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Numerical Methods and Programming, 2009, Vol. 10 (http://num­meth.srcc.msu.ru) 389
UDC 539.12: 530.145
SPECIAL STOCHASTIC REPRESENTATION OF QUANTUM MECHANICS
AND SOLITONS
Yu. P. Rybakov 1
A special stochastic representation of quantum mechanics is proposed. The basis of this representation
is the linear combination of soliton solutions to some nonlinear field equations, the particles being
identified with the soliton configurations. The wave function of a particle proves to be a vector
in the random Hilbert space. The many­particle wave function is constructed via the many­solitons
configurations. It is shown that, in the point­like limit when the proper size of the soliton configuration
vanishes, the main principles of quantum mechanics are restored. In particular, the expectation values
of physical observables are obtained as the Hermitian forms generated by self­conjugated operators
and the spin­statistics correspondence stems from the extended character of particles­solitons.
Keywords: stochastic representation, solitons, random Hilbert space.
1. Introduction. One could imagine two possible ways of constructing quantum theory of extended par­
ticles. The first one was suggested by L. de Broglie [1] and A. Einstein [2] and consists in using soliton solutions
to some nonlinear field equations for the description of the particles' internal structure. The second way sup­
poses the introduction of minimal elementary length ` 0 [3] and/or nonlocal interaction [4]. The aim of our
work is to show the consistency of the first approach with the principles of quantum mechanics (QM). To this
end, we introduce a special stochastic representation of the wave function using solitons as images of the ex­
tended particles. To realize this approach, suppose that a field OE describes n particles­solitons and has the form
OE(t; r) =
n
X
k=1
OE (k) (t; r), and the same for the conjugate momenta ú(t; r) = @L
@OE t
=
n
X
k=1
ú (k) (t; r), OE t = @OE
@t . Let
us define the auxiliary functions
' (k) (t; r) = 1
p
2
`
š k OE (k) + iú (k)
š k
'
(1)
with the constants š k satisfying the normalization condition ~ =
Z
d 3 x
fi fi ' (k)
fi fi 2
. Now we define the analog of
the wave function in the configurational space R 3n 3 x = fr 1 ; : : : ; rn g as follows:
\Psi N (t; r 1 ; : : : ; rn ) = (~ n N ) \Gamma1=2
N
X
j=1
n
Y
k=1
' (k)
j (t; r k ): (2)
Here N AE 1 stands for the number of trials (observations) and ' (k)
j is the one­particle function (1) for the
j­th trial. Earlier [5, 6] it was shown that the quantity ae N = 1
(4—) n
Z
(4—) n aeR 3n
d 3n x
fi fi \Psi N
fi fi 2
, where 4— is the
elementary volume supposed to be much greater than the proper volume of the particle ` 0
3 = — 0 Ü 4—, plays
the role of probability density for the distribution of solitons' centers.
It is interesting to emphasize that the solitonian scheme in question contains also the well­known spin­
statistics correlation [7]. Namely, if ' (k)
j is transformed under the rotation by the irreducible representation
D (J) of SO(3) with the weight J , then the transposition of two identical extended particles is equivalent to the
relative 2ú­rotation of ' (k)
j that gives the multiplication factor (\Gamma1) 2J in \Psi N . To show this property, suppose
that our particles are identical, i.e., their profiles ' (k)
j may differ in phases only. Therefore, the transposition of
the particles with the centers at r 1 and r 2 means the ú­rotation of 2­particle configuration around the median
1 Department of Theoretical Physics, Peoples' Friendship University of Russia, 117198, Moscow, 6, Miklukho­
Maklay Str., Russia; Professor, e­mail: soliton4@mail.ru
c
fl Science Research Computing Center, Moscow State University

390 Numerical Methods and Programming, 2009, Vol. 10 (http://num­meth.srcc.msu.ru)
axis of the central vector line r 1 \Gamma r 2 . Due to extended character of the particles, however, to restore the initial
configuration, one should perform additional proper ú­rotations of the particles. The latter operation being
equivalent to the relative 2ú­rotation of particles, it results in the aforementioned multiplication of \Psi N by
(\Gamma1) 2J . Under the natural supposition that the weight J is related with the spin of particles­solitons, one infers
that the many­particles wave function (2) should be symmetric under the transposition of the two identical
particles if the spin is integer but should be antisymmetric if the spin is half­integer (the Pauli principle).
Now let us consider the measuring procedure for some observable A corresponding, due to Noether's
theorem, to the symmetry group generator c
MA . For example, the momentum P is related with the generator
of space translation c
MP = \Gammai 5, the angular momentum L is related with the generator of space rotation
c
ML = J and so on. As a result one can represent the classical observable A j for the j­th trial in the form
A j =
Z
d 3 xú j i c
MA OE j =
Z
d 3 x' \Lambda
j
c
MA' j . The corresponding mean value is
E(A) j
1
N
N
X
j=1
A j = 1
N
N
X
j=1
Z
d 3 x' \Lambda
j
c
MA' j =
Z
d 3 x \Psi \Lambda
N
b
A\Psi N + O
` — 0
4—
'
; (3)
where the Hermitian operator b
A reads b
A = ~ c
MA . Up to the terms of the order — 0
4— Ü 1, we obtain the
standard QM rule (3) for the calculation of mean values. Thus, we conclude that in the solitonian scheme the
spin­statistics correlation stems from the extended character of particles­solitons. With the particles in QM
being considered as point­like ones, however, it appears inevitable to include the transpositional symmetry of
the wave function as the first principle (cf. Hartree--Fock receipt for fermions). Various aspects of the fulfillment
of the QM correspondence principle for the Einstein--de Broglie's soliton model were discussed in [5--11]. The
fundamental role of the gravitational field in the de Broglie--Einstein solitonian scheme was discussed in [7].
The soliton model of the hydrogen atom was developed in [8, 9]. The wave properties of solitons were described
in [10]. The dynamics of solitons in external fields was discussed in [11].
As a result, we obtain the stochastic realization (2) of the wave function \Psi N which can be considered as an
element of the random Hilbert space H rand with the scalar product (/ 1 ; / 2 ) = M (/ \Lambda
1 / 2 ) with M standing for the
expectation value. As a crude simplification, one can assume that the averaging in this product is taken over the
random characteristics of particles­solitons, such as their positions, velocities, phases, and so on. It is important
to recall once more that the correspondence with the standard QM is retained only in the point­particle limit
(4— AE — 0 ) for N !1.
In conclusion we emphasize that, using special stochastic representation of the wave function, we show that
the main principles of quantum theory can be restored in the limit of the point­like particles. Moreover, we
prove that the spin­statistics correspondence (the Pauli principle) can be considered as the natural consequence
of the internal structure of elementary particles.
References
1. de Broglie L. Les incertitudes d'Heisenberg et l'interpr'etation probabiliste de la m'ecanique ondulatoire. Paris:
Gauthier--Villars, 1982.
2. Einstein A. Collected papers. Moscow: Nauka, 1967.
3. Snyder H. Quantized space­time // Physical Review. 1947. 71. 38--41.
4. Yukawa H. On the radius of the elementary particle // Physical Review. 1949. 76. 300--301.
5. Rybakov Yu.P. On the causal interpretation of quantum mechanics // Found. of Physics. 1974. 4, N 2. 149--161.
6. Rybakov Yu.P. La th'eorie statistique des champs et la m'ecanique quantique // Ann. Fond. L. de Broglie. 1977. 2,
N 3. 181--203.
7. Rybakov Yu.P. Self­gravitating solitons and nonlinear­resonance quantization mechanism // Bulletin of Peoples'
Friendship University of Russia. Ser. Physics. 1995. 3, N 1. 130--137.
8. Rybakov Yu.P., Saha B. Soliton model of atom // Found. of Physics. 1995. 25, N 12. 1723--1731.
9. Rybakov Yu.P., Saha B. Interaction of a charged 3D soliton with a Coulomb center // Physics Letters. Ser. A. 1996.
222, N 1. 5--13.
10. Rybakov Yu.P., Shachir M. On Fresnel diffraction of solitons in the Synge model // Izvestia VUZov. Ser. Physics.
1982. 25, N 1. 36--38.
11. Rybakov Yu.P., Terletsky S.A. Dynamics of solitons in external fields and quantum mechanics // Bulletin of Peoples'
Friendship University of Russia. Ser. Physics. 2004. 12. 88--112.
Received November 5, 2009