We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics October 1980, Volume 73, Issue 3, pp 247–264. Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements. I will appreciate your comments.

1. Stochastic Approach to Construction of the Schrödinger Equation Solution and its Applications to
Demolition Measurements.
Ilya Gikhman
6077 Ivy Woods Court
Mason, OH 45040 USA
Ph. (513)-573-9348
Email: ilyagikhman@mail.ru
This paper represents §4 of the 6 chapter of the book
STOCHASTIC DIFFERENTIAL EQUATIONS AND ITS APPPLICATIONS: Stochastic analysis of the
dynamic systems.
 Paperback: 252 pages
 Publisher: LAP LAMBERT Academic Publishing (July 13, 2011)
 Language: English
 ISBN-10: 3845407913
 ISBN-13: 978-3845407913
The paper is presented in http://www.slideshare.net/list2do/stochastic-schrdinger-equations
The original idea to use complex coordinate space for interpretation of the quantum mechanics was first
represented by Doss, Sur une resolution stochastique de' l'equation de Schrödinger a coefficients
analytiques. Communications Mathematical Physics 73, 247-264, (1980).
Haba, J. Math. Phys.35:2 6344-6359, 1994, J. Math. Phys.39:4 1766-1787, 1998 extended Doss approach
for more general quantum mechanics problems.
Il. I. Gikhman, Probabilistic representation of quantum evolutions. Ukrainian Mathematical Journal,
volume 44, #10, 1992, 1314-1319 (Translation. Probabilistic representation of quantum evolution.
Ukrainian Mathematical Journal, Springer New York, Volume 44, Number 10, October, 1992, 1203-

2. 1208), A quantum particle under the forces of “white noise” type. Ukrainian Mathematical Journal,
Volume 45, #7, 1993, 907-914. (Translation. Ukrainian Mathematical Journal, Springer New York,
Volume 45, Number 7 / July, 1993, 1004-1011. I unfortunately did not know Doss paper at that time and
did not make reference on his original paper.
Relevant mathematical problems were discussed in: S.Albeverio, V. Kolokol'tsov, O. Smolyanov, C.R.
Acad. Sci. Paris, t. 323, Ser. 1, ( 1996 ), 661-664; V. Kolokoltsov, Lecture Notes in Mathematics v.1724,
(2000); I. Davis, O.Smolyanov, A. Truman, Representation of the solutions to Schrödinger Stochastic
Equations on compact Riemannian manifolds, Doklady Mathematics, v.62, No1, 2000, 4 – 7.
Bearing in mind a formal construction of the quantum trajectories one could study the problem related to
the quantum non-demolition continuous measurements. The correspondent problems was actively studied
in physics [M.B. Mensky, Decoherence and the theory of continuous measurements. Physics-Uspekhi 41
(9) 923-940 (1998) , 1998 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences].
Introduction.
In this paper we represent a class SPDE. It is stochastic Schrödinger equations. Two peculiarities
distinguish Schrödinger equation. The Schrödinger equation is linear with complex coefficients. The
linearity of the equation simplifies standard analytic methods are used but on the other hand, the complex
coefficients present mathematical difficulties which did not appear with the real valued partial differential
equations. We first introduce a stochastic representation of the Schrödinger equation solution. We will
show that ‘quantum’ particle associated with the Schrödinger equation represents a browning motion
in a complex coordinate space. The formula, which we present here for the solution differs from the
famous Feynman formula. It is clear that two different formulas for the wave equation should be equal.
We present next that the proof of equivalence of the complex space probabilistic representation and
Feynman formula. The last result of this section we show how choosing nonstandard type of Lagrangian
for the particle in the potential field to arrive at the probabilistic density of the particle. It looks as a
significant advantage with respect to the standard Lagrangian approach, which is used to use in quantum
mechanics.
On the first step, we recall the Feynman representation. The Lagrangian for a system of N particles with
masses m j , j =1 , 2 , .... N moving in a potential field V ( t, x )is
)t,
td
xd
,x(L = 
N
1j2
1
m j (
td
xd j
) 2
– V ( x , t ) (6.4.1)
Here x = ( x ( 1 )
, x ( 2 )
, x ( 3 )
) is appoint in 3-dimensional Euclidean space. The function x j ( t ) represents
a path of the j-th particle and
td
xd j
is its velocity along the path.

3. Let ψ ( x , s ) denote a probability amplitude of finding the particle at the location x at the time s
. The function K ( x , s | y , t ) represents the kernel of the transformation of the wave function ψ over a
period [ t , s ]. In other words K is the transition of amplitude for a quantum particle emitted at ( x , s )
and then be detected at ( y , t ). Formally, this definition could be written in the form
ψ ( y , s ) = ∫ K ( x , s | y , t ) ψ ( x , t ) d x (6.4.2)
This equation (6.4.2) is equivalent to the Schrödinger equation. The problem is to find analytical
formula for the kernel. Following a Dirac idea, Feynman suggested to represent the kernel as
K ( x , s | y , t ) =  


t
s
y)t(x
x)s(x
exp)(xd
h
i
L ( x ( r ) ,
rd
)r(xd
, r ) d r (6.4.3)
where the right hand side of (6.4.3) is interpreted as following
0ε
lim

 …  exp
h
i
S
A
xd
....
A
xd 1-N1
, A = 2
1
)
m
επ2
(
hi
(6.4.4)
and S = 
t
s
L ( x ( r ) ,
rd
)r(xd
, r ) d r is the functional of action. In Schrödinger quantum mechanics
the complex-valued ψ ( x , t ) satisfies the Schrödinger equation
i h 2
2
j
N
1j x
ψ
m2t
ψ






h
+ V ( x , t )  (6.4.5)
and | ψ ( x , t ) | 2
is the probability density of the presence of the particles at the points

4. x = ( x 1 , x 2 , … , x N ) at time t. First, let us establish the probabilistic representation of the solution of
the Schrödinger equation (6.4.5) by using diffusion in complex space. Putting
z = x + i y , i = ; x , y  E 3
define a random process
 j ( s ; t , z j ) = z j + 2
1
j
)
m
(
hi
[ w j ( s ) – w j ( t ) ] =
(6.4.6)
= x j + 2
1
j
)
m2
(
h
[ w j ( s ) – w j ( t ) ] + i { y j + 2
1
j
)
m2
(
h
[ w j ( s ) – w j ( t ) ] }
where w j ( t ) are mutually independent Wiener processes. From the formula (6.4.6) it follows that for
given point z j = xj + i y j and t > 0 the coordinate space of the random process
 j ( s ; t , z j ) is a linear manifold in the complex 3-dimensinal space. This linear manifold is the set of
direct lines in each coordinate plane x ( k )
, y ( k )
going through the point
x )k(
j , y )k(
j and having slope
4
π
, k = 1, 2, 3 ; j = 1, 2, … , N. It follows from (6.4.6) that each of the
functions  j ( s ; t , z j ) is analytic with probability 1 with respect to the complex variable z j . We can call
the complex process  j ( s ; t , z j ) by characteristics of the Schrödinger equation (6.4.5). On the other
hand the random function  j ( s ; t , z j ) represent the quantum particle movement. The values of the
wave function ψ give us the complete information about the quantum particle system. Knowledge of
the function ψ in quantum mechanics is similar to the knowledge of the trajectory in classical
mechanics.
Theorem 6.4.1. Assume that nonrandom vector and scalar functions Ψ0 ( z ) , V ( t , z ) defined for ( t , z
)  [ 0 , + ∞ )  Z3 N
= Z are analytic with respect to z and continuous in t . Then the function
Ψ ( z , t ) = E Ψ0 (  ( 0 ; t , z )) exp V (  ( s ; t , z ) , s ) d s (6.4.7)
1
N3


t
h
i
0

5. is a classical solution of the Cauchy problem
)t,z(
m2t
)t,z( 2
j
j
N
1j




hi
–
h
i
V ( x , t )  ( z , t )
(6.4.8)
 ( z , 0 ) =  0 ( z )
Recall that complex-valued function Ψ ( z , t ) is said to be analytic on an open area if it has derivative at
every point of the area. One possible way to justify analyticity is to test that the function does not
depend on z = x – i y . Other way to prove analyticity of the function is to verify the Cauchy-Riemann
equations. If the first partial derivatives of the real and imaginary parts of the complex function are
continuous at a point and satisfy the Cauchy-Riemann equations then the function is analytic at this
point. For illustrative simplicity consider a simple example to justify analyticity. Putting N = 1 , V = 0
and setting
Ψ ( z , t ) = P ( x , y , t ) + i Q ( x , y , t )
t  0 we see that
P ( x , y , t ) =
= E P ( x + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , y + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , 0 )
Q ( x , y , t ) =
= E Q ( x + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , y + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , 0 )

6. One can easy to see that first order partial derivatives of the functions P ( x , y , t ) , Q ( x , y , t ) , t > 0
are equal to the correspondent derivatives of these functions at t = 0. Hence, Cauchy-Riemann
equations
x
)t,y,x(P


=
y
 )t,y,x(Q
,
y
)t,y,x(P


= –
x
)t,y,x(Q


are valid. Applying Cauchy-Riemann equations one can easy justify that
m4
h
(
x

+
y

) 2
P ( x , y , t ) = –
m2
h
2
2
x
)t,y,x(Q


m4
h
(
x

+
y

) 2
Q ( x , y , t ) = –
m2
h
2
2
x
)t,y,x(P


Hence,
t

[ P + i Q ] =
m2
hi
[ P + i Q ]
Putting z = x + i 0 in (6.4.7) we verify that ψ ( x , t ) = Ψ ( x + i 0 , t ) is a solution of the Schrödinger
equation with initial condition ψ ( x , 0 ) = Ψ0 ( x ). Thus, Schrödinger equation can be interpreted as
the complex trace of backward Kolmogorov equation on the real subspace of the complex-valued
quantum diffusion. Recall that the wave function representing solution of the Schrödinger equation
admits the probabilistic interpretation. The feasibility of this interpretation stems from normalization
condition

7.  | ψ ( x , t ) | 2
d x = 1
taking place for each t  0. The initial condition of the Schrödinger equation is a complex-valued function
ψ0 ( x ), x  ( – ∞ , + ∞ ) sa sfying normaliza on condi on. On the other hand recall that the function Ψ
0 ( z ) in (6.4.8) was assumed analytic in the complex space. Next is a formal result that states a
possibility of a continuation of a complex-valued function defined on the real space onto complex
extension.
Theorem ( Polya, Plancherel ). In order to a complex-valued function φ ( z ) be an integer function of the
exponential type and  | φ ( x + i 0 ) | 2
dx <  , it is necessary and sufficient that following
representation takes place
 ( z ) = ( 2  ) 2
n

n
E
 ( q ) exp – i ( q , z ) d q
where  is a function from the Lebesgue space L2 ( E n
) having a compact support.
Introduce inverse time Wiener process  ( s ; t , x ) starting at the moment t from the point x
 ( s ; t , x ) = x + [ w ( s ) – w ( t ) ]
Then the complex process  describing quantum particle evolution admits representation
 ( s ; t , x ) = x + (
m2
hi
) 2
1
[  ( s ; t , x ) – x ]
Using inverse time Markov property of the process  the formula (6.4.7) can be representated as
 ( x , t ) = V ( x , t )  ×
h
i
explim
0

8. × 


exp V ( x + (
m2
hi
) 2
1
( x N – 1 – x )  , t N – 1 ) p ( t , x ; t N – 1 , x N – 1 ) d x N – 1 ×
(6.4.9)
× 


exp V ( x + (
m2
hi
) 2
1
( x 1 – x )  , t 1 ) p ( t2 , x 2 ; t 1 , x 1 ) d x 1 ×
× 


 0 ( x + (
m2
hi
) 2
1
( x 0 – x )  ) p ( t 1 , x 1 ; t 0 , x 0 ) d x 0
where  = t j + 1 – t j , j = 0 , 1 , … , N – 1 , s = t0 < t1 < …< tN = t and
p ( t , x ; s , y) = [ 2 π ( t – s ) ] exp –
)st(2
)yx( 2


Remarkably, that the wave function (6.4.9) does not coincide with Feynman path integral
(6.4.2-4). Next theorem states the equivalence of these two representations. Using this theorem, we
could see that non-relativistic quantum mechanics can be interpreted as the real-world trace of the
complex-valued dynamics. We make some technical assumptions that technically will simplify the proof.
Theorem 6.2.2. Assume that V = 0 , N = 1 and let the dimension of the coordinate space is 1. Then
representations (6.4.2-4) and (6.4.9) are equivalent.
Proof. Taking into account conditions the formula (6.4.9) could be rewritten
 ( x , t ) = E  0 (  ( s ; t , x ) ) =
(6.4.10)
= [ 2 π ( t – s ) ]   0 ( x + (
m2
hi
) 2
1
λ ) exp –
)st(2
λ 2

d λ
h
i
h
i
2
3 N

2
1


9. Introdice Fourier transforms
 ( y ) = ( 2 π ) – 1



e – i x y
 0 ( x ) d x
 0 ( x ) = 


 ( y ) e i x y
d y
and suppose that the function  ( y ) has a compact support. Then changing the order of the integration
in (6.4.10) we note that
 ( x , t ) = [ 2 π ( t – s ) ] 2
1




exp –
)st(2
λ 2

d λ 


 ( y ) ×
× exp { [ x + (
m2
hi
) 2
1
λ ) ] y } d y = [ 2 π ( t – s ) ] 2
1




 ( y ) d y ×
× 


exp { –
)st(2
λ 2

+ i [ x – (
m2
hi
) 2
1
λ ] y } d λ = [ 2 π ( t – s ) ] 2
1

×
× 


 ( y ) ei x y
d y 


exp { –
)st(2
λ 2

+ (
m2
hi
 ) 2
1
λ y } d λ =
= 


 ( y ) exp i [ –
m2
y)st( 2
h
+ x y ] d y [ 2 π ( t – s ) ] 2
1

×

10. × 


exp – =
= 


 ( y ) exp i [ –
m2
y)st( 2
h
+ x y ] d y
Taking into account equality
 ( y ) exp – i
m2
y)st( 2
h
= ( 2 π ) – 1



 0 ( λ ) exp – i [ λ y +
+ ( 2 m ) – 1
h ( t – s ) y 2
] d λ
we arrive at the formula



 ( y ) exp i [ –
m2
y)st( 2
h
+ x y ] d y =
= ( 2 π ) – 1



 0 ( λ ) d λ 


exp i [ y ( x – λ ) – ( 2 m ) – 1
h ( t – s ) y 2
] d y
Calculation of the inner integral results
( 2 π ) – 1
 exp i [ y ( x – λ ) – ( 2 m ) – 1
h ( t – s ) y 2
] d y =


d
m
sthi
y
st
2
]
)(
[
2
1 



11. = ( 2 π ) – 1



exp –
2
1
{ [
m
)st( hi
] 2
1
y – (
)st(
m
h
i
) 2
1
( x –  ) } 2

exp –
)st(2
)λx(m 2


h
i
d y = 2
1
]
)s-t(
mπ2
[
π2
1
hi
exp –
)st(2
)λx(m 2


h
i
Hence,
 ( x , t ) = 



2
1
]
)st(π2
m
[
h
exp –
)st(2
)yx(m 2


h
i
 0 ( y ) d y
This representation of the wave function is identical to the Feynman formula. Thus, the probabilistic
representation of the Schrödinger equation solution takes place in the complex coordinate space. The
quantum measurements are the real-world actions of the complex quantum distributions of the
quantum particles.
Let us consider a classical mechanic system with Lagrangian in the form (6.4.1). Let potential
function in (6.4.1) represents the ‘white noise’ external forces
V ( t , x , ω ) = F ( x , t ) + G ( x , t )

β ( t )
where
F ( x , t ) = ( f ( t ) , x ) , gj k ( t ) x j

β k ( t ) = G ( x , t )

β ( t )  
N
j
d
k1 1

12. Here  ( t ) is a d-dimensional Wiener process independent on the Wiener process w ( t ). This type of
the potential functions could be used for description of the continuous time quantum measurements.
Using the probabilistic representation, we derive the Schrödinger equation that corresponds to ‘white
noise’ potential function. Bearing in mind the complex interpretation of the quantum evolutions, we
introduce the functional
Ψ ( z , t ) = E Ψ0 (  ( 0 ; t , z )) exp
h
i

t
0
[ F (  ( s ; t , z ) , s ) d s +
(6.4.11)
+ 
t
0
G (  ( s ; t , z ) , s ) d

β ( s ) | F ] }
Filtration F is generated by the increaments of the Wiener process  up to the moment t . Let us
briefly comment the formula (6.4.11). In this formula the Wiener process  ( s ) is ndependent on the
Wiener process w ( t ). Recall that the process w ( t ) is a characteristic of the quantum particle system,
i.e. relates to the quantum world. The random process  ( s ) is the characteristic of the interaction of
the external world with the quantum system. This interaction does not be quantized and therefore
preserves its the classical mechanics form. Such a construction can be formally represented by the
conditional expectation with respect to -field generated by the observations on the external random
potential. Consider a derivation of the correspondent Schrödinger equation.
Theorem 6.3.3. Suppose that nonrandom functions Ψ0 ( z ) , F ( t , z ) , G ( t , z ) are continuous in t and
analytic with respect to z. Then the function  ( x , t ) = Ψ ( x + i 0 , t ) is a classical solution of the
Cauchy problem of the Schrödinger equation
t
 )t,x(ψ
=
m2
hi
 2
 ( x , t ) + h – 1
[ i F ( x , t ) – h – 1
G ( x , t ) ]  ( x , t ) –
– i h – 1
G ( x , t ) ]  ( x , t )

β ( t ) (6.4.12)
β
t
β
t

13.  ( x , 0 ) =  0 ( x + i 0 )
The classical solution of the Cauchy problem (6.4.12) is a random function  ( x , t ) twice continuously
differentiable in x in the sense of mean convergence and measurable for each t with respect to F and
satisfies with probability 1 the equality
 ( x , t ) –  ( x , 0 ) = 
t
0
{
m2
hi
 2
 ( x , s ) + h – 1
[ i F ( x , s ) –
(6.4.13)
– h – 1
G ( x , s ) ]  ( x , s ) d s – i h – 1

t
0
G ( x , s )  ( x , s ) d  ( s )
with probability 1 for all t at once. In the equation (6.4.13) the stochastic integral with respect to  is
interpreted as the forward time Ito integral.
Proof. We briefly outline the derivation of the equation (6.4.13). Let s = t 1 < t 2 < …< t N = t be a
partition of the interval [ s , t ] and λ = max  t j . Taking into account that the processes
 ( s ; t , z ) and  ( t ) are independent and filtration F is continuous in t we note that
Ψ ( z , t k + 1 ) – Ψ ( z , t k ) = E { Ψ (  ( t k ; t k + 1 , z ) , t k ) –
– Ψ ( z , t k ) exp
h
i
[ 
N
1j
f j ( t k ) z j  t k – 
N
1j

d
1l
g j l ( t k ) z j   l ( t k ) ] +
+ Ψ ( z , t k ) [ exp
h
i
[ 
N
1j
f j ( t k ) z j  t k –
– 
N
1j

d
1l
g j l ( t k ) z j   l ( t k ) ] | F } – 1 ] + o ( λ )
β
t
β
t

1k

14. where F β
1k  = F β
t 1k 
. Bearing in mind that -algebras F β
)1k,k[  and F β
)1k,k[   F ξ
)k,0[ are
conditionally independent, we have
Ψ ( z , t k + 1 ) – Ψ ( z , t k ) = E { [  ( t k ; t k + 1 , z ) – z ] z Ψ ( z , t k ) +
+ [  ( t k ; t k + 1 , z ) – z ] z Ψ ( z , t k ) [  ( t k ; t k + 1 , z ) – z ] *
+
+ Ψ ( z , t k )
h
i
[ 
N
1j
f j ( t k ) z j – 2
2
1
h

N
1j

d
1l
g j l ( t k ) z j   l ( t k ) ]  t k –
– g j l ( t k ) z j   l ( t k ) + o ( λ ) | F }
Passing to the limit in probability as λ  0 we arrive at the equation
Ψ ( z , t ) – Ψ ( z , s ) = 
t
s
[
m2
hi
  ( z , r ) +
h
i
F ( z , r )  ( z , r ) –
(6.4.14)
– 2
2
1
h
G 2
( z , r )  ( z , r ) ] d r –
h
i

t
s
G ( z , r )  ( z , r ) d  ( r )
Setting in this equation z = x + i 0 and taking into account that
 z Ψ ( z , t ) | z = x + i 0 =  x Ψ ( x , t )
 
d
l
N
jh
i
11

1k

15. one could note that equation (6.4.14) transforms into (6.4.13).
Theorem 6.4.4. Assume that the conditions of the Theorem 6.4.3 are fulfilled. Then for
each t  0
 |  ( x , t ) | 2
d x =  |  ( x , 0 ) | 2
d x
Proof. Rewrite the equality (6.4.14) in the differential form
 s Ψ ( z , s ) = [
m2
hi
  ( z , s ) +
h
i
F ( z , s )  ( z , s ) –
– 2
2
1
h
G 2
( z , s )  ( z , s ) ] d s –
h
i
G ( z , s ) Ψ ( z , s ) d ( s )
where  s denotes the partial differentiation with respect to s. Taking the complex conjugation in latter
equality we see that
 s )s,z(Ψ = [ –
m2
hi
 )s,z(Ψ –
h
i
F ( z , s ) )s,z(Ψ –
– 2
2
1
h
G 2
( z , s ) )s,z(Ψ ] d s +
h
i
G ( z , s ) )s,z(Ψ d  ( s )
Applying the integration by parts formula, we note that

16.  s | Ψ ( z , s ) | 2
=  s [ )s,z(Ψ Ψ ( z , s ) ] = [  s )s,z(Ψ ] Ψ ( z , s ) +
+ )s,z(Ψ [  s Ψ ( z , s ) ] +  s < )s,z(Ψ Ψ ( z , s ) > =
= [
m2
hi
)s,z(Ψ  Ψ ( z , s ) +
h
i
F ( z , s ) |  ( z , s ) | 2
– 2
2
1
h
G 2
( z , s ) 
 |  ( z , s ) | 2
] d s –
h
i
G ( z , s ) | Ψ ( z , s ) | 2
d  ( s ) – [
m2
hi
Ψ ( z , s )  )s,z(Ψ +
+
h
i
F ( z , s ) |  ( z , s ) | 2
+ 2
2
1
h
G 2
( z , s ) |  ( z , s ) | 2
] d s –
h
i
G ( z , s ) 
 | Ψ ( z , s ) | 2
d  ( s ) + 2
1
h
G ( z , s ) G ( z , s ) |  ( z , s ) | 2
d s
Setting in this equality z = x + i 0 and putting  ( x , t ) =  ( x + i 0 , t ) the right hand side could
be simplified. Indeed
 s |  ( x , s ) | 2
=
m2
hi
[ )s,(ψ x   ( x , s ) –  ( x , s ) )s,(ψ x ]
Integrating this equality in x over the sphere S R = { x : | x |  R } and applying Green’s formula we
see that

RS
t

|  ( x , t ) | 2
d x =

17. =
m2
hi

RS
div [ ψ ( x , t )   ( x , t ) –  ψ ( x , t )  ( x , t ) ] d x =
=
m2
hi

RS
[ ψ ( x , t )   ( x , t ) –  ψ ( x , t )  ( x , t ) ] n d S R
where [ q ] n denotes projection of the vector q onto external normal to the S R and
d S R is differential element of the surface of the sphere S R . The right hand side has the standard form
and it is usually assumed in quantum mechanics that it converges to 0 when R  + . This remark
completes the proof.
The Lagrangian form of the classical mechanics is the initial step of the Feynman approach and
we have followed this line. Nevertheless, it is known another form of Lagrangian that also leads to the
same classical dynamic equations. Let us recall this construction by considering a system
m 2
2
td
xd
+ r
td
xd
+ k x + a = 0
(6.4.15)
m 2
2
td
yd
– r
td
yd
+ k y + a = 0
Here m, r, k , are known constants though the next contstuctions remain correct when these constant
are functions on t. The Lagrangian of the system (6.4.15) can be written in the form
L ( x ,
td
xd
, y ,
td
yd
, t ) = m
td
xd
td
yd
+
2
r
[ y
td
xd
– x
td
yd
] –
(6.4.16)

18. – k x y – a ( x + y )
The correspondent action S then is defined as
S = 
t
s
L ( x ,
td
xd
, y ,
td
yd
, l ) d l
The variables x and y in S are assumed to be independent. One can verify that variations of the action
with respect to the variables x and y lead to the motion equations (6.4.15). Bearing in mind that the
same mechanics motion equations could be presented by the different Lagrangian functions (6.4.1) or
(6.4.16) it looks essential to justify that Feynman’s interpretation of the wave function does not depend
on a choice of the Lagrangian. For illustrative simplicity we assume that r = a = 0. Denote
I ( x , y ) = exp
h
i

t
s
L ( x ,
td
xd
, y ,
td
yd
, l ) d l
where
L ( x ,
td
xd
, y ,
td
yd
, t ) = m
td
xd
td
yd
– k x y (6.4.16)
Then the expression P corresponding to I ( x , y ) could be written as following
P = 
2
1
x
x

2
1
y
y
I ( x , y ) D x ( * ) D y ( * ) (6.4.17)

t
s

19. We will show that the propagator P has another interpretation than the wave function presented by
Feynman’s approach dealing with Lagrangian (6.4.1). Note that the right hand side of (6.4.17) is the limit
of the correspondent discrete time approximation. That is
P =
0ε
lim
 2
A
1
  …  exp
h
i
{ 
N
1j ε
)yy()xx(m 1jj1jj  
–
(6.4.18)
– k x j – 1 y j – 1  }
A
xd 1
A
yd 1
…
A
xd 1N 
A
yd 1N 
where A = ( 2  i  h m – 1
) is the normalized factor, and x 0 = x s , x N = x t . For calculation
expression in the right hand side of (6.4.18) introduce the new coordinates
x = ( 2 ) 2
1

( x l + y l )
y = ( 2 ) 2
1

( y l – x l )
l = 1, 2, … , N – 1. The inverse transformations could be presented as
x l = ( 2 ) 2
1

( x – y )
y l = ( 2 ) 2
1

( x + y )
l = 0 , 1, … N. Bearing in mind that the transition Jacobian equal to 1 we note that substitution of the
new variables into right hand side of (6.4.18) leads us to the formula
2
1
*
l
*
l
*
l
*
l
*
l
*
l

20. P ( x s , y s , s | x t , y t , t ) =
0ε
lim
 2
A
1
  ... exp
h
i
2
{ 
N
1j ε
m
( x j – x j – 1 ) 2
–
–
ε
m
( y j – y j – 1 ) 2
+ k ( x 2
1j  – y 2
1j  )  }
A
xd 1
A
yd 1
…
A
xd 1N 
A
yd 1N 
In this formula for the writing simplicity the *-symbol is omitted under the integrals for the variables x l
, y l , l = 1, ... , N – 1. Thus
P ( x s , y s , s | x t , y t , t ) = K ( x s , s | x t , t ) )t,y|s,y(K ts
Here K (  |  ) is the Feynman’s kernel. Putting x s = y s = x and x t = y t = y we arrive at the result
that starting with the Lagrangian (6.4.16) Feynman’s approach presents distribution density in the form
P ( x , s | y , t ) = P ( x , x , s | y , y , t ) = | K ( x , s | y , t ) | 2
(6.4.19)
The Lagrangian (6.4.16) corresponds to the motion equations (6.4.15) in which the first equation
describes the motion of the particle with external power given that r > 0 while the second one
describes the motion of damped particle. It looks unexpectedly that the quantum dynamics either
motions nonseparable from its time inverse counterpart.