**VOLUME 29. NUMBER 3. JULY.љ 1957**

“Relative
State” Formulation of Quantum
Mechanics*

*Palmer Physical Laboratory, Princeton
University, Princeton, New Jersey*

___________________________

*
Thesis submitted to Princeton University March
1, 1957 in partial fulfillment of the requirements for the Ph.D. degree. An earlier draft
dated January, 1956 was circulated to several physicists whose comments were helpful. Professor Niels Bohr, Dr. H.J.Groenewald, Dr. Aage Peterson, Dr. A.Stern, and Professor L.Rosenfeld are
free of any responsibility, but they are warmly thanked for
the useful objections that they raised. Most particular thanks are due
to Professor John A.Wheeler for his continued guidance and encouragement. Appreciation is also expressed to the National Science Foundation for fellowship
support.

†
Present address: Weapons
Systems Evaluation Group, The Pentagon. Washington, D.C.

1.
INTRODUCTION

The
task of quantizing general relativity raises serious
questions about the
meaning of the present formulation and
interpretation of quantum mechanics when applied to so fundamental a
structure as the space-time geometry itself. This paper
seeks to clarify the formulations of quantum mechanics. It presents a reformulation of
quantum theory in a form believed suitable for
application to general relativity.

The
aim is not to deny or contradict the conventional
formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a
new, more general and complete formulation, from
which the conventional
interpretation can be deduced.

The relationship of this new formulation to the older
formulation is therefore
that of a metatheory to a theory, that is, it is an
underlying theory in which the nature
and consistency, as well as the realm of applicability,
of the older theory can
be investigated and clarified.

The
new theory is not based on any radical departure
from the conventional one. The special postulates in the old theory which deal with observation are omitted in the
new theory. The altered
theory thereby acquires a new character. It has to be analyzed in and for itself before any identification becomes possible between the quantities of the theory and the properties of the world of experience. The identification, when made, leads back to the omitted postulates of the conventional theory that deal with observation, but in a manner which clarifies their role and logical
position.

We begin with a brief discussion of the conventional
formulation, and some of the reasons which motivate
one to seek a
modification.

2.
REALM OF APPLICABILITY OF THE CONVENTIONAL OR
"EXTERNAL OBSERVATION" FORMULATION OF QUANTUM MECHANICS

We
take the conventional or
"external observation"
formulation of quantum mechanics to be essentially
the following** ^{1}**: A physical system is
completely described by a state function

___________________________

** ^{1}**
We use the terminology
and notation of J. von
Neumann,

** Process
1:**
The discontinuous change brought about by the observation of a quantity with eigenstates

*Process***
2:**
The continuous, deterministic change of state of an isolated system with time according to a wave equation d

This
formulation describes a wealth of experience. No experimental evidence is known which contradicts it.

Not
all conceivable situations fit the framework of this
mathematical formulation. Consider for example an isolated system consisting of an observer or measuring apparatus, plus an object system. Can the change with time of the state of the total system be described by Process 2? If so, then it would appear that no discontinuous
probabilistic process like Process 1 can take place. If not, we are forced
to admit that systems which contain observers are
not subject to the same
kind of quantum-mechanical description as we admit
for all other physical
systems. The question cannot be ruled out as lying in the
domain of psychology. Much of the discussion of "observers" in quantum mechanics has to do with photoelectric cells, photographic plates, and similar
devices where a mechanistic attitude can
hardly be contested. For the following one can *limit himself to this class of problems*, if he
is unwilling to consider observers in the more familiar sense on the same mechanistic level of analysis.

What
mixture of Processes 1
and 2 of the conventional formulation is to be
applied to the case where only an approximate measurement is effected; that is, where an apparatus or
observer interacts only weakly and for a limited
time with an object system? In this case of an approximate measurement Б
proper theory must specify (1) the new state of the object system that
corresponds to any particular reading of the apparatus and (2) the probability
with which this reading will occur, von NЕumann
showed how to treat a special class ofљ
approximate measurements by the method of projection operators.** ^{2}** However, a general
treatment of all approximate measurements by the method of projection operators
can be shown (Sec. 4) to be impossible.

_______________________________

** ^{2}**
Reference 1, Chap. 4, Sec. 4.

How
is one to apply the conventional formulation of quantum mechanics to the
space-time geometry itself? The issue becomes especially acute in the case of a
closed universe.** ^{3}** There
is no place to stand outside the system to observe it. There is nothing outside
it to produce transitions from one state to another. Even the familiar concept
of a proper state of the energy is completely inapplicable. In the derivation of
the law of conservation of energy, one defines the total energy by way of
an integral extended over a surface large enough to include all parts of the
system and their interactions.

_______________________________

** ^{3}**
See A.Einstein,

** ^{4}**
L.Landau and E.Lifshitz,

How
are a quantum description of a closed universe, of approximate measurements, and
of a system that contains an observer to be made? These three questions have one
feature in common, that they all inquire about the *quantum mechanics* that is *internal lo an isolated
system*.

No
way is evident to apply the conventional formulation of quantum mechanics
to a system that is not subject to *external* observation. The whole
interpretive scheme of that formalism rests upon the notion of external
observation. The probabilities of the various possible outcomes of the
observation are prescribed exclusively by Process 1. Without that part of the
formalism there is no means whatever to ascribe a physical interpretation to the
conventional machinery. But Process 1 is out of the question for systems not
subject to external observation.^{5}

_______________________________

** ^{5}**
See in particular the discussion of this point by N.Bohr and L.Rosenfeld, Kgl.
Danske Videnskab, Selskab, Mat.-fys. Medd. 12, No. 8
(1933).

3.
QUANTUM MECHANICS INTERNAL TO AN ISOLATED SYSTEM

This
paper proposes to reward pure wave mechanics (тrocess
2 only) as a complete theory. It postulates that a wave function that obeys a
linear wave equation everywhere and at all times supplies a complete
mathematical model for every isolated physical system without
exception. It further postulates that every system that is subject to external
observation can be regarded as part of a larger isolated
system.

The
wave function is taken as the basic physical entity with *no a priori interpretation*.
Interpretation only comes *after* an
investigation of the logical structure of the theory. Here as always the theory
itself sets the framework for its interpretation.^{ 5}

For
any interpretation it is necessary to put the mathematical model of the theory
into correspondence with experience. For this purpose it is necessary to
formulate abstract models for observers that can be treated within the theory
itself as physical systems, to consider isolated systems containing such model
observers in interaction with other subsystems, to deduce the changes that
occur in an observer as a consequence of interaction with the surrounding
subsystems, and to interpret the changes in the familiar language of
experience.

Section
4 investigates representations of the state of a composite system in terms of
states of constituent subsystems. The mathematics leads one to recognize the
concept of the *relativity of states*,
in the following sense: a constituent subsystem cannot be said to be in any
single well-defined state, independently of the remainder of the composite
system. To any arbitrarily chosen state for one subsystem there will correspond
a unique *relative state* for the
remainder of the composite system. This relative state will usually depend upon
the choice of state for the first subsystem. Thus the state of one subsystem
does not have an independent existence, but is fixed only by the state of
the remaining subsystem. In other words, the states occupied by the
subsystems are not independent, but *correlated*. Such correlations
between systems arise whenever systems interact. In the present formulation
all measurements and observation processes are to be regarded simply as
interactions between the physical systems involved — interactions
which produce strong correlations. A simple model for a measurement, due to von
Neumann, is analyzed from this viewpoint.

Section
5 gives an abstract treatment of the problem of observation. This uses only the
superposition principle, and general rules by which composite system states
are formed of subsystem states, in order that the results shall have the
greatest generality and be applicable to any form of quantum theory for
which these principles hold. Deductions are drawn about the state of the
observer relative lo the state of the object system. It is found that
eИТЕriЕnces
of the observer (magnetic tape memory, counter system, etc.) are in full accord
with predictions of the conventional "external observer" formulation of quantum
mechanics, based on Process 1.

Section
6 recapituIates the "relative state" formulation of quantum
mechanics.

4.
CONCEPT OF RELATIVE STATE

We
now investigate some consequences of the wave mechanical formalism of composite
systems. If a composite system *S*, is
composed of two subsystems *S_{1}* and

*y*^{S}* =
***S**_{i,j}*a_{ij}*

From
(3.1) although *S* is in a definite
state *y***^{S}**,
the subsystems

We
can, however, for any choice of a state in one subsystem, *uniquely* assign a corresponding *relative* state in the other subsystem.
For example, if we choose *x***_{k}**
as the state for

** y**(

where
*N_{k}* is a normalization
constant. This relative state for

In
the conventional or "external observation" formulation, the relative state in *S_{2}*,

For
any choice of basis in *S_{1}*, {

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ
1

*y*^{S}* =
***S****_{i
}**—

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ
N_{i}

This
is an important representation used frequently.

*Summarizing:
There does not, in general, exist anything like a single state for one subsystem
of a composite system. Subsystems do not possess states that are independent of
the states of the remainder of the system, so that the subsystem states are
generally *correlated* with one another. One can arbitrarily
choose a state for one subsystem, and be led to the relative state for the
remainder. Thus we are faced with a fundamental *relativity* *of states*, which is implied by the formalism of
composite systems. It is meaningless to ask the absolute state of a subsystem —
one can only ask the state relative to a given state of the remainder of the
subsystem.*

At
this point we consider a simple example, due to von Neumann, which serves as a
model of a measurement process. Discussion of this example prepares the ground
for the analysis of "observation." We start with a system of only one
coordinate, *q* (such as position of a
particle), and an apparatus of one coordinate *r* (for example the position of a meter
needle). Further suppose that they are initially independent, so that the
combined wave function is *y*_{0}^{S+A}*
*=
*f**(q)**h**
(r*)
where *f**(q)
*is
the initial system wave function, and *h**
(r*)
is the initial apparatus function. The Hamiltonian is such that the two systems
do not interact except during the interval *t = 0 *to* t = T*, during which time the total
Hamiltonian consists only of a simple interaction,

*H_{I} = - i*

Then
the state

*y*_{t}^{S+A}*
(q,r) *=
*f**(q)**h**
(r - qt*)
љљљљљљљљљљљљ (5)

is
a solution of the Schrödinger
equation,

*i**·**(*d*y*_{t}^{S+A}*
/*d*t)* =
*H_{I}*

for
the specified initial conditions at lime *t = 0*.

From
(5) at time *t = T* (at which time
interaction stops) there is no longer any definite independent apparatus state,
nor any independent system state. The apparatus therefore does not indicate any
definite object-system value, and nothing like process 1 has
occurred.

Nevertheless,
we can look upon the total wave function (5) as a *superposition* of pairs of subsystem
states, each element of which has a definite *q* value and a correspondingly displaced
apparatus state. Thus after the interaction the state (5) has the
form:

*y*_{T}**^{S+A
}**=

which
is a superposition of states *y*_{q}** _{’}** =

Conversely,
if we transform to the representation where the *apparatus* coordinate is definite, we
write (5) as

*y*_{T}**^{S+A
}**=

where

*x*_{
}^{r}^{’}*
(q) = N_{r}*

and

*љ(1/N_{r}*

Then
the *x*_{
}^{r}^{’}*(q)
*are
the relative system state functions** ^{6}** for the apparatus states

_______________________________

** ^{6}**
This example provides a model of an approximate measurement. However, the
relative system state after the interaction

*E(NE**f**(q))
= NE*^{2}*f**(q)
= N**’’**f**(q)**h*^{2}*(r**’** -
qt)*

and
*E*^{2}*f**(q)
= (N**’’**/N)**f**(q)**h*^{2}*(r**’** -
qt). *But
the condition *E*^{2 }*= E *which is necessary for *E* to be a projection implies that *N**’**/N**’’**h**(q)
= **h*^{2}*(q)*
which is generally false.

If
*T* is sufficiently large, or *h**(r)*
sufficiently sharp (near *d**(r))
*then
*x*^{r}^{’}*(q)
*is
nearly *d**(q
- r**’**/T)*
and the relative system states *x*_{
}^{r}^{’}*
(q) *are
nearly eigenstates for the values *q* =
*r**’**/T*.

We
have seen that (8) is a superposition of states *y*_{r}** _{’}**,

von
Neumann's example is only a special case of a more general situation. Consider
any measuring apparatus interacting with any object system. As a result of
the interaction the state of the measuring apparatus is no longer capable of
independent definition. It can be defined only *relative* to the state of the object
system. In other words, there exists only a correlation between the states of
the two systems. It seems as if nothing can ever be settled by such a
measurement.

This
indefinite behavior seems to be quite at variance with our observations, since
physical objects always appear to us to have definite positions. Can we
reconcile this feature wave mechanical theory built purely on Process 2 with
experience, or must the theory be abandoned as untenable? In order to answer
this question we consider the problem of observation itself within the framework
of the theory.

5.
OBSERVATION

We
have the task of making deductions about the appearance of phenomena to
observers which are considered as purely physical systems and are treated
within the theory. To accomplish this it is necessary to identify some present
properties of such an observer with features of the past experience of the
observer.

Thus,
in order to say that an observer 0 has observed the event **a**,
it is necessary that the state of 0 has become changed from its former state to
a new state which is dependent upon **a**.

It
will suffice for our purposes to consider the observers to possess memories
(i.e., parts of a relatively permanent nature whose states are in correspondence
with past experience of the observers). In order to make deductions about the
past experience of an observer it is sufficient to deduce the present
contents of the memory as it appears within the mathematical
model.

As
models for observers we can, if we wish, consider automatically functioning
machines, possessing sensory apparatus and coupled to recording devices capable
of registering past sensory data and machine configurations. We can further
suppose that the machine is so constructed that its present actions shall be
determined not only by its present sensory data, but by the contents of its
memory as well. Such a machine will then be capable of performing a sequence of
observations (measurements), and furthermore of deciding upon its future
experiments on the basis of past results. If we consider that current sensory
data, as well as machine configuration, is immediately recorded in the memory,
then the actions of the machine at a given instant can be regarded as a function
of the memory contents only, and all relevant experience of the machine is
contained in the memory.

For
such machines we are justified in using such phrases as "the machine has
perceived *A*" or "the machine is aware
of *A*" if the occurrence of *A* is represented in the memory,
since the future behavior of the machine will be based upon the occurrence of *A*. In fact, all of the customary
language of subjective experience is quite applicable lo such machines, and
forms the most natural and useful mode of expression when dealing with their
behavior, as is well known to individuals who work with complex
automata.

When
dealing with a system representing an observer quantum mechanically we
ascribe a state function, *y***^{0}**,
to it. When the state

*y*^{0
}*[A,
B,љ , C] љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ
*(9)

The
symbols *A, B,љ , **у*,
which we assume to be ordered time-wise, therefore stand for memory
configurations which are in correspondence with the past experience of the
observer. These configurations can be regarded as punches in a paper tape,
impressions on a magnetic reel, configurations of a relay switching circuit, or
even configurations of brain cells. We require only that they be capable of the
interpretation: "The observer has experienced the succession of events *A, B,љ
, **у*."
(We sometimes write dots in a memory sequence, *љA,
B,љ , **у*,
to indicate the possible presence of previous memories which are irrelevant to
the case being considered.)

The
mathematical model seeks to treat the interaction of such observer systems with
other physical systems (observations), within the framework of Process 2 wave
mechanics, and to deduce the resulting memory configurations, which are then to
be interpreted as records of the past experiences of the
observers.

We
begin by defining what constitutes a "good" observation. A good observation of a
quantity *A*, with eigenfunctions
*f***_{i}**,
for a system

*y***^{S+0}** =

into
a new state

*y*^{S+0}**^{’}** =

where
*a***_{i}**
characterizes

_______________________________

** ^{7
}**It
should be understood that

From
the definition of a good observation we first deduce the result of an
observation upon a system which is *not* in an eigenstate of the observation.
We know from our definition that the interaction transforms states *f*_{i}*y***^{0}**[

*y*^{S+0}**^{’}** =

This
superposition principle continues to apply in the presence of further systems
which do not interact during the measurement. Thus, if systems *S_{1}, S_{2}, ^{. . .} ,^{
}S_{n}* are present
as well as 0, with original states

*y*^{S}_{1}^{ +
S}_{2}^{ +
. . . + S}_{n}**^{+
0 }**=

into
the final state:

*y**
*^{’}^{S}_{1}^{ +
S}_{2}^{ +
. . . + S}_{n}**^{+
0 }**=

where
*a_{i }*= (

Thus
we arrive at the general rule for the transformation of total state
functions which describe systems within which observation processes
occur:

*Rule
1:*
The observation of a quantity *A*, with
eigenfunctions *f*_{i}^{
S}**_{1}**,
in a system

*y*^{S}_{1}*y*^{S}_{2}^{. . .}*y*^{S}_{n}*y***^{0}**[

®
**S**_{i}*a_{i}*

where

*a_{i }*=
(

If
we next consider a *second* observation
to be made, where our total state is now a superposition, we can apply Rule 1
separately to each element of the superposition, since each element separately
obeys the wave equation and behaves independently of the remaining elements, and
then superpose the results to obtain the final solution. We formulate this
as:

*Rule
2*:
Rule 1 may be applied separately to each element of a superposition of total
system states, the results being superposed to obtain the final total state.
Thus, a determination of *B*, with
eigenfunctions *h*_{j}^{S}**_{2}**,^,
on

**S**_{i}*a_{i}*

into
the state

**S**_{i,j}*a_{i }b_{j} *

where
*b_{j}* = (

These
two rules, which follow directly from the superposition principle, give a
convenient method for determining final total states for any number of
observation process in any combinations. We now seek the *interpretation* of such final total
states.

Let
us consider the simple case of Б
single observation of a quantity *A*,
with eigenfunctions *f***_{i}**,
in the system

*y**
*^{’}**^{S
+ 0 }**=

There
is no longer any independent system state or observer state, although the two
have become correlated in a one-one manner. However, in each *element* of the superposition, *f*_{i}*y***^{0}**[

We
now consider a situation where the observer system comes into interaction with
the object system for a second time. According lo Rule 2 we arrive at the total
state after the second observation:

*y**
*^{’’}**^{S
+ 0 }**=

Again,
each element *f*_{i}*y***^{0}**[

Consider
now a different situation. An observer-system 0, with initial state *y***^{0}**[

*y*_{0}^{S}_{1}^{ +
S}_{2}^{ +
. . . + S}_{n}**^{+
0 }**=

We
assume that the measurements are performed on the systems in the order *S_{1}*,

*y*_{1}^{S}_{1}^{ +
S}_{2}^{ +
. . . + S}_{n}**^{+
0 }**=

(where
*a*_{i}** ^{1}**
refers to the first system,

After
the second measurement it is, by Rule 2,

*y*_{2}^{S}_{1}^{ +
S}_{2}^{ +
. . . + S}_{n}^{+
0}

=
**S**_{i,j}*a_{i }a_{j} *

and
in general, after *r* measurements have
taken place (*r* **£**
*n*), Rule 2 gives the result :

*y***_{r}** =

We
can give this state, *y***_{r,
}**the
following interpretation. It consists of a superposition of
states:

*y**
*^{’}**_{ij
. . . kљ }**=

**5***y*^{S}_{r+1}^{. . .}*y*^{S}_{n}*y***^{0}**[

each
of which describes the observer with a definite memory sequence [*a*_{i}^{1}_{,}*a*_{j}^{2}^{.
. .}*
**a*_{k}** ^{r}**].
Relative to him the (observed) system states are the corresponding
eigenfunctions

A
typical element *y*^{’}**_{ij
... k}**
of the final superposition describes a state of affairs wherein the observer has
perceived an apparently random sequence of definite results for the
observations. Furthermore the object systems have been left in the corresponding
eigenstates of the observation. At this stage suppose that a redetermination of
an earlier system observation (

We
thus arrive at the following picture: Throughout all of a sequence of
observation processes there is only one physical system representing the
observer, yet there is no single unique *state* of the observer (which follows
from the representations of interacting systems). Nevertheless, there is a
representation in terms of a *superposition*, each element of which
contains a definite observer state and a corresponding system state. Thus with
each succeeding observation (or interaction), the observer state "branches" into
a number of different states. Each branch represents a different outcome of the
measurement and the *corresponding*
eigenstate for the object-system state. All branches exist simultaneously in the
superposition after any given sequence of observations.‡
The "trajectory" of the memory configuration of an observer performing a
sequence of measurements is thus not a linear sequence of memory configurations,
but a branching tree, with all possible outcomes existing simultaneously in a
final superposition with various coefficients in the mathematical model. In any
familiar memory device the branching does not continue indefinitely, but must
stop at a point limited by the capacity of the memory.

‡
*Note added in
proof**.*
—
In reply to a preprint of this article some correspondents have raised the
question of the "transition from possible to actual," arguing that in "reality"
there is — as our experience testifies — no such splitting of observers states,
so that only one branch can ever actually exist. Since this point may occur to
other readers the following is offered in explanation.

The
whole issue of the transition from "possible" to "actual" is taken care of in
the theory in a very simple way — there is no such transition, nor is such a
transition necessary for the theory to be in accord with our experience. From
the viewpoint of the theory all elements of a superposition (all "branches") are
"actual," none <are *[added in
M.Price’s FAQ — E.Sh.]*> any more "real" than the rest. It is unnecessary
to suppose that all but one are somehow destroyed, since all the separate
elements of a superposition individually obey the wave equation with complete
indifference to the presence or absence ("actuality" or not) of any other
elements. This total lack of effect of one branch on another also implies that
no observer will ever be aware of any "splitting" process.

Arguments
that the world picture presented by this theory is contradicted by experience,
because we are unaware of any branching process, are like the criticism of the
Copernican theory that the mobility of the earth as a real physical fact is
incompatible with the common sense interpretation of nature because we feel no
such motion. In both cases the argument fails when it is shown that the theory
itself predicts that our experience will be what it in fact is. (In the
Copernican case the addition of Newtonian physics was required to be able to
show that the earth's inhabitants would be unaware of any motion of the
earth.)

In
order to establish quantitative results, we must put some sort of measure
(weighting) on the elements of a final superposition. This is necessary to be
able to make assertions which hold for almost all of the observer states
described by elements of a superposition. We wish to make quantitative
statements about the relative frequencies of the different possible results of
observation — which are recorded in the memory — for a typical observer state;
but to accomplish this we must have Б
method for selecting a typical element from a superposition of orthogonal
states.

We
therefore seek a general scheme to assign a measure to the elements of a
superposition of orthogonal states **S**_{i}*a_{i }*

We
now impose an additivity requirement. We can regard a subset

*љљљљ љљљљљљљљљљљљљљљљљљ љљљљљљn*

of
the superposition, say **S**
*a_{i}*

љљљљљљљљљљљљљљљљљљљљљљљ љљ *i = 1*

*љљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ
љљљљљљљљn*

*a**f***^{’
}**=

љљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ
љљљљљ*i =
1*

We
then demand that the measure assigned to *f***^{’}**
shall be the sum of the measures assigned to the

*љљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ
љљљљљљљљљn*

*m*(** a**)
=

*љљљљљ i = 1*

Then
we have already restricted the choice of *m* to the square amplitude alone; in
other words, we have *m*(*a_{i}*) =

To
see this, note that the normality of *f***^{’}**
requires that |

*m*(** a**)
=

Defining
a new function *g*(*x*)

*g*(*x*) = *m*(**Ö***x*)љљљљљљљљљљљљ љљљљљљљљљљљљ љљ(28)

we
see that (27) requires that

*g*(**S***u_{i}^{2}*)
=

Thus
*g* is restricted to be linear and
necessarily has the form:

*g*(*x*) = *cx*љљљљљљљ (*c* constant).љљљљљ љљљљљљљљљљљљљљљљљљљљљ љљљљ(30)

Therefore
*g*(*x*** ^{2}**) =

*m*(*a_{i}*) =

We
have thus shown that the only choice of measure consistent with our additivity
requirement is the square amplitude measure, apart from an arbitrary
multiplicative constant which may be fixed, if desired, by normalization
requirements. (The requirement that the total measure be unity implies that this
constant is 1.)

The
situation here is fully analogous to that of classical statistical mechanics,
where one puts a measure on trajectories of systems in the phase space by
placing a measure on the phase space itself, and then making assertions (such as
ergodicity, quasi-ergodicity, etc.) which hold for "almost all" trajectories.
This notion of ''almost all" depends here also upon the choice of measure, which
is in this case taken to be the Lebesgue measure on the phase space. One could
contradict the statements of classical statistical mechanics by choosing a
measure for which only the exceptional trajectories had nonzero measure.
Nevertheless the choice of Lebesgue measure on the phase space can be justified
by the fact that it is the only choice for which the "conservation of
probability" holds, (Liouville's theorem) and hence the only choice which makes
possible any reasonable statistical deductions at all.

In
our case, we wish to make statements about "trajectories" of ob-servers.
However, for us a trajectory is constantly branching (transforming from state to
superposition) with each successive measurement. To have a requirement analogous
to the "conservation of probability" in the classical case, we demand that the
measure assigned to a trajectory at one time shall equal the sum of the measures
of its separate branches at a later time. This is precisely the additivity
requirement which we imposed and which leads uniquely to the choice of
square-amplitude measure. Our procedure is therefore quite as justified as that
of classical statistical mechanics.

Having
deduced that there is a unique measure which will satisfy our requirements, the
square-amplitude measure, we continue our deduction. This measure then assigns
to the *i*,*j*,_{ }* ^{. . .} k*th element of the
superposition (24),

*f*_{i}^{
S}_{1}*f*_{j}^{S}_{2
}^{.
. .}*
**f*_{k}^{
S}_{r}*y*^{S}_{r+1}^{. . .}*y*^{S}_{n}*y***^{0}**[

the
measure (weight)

*M_{i,j, }^{. . .}_{
k}* =
(

so
that the observer state with memory configuration [*a*_{i}^{1}_{,}*a*_{j}^{2}_{,}^{.
. .}*
_{,}*

*M_{i,j, }^{. . .}_{
kљ }*=

where

*M_{i}*љ =

so
that the measure assigned to a particular memory sequence [*a*_{i}^{1}_{,}*a*_{j}^{2}_{,}^{.
. .}*
_{,}*

There
is a direct correspondence of our measure structure to the probability theory of
random sequences. *lf we regard* the *M_{i,j, }^{. . .}_{ k
}*as probabilities for the sequences then the sequences are
equivalent to the random sequences which are generated by ascribing to each term
the

Thus,
in particular, if we consider the sequences to become longer and longer (more
and more observations performed) *each*
memory sequence of the final superposition will satisfy any given criterion for
a randomly generated sequence, generated by the independent probabilities *a_{i}**

While
we have so far considered only sequences of observations of the same quantity
upon identical systems, the result is equally true for arbitrary sequences of
observations, as may be verified by writing more general sequences of
measurements, and applying Rules 1 and 2 in the same manner as presented
here.

We
can therefore summarize the situation when the sequence of observations is
arbitrary, when these observations are made upon the same or different
systems in any order, and when the number of observations of each quantity in
each system is very large, with the following result:

Except
for a set of memory sequences of measure nearly zero, the averages of any
functions over a memory sequence can be calculated approximately by the use of
the independent probabilities given by Process 1 for each initial observation,
on a system, and by the use of the usual transition probabilities for succeeding
observations upon the same system. In the limit, as the number of all types of
observations goes to infinity the calculation is exact, and the exceptional
set has measure zero.

This
prescription for the calculation of averages over memory sequences by
probabilities assigned to individual elements is precisely that of the
conventional "external observation" theory (Process 1). Moreover, these
predictions hold for almost all memory sequences. Therefore all predictions of
the usual theory will appear to be valid to the observer in almost all observer
states.

In
particular, the uncertainty principle is never violated since the latest
measurement upon a system supplies all possible information about the relative
system state, so that there is no direct correlation between any earlier
results of observation on the system, and the succeeding observation. Any
observation of a quantity *B*, between
two successive observations of quantity *A* (all on the same system) will destroy
the one-one correspondence between the earlier and later memory states for the
result of *A*. Thus for alternating
observations of different quantities there are fundamental limitations upon the
correlations between memory states for the same observed quantity, these
limitations expressing the content of the uncertainty
principle.

As
a final step one may investigate the consequences of allowing several observer
systems to interact with (observe) the same object system, as well as to
interact with one another (communicate). The latter interaction can be treated
simply as an interaction which correlates parts of the memory configuration of
one observer with another. When these observer systems are investigated, in the
same manner as we have already presented in this section using Rules 1 and 2,
one finds that in *all elements* of the
final superposition:

1.
When several observers have separately observed the same quantity in the object
system and then communicated the results to one another they find that they
are in agreement. This agreement persists even when an observer performs his
observation *after* the result has been
communicated to him by another observer who has performed the
observation.

2.
Let one observer perform an observation of a quantity *A* in the ПbjЕct
system, then let a second perform an observation of a quantity *B* in this object system which does not
commute with *A*, and finally let the
first observer repeat his observation of *A*. Then the memory system of the first
observer will *not* in general show the
same result for both observations. The intervening observation by the other
observer of the non-commuting quantity *B* prevents the possibility of any one to
one correlation between the two observations of *A*.

3.
Consider the case where the states of two object systems are correlated, but
where the two systems do not interact. Let one observer perform a specified
observation on the first system, then let another observer perform an
observation on the second system, and finally let the first observer repeat his
observation. Then it is found that the first observer always gets the same
result both times, and the observation by the second observer has no effect
whatsoever on the outcome of the first's observations. Fictitious paradoxes like
that of Einstein, Podolsky, and Rosen** ^{8}** which are concerned with
such correlated, noninteracting systems are easily investigated and clarified in
the present scheme.

** ^{8
}**Einstein,
Podolsky, and Rosen, Phys. Rev.

Many
further combinations of several observers and systems can be studied within the
present framework. The results of the present "relative state" formalism agree
with those of the conventional "external observation" formalism in all
those cases where that familiar machinery is applicable.

In
conclusion, the continuous evolution of the state function of a composite system
with time gives a complete mathematical model for processes that involve an
idealized observer. When interaction occurs, the result of the evolution in time
is a superposition of states, each element of which assigns a different state to
the memory of the observer. Judged by the state of the memory in almost all of
the observer states, the probabilistic conclusion of the usual "external
observation" formulation of quantum theory are valid. In other words, pure
Process 2 wave mechanics, without any initial probability assertions, leads to
all the probability concepts of the familiar
formalism.

**6.
DISCUSSION**

The
theory based on pure wave mechanics is a conceptually simple, causal theory,
which gives predictions in accord with experience. It constitutes a framework in
which one can investigate in detail, mathematically, and in a logically
consistent manner a number of sometimes puzzling subjects, such as the measuring
process itself and the interrelationship of several observers. Objections have
been raised in the past to the conventional or "external observation"
formulation of quantum theory on the grounds that its probabilistic features are
postulated in advance instead of being derived from the theory itself. We
believe that the present "relative-state" formulation meets this objection,
while retaining all of the content of the standard
formulation.

While
our theory ultimately justifies the use of the probabilistic interpretation as
an aid to making practical predictions, it forms a broader frame in which to
understand the consistency of that interpretation. In this respect it can
be said to form a *metatheor**Х*
for the standard theory. It transcends the usual ''external observation"
formulation, however, in its ability to deal logically with questions of
imperfect observation and approximate measurement.

The
"relative state" formulation will apply to all forms of quantum mechanics which
maintain the superposition principle. It may therefore prove a fruitful
framework for the quantization of general relativity. The formalism invites one
to construct the formal theory first, and to supply the statistical
interpretation later. This method should be particularly useful for
interpreting quantized unified field theories where there is no question of
ever isolating observers and object systems. They all are represented in a *single* structure, the field. Any
interpretative rules can probably only be deduced in and through the theory
itself.

Aside
from any possible practical advantages of the theory, it remains a matter of
intellectual interest that the statistical assertions of the usual
interpretation do not have the status of independent hypotheses, but are
deducible (in the present sense) from the pure wave mechanics that starts
completely free of statistical postulates.