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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
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@ 1991 American Institute of Physics
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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
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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
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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
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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
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[ 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
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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
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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.
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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.
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