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Numerical Methods and Programming, 2009, Vol. 10 (http://num-meth.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 eld equations, the particles being identi ed with the soliton con gurations. The wave function of a particle proves to be a vector in the random Hilbert space. The many-particle wave function is constructed via the many-solitons con gurations. It is shown that, in the point-like limit when the proper size of the soliton con guration 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 self-conjugated operators and the spin-statistics correspondence stems from the extended character of particles-solitons.

ticles. The rst one was suggested by L. de Broglie 1] and A. Einstein 2] and consists in using soliton solutions to some nonlinear eld equations for the description of the particles' internal structure. The second way supposes 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 rst 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 extended particles. To realize this approach, suppose that a eld describes n particles-solitons and has the form n n X X @ (k ) (k ) (t; r) = (t; r), and the same for the conjugate momenta (t; r) = @ L = (t; r), t = @ t . Let @ t k=1 k=1 us de ne the auxiliary functions (k ) 1 (1) '(k) (t; r) = p k (k) + i 2 k with the constants k satisfying the normalization condition ~ = d3x '(k) 2. Now we de ne the analog of the wave function in the con gurational space R3n 3 x = fr1 ; : : : ; rn g as follows:
N (t; r1; : : : ; rn) = (~n N ) 1=2
XY Z

Keywords: stochastic representation, solitons, random Hilbert space. 1. Introduction. One could imagine two possible ways of constructing quantum theory of extended par-

Nn
=1

j

k

=1

'(k) (t; rk ): j

(2)

Here N 1 stands for the number of trials (observations) and '(k) is the one-particle function (1) for the j Z 1 j -th trial. Earlier 5, 6] it was shown that the quantity N = (4_)n d3nx N 2 , where 4_ is the elementary volume supposed to be much greater than the proper volume of the particle `0 3 = _0 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 well-known spinstatistics correlation 7]. Namely, if '(k) is transformed under the rotation by the irreducible representation j D(J ) of S O(3) with the weight J , then the transposition of two identical extended particles is equivalent to the relative 2 -rotation of '(k) that gives the multiplication factor ( 1)2J in N . To show this property, suppose j that our particles are identical, i.e., their pro les '(k) may di er in phases only. Therefore, the transposition of j the particles with the centers at r1 and r 2 means the -rotation of 2-particle con guration around the median 1 Department of Theoretical Physics, Peoples' Friendship University of Russia, 117198, Moscow, 6, MiklukhoMaklay Str., Russia; Professor, e-mail: soliton4@mail.ru c Science Research Computing Center, Moscow State University
(

4_

)n

R

3n


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Numerical Methods and Programming, 2009, Vol. 10 (http://num-meth.srcc.msu.ru)

axis of the central vector line r1 r2 . Due to extended character of the particles, however, to restore the initial con guration, one should perform additional proper -rotations of the particles. The latter operation being equivalent to the relative 2 -rotation of particles, it results in the aforementioned multiplication of N by ( 1)2J . Under the natural supposition that the weight J is related with the spin of particles-solitons, one infers that the many-particles 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 half-integer (the Pauli principle). Now let us consider the measuring procedure for some observable A corresponding, due to Noether's c theorem, to the symmetry group generator MA . For example, the momentum P is related with the generator c of space translation MP = i 5, the angular momentum L is related with the generator of space rotation c ML =Z J and so on. As Z result one can represent the classical observable Aj for the j -th trial in the form a 3 c c Aj = d x j iMA j = d3x 'j MA 'j . The corresponding mean value is
E

(A)

_0 b b c where the Hermitian operator A reads A = ~MA . Up to the terms of the order 4_ 1, we obtain the standard QM rule (3) for the calculation of mean values. Thus, we conclude that in the solitonian scheme the spin-statistics correlation stems from the extended character of particles-solitons. With the particles in QM being considered as point-like ones, however, it appears inevitable to include the transpositional symmetry of the wave function as the rst principle (cf. Hartree{Fock receipt for fermions). Various aspects of the ful llment of the QM correspondence principle for the Einstein{de Broglie's soliton model were discussed in 5{11]. The fundamental role of the gravitational eld 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 elds was discussed in 11]. As a result, we obtain the stochastic realization (2) of the wave function N which can be considered as an element of the random Hilbert space Hrand with the scalar product ( 1 ; 2) = M ( 1 2 ) with M standing for the expectation value. As a crude simpli cation, one can assume that the averaging in this product is taken over the random characteristics of particles-solitons, 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 point-particle limit (4_ _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 point-like particles. Moreover, we prove that the spin-statistics correspondence (the Pauli principle) can be considered as the natural consequence of the internal structure of elementary particles.
1. de Broglie L. Les incertitudes d'Heisenberg et l'interpre tation probabiliste de la mecanique ondulatoire. Paris: Gauthier{Villars, 1982. 2. Einstein A. Collected papers. Moscow: Nauka, 1967. 3. Snyder H. Quantized space-time // 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 theorie statistique des champs et la mecanique quantique // Ann. Fond. L. de Broglie. 1977. 2, N 3. 181{203. 7. Rybakov Yu.P. Self-gravitating solitons and nonlinear-resonance 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 di raction 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 elds and quantum mechanics // Bulletin of Peoples' Friendship University of Russia. Ser. Physics. 2004. 12. 88{112. Received November 5, 2009

N N 1 XA = 1 X N j =1 j N j =1

Z

d x 'j MA 'j =
3

Z

c

_0 b d3 x N A N + O 4_ ;

(3)

References