July 1957 Issue of Reviews of Modern Physics
REVIEWS OF Mod PHYSICS
VOLUME 29. NUMBER iii. JULY. 1957
"Relative Country" Conception of Quantum Mechanics*
Hugh Everett, lIl �
Palmer Physical Laboratory, Princeton University, Princeton, New Jersey
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* Thesis submitted to Princeton University March 1, 1957 in partial fulfillment of the requirements for the Ph.D. degree. An earlier typhoon 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 gratuitous of any responsibleness, but they are warmly thanked for the useful objections that they raised. Most detail 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
T he task of quantizing full general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when practical to and then primal a structure every bit the infinite-time geometry itself. This newspaper 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 tin be deduced.
The relationship of this new conception to the older formulation is therefore that of a metatheory to a theory, that is, information technology 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 non based on any radical departure from the conventional one. The special postulates in the quondam theory which deal with observation are omitted in the new theory. The contradistinct theory thereby acquires a new grapheme. It has to exist analyzed in and for itself before any identification becomes possible betwixt 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 ascertainment, but in a style 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" conception of quantum mechanics to exist essentially the following 1 : A physical system is completely described by a state function y , which is an chemical element of a Hilbert space, and which furthermore gives information only to the extent of specifying the probabilities of the results of various observations which can be made on the arrangement by external observers. There are two fundamentally different ways in which the land function can change:
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ane We use the terminology and annotation of J. von Neumann, Mathematical Foundations of Quantum Mechanics, translated past R.T.Beyer (Princeton University Printing, Princeton, I955).
Process i: The discontinuous change brought about past the observation of a quantity with eigenstates f 1 , f 2 , , in which the land y will be inverse to the state f j , with probability |( y , f j )| 2 .
Procedure 2: The continuous, deterministic change of country of an isolated system with time according to a wave equation d y / d t = � y , where A is a linear operator.
This formulation describes a wealth of experience. No experimental evidence is known which contradicts it.
Northward ot 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 arrangement. Tin can the modify with fourth dimension of the state of the total arrangement be described by Procedure two? If so, and then it would appear that no discontinuous probabilistic process like Process ane tin take place. If not, we are forced to acknowledge that systems which contain observers are not subject to the same kind of breakthrough-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 give-and-take of "observers" in quantum mechanics has to do with photoelectric cells, photographic plates, and like devices where a mechanistic attitude can hardly be contested. For the following one can limit himself to this class of issues, if he is unwilling to consider observers in the more than familiar sense on the same mechanistic level of assay.
What mixture of Processes one and 2 of the conventional formulation is to be applied to the case where but an approximate measurement is effected; that is, where an appliance or observer interacts merely weakly and for a limited time with an object system? In this case of an approximate measurement � proper theory must specify (i) the new state of the object system that corresponds to any particular reading of the apparatus and (two) the probability with which this reading will occur, von N � umann showed how to care for a special class of gauge measurements by the method of projection operators. 2 All the same, a general treatment of all approximate measurements by the method of projection operators can exist shown (Sec. 4) to exist impossible.
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2 Reference 1, Chap. 4, Sec. 4.
How is i to apply the conventional formulation of breakthrough mechanics to the space-time geometry itself? The consequence becomes especially astute in the case of a closed universe. iii There is no place to stand outside the system to notice it. There is zilch exterior it to produce transitions from 1 country 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 full energy by way of an integral extended over a surface large enough to include all parts of the organisation and their interactions. 4 But in a closed infinite, when a surface is made to include more and more of the book, it ultimately disappears into nothingness. Attempts to ascertain a total energy for a airtight space collapse to the vacuous argument, null equals naught.
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3 See A.Einstein, The Meaning of Relativity (Princeton University Press, Princeton, 1950), third edition, p. 107.
4 L.Landau and East.Lifshitz, The Classical Theory of Fields, translated by Chiliad.Hamermesh (Addison-Wesley Press, Cambridge, 1951), p. 343.
How are a quantum description of a closed universe, of approximate measurements, and of a arrangement that contains an observer to exist fabricated? These 3 questions have one feature in mutual, that they all ask nearly the breakthrough 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 ascertainment. The whole interpretive scheme of that ceremonial 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 function of the formalism there is no ways whatsoever to ascribe a concrete estimation to the conventional machinery. But Process 1 is out of the question for systems non subject to external observation. v
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5 Meet in item the discussion of this signal past N.Bohr and L.Rosenfeld, Kgl. Danske Videnskab, Selskab, Mat.-fys. Medd. 12, No. 8 (1933).
three. Breakthrough MECHANICS INTERNAL TO AN ISOLATED Arrangement
This newspaper proposes to reward pure wave mechanics ( � rocess 2 only) as a complete theory. Information technology postulates that a wave office 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 organisation that is subject to external ascertainment can be regarded every bit part of a larger isolated system.
The moving ridge function is taken every bit the basic physical entity with no a priori estimation. Estimation just comes after an investigation of the logical structure of the theory. Here as always the theory itself sets the framework for its estimation. 5
For any estimation it is necessary to put the mathematical model of the theory into correspondence with feel. For this purpose it is necessary to codify abstruse models for observers that can be treated inside 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 translate the changes in the familiar language of feel.
Section four investigates representations of the state of a composite system in terms of states of constituent subsystems. The mathematics leads 1 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-divers country, independently of the remainder of the composite system. To whatsoever arbitrarily called state for one subsystem in that location will stand for a unique relative land for the remainder of the composite arrangement. This relative state volition normally depend upon the pick of state for the first subsystem. Thus the land of one subsystem does not accept an independent existence, but is stock-still simply by the state of the remaining subsystem. In other words, usa occupied by the subsystems are not independent, but correlated. Such correlations between systems arise whenever systems interact. In the nowadays formulation all measurements and ascertainment processes are to be regarded just equally interactions betwixt the physical systems involved � interactions which produce stiff correlations. A simple model for a measurement, due to von Neumann, is analyzed from this viewpoint.
Section 5 gives an abstruse 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 society that the results shall take the greatest generality and be applicative 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 arrangement. It is found that eastward xpe ri e nces of the observer (magnetic record retentivity, counter organisation, etc.) are in total accordance with predictions of the conventional "external observer" formulation of quantum mechanics, based on Process 1.
Section 6 recapituIates the "relative land" formulation of breakthrough mechanics.
four. CONCEPT OF RELATIVE STATE
We now investigate some consequences of the moving ridge mechanical ceremonial of composite systems. If a composite system Southward, is composed of ii subsystems S i and S two , with associated Hilbert spaces H i and H two , then, according to the usual formalism of composite systems, the Hilbert space for S is taken to be the tensor product of H 1 and H 2 (written H = H 1 Ä H two ). This has the consecuence that if the sets { ten i Southward 1 } and { h j S 2 } are complete orthonormal sets of states for S one and Due south 2 , respectively, then the full general state of S tin be written as a superposition:
y South = Due south i,j a ij x i Southward 1 h j South 2 . (i)
From (3.1) although S is in a definite state y S , the subsystems S 1 and S 2 do not possess anything like definite states independentl � of one another (except in the special case where all just ane of the a ij are aught).
We can, however, for whatever option of a state in one subsystem, uniquely assign a corresponding relative state in the other subsystem. For example, if we choose ten k as the state for S 1 , while the blended arrangement S is in the state y S given by (3.i), and so the corresponding relative state in S two , y (S two ; rel x 1000 , S 1 ), will be:
y (Southward ii ; rel 10 k , S 1 ) = North k S j a kj h j S 2 (2)
where Northward m is a normalization constant. This relative country for x k is contained of the choice of basis { ten i } (i ¹ m ) for the orthogonal complement of ten k , and is hence adamant uniquely by x thou alone. To observe the relative country in S two for an arbitrary state of Due south 1 therefore, i simply carries out the above procedure using any pair of bases for S one and S 2 which contains the desired state as one element of the footing for Due south 1 . To find states in S 1 relative to states in Due south 2 , interchange S 1 and Southward 2 in the procedure.
In the conventional or "external observation" formulation, the relative state in S 2 , y (S 2 ; rel f , Due south ane ) for a state f S 1 in S i , gives the conditional probability distributions for the results of all measurements in S ii , given that South 1 has been measured and found to be in state f S one � i.e., that f Due south one is the eigenfunction of the measurement in S ane respective to the observed eigenvalue.
For whatsoever option of basis in S 1 , { x i }, it is e'er possible to correspond the state of S, (i), as a single superposition of pairs of states, each consisting of a state from the basis { x i } in S one and its relative state in S two . Thus, from (two), (1) tin can exist written in the form:
i
y S = S i � x i S i y (Southward 2 ; rel x i , South 1 ). (three)
North i
This is an important representation used frequently.
Summarizing: There does non, in general, exist anything like a unmarried state for 1 subsystem of a blended arrangement. Subsystems exercise not possess states that are independent of the states of the remainder of the arrangement, so that the subsystem states are generally correlated with one another. One can arbitrarily cull a country for one subsystem, and be led to the relative state for the remainder. Thus nosotros are faced with a fundamental relativity of states, which is implied past the formalism of composite systems. Information technology is meaningless to ask the absolute country 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 every bit a model of a measurement process. Give-and-take of this example prepares the ground for the analysis of "observation." We outset with a system of merely 1 coordinate, q (such equally 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 contained, so that the combined wave role is y 0 Southward+A = f (q) h (r ) where f (q) is the initial system moving ridge role, and h (r ) is the initial appliance function. The Hamiltonian is such that the two systems practice not collaborate except during the interval t = 0 to t = T, during which time the full Hamiltonian consists only of a uncomplicated interaction,
H I = - i � q( d / d r). (iv)
Then the state
y t S+A (q,r) = f (q) h (r - qt ) (v)
is a solution of the Schr ö dinger equation,
i � ( d y t Southward+A / d t) = H I y t Due south+A , (6)
for the specified initial weather condition at lime t = 0.
From (5) at fourth dimension t = T (at which time interaction stops) there is no longer any definite independent apparatus state, nor any independent system land. The apparatus therefore does not indicate whatsoever definite object-system value, and nothing like process ane has occurred.
Nevertheless, we can await upon the full wave function (five) equally a superposition of pairs of subsystem states, each element of which has a definite q value and a correspondingly displaced apparatus state. Thus afterward the interaction the state (v) has the form:
y T Due south+A = 
which is a superposition of states y q ' = d (q - q ' ) h (r - q ' T). Each of these elements, y q ' , of the superposition describes a land in which the system has the definite value q = q ' , and in which the apparatus has a land that is displaced from its original country by the amount q ' T . These elements y q ' are so superposed with coefficients f (q ' ) to grade the total land (7).
Conversely, if we transform to the representation where the appliance coordinate is definite, we write (5) as
y T S+A =
where
x r ' (q) = Due north r ' f (q) h (r ' - qT) (eight)
and
(ane/N r ' ) 2 =
And so the x r ' (q) are the relative organization state functions 6 for the appliance states d (r - r ' ) of definite value r = r ' .
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6 This example provides a model of an estimate measurement. Nonetheless, the relative organisation state after the interaction x r ' (q) cannot usually exist generated from the � original system state f by the application of � n � projection operator, E. Proof: Suppose on the contrary that 10 r ' (q) = NE f (q) = N ' f (q) h (r ' - qt), where N, N ' are normalization constants. Then
Eastward(NE f (q)) = NE 2 f (q) = North '' f (q) h 2 (r ' - qt)
and E ii f (q) = (Northward '' /Due north) f (q) h ii (r ' - qt). Merely the status E 2 = Due east which is necessary for E to be a projection implies that North ' /Northward '' h (q) = h 2 (q) which is more often than not false.
If T is sufficiently big, or h (r) sufficiently sharp (near d (r)) then x r ' (q) is about d (q - r ' /T) and the relative arrangement states ten r ' (q) are nearly eigenstates for the values q = r ' /T .
We have seen that (8) is a superposition of states y r ' , for each of which the apparatus has recorded a definite value r ' , and the system is left in approximately the eigenstate of the measurement corresponding to q = r ' /T . The discontinuous "bound" into an eigenstate is thus only a relative proposition, dependent upon the mode of decomposition of the total moving ridge function into the superposition, and relative to a particularly chosen apparatus-coordinate value. Then far as the complete theory is concerned all elements of the superposition exist simultaneously, and the entire procedure is quite continuous.
von Neumann'south example is but a special example of a more than general situation. Consider any measuring apparatus interacting with any object organisation. As a outcome of the interaction the country of the measuring appliance is no longer capable of independent definition. It can be defined simply relative to the state of the object arrangement. In other words, in that location exists simply a correlation between us of the two systems. Information technology seems as if nix tin e'er be settled past 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 Procedure 2 with experience, or must the theory be abandoned every bit untenable? In gild to answer this question we consider the problem of observation itself inside the framework of the theory.
5. OBSERVATION
Nosotros accept the task of making deductions most the appearance of phenomena to observers which are considered equally purely physical systems and are treated within the theory. To reach this it is necessary to place some nowadays properties of such an observer with features of the past feel of the observer.
Thus, in society to say that an observer 0 has observed the event a , it is necessary that the state of 0 has get inverse from its former state to a new land which is dependent upon a .
Information technology 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 feel of an observer it is sufficient to deduce the present contents of the retentiveness as it appears inside the mathematical model.
Equally models for observers we can, if we wish, consider automatically functioning machines, possessing sensory apparatus and coupled to recording devices capable of registering by sensory data and machine configurations. Nosotros can further suppose that the machine is so synthetic that its present actions shall exist determined not but by its present sensory information, but by the contents of its memory also. Such a machine will then be capable of performing a sequence of observations (measurements), and furthermore of deciding upon its futurity experiments on the basis of by results. If we consider that electric current sensory data, also equally car 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 simply, and all relevant experience of the auto is independent in the memory.
For such machines we are justified in using such phrases as "the car has perceived A" or "the machine is aware of A" if the occurrence of A is represented in the retention, since the future beliefs of the machine will exist 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 organisation representing an observer quantum mechanically nosotros ascribe a land function, y 0 , to it. When the land y 0 describes an observer whose memory contains representations of the events A, B, , � we denote this fact past appending the memory sequence in brackets as a subscript, writing:
y 0 [A, B, , C] (9)
The symbols A, B, , � , which nosotros assume to exist ordered time-wise, therefore stand for memory configurations which are in correspondence with the past feel of the observer. These configurations tin can exist 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 just that they be capable of the estimation: "The observer has experienced the succession of events A, B, , � ." (We sometimes write dots in a memory sequence, A, B, , � , to signal the possible presence of previous memories which are irrelevant to the example being considered.)
The mathematical model seeks to care for the interaction of such observer systems with other physical systems (observations), within the framework of Process two wave mechanics, and to deduce the resulting memory configurations, which are and then to be interpreted as records of the by experiences of the observers.
Nosotros begin by defining what constitutes a "good" observation. A good ascertainment of a quantity A, with eigenfunctions f i , for a arrangement S, past an observer whose initial land is y 0 , consists of an interaction which, in a specified period of time, transforms each (total) state
y Due south+0 = f i y 0 [ . . . ] (10)
into a new country
y S+0 ' = f i y 0 [ . . . a i ] (11)
where a i characterizes 7 the country f i . (The symbol, a i , might represent a recording of the eigenvalue, for example.) That is, we require that the organisation country, if it is an eigenstate, shall exist unchanged, and (ii) that the observer state shall change so as to depict an observer that is "aware" of which eigenfunction it is; that is, some belongings is recorded in the memory of the observer which characterizes f i , such equally the eigenvalue. The requirement that the eigenstates for the arrangement be unchanged is necessary if the observation is to be significant (repeatable), and the requirement that the observer land alter in a fashion which is different for each eigenfunction is necessary if nosotros are to be able to call the interaction an ascertainment at all. How closely a full general interaction satisfies the definition of a good observation depends upon (one) the way in which the interaction depends upon the dynamical variables of the observer system �including retentivity variables � and upon the dynamical variables of the object system and (2) the initial state of the observer system. Given (i) and (two), ane can for example solve the moving ridge 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.
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7 It should be understood that y 0 [ . . . a i ] is a different state for each i. A more precise note would write y 0 i [ . . . a i ], but no confusion can arise if we simply let the y 0 i be indexed only by the index of the memory configuration symbol.
From the definition of a adept observation we first deduce the issue of an ascertainment 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 [ . . . ] into states f i y 0 [ . . . a i ]. Consequently these solutions of the wave equation can exist superposed to give the concluding land for the case of an arbitrary initial system state. Thus if the initial system state is not an eigenstate, only a general state South i a i f i , the terminal total state will have the form:
y S+0 ' = South i a i f i y 0 [ . . . a i ]. (12)
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 , . . . , South n are present as well every bit 0, with original states y Due south 1 , y Due south two , . . . , y South n , and the but interaction during the time of measurement takes place betwixt S 1 and 0, the measurement will transform the initial total state:
y S 1 + S two + . . . + S due north + 0 = y Due south 1 y S 2 . . . y South north , y 0 [ . . . ] (thirteen)
into the concluding land:
y ' S 1 + S two + . . . + S n + 0 = Southward i a i f i S 1 y S ii . . . y South n , y 0 [ . . . a i ] (14)
where a i = ( f i Southward 1 , y Southward ane ) and f i Southward ane are eigenfunctions of the observation.
Thus we arrive at the full general rule for the transformation of total state functions which depict systems inside which observation processes occur:
Dominion 1: The observation of a quantity A, with eigenfunctions f i S 1 , in a system Due south 1 past the observer 0, transforms the full state according to:
y Due south ane y S ii . . . y Southward n y 0 [ . . . ]
� S i a i f i S one y South 2 . . . y Southward n , y 0 [ . . . a i ] (15)
where
a i = ( f i Southward i , y S 1 ).
If we next consider a second observation to be made, where our total state is now a superposition, nosotros can employ Dominion 1 separately to each element of the superposition, since each element separately obeys the moving ridge equation and behaves independently of the remaining elements, and so superpose the results to obtain the final solution. We formulate this as:
Rule ii : Rule 1 may be practical separately to each chemical element of a superposition of full organisation states, the results being superposed to obtain the terminal total state. Thus, a conclusion of B, with eigenfunctions h j S two ,^, on S 2 past the observer 0 transforms the full state
S i a i f i S i y S 2 . . . y S due north , y 0 [ . . . a i ] (xvi)
into the land
S i,j a i b j f i S 1 h j S 2 y South 2 . . . y S northward , y 0 [ . . . a i, b j ] (17)
where b j = ( h j S 2 , y S 2 ), which follows from the application of Rule 1 to each element f i South 1 y S 2 . . . y Due south due north , y 0 [ . . . a i ], and so superposing the results with the � coefficients a i .
These two rules, which follow directly from the superposition principle, give a convenient method for determining final total states for any number of ascertainment process in any combinations. We now seek the interpretation of such terminal total states.
Let us consider the simple example of � single observation of a quantity A, with eigenfunctions f i , in the arrangement Due south with initial country y S , by an observer 0 whose initial land is y 0 [ . . . ]. The last result is, as we have seen, the superposition
y ' Southward + 0 = South i a i f i y 0 [ . . . a i ]. (xviii)
There is no longer any independent organisation state or observer country, although the two have go correlated in a one-one style. However, in each element of the superposition, f i y 0 [ . . . a i ], the object-system state is a detail eigenstate of the observation, and furthermore the observer-s � stem state describes the observer as definitely � perceiving that particular organization land . This correlation is what allows ane 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 arrangement for a 2d time. According lo Rule 2 nosotros make it at the total state after the second observation:
y '' S + 0 = S i a i f i y 0 [ . . . a i, a i ]. (19)
Again, each chemical element f i y 0 [ . . . a i, a i ] describes a system eigenstate, but this time also describes the observer every bit having obtained the same result for each of the two observations. Thus for every separate land 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 organisation state for a particular observer land is the corresponding eigenstate.
Consider at present a different situation. An observer-system 0, with initial state y 0 [ . . . ], measures the same quantity A in a number of split up, identical, systems which are initially in the aforementioned land, y S 1 = y S 2 = . . . = y S northward = Southward i a i f i (where the f i are, as usual, eigenfunctions of A). The initial total state part is and so
y 0 Southward 1 + S ii + . . . + Southward n + 0 = y S 1 y S 2 . . . y S northward y 0 [ . . . ] (xx)
Nosotros presume that the measurements are performed on the systems in the club Due south 1 , S 2 , . . . ,S n . And then the total state afterwards the first measurement is past Dominion 1,
y 1 Southward 1 + S ii + . . . + S n + 0 = S i a i f i Due south 1 y S 2 . . . y Due south n , y 0 [ . . . a i ane ] (21)
(where a i 1 refers to the get-go system, S ane ).
Afterward the second measurement it is, by Rule 2,
y 2 Due south i + South 2 + . . . + S northward + 0
= Due south i,j a i a j f i South 1 f j S 2 y S 3 . . . y South north , y 0 [ . . . a i i , a j 2 ] (22)
and in full general, after r measurements accept taken place (r £ n ), Dominion 2 gives the issue :
y r = S i,j, ... thou a i a j . . . a k f i South one f j Southward 2 y S 3 . . . y S northward , y 0 [ . . . a i 1 , a j ii ] (23)
Nosotros tin give this state, y r, the post-obit interpretation. It consists of a superposition of states:
y ' ij . . . m = f i Due south 1 f j S ii . . . f 1000 S r
5 y Southward r+1 . . . y S n y 0 [ a i 1 , a j 2 . . . a k r ] (24)
each of which describes the observer with a definite memory sequence [ a i 1 , a j 2 . . . a grand r ]. Relative to him the (observed) arrangement states are the corresponding eigenfunctions f i S 1 , f j South 2 , . . . , f one thousand South r , the remaining systems, Southward i , S 2 , . . . ,S n , being unaltered.
A typical element y ' ij ... k of the last superposition describes a country of diplomacy wherein the observer has perceived an apparently random sequence of definite results for the observations. Furthermore the object systems have been left in the respective eigenstates of the ascertainment. At this stage suppose that a redetermination of an earlier system observation (S l ) takes place. And so it follows that every chemical element of the resulting final superposition will describe the observer with a memory configuration of the form [ a i 1 , . . . a j l , . . . a k r , a j 50 ] in which the earlier memory coincides with the later � i.e., the memory states are correlated. It will thus announced to the observer, as described by a typical element of the superposition, that each initial observation on a system caused the system to "leap" into an eigenstate in a random fashion and thereafter remain there for subsequent measurements on the aforementioned system. Therefore � disregarding for the moment quantitative questions of relative frequencies � the probabilistic assertions of Process one appear to exist valid to the observer described past a typical element of the concluding superposition.
Nosotros thus arrive at the post-obit picture: Throughout all of a sequence of ascertainment processes there is only one concrete system representing the observer, yet at that place is no single unique state of the observer (which follows from the representations of interacting systems). Nonetheless, there is a representation in terms of a superposition, each element of which contains a definite observer state and a corresponding system country. Thus with each succeeding observation (or interaction), the observer state "branches" into a number of different states. Each branch represents a different issue of the measurement and the corresponding eigenstate for the object-system land. All branches exist simultaneously in the superposition subsequently whatever 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, only a branching tree, with all possible outcomes existing simultaneously in a last superposition with various coefficients in the mathematical model. In any familiar retentiveness device the branching does not continue indefinitely, but must stop at a point limited by the capacity of the retentiveness.
� Annotation added in proof . � In respond to a preprint of this article some correspondents have raised the question of the "transition from possible to bodily," arguing that in "reality" there is � as our feel testifies � no such splitting of observers states, so that only 1 branch can ever actually be. Since this signal 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 Grand.Cost'southward FAQ � Due east.Sh.]> whatsoever more than "real" than the residual. Information technology is unnecessary to suppose that all but one are somehow destroyed, since all the divide elements of a superposition individually obey the moving ridge equation with consummate indifference to the presence or absence ("actuality" or not) of any other elements. This full lack of upshot of 1 branch on some other also implies that no observer will ever exist aware of any "splitting" procedure.
Arguments that the earth picture presented by this theory is contradicted by feel, because we are unaware of whatsoever branching procedure, are like the criticism of the Copernican theory that the mobility of the earth as a real concrete fact is incompatible with the mutual sense estimation of nature considering we feel no such motion. In both cases the statement fails when it is shown that the theory itself predicts that our feel 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 world.)
In order to institute quantitative results, we must put some sort of mensurate (weighting) on the elements of a terminal superposition. This is necessary to be able to make assertions which hold for nearly all of the observer states described by elements of a superposition. We wish to make quantitative statements near the relative frequencies of the dissimilar possible results of ascertainment � which are recorded in the retentiveness � for a typical observer land; simply to accomplish this nosotros must have � method for selecting a typical element from a superposition of orthogonal states.
Nosotros therefore seek a full general scheme to assign a measure to the elements of a superposition of orthogonal states Southward i a i f i . We require a positive function m of the complex coefficients of the elements of the superposition, and then that thou(a i ) shall be the measure assigned to the clement f i . In order that this general scheme exist unambiguous we must first require that united states themselves always be normalized, so that nosotros can distinguish the coefficients from us. However, we can withal only determine the coefficients, in stardom to the states, upwardly to an arbitrary phase gene. In order to avoid ambiguities the function m must therefore exist a function of the amplitudes of the coefficients alone, m(a i ) = k(|a i |).
Nosotros at present impose an additivity requirement. Nosotros can regard a subset
n
of the superposition, say S a i f i , every bit a unmarried chemical element a f ' :
i = 1
n
a f ' = Southward a i f i . (25)
i = 1
Nosotros then demand that the measure out assigned to f ' shall be the sum of the measures assigned to the f i (i from i to n):
north
m ( a ) = S one thousand(a i ). (26)
i = 1
Then nosotros take already restricted the choice of m to the foursquare amplitude alone; in other words, nosotros take m(a i ) = a i *a i, apart from a multiplicative constant.
To see this, note that the normality of f ' requires that | a | = ( South a i *a i ) i/2 . From our remarks about the dependence of m upon the amplitude alone, we supervene upon the a i past their amplitudes u i = |a i |. Equation (26) and then imposes the requirement,
thou ( a ) = m( S a i *a i ) 1/2 = m(u i ii ) 1/2 = S m(u i ) = S m (u i 2 ) ane/2 . (27)
Defining a new function thou(x)
m (10) = thou( Ö 10 ) (28)
we encounter that (27) requires that
k ( S u i 2 ) = S thousand (u i 2 ) . (29)
Thus g is restricted to exist linear and necessarily has the grade:
g (x) = cx (c abiding). (30)
Therefore thou(x two ) = cx 2 = m( Ö x ii ) = m(10) and we have deduced that m is restricted to the class
thousand (a i ) = m(u i ) = cu i 2 = ca i *a i . (31)
We have thus shown that the but option of measure consequent with our additivity requirement is the foursquare aamplitude measure, autonomously 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 ane.)
The situation here is fully analogous to that of classical statistical mechanics, where 1 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 agree for "well-nigh all" trajectories. This notion of ''almost all" depends hither also upon the option of measure, which is in this example taken to be the Lebesgue measure out on the phase space. One could contradict the statements of classical statistical mechanics by choosing a measure out for which only the exceptional trajectories had nonzero measure out. However the choice of Lebesgue mensurate on the stage space can exist justified by the fact that it is the only selection 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 virtually "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 example, nosotros demand that the measure assigned to a trajectory at one time shall equal the sum of the measures of its separate branches at a afterward time. This is precisely the additivity requirement which we imposed and which leads uniquely to the selection of square-amplitude measure. Our procedure is therefore quite as justified as that of classical statistical mechanics.
Having deduced that in that location is a unique measure which will satisfy our requirements, the foursquare-amplitude measure, we proceed our deduction. This measure out so assigns to the i,j, . . . grandth element of the superposition (24),
f i S ane f j S 2 . . . f thousand S r y South r+1 . . . y Southward n y 0 [ a i i , a j 2 , . . . a thousand r ] (32)
the measure (weight)
M i,j, . . . m = (a i a j . . . a chiliad )*( a i a j . . . a one thousand ) (33)
so that the observer state with memory configuration [ a i i , a j 2 , . . . , a k r ] is assigned the measure out a i* a i a j* a j . . . a k* a k = M i,j, . . . thousand . We see immediately that this is a product measure, namely,
M i,j, . . . chiliad = M i K j . . . M k (34)
where
M i = a i *a i
so that the measure assigned to a particular memory sequence [ a i 1 , a j two , . . . , a g r ] is simply the product of the measures for the individual components of the memory sequence.
There is a direct correspondence of our mensurate structure to the probability theory of random sequences. lf nosotros regard the G i,j, . . . grand as probabilities for the sequences then the sequences are equivalent to the random sequences which are generated past ascribing to each term the independent probabilities M i = a i *a i . Now probability theory is equivalent to measure theory mathematically, so that nosotros tin can make use of it, while keeping in mind that all results should be translated back to measure theoretic language.
Thus, in detail, if we consider the sequences to become longer and longer (more and more observations performed) each memory sequence of the last superposition volition satisfy any given benchmark for a randomly generated sequence, generated by the independent probabilities a i *a i , except for a ready of total measure which tends toward goose egg as the number of observations becomes unlimited. Hence all averages of functions over any memory sequence, including the special case of frequencies, can exist computed from the probabilities a i *a i , except for a set of retentivity sequences of measure zero. We have therefore shown that the statistical assertions of Process i volition announced 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 capricious sequences of observations, as may be verified by writing more general sequences of measurements, and applying Rules one and 2 in the same manner every bit presented here.
We can therefore summarize the situation when the sequence of observations is capricious, when these observations are made upon the same or dissimilar systems in any guild, and when the number of observations of each quantity in each system is very large, with the following event:
Except for a set of retention sequences of mensurate nearly null, the averages of any functions over a retentiveness sequence can be calculated approximately by the utilise of the independent probabilities given by Process ane for each initial ascertainment, 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 adding is exact, and the exceptional set has measure out goose egg.
This prescription for the calculation of averages over retentiveness sequences by probabilities assigned to individual elements is precisely that of the conventional "external ascertainment" theory (Process one). Moreover, these predictions agree for near all memory sequences. Therefore all predictions of the usual theory volition appear to be valid to the observer in almost all observer states.
In particular, the doubt principle is never violated since the latest measurement upon a system supplies all possible data about the relative system state, then that there is no direct correlation between whatsoever 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) volition destroy the i-one correspondence between the earlier and later on retentivity states for the result of A. Thus for alternating observations of different quantities in that location are central limitations upon the correlations between memory states for the same observed quantity, these limitations expressing the content of the uncertainty principle.
As a last step one may investigate the consequences of allowing several observer systems to interact with (detect) the same object arrangement, too as to interact with 1 some other (communicate). The latter interaction tin can exist treated simply as an interaction which correlates parts of the retentivity configuration of one observer with some other. When these observer systems are investigated, in the aforementioned mode as nosotros have already presented in this section using Rules one and 2, 1 finds that in all elements of the final superposition:
one. When several observers have separately observed the aforementioned quantity in the object system and then communicated the results to 1 another they detect that they are in agreement. This understanding persists even when an observer performs his ascertainment after the outcome has been communicated to him by another observer who has performed the observation.
2. Let i observer perform an observation of a quantity A in the � bj � ct arrangement, then let a second perform an observation of a quantity B in this object system which does not commute with A, and finally permit the first observer repeat his ascertainment of A. And then the memory organization of the first observer will not in general testify the same result for both observations. The intervening observation by the other observer of the not-commuting quantity B prevents the possibility of any one to 1 correlation between the ii observations of A.
three. Consider the case where the states of two object systems are correlated, but where the two systems do non interact. Permit one observer perform a specified observation on the kickoff arrangement, and then let another observer perform an ascertainment on the second arrangement, and finally let the first observer repeat his observation. Then it is institute that the commencement observer always gets the aforementioned upshot both times, and the observation past the second observer has no effect whatsoever on the consequence of the first's observations. Fictitious paradoxes like that of Einstein, Podolsky, and Rosen 8 which are concerned with such correlated, noninteracting systems are hands investigated and clarified in the present scheme.
eight 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 hold with those of the conventional "external ascertainment" formalism in all those cases where that familiar machinery is applicative.
In conclusion, the continuous evolution of the state role of a composite system with time gives a complete mathematical model for processes that involve an idealized observer. When interaction occurs, the event of the evolution in time is a superposition of states, each element of which assigns a different country to the memory of the observer. Judged past the state of the memory in near all of the observer states, the probabilistic conclusion of the usual "external ascertainment" formulation of quantum theory are valid. In other words, pure Process 2 wave mechanics, without whatsoever initial probability assertions, leads to all the probability concepts of the familiar ceremonial.
vi. DISCUSSION
The theory based on pure wave mechanics is a conceptually unproblematic, 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 mode a number of sometimes puzzling subjects, such as the measuring procedure itself and the interrelationship of several observers. Objections accept been raised in the past to the conventional or "external ascertainment" conception 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 nowadays "relative-state" conception 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 applied predictions, it forms a broader frame in which to empathise the consistency of that interpretation. In this respect information technology tin can be said to course a metatheor � for the standard theory. It transcends the usual ''external ascertainment" formulation, however, in its power to deal logically with questions of imperfect ascertainment and approximate measurement.
The "relative state" formulation will utilise to all forms of quantum mechanics which maintain the superposition principle. It may therefore prove a fruitful framework for the quantization of full general relativity. The formalism invites 1 to construct the formal theory first, and to supply the statistical estimation later. This method should exist especially useful for interpreting quantized unified field theories where in that location is no question of always isolating observers and object systems. They all are represented in a unmarried 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, information technology remains a matter of intellectual involvement that the statistical assertions of the usual interpretation exercise not have the status of independent hypotheses, but are deducible (in the present sense) from the pure moving ridge mechanics that starts completely free of statistical postulates.
Source: http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html
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