Into a new state

^S+0' = _i⁰[^{. . .}_i]                               (11)

where _i characterizes⁷ the state _i. (The symbol, _i, might stand for a recording of the eigenvalue, for example.) That is, we require that the system state, if it is an eigenstate, shall be unchanged, and (2) that the observer state shall change so as to describe an observer that is "aware" of which eigenfunction it is; that is, some property is recorded in the memory of the observer which characterizes _i, such as the eigenvalue. The requirement that the eigenstates for the system be unchanged is necessary if the observation is to be significant (repeatable), and the requirement that the observer state change in a manner which is different for each eigenfunction is necessary if we are to be able to call the interaction an observation at all. How closely a general interaction satisfies the definition of a good observation depends upon (1) the way in which the interaction depends upon the dynamical variables of the observer system —including memory variables — and upon the dynamical variables of the object system and (2) the initial state of the observer system. Given (1) and (2), one can for example solve the wave equation, deduce the state of the composite system after the end of the interaction, and check whether an object system that was originally in an eigenstate is left in an eigenstate, as demanded by the repeatability postulate. This postulate is satisfied, for example, by the model of von Neumann that has already been discussed.

_______________________________

⁷ It should be understood that ⁰[^{. . .}_i] is a different state for each i. A more precise notation would write ⁰_i[^{. . .}_i], but no confusion can arise if we simply let the ⁰_i be indexed only by the index of the memory configuration symbol.

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 _i⁰[^{. . .}] into states _i⁰[^{. . .}_i]. Consequently these solutions of the wave equation can be superposed to give the final state for the case of an arbitrary initial system state. Thus if the initial system state is not an eigenstate, but a general state _ia_i_i, the final total state will have the form:

^S+0' = _ia_i_i⁰[^{. . .}_i].                       (12)

This superposition principle continues to apply in the presence of further systems which do not interact during the measurement. Thus, if systems S₁, S₂, ^{. . .} , S_n are present as well as 0, with original states ^S₁, ^S₂, ^{. . .} , ^S_n, and the only interaction during the time of measurement takes place between S₁ and 0, the measurement will transform the initial total state:

^S₁ ^{+ S}₂ ^{+ . . . + S}_n^{+ 0} = ^S₁^S₂ ^{. . .}^S_n,⁰[^{. . .}]                 (13)

into the final state:

 ^'S₁ ^{+ S}₂ ^{+ . . . + S}_n^{+ 0} = _ia_i_i ^S₁^S₂ ^{. . .}^S_n,⁰[^{. . .}_i]           (14)

where a_i = (_i ^S₁,^S₁) and _i ^S₁ are eigenfunctions of the observation.

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 _i ^S₁, in a system S₁ by the observer 0, transforms the total state according to:

^S₁^S₂ ^{. . .}^S_n⁰[^{. . .}]

 _ia_i_i ^S₁^S₂ ^{. . .}^S_n,⁰[^{. . .}_i] (15)

where

a_i = (_i ^S₁,^S₁).

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 _j^S₂,^, on S₂ by the observer 0 transforms the total state

_ia_i_i ^S₁^S₂ ^{. . .}^S_n,⁰[^{. . .}_i]                      (16)

into the state

_i,ja_i b_j _i ^S₁_j^S₂^S₂ ^{. . .}^S_n,⁰[^{. . .}_i,_j]                  (17)

where b_j = (_j^S₂,^S₂), which follows from the application of Rule 1 to each element _i ^S₁^S₂ ^{. . .}^S_n,⁰[^{. . .}_i], and then superposing the results with the Уcoefficients _i.

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 _i, in the system S with initial state ^S, by an observer 0 whose initial state is ⁰[^{. . .}]. The final result is, as we have seen, the superposition

 ^{'S + 0} = _ia_i_i ⁰[^{. . .}_i].                    (18)

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, _i⁰[^{. . .}_i], the object-system state is a particular eigenstate of the observation, and furthermore the observer-sХstem state describes the observer as definitelyХ perceiving that particular system state. This correlation is what allows one to maintain the interpretation that a measurement has been performed.

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:

 ^{''S + 0} = _ia_i_i ⁰[^{. . .}_i,_i].               (19)

Again, each element _i⁰[^{. . .}_i,_i] describes a system eigenstate, but this time also describes the observer as having obtained the same result for each of the two observations. Thus for every separate state of the observer in the final superposition the result of the observation was repeatable, even though different for different states. This repeatability is a consequence of the fact that after an observation the relative system state for a particular observer state is the corresponding eigenstate.

Consider now a different situation. An observer-system 0, with initial state ⁰[^{. . .}], measures the same quantity A in a number of separate, identical, systems which are initially in the same state, ^S₁  =^S₂ = ^{. . .} = ^S_n = _ia_i_i (where the _i are, as usual, eigenfunctions of A). The initial total state function is then

₀^S₁ ^{+ S}₂ ^{+ . . . + S}_n^{+ 0} = ^S₁^S₂ ^{. . .}^S_n⁰[^{. . .}]       (20)

We assume that the measurements are performed on the systems in the order S₁, S₂, ^{. . .} ,S_n. Then the total state after the first measurement is by Rule 1,

₁^S₁ ^{+ S}₂ ^{+ . . . + S}_n^{+ 0} = _ia_i_i ^S₁^S₂ ^{. . .}^S_n,⁰[^{. . .}_i¹]       (21)

(where _i¹ refers to the first system, S₁).

After the second measurement it is, by Rule 2,

₂^S₁ ^{+ S}₂ ^{+ . . . + S}_n^{+ 0}

= _i,ja_i a_j _i ^S₁_j^S₂^S₃ ^{. . .}^S_n,⁰[^{. . .} _i¹_, _j²]     (22)

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

_r = _{i,j, ... k} a_i a_j ^{. . .} a_k _i ^S₁_j^S₂^S₃ ^{. . .}^S_n,⁰[^{. . .} _i¹_, _j²] (23)

We can give this state, _r, the following interpretation. It consists of a superposition of states:

 ^'_{ij . . . k}  = _i ^S₁_j^S₂ ^{. . .} _k ^S_r

5^S_r+1 ^{. . .}^S_n⁰[_i¹_, _j^{2. . .} _k^r] (24)

each of which describes the observer with a definite memory sequence [_i¹_,_j^{2. . .} _k^r]. Relative to him the (observed) system states are the corresponding eigenfunctions _i^S₁,_j^S₂, ^{. . .} ,_k^S_r, the remaining systems, S₁, S₂, ^{. . .} ,S_n, being unaltered.

A typical element ^'_{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 (S_l) takes place. Then it follows that every element of the resulting final superposition will describe the observer with a memory configuration of the form [_i¹_, ^{. . .}_j^l_, ^{. . .}_k^r_,_j^l] in which the earlier memory coincides with the later — i.e., the memory states are correlated. It will thus appear to the observer, as described by a typical element of the superposition, that each initial observation on a system caused the system to "jump" into an eigenstate in a random fashion and thereafter remain there for subsequent measurements on the same system. Therefore — disregarding for the moment quantitative questions of relative frequencies — the probabilistic assertions of Process 1 appear to be valid to the observer described by a typical element of the final superposition.

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 _ia_i _i. We require a positive function m of the complex coefficients of the elements of the superposition, so that m(a_i) shall be the measure assigned to the clement _i. In order that this general scheme be unambiguous we must first require that the states themselves always be normalized, so that we can distinguish the coefficients from the states. However, we can still only determine the coefficients, in distinction to the states, up to an arbitrary phase factor. In order to avoid ambiguities the function m must therefore be a function of the amplitudes of the coefficients alone, m(a_i) = m(|a_i|).

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

                              n

of the superposition, say  a_i_i, as a single element ^':

                           i = 1

                                                                                                        n

^' =  a_i_i .                         (25)

                                                                                                     i = 1

We then demand that the measure assigned to ^' shall be the sum of the measures assigned to the _i (i from 1 to n):

                                                                                                         n

m() =  m(a_i).                                  (26)

      i = 1

Then we have already restricted the choice of m to the square amplitude alone; in other words, we have m(a_i) = a_i*a_i, apart from a multiplicative constant.

To see this, note that the normality of ^' requires that || = (a_i*a_i)^1/2. From our remarks about the dependence of m upon the amplitude alone, we replace the a_i by their amplitudes u_i = |a_i|. Equation (26) then imposes the requirement,

m() = m(a_i*a_i)^1/2 = m(u_i²)^1/2  =  m(u_i) =  m(u_i²)^1/2.     (27)

Defining a new function g(x)

g(x) = m(x)                            (28)

we see that (27) requires that

g(u_i²) =  g(u_i²) .                                        (29)

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

g(x) = cx        (c constant).                                (30)

Therefore g(x²) = cx² = m(x²) = m(x) and we have deduced that m is restricted to the form

m(a_i) = m(u_i) = cu_i² = ca_i*a_i.                (31)

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, ^{. . .} kth element of the superposition (24),

_i ^S₁_j^S₂ ^{. . .} _k ^S_r^S_r+1 ^{. . .}^S_n⁰[_i¹_,_j²_,^{. . .} _k^r]       (32)

the measure (weight)

M_i,j, ^{. . .} _k = (a_i a_j ^{. . .} a_k)*( a_i a_j ^{. . .} a_k)                  (33)

so that the observer state with memory configuration [_i¹_,_j²_,^{. . .} _,_k^r] is assigned the measure a_i*a_ia_j*a_j ^{. . .} a_k*a_k = M_i,j, ^{. . .} _k. We see immediately that this is a product measure, namely,

M_i,j, ^{. . .} _k  = M_i M_j ^{. . .} M_k                            (34)

where

M_i  = a_i *a_i

so that the measure assigned to a particular memory sequence [_i¹_,_j²_,^{. . .} _,_k^r] is simply the product of the measures for the individual components of the memory sequence.

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 independent probabilities M_i = a_i*a_i. Now probability theory is equivalent to measure theory mathematically, so that we can make use of it, while keeping in mind that all results should be translated back to measure theoretic language.

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*a_i, except for a set of total measure which tends toward zero as the number of observations becomes unlimited. Hence all averages of functions over any memory sequence, including the special case of frequencies, can be computed from the probabilities a_i*a_i, except for a set of memory sequences of measure zero. We have therefore shown that the statistical assertions of Process 1 will appear to be valid to the observer, in almost all elements of the superposition (24), in the limit as the number of observations goes to infinity.

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 Rosen⁸ which are concerned with such correlated, noninteracting systems are easily investigated and clarified in the present scheme.

⁸ Einstein, Podolsky, and Rosen, Phys. Rev. 47, 777 (1935). For a thorough discussion of the physics of observation, see the chapter by N.Bohr in Albert Einstein, Philosopher-Scientist (The Library of Living Philosophers, Inc., Evanston, 1949).

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 metatheorХ 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.