ, ζ(τ − Δτ))

Δτ



, (ζ(τ ) − ζ(τ − Δτ))

≤ 0



lim

Δτ↓0



Z(τ + Δτ, ζ(τ + Δτ)) − Z(τ,ζ(τ))

Δτ



, (ζ(τ + Δτ ) − ζ(τ))

≥ 0



15.3 Relativistic Stochastic Mechanics 379

(a modiﬁcation of (8.22)and(8.23), respectively, by analogy with (7.46)and

(7.47)).Takingintoaccount(8.24)and(8.25), one can easily derive

Z(τ,ζ(τ)) =

∂

∂τ

Z +(Y

, ∇)Z +

2Z,

(τ,ξ(τ)) =

∂

∂τ

Z +(Y

−

, ∇)Z −

2Z, (15.43)

where 2 = −

∂

∂x

∂

∂x

∂

∂x

∂

∂x

is the wave operator (the d’Alembertian,

cf. (8.24)and(8.25)).

Deﬁnition 15.37. The vector

−

)ξ(τ) is called the 4-acceler-

ation of the process ξ at τ.

Below we use the ordinary quadratic mean derivative D

introduced in Def-

inition 8.10. The introduction of a special relativistic quadratic mean deriva-

tive is not necessary in the problems under consideration for the following

reasons.

Introduce “relativistic” increments of the process ζ(τ) as follows:

ζ(τ )=(ζ

(τ + Δτ) − ζ

(τ),

ζ(τ ) −

ζ(τ − Δτ))

and

−

ξ(τ)=(ζ

(τ) − ζ

(τ − Δτ),

ζ(τ + Δτ) −

ζ(τ ))

which are related to ordinary increments in the same way as relativistic for-

ward and backward mean derivatives are related to ordinary forward and

backward mean derivatives.

Lemma 15.38 Let ζ(τ ) be an Itˆo diﬀusion type process on M

whose dif-

fusion term is given by a Borel measurable ﬁeld of linear operators A(x):

→ M

that is independent of the past and so depends only on x ∈ M

Denote by D

the ordinary quadratic mean derivative as in Deﬁnition 8.10.

Then

ζ(τ ) = lim

Δτ→+0

ζ(τ ) ⊗Δ

ζ(τ )

Δτ

= lim

Δτ→+0

−

ζ(τ ) ⊗Δ

−

ζ(τ )

Δτ

= A(ζ(τ))A

∗

(ζ(τ )). (15.44)

The proof of Lemma 15.38 is based on the fact that in the case under

consideration the diﬀusion term of ζ(τ ) and of the process with inverse time

direction coincide. Note that in (15.44) we do not use conditional expectation

since A(x) is Borel measurable and does not depend on the past.

We should recall (see Remark 13.20) that in relativistic dynamics the 4-

force necessarily depends on the 4-velocity. In addition, by deﬁnition it is

independent of the proper time τ since it is an absolute object (independent of

a reference frame: recall that proper time is well-deﬁned only in the reference

380 15 The Newton-Nelson Equation

frame of an observer). This is why, below, we consider a force ﬁeld on M

only

in the form ¯α =¯α(x)X where ¯α(x)isa(1, 1)-tensor, i.e., a linear operator

in M

, depending on x ∈ M

(an analog of ¯α

from Section 15.1.2).

Deﬁnition 15.39. (cf. Deﬁnition 15.2)Therelativistic Newton-Nelson equa-

tion with 4-force ¯α(x)X is the system in the Minkowski space M

of the form



−

+ D

−

)ξ(t)=¯α(ξ(τ))v

(τ)

ξ(t)=



(15.45)

where m is the mass (rest mass) of a particle.

We shall look for solutions of (15.45) among all Itˆo processes. From (15.45)

one can easily see that the stochastic-mechanical world line of a relativistic

particle with rest mass m is an Itˆo process ξ(τ )inR

of the form (8.15) with



such that it satisﬁes the ﬁrst equation of (15.45) as a process in

Recall that everywhere in this Subsection we assume m = 1 without loss

of generality.

Theorem 15.40 For ξ(τ) as above the following equality holds

(DD

∗

+ D

∗

D)ξ(τ)=

−

+ D

−

)ξ(τ). (15.46)

Proof. This statement is proved by straightforward calculation. Indeed, by

formulae (15.38), (15.39), (15.41)and(15.42)wehave

−

ξ(τ)=



∗

(τ),D

∗

ξ(τ)



and

−

ξ(τ)=



∗

(τ),D

∗

ξ(τ)



which leads to (15.46). 

Let the force ﬁeld ¯α(x)X on M

satisfy Condition 15.7 (with ¯α

(τ,x)=0

and ¯α

=¯α) relative to the Euclidean metric of R

. Then for any initial

forward derivative a

∈ M

and for the function t

(τ) deﬁned by formula

(13.13), we can apply the construction of a(τ )andθ of Section 15.1.2 and so

obtain the corresponding Itˆo process ξ(τ)inR

The following statement is a simple corollary of Theorems 15.9 and 15.40.

Theorem 15.41 For τ ∈ (t

,l) the process ξ(τ ) in M

, mentioned above,

satisﬁes (15.45) and so it is a stochastic-mechanical world line.

Let the tensor ﬁeld α(x) be a closed 2-form, i.e., α =dω where ω is a

1-form on M

(a particular case is an electromagnetic ﬁeld F =dA,see

Section 13.1.5). The trajectory of the stochastic-mechanical system with 4-

force ¯αX on M

, where as usual ¯α is physically equivalent to α with respect

15.3 Relativistic Stochastic Mechanics 381

to (·, ·), is connected with the solution of the corresponding Klein-Gordon

equation, so there is a link between relativistic stochastic mechanics and

relativistic quantum mechanics. This is a generalization of the construction of

Section 15.1.1 for the non-relativistic case. We describe it brieﬂy, following the

scheme of Guerra and Ruggiero [55, 142] developed for the electromagnetic

ﬁeld. In order to avoid confusion, in the case of an electromagnetic ﬁeld we

assume (without loss of generality) that the electric charge e of the particle

is equal to 1.

Let ξ(τ ) be a stochastic-mechanical world line. Let us make the additional

assumption that there exists a function S such that σ

Grad S =¯v

+¯ω

where ¯ω is the vector physically equivalent to ω with respect to (·, ·)and

Grad is deﬁned in Lemma 15.35. Introduce the complex-valued function Ψ =

exp(R+iS)whereR =

log ρ

. Then, taking into account Lemmas 15.35 and

15.36, imitating the proof of Theorem 15.4 one can deduce that Ψ satisﬁes

the equality

i h

∂Ψ

∂τ



∇−



¯ω,∇−



¯ω



Ψ (15.47)

where in the right hand side we have the formal inner product (with respect

to (·, ·)) of diﬀerential operators, considered as formal vectors (cf. Lemma

8.20 where the deﬁnition of ∇ is also given). Note that (∇, ∇)=2.

Suppose that Ψ(τ, x)=exp(−i

2

τ)ϕ(x). Then from (15.47) it follows that

ϕ(x) satisﬁes the equality



∇−



¯ω,∇−



¯ω



−



ϕ =0. (15.48)

Formula (15.48) is the Klein-Gordon equation. In particular, if ω =0,(15.48)

takes the form



2 −



ϕ =0. (15.49)

The inverse construction is more straightforward. We describe it brieﬂy

according to [237, 238] where it was derived for the electromagnetic ﬁeld.

Let a complex-valued function ϕ on M

satisfy (15.48). Represent it in the

form ϕ(x)=exp(R +iS) and consider the vector ﬁelds ¯v(x)=σ

GradS − ¯ω

and ¯u(x)=σ

Grad R =

(Grad log ρ)=

Grad ρ

on M

for ρ = ϕ ¯ϕ.

Theorem 15.42 Equation (15.48) is equivalent to the following system:

div ¯u +(¯u, ¯u) − (¯v, ¯v)=1, (15.50)

div ¯v +2(¯u, ¯v)=0. (15.51)

To prove Theorem 15.42 one should substitute ϕ =exp(R +iS)into

(15.48) and then, after natural transformations similar to those in the proof

of Theorem 15.4, separate real and imaginary parts.

382 15 The Newton-Nelson Equation

The vector ﬁelds ¯v and ¯u generate the vector ﬁelds Y

(x)=¯v(x)+¯u(x)=

Grad S +GradR +¯ω and Y

−

(x)=¯v(x) − ¯u(x)=GradS − Grad R +¯ω.

Having made the coordinate decompositions Y +(x)=(Y

)andY

−

(x)=

−

), introduce the vector ﬁelds Y (x)=(Y

−

)andY

∗

(x)=(Y

−

Consider the diﬀusion process ξ(τ )onR

satisfying the Itˆo equation

dξ(τ)=Y (ξ(τ))dτ + σ dw(τ ) (15.52)

where σ>0isasabove.

Lemma 15.43 For ξ(τ) from (15.52) D

ξ(τ)=Y

(ξ(τ)), D

−

ξ(τ)=

−

(ξ(τ)) and so ¯v

=¯v and ¯u

=¯u.

The assertion of Lemma 15.43 is a trivial consequence of the construction.

Theorem 15.44 ξ(τ) from (15.52) satisﬁes the Newton-Nelson equation

(15.45) with ¯αX physically equivalent to dω(·,X) with respect to (·, ·).

Proof. By straightforward calculations of D

−

ξ(τ)andD

−

ξ(τ)oneob-

tains the formula

−

+ D

−

)ξ(τ)=−

∇(div ¯u +(¯u, ¯u) − (¯v, ¯v)) + ¯αv (15.53)

where ¯α is physically equivalent to dω(·,X) with respect to (·, ·). From (15.50)

it follows that the ﬁrst summand on the right-hand side of (15.53) vanishes

so that (15.53)isequalto(15.45) with the above ¯α. 

Theorem 15.45 Let the Klein-Gordon equation (15.48) be well-deﬁned for a

given 4-force ¯α and let ξ(τ ) be a solution of the corresponding Newton-Nelson

equation (15.45).Then

E((v

)

+(u

)

)=−1. (15.54)

Proof. Clearly (v

)

+(u

)

=(−div u

+(v

)

−(u

)

)+( div u

+2(u

)

Direct calculations show that ( div u

+2(u

)

∇

where ∇

= ∇·∇

and so



div u

+2(u

)





∇

dλ =



∇

dλ.

Note that ξ(τ), as a solution of (15.52), is a diﬀusion process in R

.Thusthe

Kolmogorov-Fokker-Planck equation

∂

∂τ

∇

− div(ρ

Y )

is valid for ρ

where Y is as in (15.52). By construction ρ

is independent of τ.

Hence E( div u

+2(u

)

)=σ



div(ρ

Y )dλ where the latter integral is

15.3 Relativistic Stochastic Mechanics 383

equal to zero by a standard application of the divergence formula. So (15.54)

follows from (15.50). 

Remark 15.46. Equality (15.54) may be considered as the characteristic

feature for τ to be a proper time along a stochastic-mechanical world-line

ξ(τ) (cf. Deﬁnition 13.12). This idea, as well as the proof of (15.54), was

apparently ﬁrst suggested by Zastawniak [237].

15.3.2 Stochastic mechanics in the space-times of

general relativity

In this subsection M

is a 4-dimensional Lorentz manifold with metric (·, ·)

whose signature is (−, +, +, +) (see Section 13.1.1). For the sake of simplicity

we assume from now on that M

is orientable and oriented in time. In other

words, a well-deﬁned ‘future’ time direction is speciﬁed in every tangent space

, m ∈ M

Consider the principal bundle L(M

) with structural group L

−

, the proper

orthochronous Lorentz group (see [72]). The action of L

−

on Minkowski space

preserves the standard and time orientation. The bundle L(M

) is a sub-

bundle of the principal bundle of Lorentz-orthonormal frames. Denote by H

the restriction of the Levi-Civit´a connection to L(M

)andbyV the vertical

distribution on L(M

). As on OM in Section 2.7, the bundles H and V over

L(M

) are trivial. In particular H is trivialized by the basic vector ﬁelds on

L(M

The generalization of Itˆo processes to the Lorentz manifold M

requires

some modiﬁcation with respect to the case of Minkowski space. Choose a

point m

∈ M

and a Lorentz orthonormal frame b in T

. Introduce a

Euclidean structure in T

by setting b to be orthonormal in the Euclidean

sense. We may now consider a Wiener process w(τ)inT

as well as Itˆo

processes with this w(τ). One can easily show that the entire construction of

Itˆo developments on manifolds can be clearly generalized to the above case

of processes on the Lorentz manifold M

by using connections on L(M

)

instead of on OM. Those developments will be called Itˆo processes on the

Lorentz manifold M

The above parameter τ will play the role of proper time. So for a process

ξ(τ) we may expect that (15.54) is fulﬁlled.

In order to avoid any possible confusion we assume in this section that

is stochastically complete. This means that the development of a Wiener

process in the above-mentioned sense exists for τ ∈ [0, ∞) (see Section 7.6.2).

Unfortunately nothing like the criterion of stochastic completeness of Theo-

rem 7.80 is known for Lorentz manifolds.

Itˆo processes on M

, which are the developments of Itˆo processes of diﬀu-

sion type of the form z =



a(s)ds + σw(τ) with σ =





, are of particular

384 15 The Newton-Nelson Equation

interest to us. Let ξ(τ) be such a process in M

.Inachart(U, ϕ)wecan

apply formulae (15.36)and(15.37) to deﬁne relativistic forward and back-

ward, respectively, mean derivatives D

ξ(τ)andD

−

ξ(τ) and represent them

as compositions of

(τ,m)and

−

(τ,m) (determined in the chart (U, ϕ))

with ξ(τ ) (cf. Section 9.1). As in Section 9.1 we determine X

(τ,m)and

−

(τ,m) by formulae X

(τ,m)=Y

(τ,m)

and X

−

(τ,m)=Y

∗

(τ,m)

respectively, for any m ∈ M

where n denotes the calculations in the nor-

mal chart at m. Then we deﬁne the relativistic forward and backward mean

derivatives of ξ on M

with respect to H by the formulae

ξ(τ)=X

(τ,ξ(τ)),D

−

ξ(τ)=X

−

(τ,ξ(τ)) (15.55)

(cf. Deﬁnition 9.1).

The derivative

+ D

−

) is called the relativistic symmetric

mean derivative. Consider the vector ¯v

(τ,m)=

(τ,m)+X

−

(τ,m));

¯v

(τ,ξ(τ)) =

ξ(τ) is called the relativistic current 4-velocity of the pro-

cess ξ(τ);

− D

−

) is called the relativistic antisymmetric mean

derivative. Consider for ξ(τ) as above the vector ¯u

(τ,m)=

(τ,m) −

−

(τ,m)); ¯u

(τ,ξ(τ)) =

ξ(τ) is called the relativistic osmotic velocity of

ξ(τ).

Let Y (τ,m)beaC

-smooth vector ﬁeld on M

. Making the same modiﬁca-

tion for formulae (9.15) as above in this section, we can deﬁne the covariant

relativistic mean derivatives D

Y (τ,ξ(τ)) and D

−

Y (τ,ξ(τ)). Here we use

parallel translation with respect to the Levi-Civit´a connection on L(M

We can also apply the Levi-Civit´a connection on L(M

) (i.e., its normal

charts on M

, parallel translation of vectors in TM

, etc.) to deﬁne Dξ(τ),

∗

ξ(τ), D

η(τ), D

∗

η(τ), DY (τ,ξ(τ)) and D

∗

Y (τ,ξ(τ)) as in Chapters 8

and 9. In spite of the fact that these objects diﬀer from those in the above

sections, since the connections are diﬀerent, we use the old notation without

any ambiguity.

Let us pick a Lorentz orthonormal frame in T

at some m and represent

a vector in terms of its coordinates with respect to the frame X =(X

where X

denotes the ‘time-like’ component and

X the 3-dimensional ‘space-

like’ component. Note that this coordinate decomposition is covariant with

respect to Lorentz transformations in T

for D

ξ(τ)andD

−

ξ(τ) and not

covariant for Dξ(τ )andD

∗

ξ(τ). Nevertheless v

(τ) is covariant, since one can

can easily see that v

(τ)=¯v

(τ). Indeed, direct calculations show that both

(τ)and¯v

(τ) have the same coordinate decomposition: (D

(τ),D

ξ(τ ))

(cf. Section 15.3.1).

As in Section 15.3.1, for the osmotic velocity ¯u

(τ)wehave

¯u

(τ)=



(τ), −D

ξ(τ )



while u

(τ)=(D

(τ),D

ξ(τ )). The following formula holds:

15.3 Relativistic Stochastic Mechanics 385

¯u

(τ,x)=

Grad log ρ

(τ,x), (15.56)

where Grad is the gradient calculated with respect to the Lorentz metric.

Deﬁnition 15.47. The vector

−

)ξ(τ) is called the 4-acceler-

ation of the process ξ at τ.

Theorem 15.48 For an Itˆoprocessξ(τ) in M

as above the following equal-

ity holds

(DD

∗

+ D

∗

D)ξ(τ)=

−

+ D

−

)ξ(τ). (15.57)

As for formula (15.46), this statement is proved by direct calculation.

Indeed,

−

ξ(τ)=(D

∗

(τ), D

∗

ξ(τ ))

and also

−

ξ(τ)=(D

∗

(τ), D

∗

ξ(τ )),

which yields (15.57).

Here we consider the relativistic Newton-Nelson equation of the form



−

+ D

−

)ξ(τ)=¯α(ξ(τ), ¯v

(τ,ξ(τ))),

ξ(τ)=



(15.58)

where ¯α is a 4-force, i.e., it does not depend on τ and necessarily depends on

4-velocity. This is why we consider below the force ﬁeld on M

in the form

¯α(m, X)=¯α

(m)X,where¯α

(x)isalinearoperatorinT

,asabove.

There is no



Ric term in (15.58) (for the same reasons that such a term

appeared in (15.29); see Section 15.2.1). In the relativistic case a natural

relation between the Newton-Nelson equation and the Klein-Gordon equation

is established (for Minkowski space this was illustrated in Section 15.3.1,

see also, e.g., [55, 142, 189, 190, 228, 237, 238]). But in the Klein-Gordon

equation (unlike the Schr¨odinger equation on a Riemannian manifold) the

wave operator 2 = ∇

∗

·∇is involved, where ∇ is the covariant derivative of

the Levi-Civit´a connection on M

(see, e.g., [49]). Since we need not obtain

Δ =dδ + δd in the corresponding equation of Schr¨odinger type, here we do

not include the



Ric term in (15.58) (cf. Section 15.2.1).

Deﬁnition 15.49. An Itˆo process ξ(τ )onM

of the type mentioned above

is called a trajectory of a relativistic stochastic-mechanical system with force

ﬁeld ¯α(m, X) if it satisﬁes (15.58).

Let M

be stochastically complete. Specify a point m

∈ M

and con-

sider the Itˆo development R

W (τ) of a Wiener process W (τ)inT

described above in this section. Choose a Lorentz-orthonormal frame b in