Equations of the de Broglie Wave Field and Their Relationship to Riemann'sCurvature TensorJ. Kulhánek Citation: Journal of Mathematical Physics 11, 3199 (1970); doi: 10.1063/1.1665115 View online: http://dx.doi.org/10.1063/1.1665115 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/11/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The de Broglie relationship—Fact and fiction Am. J. Phys. 60, 1065 (1992); 10.1119/1.16950 de Broglie phase waves, of a truck? Am. J. Phys. 48, 335 (1980); 10.1119/1.12088 Dispersion relation for de Broglie waves Am. J. Phys. 48, 283 (1980); 10.1119/1.12140 Doppler shifted de Broglie wave Am. J. Phys. 46, 309 (1978); 10.1119/1.11346 SpaceTime of the de Broglie Wave Field J. Math. Phys. 12, 1535 (1971); 10.1063/1.1665770

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THE PROPER VIBRATION OF THE SPACE 3199

vibrations of a spherical surface12 are characterized by discrete eigenfrequencies. The discreteness is a consequence of the fitting of wavelengths into a finite span.

The vibrations of 3-space with k > 0, which are considered in this paper, are similar to a pulsating sphere. This particular motion of a spherical surface is characterized by eigenfrequency which is zero.12 The assumed isotropy of 3-space ruled out the higher eigenfrequencies. However, it is possible that the eigenfrequencies of nonisotropic and finite 3-space might provide an adequate description of the observed atomicity of matter and light.

1 E. Schrodinger, Proc. Roy. Irish Acad. 46A, 25 (1940). 2 B. C. Mukerji, Z. Physik 101, 270 (1936). 3 E. Schrodinger, Commentatones Pontificiae Acad. Sci. 2, 321

(1938). 4 E. Schrodinger, Physica 6, 899 (1939). 5 P. O. Miiller, Physik. Z. 40, 366 (1939). 6 L. Infeld and A. E. Shild, Phys. Rev. 70, 410 (1946). 7 J. Kulhanek, Nuovo Cimento Supp. 4,172 (1966). 8 R. Courant and D. Hilbert, Methods of Mathematical Physics

(Interscience, New York, 1962), Vol. II. 9 J. Kulhanek, Nuovo Cimento 63B, 497 (1969).

10 Explanatory footnote: The constant Je is defined as Je2 = gat5YaYt5 where Ya are components of 4-momentum. When the signa­ture of first metric form is +2, we put Je2 = _Je2 and consider JC as the rest mass. When the signature is -2, we put Je = Je2 and consider Je as the rest mass.

11 H. P. Robertson, Rev. Mod. Phys. 5, 62 (1933). 12 E. J. Konopinski, Classical Descriptions of Motion (Freeman,

San Francisco, 1969), p. 407.

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 11, NUMBER 11 NOVEMBER 1970

Equations of the de Broglie Wave Field and Their Relationship to Riemann's Curvature Tensor

J. KULHANEK

Physics Department, University of Windsor, Windsor, Ontario, Canada

(Received 10 April 1970)

The equations of the de Broglie wave field (field equations) [J. Kulhanek, Nuovo Cimento Supp. 4, 172 (1966) I under special conditions require a very particular geometry together with a specific inter­pretation of the curvature scalar. The purpose of the present paper is to show that the same condition turns the conservation law (which is a consequence of the field equations) into an identity and that the Rainich [Nature 115, 498 (1925)1 decomposition of Riemann's curvature tensor gives only one component.

A SPECIAL CASE OF THE FIELD EQUATIONS

The general form of the field equations1 is

Je2 Rab - tRgab - h2 gab

= - (R + 4 Je2

2

) (tgab - XaX/,) , (1) h ,

where gab = gba' Je2 is the rest mass, h is Planck's constant, and the xa are components of the unit 4-vector normal to the 3·wavesurface of the de Broglie wave. On the left-hand side of (1) is a well-known tensor, whose covariant divergence vanishes. Thus, from (1) we have the conservation equations,

gbf ( R + 4 ~:) (tgab - XaXb) l! = O. (2)

It is well known that the quantity Rik;i;k is the scalar curvature of a 3-dimensional space which is perpendicular to ;a. If we put ;a == xu, then from (1) it follows that

RabxaXb = R + 3 (Je2/ h2) • (3)

Thus quantity R + 3 (Je2/h2) represents scalar curvature of the 3-wavesurface of the de Broglie wave.

In the case when we assume that

R + 4 (Je2/ h2) = 0, (4)

then the conservation law (2) is trivial and Eqs. (1) are red uced to

(5)

From Eqs. (3) and (4) we have that

(6) or

(7)

The scalar curvature of the 3-wavesurface of the de Broglie wave is given as -Je2/h2 or iR.

Rainich2 showed that Riemann's curvature tensor Rab ek in the 4-dimensional space can be decomposed into two parts which have different properties of symmetry. Considering the Riemannian curvature in the 2-direction defined by the unit bivector Vab and

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3200 J. KULHANEK

the Riemannian curvature in the perpendicular 2-direction defined by the unit bivector j7ab, we can then write2

(8)

For the 2-directions Vab and vub the component Sik 1m defines the same sign curvature while Aik 1m

defines curvature of the opposite sign. We can write that

and

(10)

Now, in Ref. 2 it is shown that

Aik 1m = t(gi/Kkm + gkmKa - gimKkl - gkIKim), (11)

where we denote

Kim == Rim - !Rg1m •

From (11), (12), and (5) it follows that

(12)

Aik 1m = 0. (13)

We see from (11) that (13) and (5) are equivalent. The field equations (1) with condition (2) imply that Riemann's tensor is equal to the component Sik 1m

of the Rainich decomposition (8).

1 J. Kulhanek, Nuovo Cimento Suppl. 4, 172 (1966). 2 C. Y. Rainich, Nature 115,498 (1925).

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 11, NUMBER 11 NOVEMBER 1970

Calculation of Correlation Functions of Solutions of a Stochastic Ordinary Differential Equation

J. A. MORRISON Bell Telephone Laboratories, Inc., Murray Hill, New Jersey 07974

(Received 27 May 1970)

In this paper we use the "smoothing method" to calculate the correlation functions of the solutions of the equation

d 2u dz2 + P~[1 + 1jN(z)]u = 0,

s~tisfying nonstochastic initial conditions, ~.here N(z) is a r~l, wide-sense stationary stochastic process With zero ~ean and Po and 1j « 1 are posItive constants. It IS shown that an appropriate application of the smoothIng method lea~s to the e:,~ct results in the case when N(z) is the random telegraph process. Mor~over, under approprIate co~dltl?ns on the general process N(z), approximate expressions are obtaIned for the correlation functions In terms of the first- and second-order moments of the solutions and approximate expressions are given for these moments. '

1. INTRODUCTION AND SUMMARY

In a previous paperl we used the "smoothing method" to calculate approximately the first- and second-order moments of the solutions of the stochas­tic differential equation (1.2). Our interest was in the propagation of an electromagnetic wave through a randomly stratified dielectric slab, but Eq. (1.2), in which the stochastic process N(z) satisfies (1.1), also corresponds to a harmonic oscillator with a random spring and arises in many other contexts. In this paper we apply the smoothing method to calculate approximately the correlation functions of the solu­tions of (1.2). We also show that the results are in fact exact in the particular case in which N(z) is the random telegraph process T(z) , defined in Sec. 3. We derived the exact results in this case in an earlier paper.2

The smoothing method3 for calculating the ex­pected value of the solution of a linear stochastic equation has been developed extensively by Keller4- lo

and Bourret.ll- l6 In Sec. 2 we give an outline of the method in a form which is appropriate for our purposes, and is close to that given by Keller. How­ever, our application of the smoothing approxima­tion to the calculation of the correlation functions of the solutions of Eq. (1.2) differs considerably from that proposed by Keller.8 - lo In our approach it is necessary to solve only ordinary differential and integro-differential equations, whereas the scheme proposed by Keller leads to partial differential and integro-differential equations.

Let N(z) be a real, wide-sense stationary stochastic process with

(N(z» = 0, (N(y)N(z» = r(y - z), (1.1)

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