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THEORY OF REALITY
Henry Pierce Stapp
Lawrence Berkeley Laboratory
University of California
Berkeley, California 94720
April 29, 1975
. ABSTRACT
Bell's theorem is used to guide the formulation of a unified
theory of reality that incorporates the basic principles of relativ-
istic quantum theory.
I ..
INTRODUCTION
Quantum theory is a theory of observations; the realities it
deals with are certain observations of scientists who use the theory.
These observations are only a small part of reality. Consequently quantum
theory, considered as a theory of reality, is incomplete, Prevailing
opinion holds, in fact, that no complete theory of reality can ade-
quately describe quantum phenomena. This opinion stems from the long
history of failures of attempts to achieve this end.
It is not clear, however, whether these failures arise from
an inadequacy of the reality concept, or merely from a breakdown of
the classical idea of causal space-time development. Bohr often
emphasized the breakdown of this classical idea in the realm of quantum
phenomena, and his point has now been strikingly verified and clarified
by the work of J. S. Bell.
(1)
Bell's work was originally formulated in the restricted
framework of hidden-variable thecry. However, it was soon realized
(2)
* Work supported by U.S. Energy Research and Development Administration.
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they what; Bell had established was the following profound result:
--
The statistical predictions of quantum theory are incompatible
with the principle of local causes.
.
The principle of local causes asserts that what happens in
one space-time region is approximately independent of variables subject
to the control of an experimenter in a far-away space-like-separated
region.
This principle holds in relativistic quantum theory at the
level of statistical predictions.
However, the character of these
predictions is such that the principle must fail at the level of the
individual events. The statistical predictions from which this result
follow come directly from the basic principles of quantum theory, not
from the detailed dynamics, and they have been experimentally tested
and confirmed.
(3)
Bell's theorem shows that no theory of reality compatible with
quantum theory can allow the spatially separated parts of reality tc
be independent :
These parts must be related some way that goes beyond
the familiar idea that causal connections propagate only into the
forward light-cone. This conclusion will guide our thoughts.
The first task of any general theory of reality is to formulate
the connection between the experiential or psychic aspects of reality
and the material or space-time aspects.
The debate
between Bohr and
Einstein
(4)
pointed to the importance of this question, for Einstein
appealed finally to the need for a comprehensible understanding of
space-time relations, whereas Bohr appealed ultimately to the primacy
of experiential relations. A unified theory of reality must bring these
two aspects of reality into one coherent scheme.
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A unified theory of reality has been formulated by Alfre:
North Whitehead.
(5)
According to this theory reality consists of
discrete events.
Each event .has a location, which is a finite space-
time region.
It also has certain experiential characteristics.
To support the idea that experience comes in discrete units
Whitehead cites the authority of William James, who writes:
(6)
"Either your experience is of no content, of no change, or it
is of a perceptible amount of content or change. Your acquaintance :
with reality grows literally by buds or drops of perception.
Intellec-
tually and on reflection you can divide these into components, but as
immediately given they come totally or not at all."
To support the idea that physical processes consist of dis-
crete events one may cite the authority of Niels Bohr:
(7)
"( The essence of quantum theory ) may be expressed in the so-
called quantum postulate, which attributes to any atomic process an
essential discontinuity, or rather individuality, completely foreign
to the classical theories and symbolized by Planck's quantum of
action. "
A reality consisting of discrete events seems hopelessly
fragmented and pluralistic. Yet Whitehead's reality is unified.
£
This
unity is achieved by considering each event to be a process in which
all prior events are brought together, or "prehended", in a new
pattern.
Reality thus becomes the process of creation, in discrete
individual steps, of an ever-growing web of relations between things
that are parts of this same process. Mental events are a part of
this general world process, and they afford an illustration of how
events can be processes that bring together prior events in new
patterns.
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Fach event in the world process prehenis in some particiit
way every prior event, and hence contains within itself, in a certain
Sense, the whole of creation. :
Whitehead chose a model that did not attain the full unity
just described. He believed that relativity theory required space-like-
separated events to be causally independent, and hence decreed that
each event prehend, not all of creation, but only those events whose
.
locations lay in its backward light-cone.
This mutilation of the
model destroys its natural unity and logical simplicity. Moreover, it
is incompatible with quantum theory, by virture of Bell's theorem.
Thus it must be undone. The result is a philosophically attractive
unified model of reality that provides a natural setting for relativ-
istic quantum theory.
II.
THEORY OF EVENTS
In this section a physical theory of events is erected on the
model of reality described above.
This theory incorporates the basic
principles of relativistic quantum theory. The theory is set forth
in eight assumptions or postulates, which have physical, metaphysical,
and mathematical aspects.
The guiding principle is maximal simplicit ;:
The aim is to use the simplest and most economical metaphysical and
mathematical structures consistent with what we know from experience.
The postulates are as follows :
1.
The creative process. There is a creative process that
consists of a well-ordered sequence of individual creative acts called
events.
Remark
This assumption affirms that there is actual creation, i.e.,
a real coming into being, or a coming into existence, and that the
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process of creation can be decomposed into a sequence of indivi !vai
acts.
Whatever is created exists, and nothing else exists. Nothing
passes out of existence.
At the end of each creative act the whole of
creation is settled and definite:
all that exists is unambiguously
fixed.
This simple logical structure can be contrasted with ones in
which all of creation, past, present, and future exists, and is fixed,
and change is some sort of illusion. . It may also be contrasted with
..
ones in which the creative process is not a single linear process but
rather a multiple process that proceeds somehow independently in
different space-time regions, sothat what exists is not globally well-
defined but depends on the space-time point from which the determination
of what exists is made. (These models bifurcate nature:
they posit
either changing experiences of a pre-existing world or a changing world
in pre-existing space-time. )
2. Space-time location. Each event has characteristics that
define an associated region in a four-dimensional mathematical space.
This mathematical space is called the space-time continuum, and the
region in this space associated with an event is called its location.
Remark
Space-time has no independent existence in this theory.
Rather each event has characteristics that can be interpreted as a
region in a certain mathematical space. For physical applications this
metaphysical distinction is unimportant, and one may imagine a pre-
existing space-time continuum with the events scattered through it.
Definition
An event is prior to another if it occurs earlier in the
sequence of creative acts described in (1).
It is subsequent if it
occurs later in this sequence.
3.
Conservation of momentum-energy.
Among the events prior
to a given event are some events called its antecedents. Any event is
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. successor to each of its antecedents.
The location of each event
-
is connected to the location of each of its antecedents by a directed
geodesic (a directed straight line in space-time ) that runs from the
location of the antecdent to the location of the successor. Each
geodesic is associated with a real mass-value m, and also with a
momentum-energy vector p = mv, where v is the four-velocity defined
by the direction of the geodesic. The sum of the momentum-energy
vectors associated with the geodesics coming into the location of a :
given event from the locations of its antecedents is equal to the sum
of the energies associated with the geodesics going out from the
location of the event to the locations of its successors.
Remark This physical assumption, like those that follow, is holistic
rather than mechanistic; it is formulated as a mathematical condition
on the overall space-time structure of what emerges from the process
of creation, not as a dynamical law that governs the detailed way in
which reality unfolds.
Definition A system is a local space-time pattern of events.
4. Lorentz Invariance. Probabilities are determined by local
conditions :
under suitable conditions of isolation the statistical
behavior of ensembles of systems defined by local specifications do not
depend on the Lorentz frame used to relate the local specifications
to global space-time.
Remark The isolation condition requires a local system to be isolated
in the sense that outside sources of energy are negligible. The
assumption is that under this condition of isolation ensembles of
subsystems defined by local specifications exhibit the type of
behavior characterized by probability functions.
Moreover these
probability functions are invariant under Lorentz transformations.
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Thus if A represents the local specifications that characteriso an
initial ensemble and
B
represents the local specifications that
define a final ensemble and
P[A; B]
is the probability that
B
holds under conditions A, then
P[A; B]
is independent of the
Lorentz frame used to relate the space-time coordinates occurring in
the local specifications
A
and B to physical space-time points.
5. Scattering formalism. The statistical results of scatter-
ing experiments can be described by the formalism of classical rela-
tivistic statistical mechanics, with the geodesics identified with
the trajectories of classical point particles.
Remark
In the classical description each beam of initial particles
is described by a probability or weight function w(p,x) and the
detection system for each of the final particles is described by an
efficiency function e(p,x). The expression
[a30 a3x w(p,x) e(p,x)|
=
0
Pf w, e]
(1)
x =t
gives the probability that a particle in the beam described by
W
will be detected by the system described by e. (The time t can be
chosen arbitrarily. ) For a scattering of m particles into
n
particles the expression
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m
.
PÅ"1,"2, ... ,"; e1,e2 .... ,en]
i=1
1
3
d' P.
3.
a Xi
×
d'p. d'x; e;(pj,x;)
n
j=1
.
×
S(P1,X], ... , Pm>m' Pj, x's ·, Ph, x" )
.
=+
1
0
-
x
X;
j
0
=t
(2)
gives the probability that if the initial beams are described by the
weight functions
and the final-particle detection systems
are described by the efficiency functions
el .... ,en
then all
n
final particles will be detected. (The times
ti
and
t, can be
chosen arbitrarily. )
Each function
w. (p,x) is a real function of the real mass-
shell momentum-energy vector p and the real four-vector
x.
It
satisfies, for any ),
.
w. (p,x) = w.(p,x + Ap) .
(3)
This condition arises from the fact that all the particles of momentum
p move in the direction defined by p = mv; i. e., along p.
.
Functions satisfying (3) can be constructed by specifying
w(p, x) at some time, say x = t, and then forming
w(p,x)
=
[à3x' d(Xpº) w(p, x' ) 84(x' - x - Xp)
1.01
-
x =t
(4)
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Another way of constructing solutions to (3) is to write, for any
complex function w(p) and any real constant n,
4,
d4a
w(p,x)
11
( 2 Tr)-
13
V Nv - a) 4* (Mv +a)
(5)
-iqx/n
M
NH
e
8 ( q.v) (
-
,
where v = p/m and M = (m'
Im2 -42)*
6. The quantum assumption.
The functions w(p,x) occurring
in nature are sums of functions of the form (5), with different func-
tions
v(p) but with the same constant h.
This constant is Planck's
constant. The analogous formula holds for e(p,x).
Remark
This assumption allows the scattering formula (2) to be trans-
cribed into quantum mechanical form.
The S-matrix
(8)
S(PL, .,Pm; Pj, ... ,Ph)
is then defined in terms of the function
S(P1, ... , x')
appearing in (5).
Conservation of probability implies
the unitarity of S(p, ... , ph).
7. Macrocausality.
(9)
Momentum-energy is transferred over
macroscopic distances only by the stable systems: an event having an
incoming geodesic not positive time-like or with mass not that of a
stable system has a probability to occur that falls off exponentially
under space-time dilation. The size of the location of an event has
a finite bound that depends only on the incoming geodesics.
Remark
This macrocausality condition entails that the S-matrix
S(P. . ... , P. ) be an analytic function at all real points
(P1, ·, P')
except those lying on a set of well-defined surfaces called the
positive-a Landau surfaces. The rule of continuation around each of
these singularity surfaces is also determined.
(9)
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(10)
8.
Maximal analyticity.
The analytic continuation of
the S matrix to complex
(P, ... , p") has only those singularities
that are required by the unitarity conditions.
Remark
Maximal analyticity is a principle of economy; it asserts that
the S matrix has no unnecessary singularities. Or it is a principle
of simplicity; it asserts that the S matrix has the simplest possible
analytic structure. Any useful physical theory must be based on some
..
principle of economy or simplicity. There is no theoretical or
experimental evidence for any singularity not required by unitarity.
It seems entirely possible that the general principles of
Lorentz invariance, unitarity, macrocausality, and maximal analyticity
. may determine in principle a unique complete relativistic quantum
theory of elementary particles.
(10)
A few constants may have to be
determined empirically, at least in practice.
If this theory is carried over to the nonrelativistic limit,
where particle-creation is excluded, then it yields
(11)
the Schroedinger
equation, and hence the concept of equations of motion.
And the
Schroedinger form of quantum theory reduces, in appropriate contexts
and limits, to classical physics. It thus appears that all of
physics can emerge from the eight assumptions listed above, together,
perhaps, with a few empirical constants.
III.
BELL'S THEOREM AND THEORY OF EVENTS
The noncausal structure of events demanded by Bell's theorem
is incomprehensible in the framework of ordinary ideas, but is a
natural consequence of the theory of events described above.
In the simplest cases involving Bell's phenomena there are
three ( scattering ) events E, E1, and E2.
Their locations
Lo, LI'
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.
and
lie in three well-separated experimental areas Ap, A, and
A2.
Experiment
EO
is an antecedent of both E, and E. Thus there
is a timelike geodesic from Lo to Lg
and another from
Lo to L2,
as shown in Fig. 1. An experimenter in
A1
can choose to perform
experiment
E11
or experiment E.
An experimenter in A, can
choose to perform experiment
E
21
or experiment
E
Suppose E.
is the event (result ) that occurs in experiment
Enj
if the experi-
22°
'1jk
menter in
A2
does experiment
En .
Suppose
E
is the event
:
2jk
( result ) that occurs in E
2j
if the experimenter in
does experi-
ment.
E
The ordinary idea of causality (i.e., the principle of
1k
local causes ) demands that the E
ijk
be independent of k. But
Bell's work shows this requirement to be
incompatible with the
statistical predictions of quantum theory.
According to the theory of events one of the two events
E.
1
or
E2
is prior to the other. Suppose E.
is the prior event.
When
it occurs the possibilities for events in
A2
are radically changed.
For example, if the locations
To, In, and I2
are effectively points
( compared to the large distances between them) then the two locations
Io and
L1
determine the geodesic
L Ly, and hence the energy-
momentum carried from Lo
to Ly.
This fixes in turn the momentum-
Lo to L2, which fixes this
energy available for the geodesic from
geodesic itself, assuming that the two geodesics exhaust the momentum-
energy available from
. '.Thus after E
occurs the event in
A2
is required to lie on a fixed geodesic that is determined by the
events Eo
and E1 .
At this stage only space-time and momentum-energy considera-
tions have been introduced, and Bell's phenomena do
not enter.
The
correlations between the events in
A1
and
A2
are just those
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L,
2
Q Az
A:
TIME
-
.
Lo
SPACE
Ao
Fig. 1. Space-time picture of Bell's phenomena.
expected from classical ideas :
the course of events in
Az
is
correlated to what is observed in
A, but not on decisions made by
the experimenter in
Ay.
Though the results at this stage are similar to those of
classical particle theory, the logical structure is different. In the
classical theory what happens in
A2
is determined by what happens in
the earlier region A , whereas in the theory of events the possi-
bilities for E2
are limited jointly by the prior events E
and
En.
This logical difference becomes important in experiments involving
spin, which are the ones in which Bell's phenomena occur.
Suppose the geodesics
and
LOL2
are associated with
spin
representations of the Lorentz group. Just as before the
possibilities for E
2
are limited jointly by the prior events
E0
and E
1
Part of the information determined by E and El
is rep-
resented by the momentum-energy four-vector associated with the
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geodesic Lot1
However, these two events E and E.
determine also
another vector associated with the geodesic
LoLy, namely a spin vector
associated with the corresponding spin space.
The spin vector and the momentum-energy vector associated with
are both determined jointly by E.
and E . Thus it would be
unnatural, in the framework of the theory of events, to treat them
differently. It is accordingly assumed that these two vectors should
be treated in the same way.
Treating the spin and momentum-energy vectors in the same way
leads to very different effects with respect to the ordinary idea of
causality. This difference stems from the fact that the two experi-
menters can independently manipulate the directions of the two spin
vectors, modulo signs, but cannot do this with the two momentum
vectors, without disrupting the experiment. For the two momentum
vectors are required by the conservation laws to be essentially
parallel, whereas the two spin vectors, modulo signs, can be indepen-
dently fixed by the two experimenters.
The spin vector associated with L L., like the momentum vector,
is determined by events E.
and E. . But the experimenter in A can,
by choosing the experiment to be performed, fix this spin vector, up to
a sign. Thus, in the theory of events, the event E_ depends on what
the experimenter in
A1
decides to do. (This effect is contrary to the
ordinary idea of causality, but conforms to the requirements imposed
by Bell's theorem.
The theory of events does not conform to the ordinary idea of
causality. But it provides an alternative possible space-time picture
of causality. This picture arises by regarding the geodesic associated
with a spin-J representation of the Lorents group as a conduit. of
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spin-J. information.
This information flows from an event both forward
to its potential successors and backward to its antecedents. For
example, the determination in, event
E1
of the spin vector associated
with geodesic
is viewed as being instantly communicated along
LL1 to Io, where it can be tapped by geodesic LL,, in the assess-
ment of a possible successor to E.
having location
L2.
IV. CONCLUSIONS
The basic properties of relativistic quantum theory emerge in
a natural way from a logically simple model of reality. In this
model there is a fundamental creative process that proceeds by
discrete steps. Each step is a creative act or event.
Each event is
associated with a definite space-time location. The fundamental
process is not local in character, but it generates local space-time
patterns that have mathematical forms amenable to scientific study.
This theory of reality reconciles the positions of Einstein
and Bohr.
It conforms to Einstein's view that the complete basic
theory should be a complete theory of reality rather than a theory of
observations; i.e., it should describe "any real ( individual) situation
(as it supposedly exists apart from any act of observation). "(12)
The model described above attempts to do exactly that.
In the model
everything that exists is perfectly definite: Schroedinger's cat is
either dead or alive, not both, independently of any act of observation,
or of any choice of space-time perspective. On the other hand, the
theory is probably useless in the realm of atomic physics, and for
essentially the reasons advanced by Bohr, namely that, "The element of
wholeness, symbolised by the quantum of action and completely
foreign to classical physical principles, ... makes recourse to a
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statistical mode of description imperative as regards to the expecta-
tions of the occurrence of individual quantum effects in one and the
same experimental arrangement. "
„(13)
This probable lack of utility of the model in the realm of
atomic physics does not necessarily mean that the model has no uses
at all.
In the realm of elementary particle physics the quantum
theoretical principles, though perhaps sufficient in principle, are
difficult to apply, and the insight provided by a model of the under-
lying reality might be useful. More important would be the possible
uses in those realms of science where the approximations essential to
the applicability of quantum theory fail.
Bohr often stressed that,
the wave function of a system has meaning only to the extent that the
system can be regarded as isolated from the rest of the world,
(14)
i.e., only in those situations where the possible outside sources of
energy-momentum can be ignored. When this idealization is inapplicable
the wave function of the system is not definable, and even if it could
be defined it would be undergoing continual quantum jumps, and no
adequate theory of quantum jumps exists.
No system is completely isolated from the rest of the world,
except the whole world, which cannot be treated by quantum theory since
there is no outside "observer". And most systems of interest are
not even approximately isolated from the rest of the world. One
class of systems of special interest to man are living systems.
These require interactions with their environments to sustain life,
and consequently, as emphasized by Bohr,
(15)
they cannot be fully
described by quantum theory.
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Unity of understanding is a natural goal of thought. In attempt-
ing to unify the various branches of science and knowledge such as
physics, biology, psychology, sociology, philosophy, etc., some
overarching conceptual framework is required.
It is reasonable to
begin with the logically simplest model of reality that is consistent
with all we know. The theory of events outlined above is a logically
simple model of reality that is apparently consistent with all we
know.
Taken in conjunction with Whitehead's theory of process it is,
as far as I know, the only existing model of all of reality that
incorporates the basic principles of relativistic quantum theory.
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REFERENCES
1. J. S. Bell, Physics (N.Y. ) 1, 195 ( 1964).
2.
H. P. Stapp, Correlation Experiments and the Nonvalidity of
Ordinary Ideas About the Physical World, Berkeley ( 1968 ) and
Phys. Rev. D3, 1303 (1971). The principle of local causes is
introduced and analyzed in these works, where it is tacitly
· assumed that counter efficiences are not limited in principle.
This assumption is made also in the present work. For a discussion
of this point see J. F. Clauser and M. A. Horne, Phys. Rev. DIO,
526 (1974), and references cited there.
3. S. J. Friedman and J. F. Clauser, Phys. Rev. Lett. 28, 933 (1972).
4.
N. Bohr and A. Einstein in Albert Einstein:
Philosopher-Scientist
( Tudo Publishing Co., New York, 1951).
5.
A. N. Whitehead, Process and Reality ( Macmillan Co., New York,
1929 ).
6.
William James, quoted in Ref. 5.
7.
N. Bohr, Atomic Theory and the Description of Nature ( Cambridge
University Press, England, 1934), p. 53.
8. H. P. Stapp, Foundations of S-matrix Theory. I. Theory and
Measurement, Lawrence Berkeley Laboratory LBL-759 Rev. (1972),
or D. Iagolnitzer, Introduction to S-matrix Theory, C. E. N. - Saclay,
1973.
9. D. Iagolnitzer and H. P. Stapp, Commun. Math. Phys. 14, 15 (1969);
and D. Iagolnitzer, Ref. 8.
10.
G. F. Chew, S-matrix Theory of Strong Interactions (W. A.
Benjamin, Inc., New York, 1961), and The Analytic S-matrix (W. A.
Benjamin, Inc., New York, 1966); H. P. Stapp, Phys. Rev. 125,
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2139 (1962); J. Gunson, J. Math. Phys. 6, 827 and 845 (1965)
(preprint in 1962).
11.
R. Blankenbecler, M. L. Goldberger, N. N. Khuri, and S. B. Treizen,
Annals of Phys. 10, 62 (1960).
12.
A. Einstein, Ref. 4, p. 667.
13.
N. Bohr, Essays 1958-1962 or Atomic Physics and Human Knowledge
( Wiley, New York, 1963 ), p. 60. . See also .H. P. Stapp, Am. J.
Phys. 40, 1098 (1972), p. 1108.
14.
N. Bohr, Ref. 7, p. 54. See also Ref. 2, p. 1308.
15.
N. Bohr, Atomic Physics and Human Knowledge ( Wiley, New York, 1953 ),
p. 10.
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