Foundations of Physics, Vol . 28, No. 8, 1998

The Feynman Path Integrals and Everett’s Universal

Wave Function

D. Bar1

Received August 6, 1997; revised January 14, 1998

We study here the properties of some quantum mechanical wave functions , which,in contrast to the regular quantum mechanical wave functions , can be predeter-mined with certainty (probability 1) by performing dense measurements (3) (orcontinuous observations). These specific `̀ certain’’ states are the junction pointsthrough which pass all the diverse paths that can proceed between each two suchneighboring `̀ sure’’ points. When we compare the properties of these points to theproperties of the well-known universal wave functions of Everett(2) we find astrong similarity between these two apparently uncorrelated entities, and in thisway find the same similarity between the Feynman path integrals and Everett’suniversal wave functions .

1. INTRODUCTION

In this article we are going to show a strong equivalence between Feynman’spath integrals( 1) and Everett’s universal wave functions.( 2) We concentratehere upon the physical interpretation of the Feynman’ s paths as formulatedby Aharonov and Vardi.( 3) This formulation exploited the interestingproperty of continuous observation (or dense measurements (4 ) ) in quantummechanics (QM), namely that we can, by performing dense measurementsupon a quantum system, bypass the well-known reduction of the wavefunction. In this way we can anticipate with certainty (with probability 1)the results of these measurements. This property emerges only because ofthese dense observations, so the observer has to become an integral andinseparable part of the object system. This important fact reminds us of the`̀ relative state’’ of Everett. (2 ) In the first section we try to clarify and under-stand these `̀ sure’’ states, and in the second section we show a strong

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0015-9018/98/0800-1383$15.00/0 Ñ 1998 Plenum Publishing Corporation

1 Department of Physics, BAR Ilan University, Ramat-Gan, 52900 Israel.

similarity between these `̀ sure’’ states and the universal wave functions ofEverett.( 2) It is, of course, true that Everett’s interpretation and theAharonov± Vardi construction rest heavily on the notion of superpositionin quantum mechanics and must therefore have many features in common.What we emphasize here is the striking similarity in detail between theseformulations which suggests a categorically common source. We can there-fore consider these two apparently different wave functions as representingthe same physical entity.

2. THE `̀ SURE’’ STATES

According to Aharonov and Vardi it is possible to define paths inHilbert space which connect any two states. That is, if we measure con-tinuously and successively a dense set of projection operators, whichcorrespond to those states that define a path, we pass with probability 1from the initial state to the final state through the whole intermediatestates. Aharonov and Vardi consider the following set of projectionoperators.

CÃ n= | n ñ á n | ( 1)

where | n ñ is the minimum wave function:( 5)

| n ñ = C exp 5 2 1(x 2 x( tn ) )

2 Dx 22

6 exp 1ia

p( tn) x 2 ( 2)

It has been proven ( 3) that if the initial state was a localized wave functionabout x( t0 )= x0 , and the set of operators CÃ n was measured [passing to thelimit of n ® ¥ ], the initial state was evolved along the eigenfunctions of CÃ n

with probability 1.To clarify the last statements we will make use of Fig. 1 which depicts

in Hilbert space a small number of Feynman paths all of which start froma given initial state and finish at a given final state.

The left path is the trajectory, which we want to define according toAharonov and Vardi’s method. On this path we mark n different pointswhich signify n different states (except the given initial and final states).Now if we want to move with certainty (probability 1) from the initial stateto the final state along the left desired trajectory, we have of course tomove through the n assigned states, and as we say above to be sure thatwe move through these definite points we have to do dense measurementswith respect to each such point. Of course when we pass to the limit n ® ¥

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Fig. 1. Four Feynman pathswhich start and end at given com-mon points. On the left path thereare n different points signifying ndifferent states.

then we move with certainty along all the desired left route. For the presentwe don’t take such a limit and concentrate only upon the n allotted states.Then between each two such close points the state of affairs is exactly thesame as the state of affairs between the initial and final states [see Fig. 2].On the left path we draw between each two neighboring points several littleFeynman paths connecting these points. As we have mentioned, to comecertainly to any one of the n assigned states we have to perform densemeasurements, that is, to repeat the same measurement infinite times in afinite time. The amplitude to go from the initial state to the first `̀ sure’’

Fig. 2. The same four Feynmanpaths as in Fig. 1, except that inthis case several little Feynmanpaths are drawn between each twoneighboring points of the n pointsof the left path.

1385The Feynman Path Integrals and Everett’s Universal Wave Function

state is the sum over all possible values of amplitudes to go from the initialstate to the first `̀ sure’’ state( 6)

Y 1st= +i

w i ( 3)

On the left side of the above equation we have the mentioned amplitude.On the right w i signifies the amplitude of a certain path from the initialstate to the first `̀ sure’’ state. In the same manner the second `̀ sure’’ stateis the second point [ from the initial state] of the n assigned points. Theamplitude to go from the initial state to this second point via the first`̀ sure’’ point is the following conditional probability:( 7)

Y 2nd = +j

+i

w j w i ( 4)

w j on the right side of the equation is the amplitude of a certain path fromthe first `̀ sure’’ point to the second `̀ sure’’ point, and the double sum givesthe sum over all paths that begin in the initial state, go through the first`̀ sure’’ state, and end in the second `̀ sure’’ state. Repeating the same proce-dure n times, we get for the amplitude to go from the initial state to the n th`̀ sure’’ state via the other (n 2 1) `̀ sure’’ states

Y n th = +k

+l

+h

. . . +j

+i

w k w l . . . w j w i ( 5)

n summations

wk on the right is the amplitude of a certain path from the (n 2 1) st `̀ sure’’state to the n th `̀ sure’’ state. The last equation gives us the sum over allpaths that begin in the initial state and go through all the n assigned points( states) to the n th `̀ sure’’ state. Needless to say, if the number n of theassigned points goes to infinity (n ® ¥ ) , then all those different pathsbecome one path, which is the `̀ sure’’ defined Feynman’s path. This can beshown as follows: When we suppose that n is finite, then the w ’s in (5) arethe amplitudes of the certain paths between each two close `̀ sure’’ points.For instance, w k is the amplitude of a certain path from the (n 2 1) st `̀ sure’’point to the n th `̀ sure’’ point. When n ® ¥ each two close sure pointsbecome essentially almost the same point and the w ’s in ( 5) become theconditional amplitudes for states ( and not for paths). We can thereforewrite for the final state instead of ( 5)

Y final= limn ® `

+n

+n { 1

+n { 2

. . . á final | n ñ á n | n 2 1 ñ á n 2 1 | n 2 2 ñ . . . á 1 | initial ñ

= á final | initial ñ ( 6)

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The last result signifies the `̀ sure’’ single path from the initial state to thefinal states [ in (6) we use different letters and symbols from those usedin (5) ].

3. THE UNIVERSAL WAVE FUNCTIONS OF EVERETT AS

``SURE’’ STATES

We now show that, from Everett’s point of view, the mere act of mea-surement doesn’t reduce the wave function as supposed by the ordinaryQM. If the initial state was an eigenstate of an operator, then of course weget the same state after a measurement of the physical quantity representedby the above operator. In Everett’s formalism the initial total state Y

S + 0=w i Y

0[ ...] is transformed after the interaction into the total state Y S + 0 9 =w i Y

o[ ...ai] , where w i is the eigenstate, Y 0[ ...] signifies the observer’s statebefore the measurement, and Y 0[ ...ai] signifies the observer’s state afterthe measurement. ai stands for recording of the eigenvalue by the observer.

Now suppose the initial state was not an eigenstate but the superposi-tion å i ai w i ; then the initial total state will be

YS + 0= +

i

a i w i Y0[ ...] ( 7)

After the measurement the final total state will be

YS + 0 9 = +

i

a i w iY0[ ...ai] ( 8)

We see that at the level of the superposition nothing changes. The onlychange is at the level of the separate terms of the superposition, and thischange is of course unpredictable as required by ordinary QM. [Here wepoint out that both the theory of Everett and that of Aharonov and Vardiuse the theory of J. Von Neumann ( 8) as concerns how to deal with the totalsystem of (object system+ observer system).]

Now suppose that we continue our measurements and want tomeasure another physical quantity represented by another operator B. Thestate with which we begin is the former state obtained after our firstmeasurement.

YS + 0= +

i

a i w i Y0[ ...ai] ( 9)

1387The Feynman Path Integrals and Everett’s Universal Wave Function

We expand the function w i in the eigenfunctions of B and get ( after changingthe order of the two summations)

Y S + 0= +j

+i

b ij a i w j Y0[ ... ai] ( 10)

This is our new total initial state. After the measurement of B we get thefollowing total wave function:

YS + 0 9 = +

j+j

b ij ai w j Y0[ ...ai b i] ( 11)

Again we see that the only change is at the level of the separate terms ofthe superposition and nothing changes at the level of the superpositionitself. We can further continue along the same line and measure other dif-ferent operators, so that after n different measurements we get the followingwave function:

YS + 0= +

k+l

+h

. . . +j

+i

d l k chl . . . bija i w k Y0[ai b j . . . f l dk] ( 12)

n summations

where

b ij= á w j | w i ñ , chl = á w l | wh ñ , dl k = á w k | w l ñ , a i= á w i | Y S ñ( 13)

and w i , w j , w l , w k , w h are eigenfunctions of different operators. We obtainthe expression for a i from (9). The expression in (12) gives the analyticexpression of the universal wave function of Everett ( 2) corresponding toone object system and n different measurements and that without charac-terizing and specifying the form of the n different operators which we havemeasured in order to achieve the expression (12).

Now suppose that these n different operators are of the form (1)[CÃ n= | n ñ á n | ] and their n different eigenfunctions differ from each otheronly on a time scale like the eigenfunctions in (2). In this specific situationwe can compare (12) with (5). Equation (5) signifies the situation in whichall the diverse paths pass through the `̀ sure’’ points. Each `̀ sure’’’ point isa junction point through which all the different paths pass. The `̀ sure’’points ( states) are classical in the sense that they don’t change by anyexperiment whatever. We can say exactly the same things about the expres-sion (12) which corresponds to the situation in which all diverse points

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( states) are represented by the same superposition. These superpositionsare classical in the sense that they don’t change by any experimentwhatever.

These equivalences lead us to the conclusion that the universal wavefunctions of Everett are identical with the `̀ sure’’ points of Aharonov andVardi. The only apparent difference is that in Everett’s theory we deal withpoints (states) and not with paths of states which begin and terminate inthe `̀ sure’’ points as in Aharonov and Vardi’s theory. But this is not anessential difference because we can regard the Everett universal wave functionfrom (12) as a sum over paths. That is, each term in (12) can be regardedas some path of states. This is assured by the fact that each term in (12)has its unique observer with specific eigenfunctions corresponding to every-one of his recorded eigenvalues achieved as the result of his experiments.Moreover, it is obvious that the first measurement, which initiated all thebranching in (12), was performed upon some prepared state of some physi-cal system. So we can regard this initial state as the common given initialstate of all the latter split observers, that is, the common initial point of allthe diverse paths corresponding to all the different terms in (12).

The final state is, of course, the superposition given by (12), whichdoesn’t get reduced by any additional experiment. It can therefore be regardedfrom this point of view as the sure final given state common to all thediverse paths represented by the different terms of (12).

We must, however, note here that when we wrote above that theEverett universal wave function was actually a sum over paths we didn’tmean to say that it was like the Feynman sum over paths. The reason isthat each Everett term (path) has his unique observer, and each suchobserver can achieve, after a sequence of measurements, a certain specificstate through a multitude of different possible paths which are, of course,his specific Feynman paths. That is, each observer in each term in (12) hashis specific Feynman paths, but what we actually have in each separateterm of ( 12) is only one realization of one Feynman path, a realizationarrived at by the specific observer of this term. So the universal wave func-tion of Everett is actually a sum of a multitude of realizations of specificphysical paths ( and not a sum of probable paths) where each realizationconstitutes a physical world of its own. That is, Eq. ( 12) is actually a sumof sure physical paths from any different physical worlds. This sum, initself, constitutes a sure state that doesn’t get reduced by performing anyadditional experiment as we have remarked above.

The similarity of this sum (12) to the sure state resulting from theworks of Feynman, Aharonov, and Vardi originate in the fact that their surestate involves a great number of repetitions of the same experiment, and thisis exactly identical to performing the same measurement simultaneously

1389The Feynman Path Integrals and Everett’s Universal Wave Function

by a multitude of different observers. That is, we get a sure state in twoequivalent ways: ( 1) Repeating the same measurement a great number oftimes [actually infinite times] in a finite time. ( 2) Performing the samemeasurement simultaneously by a great number of observers. In the firstcase we get a sure physical state which is classical in the sense that we canpredetermine it before performing it.(9 ) In the second case we get a surestate in the form of a superposition which persists and doesn’t get reducedafter an additional experiment. That is, the transition from (7) to (8) afterperforming the experiment could be interpreted as if each observer has per-formed this experiment. We can say the same thing also for the transitionfrom (10) to (11). We must note that from the point of view of the super-position there is no difference between the situation where a specificmeasurement was performed once, and the situation where this specificmeasurement was done two, three, or an infinite number of times. ( 2) Wecan see this from the expressions below.

The state after one experiment: å i a i w i Y0[ ...ai]

The state after two identical experiments: å i a i w i Y0[ ...ai ai]

The state after three identical experiments: å i a i w i Y0[ ...ai ai ai]

The state after infinite identical experiments: å i a i w i Y0[ai ai ...ai ai]

From the point of view of the superposition, infinite identical measure-ments are therefore equivalent to one single measurement. The most impor-tant point is that the universal wave functions are identical with the `̀ sure’’states (points). So (5) can be identified with (12). This identification can bestrengthened by noting that (12) was achieved after performing n differentmeasurements of the projection operators (1); we get to (5) also after per-forming n different measurements (where each one of these n differentexperiments was repeated continuously infinite times). We can identify ai in( 12) with w i in ( 5) and b ij , chl , and d l k in ( 12) with respectively w j , w l , andwk in ( 5) because, according to what we have just said, they have the samemeaning in both expressions. The appearance of the eigenfunction w k in( 12) and not in (5) stems from (7) [compare (7) with (3) ]. Also in (12) wehave the observer represented by Y

0[ ...]. In essence we see a strong equiv-alence between the two discussed theories not only from the computationalside [compare (12) to (5) ] but also from the interpretational side. That is,Everett considers a simultaneous existence of physical paths, and this ledhim to speak about a many-world interpretation. Aharonov and Vardi con-sider the realization of only one Feynman path. They didn’t discuss thesimultaneous realization of more than one Feynman path. But of course ifthey did discuss such simultaneous realizations then the many-world inter-pretation of Everett would seem to be appropriate.

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4. CONCLUSION

We have shown in the last section that we can regard the universalwave functions of Everett as representing the `̀ sure’’ states of Feynman,Aharonov, and Vardi. This has to be so if we wish to look upon these twotheories as providing a good description of nature. When several physicaltheories deal accurately with the same physical domain they should beequivalent [at least with respect to their primary physical entities like theuniversal wave functions of Everett, and the `̀ sure’’ states of Feynman,Aharonov, and Vardi], otherwise one of them is likely to be false. In aseparate article( 10) the writer has identified the universal wave functions ofEverett with the `̀ hidden variables’’ of Bohm and Bub. (11) We can thereforeidentify the `̀ sure’’ states with the hidden variables which are also `̀ sure’’and classical in the sense that we can predetermine them with probabilityone.

ACKNOWLEDGMENTS

I wish to thank L. P. Horwitz for discussions of the subject and for hisreading and criticizing the manuscript.

REFERENCES

1. R. P. Feynman, Rev. Mod. Phys . 20, 367, 1948. See also R. P. Feynman and A. R. Hibbs,in Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

2. Hugh Everett, Rev. Mod. Phys . 29 , 454, 1957.3. Y. Aharonov and M. Vardi, Phys. Rev. D 21, 2235, 1980.4. See Ref. 3 and references therein.5. Leonard I. Schiff, Quantum Mechanics, 3rd edn. (McGraw-Hill, New York, 1968) .6. Equation (3) can be understood in the following way: Suppose the parameter specifying

the initial state is h, and that specifying the first `̀ sure’’ state is j; then the amplitude togo from the initial state to the first `̀ sure’’ state is w hj= å i w hiw ij . By comparing this equa-tion to (3) we find Y 1st= w hj , w hiw ij= w i .

7. Equations (4) and (5) can be understood along the same line of reasoning( 6) as we haveapplied for the understanding of (3) .

8. J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton UniversityPress, Princeton, 1955).

9. Even the given initial and final states which are the starting and final points of the originalFeynman paths cannot be achieved without these dense measurements.

10. D. Bar, Found. Phys. Lett . 10, 1, 99, 1997.11. David Bohm and J. Bub, Rev. Mod. Phys. 38, 453, 1966.

1391The Feynman Path Integrals and Everett’s Universal Wave Function