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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 contravariant components A.~ of the Ricci? principal directions are solutions of the homogenous systems

    (9)

    (no summation over k), where Pk is any root of the determinant

    DIIRii + p,.,gijll = 0 (10) of the matrix of system (9). When the roots of (10)

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  • SPACE-TIME OF THE DE BROGLIE WAVE FIELD 1537

    are simple, the principal directions are uniquely determined by (9), and any two of these directions at a point are orthogonal. When a root of (10) is multiple, say of order s, and the elementary divisors are simple, the directions corresponding to this root are linearly expressible in terms of s mutually orthogonal direc-tions, which are orthogonal also to the directions corresponding to any other root. Hence, when all the elementary divisors are simple, an orthogonal n-tuple of principal directions can be found. Equation (9) may be replaced by

    4

    Rij = - ! PkAkiAkj, k~1

    (11)

    where Aki = gi;A~. Hence, substituting Rab from (1) into (9), we obtain

    [-Je2/h2gab + (R + 4J,2fh2)xaXb + Pkgab]A% = O. (12)

    Now, if we assume that xa = A~, and since xa is unitary, we have thatgabA~A~ = 1 and from (12) we get

    IV. SYNCHRONOUS COORDINATE SYSTEM

    The unitary 4-vector ga4(g44)-1 is tangent to the lines .x4 and represents the normal 4-vector of the 3-space (usually called5 chronometrical 3-space) which is everywhere perpendicular to the lines . .x4. Ifwe assume that

    (I 8)

    the chronometrical 3-space is identical, at least locally, with the 3-wave surface of the de Broglie wave. Equations (18) are the coordinate conditions specifying a synchronous coordinate system.

    Substituting (18) into (I), we get for the components of the Ricci tensor

    Rab = (ga~b4/g44)(R + 3Je2/h2) - (Je2/h2) X (gab - ga~b4/g4J. (19)

    The conservation law (4), with the help of (18), gives

    (}~4 .Jy (R + 4Je2/h 2)' = 0, (20)

    Hence, (13) where y = det II yp II and yp = gp - g4gP4/g44 ' at, {J =

    1, 2, 3. Mter integration we get from (20), for the (14) scalar curvature R,

    Now if A~ are components of any vector orthogonal to xa , then from (12) we get

    (15)

    Since every vector orthogonal to xa satisfies this con-dition it follows that PI is a triple root of (12) and the elementary divisors are simple. Hence we see that the unit 4-vector xa normal to the 3-wave surface is the principal direction determined by the simple root PI given by (14). The triple root PI is always constant, as follows from Eq. (15). For the scalar curvature of space-time, from Eqs. (14) and (15) we obtain

    (16)

    and PI and PI are the mean curvatures in the direc-tions xa and AL respectively. When PI and PI are equal, we have from (14) and (15)

    R + 4Je2/h2 = O. (17) The principal directions are indeterminate and thus space-time cannot be identified with the one manifold of "perfect fluid" noted above. Under condition (17), Eqs. (1) reduce to the special form

    Rab - }Rgab = 0

    and the conservation law (2) is fulfilled identically.9

    R = -4Je2/h2 + F2(X')/y, (21) where F(XZ") is a function of the variables xl, X2, r. Substituting from (19) into (9), we obtain

    [(ga~b4/g4J(R + 3Je2/h2) - Je2Jh2Yab - gabPk]A~ = 0 (22)

    (no summation over k), which for the single root PI and values A~ given by

    A~ = b:/(g44)' becomes

    (23)

    Hence, the simple root

    PI = -(R + 3Je2Jh2), (24) and, using (21), we can write

    (25)

    In order that the elementary divisors corresponding to the triple root PI be simple, it is necessary and sufficient that (9) for PI be satisfied by the values (1,0,0,0), (0, 1, 0, 0), (0, 0, 1, 0) for the covariant components of A!. If we write (9) in the form

    (26)

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  • 1538 J. KULHANEK

    the conditions are the synchronous coordinate system is given by

    R~ + PI'Y; = 0, IX = 1,2,3, j ,= 1,2,3,4, (27) and thus from (27) and (19) we have

    PI = Je2/h2 (28)

    We see that PI and PI are equal only when in (21) the function F(xa) = 0. When F(;x") = 0, the field equations (1) in the synchronous coordinate system are equivalent to the generally covariant equations9

    Rab - iRgab = 0, R + 4Je2/h2 = 0. In combining (21) with (7) one sees immediately that the total energy density e of the "perfect fluid" in

    JOURNAL OF MATHEMATICAL PHYSICS

    e = r + p = 2 (xa)/y. (29) Thus e is a function of the components of the metric field ga.P' IX, fJ = 1, 2, 3, and does not depend on the first and second derivatives. The same is true for the proper material density r and pressure p.

    Permanent address: University of Windsor, Windsor, Ontario, Canada.

    1 A. Eddington, Relativity Theory of Protons and Electrons (Cambridge U.P., Cambridge, 1936).

    E. Schrodinger, Letters on Wave Mechanics (Vision, London, 1967), p. 53.

    3 J. Kulhanek, Nuovo Cimento SuppJ. 4, 172 (1966). J. Kulhanek, Nuovo Cimento 63B, 497 (1969). 5 A. L. Zelmanov, DokI. Akad. Nauk. USSR 107,15 (1956). 6 L. P. Eisenhart, Trans. Am. Math. Soc. 26, 205 (1924). 7 C. Ricci, Atti Reale 1st. Veneto 63, 1233 (1904). 8 J. L. Synge, Proc. Math. Soc. (London) 43, 376 (193...