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Establishing the Riemannian structure of space-time and free matter waves by means of light rays Jiirgen Audretsch and Claus Lammerzahl Fakultiitftir Physik der Universitiit Konstanz, Postfach 5560, D-7750 Konstanz, Germany (Received 13 November 1990; accepted for publication 6 February 1991) As compared with axiomatic schemes based on the paths of massive point particles, a constructive approach based on the classical limit of the wave mechanics of matter fields has a particular advantage: The locally approximately plane waves do not only define matter rays, but can in addition be subject to interference measurements. These, in contrast to the free-fall experiments, are sensitive to the mass. This makes it possible to further specify the geometry of space-time. Light rays reveal a conformal structure of the space-time manifold. It will be shown that if, in addition, in each point of space-time locally approximately plane matter waves can be realized that are free and massive, the space-time is a Riemann space. I. INTRODUCTION A. Space-time axiomatics The pseudo-Riemannian manifold of General Relativity is commonly accepted as the best mathematical model to describe space-time and the geometrized gravitation. A physical axiomatics of space-time should not postulate this particular geometrical structure from the beginning, but should make it a derived concept. A physical explanation and motivation should be given in basing the axiomatics on more fundamental physical experiences and in relating these experiences with the various geometrical structures of a Riemannian manifold. In this constructive approach one starts in a geometry-free way. The procedure is to discover and to describe by means of the behavior of appropriately selected physical systems (called primitive objects) in particular physical effects (taken as basic experiences) the geometrical structure of space-time. Ehlers, Pirani, and Schild (EPS) ’ have proposed a constructive space-time axiomatics based on light rays and freely falling classical point particles as primitive objects. For a further elaboration of this approach, see Coleman and Korte.’ Based on these primitive objects and appropriate basic experiences it is possible to construct a manifold and to provide it with a compatible conformal and projective structure thus assigning a torsion-free Weyl geometry to spacetime. In this scheme, geometry remains non-Riemannian, the transport of time intervals will in general be path dependent. An additional Riemannian axiom breaking the internal coherence of the scheme is needed to exclude the second clock effect. This is a deficiency of the EPS approach. In enlarging the EPS scheme by including additional experiences, this axiomatics can be completed in a twofold way: If the massive particles carry in addition a polarization direction, it has been demonstrated by Audretsch and Lgmmerzahl” that the space-time manifold can be endowed with a totally antisymmetric contorsion. Furthermore, the decisive final step from a Weyl geometry to a Riemann-Cartan geometry can be done by adding rudiments of quantum mechanics to the scheme. The reason for this is that quantum mechanics must include the classical mechanics of freely falling point particles as a limiting case. It has been demon2099 J. Math. Phys. 32 (8), August 1991 strated by Audretsch4 (see also Audretsch et al.’ ) that the self-consistency requirement that this limiting case should agree with the behavior of classical test particles as it is postulated in the EPS scheme, reduces the Weyl-Cartan space to the intended Riemann-Cartan space. That it should be possible to obtain this reduction in discussing quantum mechanical wave equations for massive particles has already been conjectured by Ehlers.6 In connection with Ref. 4 it has been stressed by Kasper’ and Ehler? that it is the appearance of a mass function in the quantum mechanical framework in Weyl space that makes it possible to introduce a Riemannian structure. 6. Matter waves as primitive elements In the following we want to contribute to a deeper understanding of the role of nonclassical matter in space-time axiomatics. Our intention is to propose postulates that lead not to the preliminary stage of a Weyl structure first, but directly to a Riemannian structure of space-time. To do so, we will not enlarge the EPS scheme, as done in all the articles cited above, but instead modify it. Torsion will thereby not be discussed. We overtake the conformal structure of the EPS scheme and replace the part referring to freely falling test particles by considerations related to the physics of free plane wave matter waves. In this sense we will introduce rudiments of quantum mechanics from the very beginning. The physics of free fall, which is in the EPS scheme a postulated behavior, will now be one of the consequences. Following this general idea, it has already been shown by Audretsch and Ltimmerzahl’ that space-time, as it is defined by light rays and free matter waves, is a Riemann space. Free matter waves are thereby not introduced as a limit of quantum mechanical field equations, but as a description of matter in its own right. The basic experiences refer essentially to interference experiments. Below we want to improve this type of approach in considering plane matter waves as a particular limiting case of wave mechanics defined by a general field equation in a manifold with a conforma1 structure. Accordingly, we refer to quantum theory in the SchrGdinger picture and position representation. The 0022-2488/91/082099-07$03.00 @ 1991 American Institute of Physics Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 2099 full mathematical scheme of quantum mechanics is not needed. Candidates for the corresponding relativistic field equation could be the generalized Dirac or Klein-Gordon equation. For our use they must be formulated in a (3 + 1) -manifold with a conformal structure. But this generalization of the special relativistic equation leads to difficulties and ambiguities with regard to the covariant derivatives and the mass parameter. In addition it seems to be difficult to justify operationally with reference to basic physical experiences the particular generalized form of the relativistic field equations. To overcome all these problems of physical motivation, we follow a different strategy: We take as field equation for the vector valued complex field the most general linear system of partial differential equations of arbitrary order r. A physical justification for this description of matter has already been given in Audretsch and Limmerzahl’” in a constructive axiomatic scheme based on experiences which are read off from the large amount of experiments one has made in the framework of quantum mechanics with matter waves. It is demanded that these fields obey a superposition principle and show a deterministic evolution which is local. This paper’” accordingly forms a part of the space-time axiomatics based on matter waves as test fields and preceedes the arguments given below. With regard to the conformal structure, we will still make use of the results of the first part of the EPS axiomatics, which uses light rays as primitive elements. It therefore remains the task to derive the conformal structure within our overall scheme in relating it to the behavior of matter waves, i.e., to the general linear field equation mentioned above. See Lammerzahl” for a possible approach. This would close a gap, making the considerations below the final part of one coherent scheme. C. Geometry and quantum theory The fact, that our approach will be based on very general matter waves as compared to massive point particles, has disadvantages and advantages: Matter fields, although defined in configuration space, will in general not represent localized parts of physical reality. They are not directly related to local physical experiments like, for example, electric fields or point particles, the quantum mechanical probability interpretation prevents this. Nevertheless, there are today many experiments that refer rather closely to the matter fields mainly in the framework of interference experiments. This disadvantage is largely balanced by the fact, that a space-time axiomatics based on matter fields give a more plausible answer to the question: Is there a classical spacetime underlying microphysics (above the Planck-limit), does it have a Riemannian structure, and how can this operationally be justified? An axiomatics based on light rays and point particles is well adjusted to domains that may be explored with the help of radar signals and satellites. It needs different primitive elements and related fundamental experiences to give physical reasons to describe quantum mechanical and quantum field theoretical processes in strong gravitational fields on the bases of a Riemannian geometry of 2100 J. Math. Phys., Vol. 32, No. 8. August 1991 space-time. It is the aim of this paper to contribute to this approach. II. THE CONFORMAL METRICAL STRUCTURE A. The eonformal structure In our approach to the space-time structure we will overtake from EPS only the axiomatic introduction of a differentiable manifold as arena for the physical events and the introduction of the conformal structure. The latter is established by means of light rays, which introduce the causal relations of events. That means that we have a four-dimensional differentiable manifold & (for ways to establish such an +,r? see, e.g., EPS’ or Schroter’ ) as set of event called space-&m with a light cone structure on it, which is related to an equivalence class of conformally related metrics: [ g;w (x) I : = &(x)&,(x) = en’“‘g;,(x),s2EiR), (2.1) with det gi, (x) #O. We chose the signature to be - 2. A metric can be constructed by means of light rays and a particle path with some parametrization. By chasing another parametrization or another particle path, the construction yields the same metric up to an overall factor thus leading to the above conformal equivalence class of metrices. With this class of metrics one can distinguish between spacelike, lightlike, and timelike vectors and onecan introduce the notion of an angle, especially of orthogonality. Certain types of measurement are therefore possible already at this stage of the development of the axiomatic scheme. We will make use of this fact. Any transformation of a tensorial object of the kind A .+A ‘(x) = p(A)-)/4 (x) (2.2) in combination with a respective transformation of the metric g,T, -& mation (x) = p(x) g(x) c is called a conformal transfor- with w(A) being the weight of the tensor A; cu(gi,.) = 1. All physical equations have to be written in a conformaily covariant way. 6. The light rays For light rays with tangent vector l@(x) we have giV (x)Z~(x)/ “(x) = 0. Furthermore, it can be shown that the light rays are derivable from the gradient of some scalar weight function with w(T) =o: T(x) I’“(x) -gV(x)d, &‘(x)g$ T(X) whereby gkx) is defined by (x) = 6. This leads to the eikonal equation, i.e., to the Hamilton-Jacobi s;;y(x,a, equation for light: T(x)d, T(x) = 0. (2.3) By partial derivation of this equation we arrive at the equation for the propagation of light rays: J. Audretsch and C. LiImmerzahl Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 2100 lq‘l~+ IVP p I;‘p-,y (2.4) with P c: = +&(a& 11 v which is a conformally + a&?:,.- c&g:,1, invariant equation. B. The classical limit C. Local Lorentz transformations Having a class of metrics gi,. available, it is possible to construct in every point orthotetrads & which are normalized with respect to the chosen metric: qmn = gi,,&i;, *=diag( + I,- l,- I,- 1). The tedrads .& are of 77lVtl8’ weight - 4. By means of the orthotetrads one can define Lorentz vectors A “‘: = ::A p(,etc. These tedrads can be used to define local Lorentz transformations as those transformations leaving the length of Lorentz-vectors invariant: = vn L k L i. Operationally, the orthotetrads represent 77mn one timelike and three spacelike orthogonal directions, to which measurements may be related. b All considerations below will also be local and refer to an event and its neighborhood. III. WAVE MECHANICS AND ITS CLASSICAL LIMIT A. Field equations As primitive objects we take fields p, represented by vector-valued complex functions p: M -+ C”:-(x), with which the following basic experiences can be made: The field p(x) shows a deterministic evolution that is local and obeys a superposition principle. We postulate this behavior and call these fields matterfields because they will be the nongeometrical fields on the manifold. The respective set is denoted by 3. Typical representatives of these fields will be found in the context of quantum mechanics in the Schriidinger picture and the space representation. Accordingly, we will refer to the physics of the fields .F as wave mechanics. It has been shown in Ref. 10 that the elements ofY obey a linear system of partial differentialfield equations: j~o~,...“(x)(~lll...~~,~)(x) =O, (3.1) with gp: = - ia,, and appropriate functions f”“‘J(x) which are complex s x s matrices in C ’(&,@“). One set of f”“‘J(x) refers to a particular type of fields. This type is in a physical realization for example related to one sort of quantum objects like neutrons or electrons in certain situations that may be characterized by the presence of external fields. These external fields are contained in the functions f”‘yx). Equation (3.1) is written down in an arbitrary coordinate and C’ base system. By performing a coordinate and base transformation XI-+X’ =f(x), w,‘= hq, with &Gl(s,C), we get a new differential equation of the same 2101 J. Math. Phys., Vol. 32, No. 8, August 1991 type with new coefficients J”;““‘; which can be calculated from the old one. Thereby all coefficients but the one of heighest order f’““l’(x) transform inhomogeneously whereby the inhomogeneous terms depend on second derivatives off resp. on derivatives of A. We base our axiomatic scheme on basic experiments that can be made in the classical limit of wave mechanics. By classical limit we denote the physics of locally approximately plane-wave solutions of (3.1). Such a solution can be decomposed approximately according to p = ae is into a “slowly varying” amplitude a and a phase S, so that all terms containing at least one derivative of the amplitude or at least the second derivative of the phase can be neglected. Formally this means we have the following definition. Dejnition: A solution p of (3.1) is a locally approximately plane matter wave in x&-briefly called plane matter wave-if within an appropriate neighborhood of X&J? there is a field of c’bases and a coordinate system * as well as functions S&‘(,,M,R) and a&‘(,,&,@‘) so that it can be represented as p(x) = a(x)e”‘“‘, (3.2) with Il~~‘“)“‘~A~(~~*,,...a,ils, (a& IAl . . dq.9 t ~~-~p,a/l$ll~~~,(3.3) *. . %A, , -.*Jpk,S)~pq+ with k, = 1, k, = q for allj = l,..., r, and q <j or at least for one l< A<l: k, - k A _ ,>2, if a is represented with respect to the field of @ ”bases. S is of weight zero. Hereby, llalj is some norm in C’. Equations that are valid only in this special base and coordinate systems are marked byan*. It should be noted that our classical limit in contrast to the WKB scheme is no expansion with respect to the Planck constant fi, because it remains unspecified if and in which way the field equations (3.1) contain such quantity. C. The Hamilton-Jacobi equation We are looking for equations that must be valid in a space-time region if it admits plane matter waves. Inserting (3.2) into the field equation we get in the coordinate system * using (3.3) j$o G?(x)pp, ’* ‘pcl,a l 0, (3.4) where PP *=a,s . is called the momentum of the plane wave. The coefficients l<&“‘(x) are defined to be equal to the coefficients f”““‘(x) of the field equation (3.1) in the special base and coordinate system *. The transformation properties of the coefficients J,‘(\‘,“‘(x) can be derived by means of the requirement, that after a base and coordinate transformation the transformed J. Audretsch and C. LPmmerzahl 2101 Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp equation (3.4) must have the transformed amplitude a’ as solution. At first we consider coordinate transformations. a is invariant under coordinate transformations, pp is a covariant vector and no derivatives occur in (3.4). Accordingly, it is clear that the $‘;‘P1(x) transforms like a tensor under coordinate transformations. For C-base transformations the above requirement means that an amplitude a’ obtained with respect to a in a new Chase by means of a linear transformation M’ = Aa solves the equation i j=O y;E;“‘“‘( x)ppl * * *pp,a’= 0 while a solves (3.4). Inserting a’ one gets 2 $;++)pr, j-0 - - -ppla = 0 e i y;‘d;“‘“wP~, * * *p&a j=O = 0. Therefore i j=O y;gywp,, - +.p/,,A = K i $‘,“%)P~, * * ‘Pgi j=O with an arbitrary nonsingular matrix K. We can chose K = A. Since this is true for allp we deduce y;‘d;+(x) = A<;;“‘“,(x)A - ‘. Therefore, the coefficients “/&‘P1(x) of (3.4) transform homogeneously and agree with the coefficients “/l’““l’(x) of (3.1) in the special base and coordinate system in which (3.3) is valid: $‘;‘P1(x) r f”“P1(x). To sum up, we have the result that (3.4) transforms homogeneously under base and coordinate transformations. In such regions where there is a classical limit, the amplitude a cannot vanish, so that the solvability condition = det i y$;“J(x)p~, . . -pp, = 0, (3.5) ( j=O > must be fulfilled. This equation, the Hamilton-Jacobi-equation, is a polynomial of order rs in the momentump, . Note that it does not contain derivatives of pp. Equation (3.5) corresponds to the eikonal equation of geometrical optics. In a sense we treat light and plane matter waves on the same footing. But with regard to plane matter waves we refer in addition to interference experiments. For given p@, ,G = 1,2,3, Eq. (3.5) can be (not necessary uniquely) solved forp, = f( x,p$ ). Equation (3.5) is a complex equation and is invariant against coordinate transformations and transformations of the c‘ bases. H(x,p): Having already established a conformal structure on the manifold and the related equivalence class of metrics r 7 giV (x) , some elementary measurements are operationally I I possible. For two different plane matter waves Q,and P ’the ratio of the related Lorentz componentsp,/p: has an invariant meaning and represents a measuring quantity related in the usual way to the phase function S(x) and the succession of hyperplanes of constant phase. For index a = 0 it comesponds to a ratio of frequencies, for a = & it COn%spon& to a ratio of wavelengths and a propagation direction relative to the orthotetrad. On the basis of this, it is possible to formulate in postulates 1 and 2 below two basic experiences made with a sub class ACf of 3 containing the plane matter waves that me free. The first basic experience is the following: In an event and its neighborhood it is possible to find a plane matter wave and to arrange the experimental setup in such a way, that active Lorentz transformations (rotations and boosts with regard to a given orthotetrad) transport the whole arm rangement including the plane wave into an equally possible arrangement. As in corresponding experiments with free point particles (as opposed to interacting particles), this may in practice need some shielding. Loosely speaking one could say that the following is demanded: If all the direction dependent external influences that can be eliminated are indeed eliminated, then that what remains as structure allows that an active Lorentz transformation of the experimental setup leads to one which can also physically be realized. This characterizes the “remaining structure” which is related to the geometry of space-time. Postulate I: Given an event and a free plane matter wave with momentum p,, then the momentum pa* in this event active Lorentz transformation obtained by an p,HpOf : = L tpb belongs to an equally possible free plane matter wave: ii(x,L fulfilling ipb) .= 0 VL i fulfilling ii(x,p, L :L $jM = rlaC,Vpb ) = 0. Therefore, &x,p) = O&x,Lp) = 0. For free plane matter waves the respective Hamilton functions must be invariant under Lorentz transformations. This has the important consequence that according to the fundamental theorem of vector invariants of the Lorentz group, H(x,p) can only be a function of 2”( x )p,,p, . In this case the polynomial Hamilton-Jacobi ture mr/2 H(x,p) = n equation ( 3.5) must have the strucE W”(X)P~P~ - v,,, (xl), (3.7) k-1 D. Local Lorentz isotropy with some complex scalar functions Vi (x) that do not de- By means of the orthotetrads defined in Sec. II it is possible to formulate the Hamilton-Jacobi equation in terms of pend on p; w( V, kj ) = ~(2”) thep,: = kp,,, that is Hx,pP 1 = H(x,:;p, 1 = :%x,p, This equation can be solved forp,,, 2102 1. : =j(x,pb), J. Math. Phys., Vol. 32, No. 8, August 1991 (3.6) B = 1,2,3. = - 1. These functions de- pend in a complex way on the f”““‘*‘(x) of (3.1). Anticipating results below, we will call the vCkj (x) scalar mass potentials of the field a(x) . Whenever locally approximately plane-wave solutions of (3.1) are possible that are in addition free, they must fulfill (3.7) and these mass potentials J. Audretsch and C. Li’immerzahl 2102 Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp [ 1 of metrics they must exist. Together with the class 2”(x) determine the phase functions S(X) as solutions of (3.5). The functions 2”(x) and v(k) (x) together characterize I 1 the geometry of the classical limit of ( 3.1) . The role of the following postulate is to further specify the mass potentials, thus simplifying the structure of this geometry. By grouping together identical factors, we can write H(x,p) = fJ (H(i) (X,P))““‘~ ,=I with tentials p(k) (x) in the classical limit. This makes it possible that the mass ratio between different fields Vcl, (x)/Fe, (x) could still be space-time dependent. Also this effect has never been observed. Accordingly, we demand the universality of the mass function. Postulate 2: For free plane matter waves the ratio of any two scalar mass potentials proves to be constant. For Vci, (x) = 0 the relation H,,, (x,p) = 0 takes the form (2.3) describing light rays. Fields with at least one V~i, (x) #O will be called massive. Because of postulate 2 one can take for massive fields one of the potentials as universal function Vco, (x) #O and write for the other potentials (4.1) with real positive constants rnci) of weight w( m ci) ) = 0. The 1 v(i) H(,, (x,p)’=~“‘(x)p,p,, - V(i,(X) = 0, (3.8b) whereby the powers have to fulfill Since 2”(x) and p/l are real, V,, (x) must be real too. Since the highest-order part of the inhomogeneous polynomial (3.5) must coincide with the characteristic equation of (3.1), we remark that the characteristic equation is (~“k,k,.)“‘2 = 0. IV. CONSTANCY OF THE RATIOS OF MASS POTENTIALS AND RIEMANN SPACE Postulate 1 is not sufficient to single out “free” waves, because only direction dependent influences are excluded. From the point of view of Minkowskian physics the potentials Vc,, (x) may still contain in addition to mass parameters the contributions from isotropic external fields. To complete the characterization free, we must describe the physics obtained after a successful shielding of these directional independent influences too. The influence which is commonly called the gravitational one, cannot be shielded and is therefore contained in the geometry of free plane matter waves. It is well known and has been demonstrated in the COW type experiments (for a review see, e.g., Werner I2 ), that interference of plane matter waves in gravitational fields lead to mass dependent results. This is in contrast to the behavior of free test particles on which, according to the equivalence principle, mass has no influence. This sensitivity with regard to mass make matter waves superior to test particles as primitive objects in a space-time axiomatics. Returning to our axiomatic scheme this means that additional information can be extracted from interference phenomena. One type of physical field e, characterized by the respective field equation (3.1) may have different scalar mass potentials V,,, (x), V,,, (x),... . We take as basic experiences with free plane waves that the ratio V(,, (x)/VU, (x) of these potentials, which is of weight zero, turns out to be the same in every space-time event. However, there may be other physical fields @ in the world fulfilling a differential equation (3.1) with other coefficients ?“.“I, leading to different po2103 J. Math. Phys., Vol. 32, NO. 8, August 1991 (xl ( = weights of&,(x) m:i) 1 v(o) (Xl 19 and VO, (x) agree. It is the universality of the relation (4.1) that guarantees the absence of external (in the sense of nongravitational) influences. Note that negative Vci, (x) indicating tachyonic behavior are not excluded. The constants mci, may be called musses. One field equation ( 3.1) can lead to several masses. Supplementing the postulates for the conformal structure by postulates expressing basic experiments with plane matter waves that are free and massive, amount to the result: The space-time manifold J? is endowed with a class of met- [ 1 rics &, (x) and a universal mass function F’,,, (x). Taking the respective behavior under Weyl transformations into account, we can therefore define a unique metric g,,(x): = [w-(O) (x&,(x), (4.2) and the manifold <LXbecomes a Riemann space. The family of phase surfaces S(x) = const. defines the dual concept of a worldline which is everywhere orthogonal to the surfaces. We call this a matter ray. Its tangent vector is given by t a-gafipn= - gafiaDs. From Hci, (xq) (4.3) = 0 we obtain with (4.2) rw)P,P, - 4,) = 0. (4.4) Equations (4.4) and (4.2) imply that rays follow a Riemann geodesic: at x+a i :A I tKtA-t”. (4.5) Here, { } denotes the Christoffel symbol. Equation (4.5) supplements the propagation equation (2.4) for light rays. In this sense light rays and the classical limit of wave mechanics define the affine structure of the Riemann space. V. MATTER RAYS AS PATHS OF WAVE PACKETS In the approach above, operational realizations of physical interpretations refer to the surfaces of equal phase S(X) = const. in the neighborhood of an event as represented by the momentump, = - a$. Accordingly, it is characteristic to an approach based on matter waves that physics refers to the cotangent space. The rays that represent paths J. Audretsch and C. Uimmerzahl Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 2103 revealing the affine structure of the Riemannian space-time, still lack a physical realization within wave mechanics. We want to supplement this and to provide at the same time the basis of an alternative approach that could replace Sec. IV in our axiomatic scheme. To do so we have to study wave packets. Wave packets are constructed in superposing locally approximately plane waves (3.2) characterized by (3.3) as they are determined by (3.4) and the Hamilton-Jacobi equation (3.5). We take waves corresponding to values ofp out of a small p-interval and show that SH( xg)/dp, can be interpreted as group velocity of the resulting wave packet. The following considerations are quite general and do rely only on the following conditions: (i) the Hamilton function N( xg) is solvable for p0 = pO (x,pi, ) and (ii) p0 is diffentiable with respect top,. [In the case of (3.8) these conditions can easily be achieved by defining H ‘(x,p) = nc~ 1 H(i) (x,p)a ] A plane wave can be written as p(x) = a(x,p)exp( - i .J; pP dX) where in the case of contractible regions the path of integration is arbitrary and p resp. a are solutions of (3.5) resp. (3.4). For an x lying in the vicinity of an event with coordinates E we may approximate [a?: = a, + app,(a/apv)]: p(x) zp(% + !a&9 cfm+ = a@(~,,i) f [icaps, Wa@W,E) - ipp (R)a(p(Z),Z)Sx~ ze - “,J”‘“h@(ji),& (5.1) Since Q”is assumed to be a locally approximately plane-wave characterized by (3.3), the derivative of the amplitude can be neglected with respect to the amplitude. Here, p(Z) is a solution of H(E,p(E)) = 0. A complex solution pO may give an additional factor e lmp&” which according to the sign describes damping, so that after some time the waves disappear, or exponential growth. We exclude these solutions. A wave packet is a superposition of local plane waves momentum p lying within an with interval I: = [jj - Ap(X),P + Ap(Z3) 1: P(X) = A(S 4 &(33) sI x e - i<i&% + 4@))stid sspe (5.2) A( Sp) is a momentum weight factor as for example a Gauss function, which will be left unspecified, We separate in (5.2) the factor e - ri%,cn’sfiand make use of the conditions (i) and (ii) above in writing 8P* (~~)sp$3x” + 6p$xt 6p,Sxfi = dP, This results in the relation 2104 J. Math. Phys., Vol. 32, No. 6, August 1991 -t-SpGLZ) aP0 ap (9 )&co + Sxfi P Xexp d 3Sp Xexp( - IP~SX’“), (5.41 which is of the structure q(x) = B(&Z,Sx)e-“u6ti. (5.5) Here the second factor represents the supporting local plane wave for this wave packet, while the first determines the shape of this packet. For determining the motion of this wave packet, we consider the set of events for which B@,K,Sx) is constant. This especially includes the motion of the maximum of the wave packet. These events are given by the condition that the factor in B containing the space-time variable Sx is constant: const = aP0 (pfi 1 sx* + SXE”. (5.6) dPli The tangent vector to the corresponding path is ur” d&xc’ =-z with some parameter T. Because of (5.6) we have &I -vO+d=O. ap, On the other hand we can infer from 0 = (d ““/a& (that is regardingp, as function ofpc, ) + (d~a)(p(X),X)]& 32(p(2),E) $0) = I I A(S o-afi&o-+-. (5.7) (5.8) ) N( xg) m (5.9) 8Pcl dP, aPa We fix the parameter T in demanding g=.-- 1 aI (5.10) r ?.po with an appropriate r. Herewith and combining (5.8) and (5.9) we then finally obtain 1 mx,p) (5.12) I H(X.P)= 0 ah By the procedure above the paths of wave packets defined as paths of characteristic points like the maximum are locally defined; d of (5.1 f ) is the tangent vector to the respective wordlines and represents thegroup velocity of the wave packet. Because of Postulate 1 that introduces the local Lorentz invariance we have in our scheme the particular HamiltonJacobi function of the structure (3&S). Accordingly, we fmd a group velocity and the related path structure for every scalar mass potential Vti, (x) : zfY'(xs) =I u”, (1)(xp) 9 =_L aH’(xdJ) Y aJ-5 I H(,,Lw = 0 (5.12) It turns out to be related to the momentum pF according to (5.3) lJ;i) (xtP) -~~'(X)P,IH,,,(*.p~=O' (5.13) This represents the intended mapping between cotangent J. Audretsch and C. LSLmmerzaht Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 2104 and tangent space. In Sec. IV we reached the conclusion that in the free case the scalar mass potentials Vci, (x) is essentially the only one. The matter rays introduced in (4.3) obtain a physical realization by the paths of characteristic points (e.g., the maximum) of wave packets: We have t p - d. VI. AN ALTERNATIVE APPROACH The interpretation of Postulate 2 has been based on p,, and therefore on the phase function S(x). In Sec. V we have established a relation to the motion of wave packets. This makes it possible to reformulate the Postulate 2 with reference to wave packets, thus giving these postulates a new operational meaning and illustration of its physical content. There may be more than one scalar mass potential I’(,, (x). To each of these we can attribute with the help of the class of metrics function gy;; (xl: = [l/V,/, [ 1 of the conformal structure a k”(x) (6.1) (x)]2w, of weight zero. Specifying the group velocity according to (6.2) VT,, = g;fP@ = - lT%w we obtain with (3.8b) for the paths of the wave packets a geodesic equation P v1’,,4.C) + i VTI (1)Lq, q;, = 0, whereby the Christoffel symbol {‘:,}, i) is built from the @l,‘;. Wave packets follow Riemannian geodesics but for every scalar mass potential a different one. Because a negative Vci, is allowed, tachyonic wave packets are formally not excluded. For all the consequences below, it will be enough that the timelike paths of wave packets can physically be realized. We add now as basic experience that free wave packets obey the weak equivalence principle (universality of free fall) in demanding the following postulate. Postulate 2’: Wave packets out of masssive free plane matter waves follow the same paths. For two scalar mass potentials V(,, (x) and V,, (x) we have Eq. (6.3) for i and i. The relation between the two Christoffel symbols is + (&d,A + &P,A - ~,dP~p~)~ (6.4) with A: = In ( V,,,/V,, ). If equations with different indices should lead to the same path, we have to demand @ “a,. In ( V,,, /VU, ) = ad for all v”. This can be true only if (6.5) d,,W(,,/Vcj,) =o. We have therefore obtained Postulate 2 of Sec. IV as a consequence. The final result, that the space-time is a Riemann space follows now directly. 2105 J. Math. Phys., Vol. 32, No. 8, August 1991 VII. RESULT We base our analysis on basic experiences with fields showing a deterministic evolution which is local and obeys a superposition principle. These fields are called matter fields. Light rays reveal a conformal structure of the spacetime manifold. If in addition in each point of space-time locally approximately plane matter waves can be realized which are free and massive, then the space-time is a Riemann space. In other words, light rays and free massive plane matter waves together define a Riemannian geometry. Quantum mechanical fields related to neutrons, electrons, and other elementary particles have these properties and show a classical limit as described in the postulates above. The class of primitive objects of our axiomatics is therefore physically not empty. Postulates 1 and 2 refer to matter waves that are free. It needs therefore an additional hypothetical extrapolation to attribute a Riemann geometry to regions of space-time in which nonfree matter waves are influenced by external fields (for example, in the interior of an atom). Although this extrapolation is not contained in our postulates, it seems to be plausible on the background of our scheme: external fields will be introduced as causes for deviations from the free behavior. The latter must therefore be presented in the field equations. This makes it plausible to base all quantum mechanical and quantum field theoretical processes on a Riemann geometry of space-time. ACKNOWLEDGMENTS This work was supported in part by the Deutsche Forschungsgemeinschaft and the Commission of the European Communities, DG XII. We thank members of the Relativity Groups in Brussels and Buenos Aires for valuable and inspiring discussions. ’J. Ehlers, F. A. E. Pirani, and A. 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Ehlers, “Einfuhrungder Raum-Zeit Struktur mittels Lichtstrahlen und Teilchen,” in Philosophie und Physik der Raum-Zeit, edited by J. Audretsch and K. Mainzer (BI-Wiss-Verlag, Mannheim, 1988), p. 145. 9 J. Audretsch and C. Lammerzahl, Wiss. Zeitschrift der Universitlt Jena, Naturwiss. Reihe, Heft 1 ( 1990). “‘J. Audretsch and C. Lammerzahl, J. Math. Phys. 32, 1354 (1991). ” C. Lammerzahl, “The Geometry of Matter Fields,” in Quantum Mechanics in Curved Space-Time, edited by J. Audretsch and V. deSabbata (Plenum, New York 1990). “S . A . Werner, “Neutron Interferometry: Macroscopic Manifestations of Quantum Mechanics, ” in Quantum Mechanics in Curved Space-Time, edited by J. Audretsch and V. deSabbata (Plenum, New York, 1990). J. Audretsch and C. Ltimmerzahl 2105 Downloaded 29 Jul 2007 to 134.102.236.219. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp