# Space-Time of the de Broglie Wave Field

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SpaceTime of the de Broglie Wave FieldJ. Kulhnek Citation: Journal of Mathematical Physics 12, 1535 (1971); doi: 10.1063/1.1665770 View online: http://dx.doi.org/10.1063/1.1665770 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/12/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A note on wave equation in Einstein and de Sitter space-time J. Math. Phys. 51, 052501 (2010); 10.1063/1.3387249 An Introduction to Quantum Field Theory in de Sitter spacetime AIP Conf. Proc. 910, 218 (2007); 10.1063/1.2752481 Scalar field propagators in antide Sitter spacetime J. Math. Phys. 26, 2050 (1985); 10.1063/1.526878 Doppler shifted de Broglie wave Am. J. Phys. 46, 309 (1978); 10.1119/1.11346 Equations of the de Broglie Wave Field and Their Relationship to Riemann's Curvature Tensor J. Math. Phys. 11, 3199 (1970); 10.1063/1.1665115

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UNIT ADJOINT TENSOi{ OPERATORS IN SO(n) 1535

ACKNOWLEDGMENTS

I wish to thank Dr. B. Gruber, M. Lorente, J. Saldanha, and B. Han for valuaBle discussions and Dr. V. Newton for checking the results of Sec. IV.

1 J. D. Louck and L. C. Biedenharn, J. Math. Phys. 11, 2368 (1970).

JOURNAL OF MATHEMATICAL PHYSICS

2 L. C. Biedenharn, J. Math. Phys. 4, 436 (1963). 3 I. M. Gel'fand, Mat. Sb. 26,103 (1950). 4 G. Racah, Atti Accad. NazI. Lincei, Rend., Classe Sci. Fis.,

Mat. e Nat. 8, 108 (1950). B. Gruber and L. O'Raifeartaigh, J. Math. Phys. 5, 1796 (1964). 6 J. D. Louck, Los Alamos Scientific Laboratory, Report LA2451,

1960. 7 M. K. F. Wong, J. Math. Phys. 8, 1899 (1967). 8 K. T. Hecht, Nucl. Phys. 63, 177 (1965).

VOLUME 12, NUMBER 8 AUGUST 1971

Space-Time of the de Broglie Wave Field

J. KULHANEK University of Vienna, Institute for Theoretical Physics, Vienna, Austria

(Received 1 October 1970)

A geometrical characterization is given of the space-time with a metric satisfying the equations of the de Broglie wave field. To do this, we exploit a formal analogy with a "perfect fluid" whose energy and pressure depend on the components of the metric field and its derivatives. To determine the properties of the space-time, we use the Ricci principal directions.

I. INTRODUCTION

Following the lines of thought inaugurated by Eddington,! we can say that in the finiteness of space we have the clue to atomicity. The argument runs as follows: Atomicity is merely the oldest, best known and still most important expression of the inherent discontinuities in nature. Generally speaking, the latter are accounted for in wave mechanics by means of a close analogy with the discontinuous sets of proper modes of vibrating systems. However, only a finite system possesses discontinuous proper modes. So, if the space were infinite, it would be hard to explain atomicity along these lines because its proper vibrations would form a continuous sequence.

Perhaps SchrOdinger was the first2 who believed that the "P waves are to be identified with waves repre-senting disturbances of the metric field of space-time. But then, of course, the components of the metric field cannot be a solution of Einstein's equations because of the inherent dualism of field and source of field (energy momentum tensor). This dualism for "P waves is hardly acceptable, and, as a matter of fact, the Schrodinger, Klein-Gordon, or Dirac equations do not contain any term describing the source of the "P-fie1d.

We assume that matter waves are described by a metric field which satisfies the field equations of the de Broglie wave field.3.4 These equations are similar to the Einstein equations but do not contain any terms which We could call sources. Let us note that from the field equations it follows that the second

derivatives of the metric field have essential dis-continuities on the characteristic 3-surfaces. These characteristic 3-surfaces represent a history of a two-wave surface of a wave which has a phase velocity identical with the phase velocity of an ordinary de Broglie wave. The wave with the phase velocity of a de Broglie wave is propagated in a 3-space (usually called chronometrical 3-space5) everywhere perpen-dicular to the lines .x4.3 In a synchronous coordinate system (say the system co moving with the particle) the characteristic 3-surfaces and chronometrical 3-space coincide.

The field equations impose conditions which are to be satisfied by space-time. In order to characterize the space-time, we use the notion of principal direction introduced by Ricci.6 The method we use is the same as in Ref. 7 . We exploit the fact that the field equations have on the right-hand side a tensor formally identical with the energy-momentum tensor of a "perfect fluid." However, the proper density r and the pressure p of this "fluid" are functions of the metric field and its first and second derivatives. The pressure-density equation r = p holds. It then follows from the conservation equations, which are a consequence of the field equations, that there is a property of space-time represented by a function which is conserved during the evolution of the space-time. This con-served scalar function is called the index functionS (in analogy with the refractive index of a transparent medium) and it is a function of the components of the metric field and its first and second derivatives. The

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1536 J. KULHANEK

stream lines satisfy the variational principle, formally identical with Fermat's optical principle, where, however, the refraction index of a transparent medium is replaced by the index function. If Ricci's principal directions are indeterminate, the space-time cannot be considered as a space-time of a "perfect fluid."

II. IDENTIFICATION WITH PERFECT FLUID

We can write the field equations3.4 in the form

Je2 Rao - iRgao - h2 gao

= - (R + 4 ~:)Ogab - XaXb), (1) where Je is the rest mass, h is Planck's constant, and xa are the components of the unit 4-vector normal to the 3-wave surface of the de Broglie wave. The left-hand side of (1) is a well-known tensor whose covariant divergence vanishes, and we have from (1) four conservation equations

The normal vector xa has unit magnitude, and we can rewrite (2) as

xa = _ !(R + 4 Je2)OR xa. (3) ia 2 h2 oxa

Equation (3) implies that

- ( - g)! R + 4 - xa = 0 a ( Je2)!

oxa h2 (4)

or (5)

Whenever a relation in the form of Eq. (5) holds, there exists a conserved integral. Consequently, in our case the value of the integral

f (R + 4Je2/h2)!X4( - g)! dx1 dx2 dx3 is a constant, independent of time. The energy tensor of a "perfect fluid" has the form

Tab = (r + p)xaxo - pgab, (6) where r is the proper energy density and p is the pressure. On comparing (6) with the right-hand side of (1), we have

r + p = R + 4Je2Jh2, p = i(R + 4Je2/h2). (7) Combining (7), we obtain

r = HR + 4Je,2fh2). (8)

On account of the foregoing identification, the tensor on the right-hand side of (1) represents a "perfect fluid" for which r = p is true. From (7) and (8) we see that rand p are functions of the components of the metric field and its first and second derivatives. The index functionS 11 may be written in the form

l1=expf~' r+p

and characterizes the space-time property which is conserved during the evolution of space-time. From the foregoing equation, using the pressure density relation r = p, we obtain

11 ='(2)-!(R + 4Je2/h2)l. The stream lines of our "perfect fluid" satisfy the variational principles

b f(R + 4Je2/h 2)! ds = o.

This variational principle is formally identical with Fermat's optical principle in a transparent medium of refractive index 11. But the problem of determining the stream lines is not the same as that of finding the rays in a given medium, even after allowing for the fact that we are dealing with a curved manifold offour dimensions with an indefinite metric form. In the optical problem the refractive index is supposed to be known, whereas in our problem 11 is not known: For its determination the field equations must be solved, since 11 depends on the components of the metric field and its first and second derivatives. If we write

dS2 = (R + 4Je2Jh2) ds2, the variational principle has form

b f ds = 0 and the streamlines are geodesics of a metric ds2 conformal to the metric ds2 of the space-time.9

Ill. GEOMETRIC CHARACTERIZATION OF SPACE-TIME

The contra

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