Journal of Modern Physics, 2013, 4, 963-967
http://dx.doi.org/10.4236/jmp.2013.47129 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Emergence of Space-Time and Gravitation
Walter Smilga
Geretsried, Germany
Received May 1, 2013; revised June 3, 2013; accepted June 29, 2013
Copyright © 2013 Walter Smilga. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In relativistic quantum mechanics, elementary particles are described by irreducible unitary representations of the Poin-
caré group. The same applies to the center-of-mass kinematics of a multi-particle system that is not subject to external
forces. As shown in a previous article, for spin-1/2 particles, irreducibility leads to a correlation between the particles
that has the structure of the electromagnetic interaction, as described by the perturbation algorithm of quantum electro-
dynamics. The present article examines the consequences of irreducibility for a multi-particle system of spinless parti-
cles. In this case, irreducibility causes a gravitational force, which in the classical limit is described by the field equa-
tions of conformal gravity. The strength of this force has the same order of magnitude as the strength of the empirical
gravitational force.
Keywords: Conformal Gravity; Quantum Gravity; Emergence of Space-Time
1. Introduction
As a general rule of relativistic quantum mechanics, not
only elementary particles, but also compound systems of
particles are described by irreducible unitary representa-
tions of the Poincar é group, as long as no external forces
act on the system.
Within a two-particle state, irreducibility of the repre-
sentation that describes the center-of-mass kinematics,
causes a correlation of the individual particle momenta.
In a previous article [1], the author has shown that for
spin-1/2 particles, the quantum mechanical formulation
of this correlation takes on the structure of the electro-
magnetic interaction, as described by the perturbation
algorithm of quantum electrodynamics. The coupling
constant, derived from the geometrical properties of this
correlation, was found to be in excellent agreement with
the experimental value of the electromagnetic fine-
structure constant. This agreement emphasizes the crucial
role of irreducibility for the kinematics of quantum me-
chanical multi-particle systems.
Irreducible representations of the Poincaré group are
labeled by the values of two Casimir operators and
(see, e.g., [2]) P
W
,Pp p


p
 (1)
where

is the total 4-momentum of the system, and
1
,withWww w
which refers to the angular momentum of the system.
,
2Mp

 


p pm
12
ppp
 (2)
Whereas the previous article was primarily based on
the first Casimir operator P, the present article will
concentrate on the second Casimir operator W. This
operator is related to the intrinsic angular momentum of
the two-particle system, generated by the relative motion
of the particles.
Let 1 and 2 be the 4-momenta of two particles,
for simplicity with equal masses , so that


12
,qpp
(3)
denotes the total momentum and


p q
0.pq
(4)
the relative momentum. Then and satisfy
(5)

Based on Equation (5), a two-particle system can be
described by a total momentum and a spacelike
momentum , perpendicular to the timelike vector .
“Perpendicular to a timelike vector” means that is
allowed to rotate by the action of an SO(3) subgroup of
the Lorentz group. So the kinematics of the relative
momentum is restricted to a 3-dimensional subspace of
space-time.
p
q p
q
P

2
22
12 ,pppM 
For an irreducible two-particle representation, we ob-
tain from the constancy of the Casimir operator
(6)
C
opyright © 2013 SciRes. JMP

W. SMILGA 964
M
is the “mass” of the two-particle system, and where
222
40,M
22
22M m
2
12
22 .ppM
p p
p
q
W
ml
qm (7)
and further
12
pp (8)
and
pp (9)
Equations (8) and (9) correlate the particle momenta
by fixing the angle between them and with respect to the
total momentum . Rotations with rotational axis
preserve these angles. Since these rotations leave in-
variant, they can be related to an independent, internal
degree of freedom, described by an action of SO(2) on
the relative momentum .
In the quantum mechanical description, this SO(2)
symmetric degree of freedom corresponds to the internal
angular momentum of the two-particle system. In an
irreducible representation, the second Casimir operator
has a well defined value, which requires that also the
value of this angular momentum is well-defined. From
the quantum mechanics of angular momentum, we know
that for large quantum numbers the property of spherical
symmetry does not fade away, but is preserved in the
sense of an SO(2) symmetry. In [3] we find an approxi-
mation to the spherical harmonics for large angular
momenta , which for results in the probability
distribution
l
 

2
sin ,
l
A 
2
,d
ll
Y   
 (10)
describing a clearly defined closed circular orbit.
Therefore, within an irreducible two-particle represen-
tation of the Poincaré group, the existence of a classical
“Newtonian” limit, where the relative motion of the
individual particles is straight and uniform, must be put
into question.
In the sense of Newton’s first law [4], Corpus omne
perseverare in statu suo quiescendi vel movendi uni-
formiter in directum, nisi quatenus a viribus impressis
cogitur statum illum mutare, a circular orbit is to be
understood as the result of an attractive force between the
particles. Such a force of apparently universal character
has not been seen in the experiments of particle phy-
sics—or perhaps, for some reason, it has been ignored.
This article is intended to find out more about this
force, which obviously is the outcome of a combination
of quantum mechanics and relativistic invariance.
2. Parameter Space-Time vs. Physical
Space-Time
Our analysis starts with a review of the role of space-
time within the formalism of quantum mechanics.
Given an elementary particle, described by an irre-
ducible representation of the Poincaré group in a state
space with eigenstates p p of the 4-momentum ,
then states “in space-time” can be defined by superpos-
ing these momentum eigenstates:

3
3
2
0
d
,2e, 1,2,3,
2
ipx
tpk
p


p x (11)


,
x
with parameters tx. A detailed discussion of
these states can be found in [5]. See also [6].
x
form a parameter space (The parameters
x
space)
with the same metric as the energy-momentum space (
space). The states p
,txtxx are “localized” (within a
Compton wave length) at time 0 at the point of
three-dimensional space. So we can say that the
x
space has also a “physical” meaning in the sense that it is
a space in which (isolated) particles can be physically
placed.
Note that Definition (11) does not require a prior
existence of space-time. It rather defines space-time on
the basis of the momentum eigenstates. We also define a
position operator in three-dimensional space by
.
kk
Xi
p

   (12)
The definition of a corresponding “time” operator does
not make sense, because the states (11) cannot be “loca-
lized in time.” Therefore, time is not an observable, but
merely a parameter. By Definition (11), space-time is
derived as a property of matter, just as momentum is
considered a pro perty of matter.
The relation between
x
space and space contains
Planck’s constant . This is the result of having in-
dependent scales for
p
h
x
and . We can avoid this con-
stant by replacing by the wave vector , defined
by
p
pk
pk

, which in this context may be a more natural
choice.
Now consider two elementary particles, described by
an irreducible two-particle representation of the Poincaré
group. Because of the constraints from the two-particle
mass shell relation Equation (6), it is not possible to
simultaneously construct localized states for each particle.
Therefore, when two or more particles are considered,
the possibility of individually placing the particles in
x
space may be lost, but
x
space still can serve as a
useful parameter space, e.g., for wave functions. So we
have to be careful not to mix up parameter space-time
with physical space-time. In the following, physical
space-time will be understood in the sense of the
expectation value of the position operator of Equation
(12).
As a pure mathematical construct, (parameter)
x
space is not limited by any “physical” scale, such as the
Planck length. So it does not make sense to try its

W. SMILGA 965
“quantization” at Planck scales, in the hope of finding a
road to quantum gravity. On the other hand, physical
space-time is quantized right from the beginning, as it
has been defined by the expectation values of the posi-
tion operator. This means that the classical concept of
space-time may break down at scales where quantum
effects become noticeable, and this happens not at Planck
scales, but already at atomic scales.
There is a wide-spread opinion that the difficulties of a
quantum theory of gravitation result from the fact that
quantum mechanics is defined on space-time, while in
quantum gravity, this very space-time continuum “must
be quantized.” This opinion, obviously, does not make
the necessary distinction between parameter space-time
and physical space-time.
In contrast to parameter space-time, physical space-
time has a natural scale. A scale is, e.g., given by the
Bohr radius
e
cm


m
ij ji
(13)
of the electron in a hydrogen atom. This scale is deter-
mined by the electromagnetic interaction, which in [1]
was shown to be a property of the irreducible represen-
tations of the Poincaré group, and by the electron mass
e. So this mass takes over the role of the (hypothetical)
Planck mass in characterizing a “smallest length.”
3. Geometry of Physical Space-Time
Within an irreducible two-particle representation, the
motion of the particles relative to each other is deter-
mined by a well-defined angular momentum. The asso-
ciated Casimir operator W is a constant of the motion.
Quantum mechanics describes this angular momentum in
(parameter) space-time by spherical functions, which in
the limit of large quantum numbers describe probability
distributions with the shape of circular orbits.
The circular orbits of a quasi-classical two-particle
system, resulting from a well-defined angular momentum,
can be described by the sem i-classical expression
x
pxpn (14)
or by
,
tn
pr 
p
(15)
where t is the momentum in the tangential direction.
In words, the tangential momentum is proportional to the
curvature of the orbit. Since there are no external forces
to keep the particles on these orbits, we are led to the
alternative interpretation that physical space-time, in
contrast to parameter space-time, has in general a curved
metric.
This curvature is not obtained by an active defor-
mation of a predefined space-time continuum, but by the
ab initio construction of (physical) space-time from (an
entangled superposition of) momentum eigenstates with-
in an irreducible two-particle representation. Viewed in
this way, it appears more or less trivial that the dis-
tribution of energy-momentum in space-time should be
reflected in the metric of physical space-time, and it
would be surprising if it were not.
The connection between energy-momentum and
space-time is given by the factor e in the states
(11). This factor is invariant under two simultaneous
conformal transformations
ipx
1
x
x



.pp
(16)
and (17)

1

By these transformations, not only parameter space-
time, but also physical space-time, are subjected to a
scaling that changes any probability distribution in space-
time by a scaling factor


, but keeps the form of this
distribution invariant. The symmetry defined by these
transformations means that the linear size of a structure
in space-time is inversely proportional to the magnitude
of the energy-momentum that defines this structure.
Accordingly, a curvature associated with this structure is
directly proportional to the energy-momentum.
This especially applies to the curvature of the quasi-
classical orbit of two particles within an irreducible re-
presentation of the Poincaré group. Following Newton’s
first law, we can describe this orbit as the result of a
force that acts perpendicularly to the velocity vectors of
the particles. This force generates a space-like linear
momentum perpendicular to their actual velocities. (Re-
member that the kinematics of the relative momentum is
a matter of a 3-dimensional subspace of space). Such
a momentum is described by the momentum flux
p


,,1,2,3
ik
Tikik
T of the energy-momentum tensor

 ii
ik
00
Tii
T
. The diagonal elements T obviously do not
contribute to the centripetal force. Therefore, the devi-
ation of the particles’ kinematics from a straight uniform
motion can, in principle, be deduced from the traceless
part of the energy momentum tensor. (Although Lorentz
transformations may transform the components of T
into and T, these transformations leave the trace
of


invariant). Because the total linear momentum
is conserved, the second particle must contribute a flux
of linear momentum that is opposed to the flux of the
first. Metaphorically speaking, both particles exchange
momentum.
With this in mind, we now try to express the centri-
petal forces by a non-Euclidean metric of space-time.
Consequently, we have to look for a relation between the
curvature of space-time and the traceless part of the
energy-momentum tensor, as the “cause” of the curvature.

W. SMILGA 966
(Einsteinian gravity, which was set up with the goal of
replacing Newtonian gravity, uses the trace of the
energy-momentum tensor instead. Both approaches ar e in
a sense complementary, as far as spherically symmetric
solutions are concerned [7]). According to what has been
said above about conformal scaling, the curvature must
be proportional to the scaling of the momentum. There-
fore, the curvature experienced by the second particle
must be proportional to the traceless part of the energy-
momentum tensor of the first particle, and vice versa.
A curvature tensor that can be set proportional to a
traceless energy-momentum tensor, must itself be trace-
less too. Such a tensor is the Weyl tensor C


R
, which
is the traceless part of the Riemann curvature tensor


W
. From the Weyl tensor, a traceless “gravitation
tensor”


T
can be derived [8]. This tensor can then be
put into relation with the traceless part of the energy-
momentum tensor


.
Examples of traceless energy-momentum tensors,
based on different models of matter, can be found in [8].
Here we simply subtract the trace from the energy-
momentum tensor to make it traceless. This leads to the
field equations of c onformal gravity
1
2
T Tg
 





G
conf
WG
 
 (18)
with a “gravitational constant” .
conf
Conformal gravity has gained interest in recent years
because it may solve the problems usually associated
with “dark matter” and “dark energy” [7,8] without
additional ad hoc assumptions. Within the scale of our
solar system, conformal gravity is known to deliver the
same results as Einstein’s theory of general relativity,
which is based on the Riemann curvature tensor, rather
than on the Weyl tensor [7]. The problem of “ghosts,”
which has been encountered in “quantized” versions of
conformal gravity [9,10], does not exist for the classical
version.
4. The Gravitational Constant
In [1], the electromagnetic coupling constant

was
calculated from the geometry of the parameter space
associated with an irreducible two-particle state space of
spin-1/2 particles. The same calculation, done for spin-
less particles, results in a coupling constant of 4

.
There is, however, a crucial difference between quan-
tum electrodynamics and gravitation. Whereas in quan-
tum electrodynamics it makes sense to consider an iso-
lated two-particle system, this is an unrealistic configura-
tion in gravitation. There is no way to set up a “neutral”
environment or to “shield” gravitation. Therefore, an ex-
perimental setup for a “scattering experiment” in analogy
to electron-electron scattering must always take into
account the whole environment. This means we have to
take into account at least 1080 heavy particles, which is
the estimated number of protons in the (observable)
universe [11].
A gravitational scattering experiment of an (electri-
cally neutral) particle of, say, the mass of the proton,
includes at first the selection of a second particle from
1080 available particles. This is followed by a transition
from the “incoming” two-particle pure product state to an
irreducible (entangled) two-particle state. Finally, we
have a transition to an “outgoing” two-particle pure pro-
duct state, which is the quantum mechanical description
of measuring the individual momenta of the particles
after the scattering has taken place. Note that there are
two transitions between pure product states and en-
tangled state, but only one selection.
The following is an attempt to quantum mechanically
describe the “selection process.” The selection of a part-
ner particle will be considered as a “transition” from an
“incoming” one-particle state (of the first particle) to a
two-particle state. For the first particle, there are
independent ways to form a two-particle state. Let us
describe the corresponding quantum mechanical transi-
tion amplitude by a state in a 10 -dimensional state
space. Then the states of this state space have to be
normalized by the factor
80
10
80
40
110 .
This normalization ensures that the total transition
probability from a specific incoming (one-particle) state
to an outgoing (one-particle) state, through any interme-
diate two-particle state, equals unity. On the other hand,
the field equations (18) describe the contribution of only
a specific second particle, characterized by its energy-
momentum tensor at a point
x
, to the curvature of
space-time. Accordingly, the scattering process contains
only the transitions up to the outgoing two-particle pro-
duct state. For this reason, the “selection amplitude”
enters only once. The normalization factor in this ampli-
tude leads to an additional factor to the two-particle
coupling constant 40 0 of 11 .

4
This results in an estimate of the “gravitational cou-
pling constant.” It matches the empirical strength of the
gravitational interaction, which, between two protons, is
37 orders of magnitude weaker than the electromagnetic
interaction (or 43 orders, between two electrons) [12].
This weakness explains why in the experiments of parti-
cle physics the gravitational interaction can be ignored.
5. Quantum Gravity
The field equations (18) describe a classical theory of
gravitation. What, then, is their quantum mechanical
analogue? Since we just have sketched a connection
between quantum theory and classical conformal gravity,
we are able to give an answer to this question: The quan-
tum mechanical basis of conformal gravity is nothing

W. SMILGA 967
cconf
G
other than an irreducible two-particle representation of
the Poincaré group. In other words, there is no specific
“quantum gravity” apart from the common rules of
relativistic quantum mechanics. The situation is similar
to quantum electrodynamics, as discussed in [1]: Gravity
emerges from the restrictions on the two-particle state
space imposed by the condition of irreducibility.
6. Conclusions
Reasons have been given as to why gravitation can be
understood as a basic property of relativistic quantum
mechanics, more precisely, as a property of the irreduci-
ble two-particle representations of the Poincaré group.
Gravitation is not provided by “coupling” to an “external
field.” Rather it is the outcome of correlations within the
quantum mechanical state-space of matter resulting from
the condition of irreducibility. These correlations lead to
the equations of classical conformal gravity. In short,
gravitation is a quantum mechanical property of matter.
Physical space-time turns out to be just another quan-
tum mechanical property of matter. Its geometry in the
large is determined by the equations of conformal gravity.
Its scale in the small is defined by the electromagnetic
interaction and by the masses of the particles involved in
this interaction. Together, the electromagnetic and gravi-
tational interactions provide the basis for building ex-
tended atoms, molecules, and macroscopic bodies, to fill
up space-time. The electromagnetic interaction provides
photons, which can be used to unveil the geometry of
space-time to an observer. Needless to say, the electro-
magnetic interaction establishes a causal structure in
space-time. It is these interactions that make the differ-
ence between parameter space-time and physical space-
time. Therefore, the emergence of physical space-time
goes in parallel with the emergence of interactions.
The validity of classical space-time ends at scales
where quantum mechanics becomes effective. These
scales are related to the electron mass, rather than to the
Planck mass. There is no room for the latter, because it is
not possible to construct a mass from , , and .
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