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The Transactional Interpretation of Quantum Mechanics

by John G. Cramer

2.4 Nonlocality: How are Correlations of Separated Parts of the State Vector Arranged?

The problem of nonlocality is closely related to that of the collapse of the SV. The problem in a simple form was first raised by Einstein (1928) at the 5th Solvay Conference. Later it was presented in a more subtle form as one of the criticisms of quantum mechanics by Einstein, Podolsky, and Rosen (1935). Einstein (1949) stated the problem thus: "But on one point we should, in my opinion, absolutely hold fast: the real factual situation of system S1 is independent of what is done with system S2, which is spatially separated from the former." Since these words were written the thrust of the nonlocality problem has been sharpened considerably through theoretical and experimental investigations, but the issue remains essentially the same.

For the purposes of the present discussion we will distinguish between two kinds of nonlocality. Nonlocality of the first kind arises from the interpretation of the SV as a physical wave. When the SV collapses the change implicit in the collapse occurs at all positions in space described by the SV at the same time. A physical wave undergoing such a change would seem to require faster-than-light propagation of information. Indeed, even the phrase "at the same time" is only meaningful relativistically in a particular inertial reference frame. It was this kind of problem which was the basis of Einstein's (1928) original objections to quantum mechanics. Similar nonlocality problems brought about the rejection of Schrödinger's semiclassical interpretation {footnote 11}.

CI4 was constructed to avoid difficulties with nonlocalities of the first kind by denying the physical reality of the SV and identifying it instead with "our knowledge of the system". Therefore, when a measurement is made showing that a photon is located at point A (and not at B or C), our knowledge of the photon's location abruptly changes and the magnitude of the SV's value must suddenly drop to zero at B and C, although no spatial propagation, according to CI4, is associated with that abrupt change.

CI4 works well in this context. Its effectiveness may, however, reflect the naive statement of the nonlocality problem, which seems to require attribution of physical reality to the SV. But the intrinsic nonlocality of the QM formalism runs deeper than this, as becomes clear when more complicated situations are considered which involve separated measurements of parts of a correlated system. In that situation definitions of the SV become irrelevant because real measurements are involved. This leads to a nonlocality of the second kind, which is associated with the enforcement of correlations in spatially separated measurements.

2.4.1 EPR Experiments

This kind of nonlocality is demonstrated by the Freedman-Clauser experiment (1972) illustrated by Fig. 1a. Here excited calcium atoms undergo a 0+ to 1- to 0+ atomic cascade and provide a pair of photons, assumed to be emitted back-to-back, which are in a relative L=0 angular momentum state. Because of angular momentum conservation these photons are required to have identical helicities or linear combination of helicities, i.e., they must be in identical states of circular or linear polarization. For this reason the SV of the two photon system permits the photons to be in any polarization state, provided only that both are in the same state. Experimentally this means that if the photons are transmitted through perfect polarizing filters before detection, they must be transmitted with 100% probability if the polarizations of the filters select matching states and 0% if the filters select orthogonal states, no matter what orientation or polarization selectivity the filters have.

The Freedman-Clauser (FC) experiment employs linear polarizing filters and measures the coincident transmission yield of the two photon detectors when the principal axes of the two filters are set at angles thetaA and thetaB, which are varied independently. Quantum mechanics predicts that the experimentally observed yield will depend only on the relative angle thetarel = (thetaA - thetaB) between the two principal axes, and further that for ideal filters the yield will have the normalized angular dependence:

R[thetarel] = Cos2(thetarel)          [1]

Note that this is just the expression for Malus' law, which gives the transmission probability of a single photon (or a beam of unpolarized light) through two crossed linear polarizers with an angle thetarel between their principle axes. Thus the coincidence rate predicted when there is one polarizing filter in arm A and one in arm B of the experiment (Fig. 1a) is the same as if the photon in arm A went direct and unfiltered to its detector while the photon in arm B went through both polarizing filters in succession before reaching its detector (Fig. 1b). The coincidence rate is also the same as if one of the photons on encountering its polarizer, reached across with "spooky action at a distance" and placed the other photon in the same state.

The Freedman-Clauser (FC) experiment (1972) was the first definitive experimental test of the Bell inequality (1964, 1966), which for local theories with CFD places limits on the strength of changes in the polarization correlation function when the polarimeters differ in alignment by an increasing amount. A detailed discussion of the Bell inequality is beyond the scope of the present review, and we refer the reader to the original papers of Bell (1964, 1966), the review by Clauser and Shimony (1978), and Herbert (1975), d'Espagnat (1979), and Mermin (1981, 1985). The actual FC experiment used non-ideal filters and consequently had a rather more complicated expression for the correlation function than that given in [1]. The measured function was found to be in excellent agreement with the quantum mechanical prediction and to show a 6 standard-deviation violation of the limit imposed by the Bell inequality. A more recent series of similar experiments by Aspect et al (1982a,b) has demonstrated consistency with quantum mechanics and a 46 standard-deviation violation of the Bell inequality. These results indicate, assuming CFD, that the predictions of all local theories (see Section 1.0) are inconsistent with experimental observation.

To illustrate that a nonlocality of the second kind is exhibited by the FC result, let us consider the local modification of quantum mechanics. Furry (1936a,b) suggested this modification as a way of clarifying the content of the Einstein-Podolsky-Rosen (1935) criticism of quantum mechanics. Furry suggested that quantum mechanics would become a local theory if, when two parts of the system (like the two photons in the FC experiment) separate and become isolated from the possibility of speed-of-light contact, the SV describing them immediately collapses into a definite but random state. In the FC case the SV would collapse into a definite but random state of linear polarization shared by the two oppositely directed photons. This modified version of quantum mechanics would be a local theory because the Furry condition would satisfy the definition of locality given in Section 1.0. The correlated state of the two photons would be only the result of "memory" of the correlation which had existed before they became separated. The Furry modification has no effect on many of the predictions of conventional quantum mechanics. But does it significantly modify the predictions for the FC experiment which are predicted by QM and observed in the experiment? And do the Furry predictions obey Bell's Inequality? The answers to both questions are yes.

2.4.2 Furry's Modification

We cannot readily modify quantum mechanics so that it becomes local in this way. We can, however, simulate the Furry modification within the FC experiment by placing near the source an additional pair of aligned linear polarizing filters which are rapidly and randomly changed. By this mechanism each pair of photons emerging from the source will be placed in definite and identical but sequentially random states of linear polarization as the photons are transmitted through these filters near the source. This arrangement is illustrated in Fig. 1c.

The QM prediction for this case can easily be obtained by calculating the predicted rate of two-photon detection for a particular orientation angle (phi) of the randomizing filters and then averaging over all possible values of (phi). The result of this calculation is:

Rf[thetarel] = (1/8)[1 + 2 Cos2(thetarel)]          [2]

Fig. 1d compares the functions R[thetarel] (labelled "Malus") and Rf[thetarel] (labelled "Furry"). The angular dependence of Rf[thetarel] is weaker than that of R[thetarel]. In particular, Rf[thetarel] has a maximum value of 3/8, a minimum value of 1/8, and cannot go to zero for any value of thetarel. The Rf[thetarel] correlation satisfies Bell's Inequality but is inconsistent with the FC results and with quantum mechanics.

Thus the SV of the photons cannot be described as in a definite but random state. Rather the SV must contain components which describe the photons as being in all possible states of polarization. Only when at least one of the two photons is detected is the SV allowed to collapse into a definite state of polarization, which must be the same for both photons. Until the detection(s) takes place the polarizations of the photons must remain in states which are connected but not specified, in a way which is inconsistent with locality. It is this connectedness which is addressed by the Bell inequality and which cannot be explained away by the "our knowledge" definition of the SV. It is this which we have called nonlocality of the second kind.

The Furry modification of QM is only one example of a local theory. Other local theories can give a variety of predictions, including transmissions of 100% when theta=0o and 0% when theta=90o, but none can reproduce fully the FC result or the QM prediction. Bell's Theorem demonstrates that no realistic local theory, no matter how cleverly contrived, can reproduce the experimental result of the FC experiment.

2.4.3 Superluminal Communication?

One might be tempted to think that the connectedness between the two measurements of the FC experiment, i.e., their nonlocal correlation, might be exploited for nonlocal communication to transmit messages instantaneously from one arm of the experiment to the other. Perhaps, for example, one observer could telegraph a message in Morse code by rotating his polarimeter. It has been demonstrated (Eberhard, 1977, 1978; Ghirardi et al, 1979, 1980; Mittelstaedt, 1983) that no such observer-to- observer communication is possible, essentially because the QM operator corresponding to any measurement done on the right photon commutes with the operator for any measurement on the left photon. The nonlocal character of the connectedness is a subtle one which permits the instantaneous enforcement of correlations across spacelike separations but does not permit signalling. See Section 4.5 for further discussion of this point.

As previously mentioned in Section 2.1, the CI4 "knowledge" interpretation of the SV may be applied in two different ways:

Application of these variants of the "knowledge" interpretation to the FC experiment allows us to demonstrate that alternative CI4a above, which may appear to be the more reasonable of the two, leads to a relativity/causality paradox.

As a gedanken experiment consider a "stretched" version of the Freedman-Clauser experiment in which the two arms of the apparatus are lengthened to very large distances. Let us also assume the use of 100% efficient linear polarimeters of the type used by the Aspect group that split the incident beam into two orthogonal polarization states so that a given photon is always detected by one or the other of a pair of photomultiplier detectors sensing the two orthogonal states, i.e., linear polarization states parallel and perpendicular to the principal axis of the polarimeter.

We assume that a previous arrangement has been made with an assistant at the light source to direct a pair of correlated photons to the two measurement sites, with the photons leaving the source apparatus at a well defined time T. The left photon travels to the location of the left observer, who sets his polarimeter angle theta1 and makes a measurement that we will call M1. Similarly, the right photon travels to the location of the right observer, who sets his polarimeter to another angle theta2 and makes measurement M2. Each observer always knows in advance when a photon will arrive and always obtains a definite result from each polarization measurement. Let us consider the SV collapse event that occurs as a result of one or the other of these measurements, under assumption CI4a that there is some universal SV which is a representation of the overall knowledge of the system.

We choose to describe the experiment as it occurs some inertial reference frame F1 in which the measurement event M1 occurs earlier in time sequence than does event M2. Accordingly, CI4a tells us that event M1 alters the SV describing the entire system because that measurement alters "our knowledge of the system". The formalism of QM requires that the SV collapse to a state that is consistent with the results measurement M1, and from CI4a this collapse is triggered by a local event occurring at the location and time of M1. Later, when the other photon reaches the right polarimeter and measurement M2 is made, the system is already in a definite quantum mechanical state, determined by the result of M1. Therefore, measurement M2 produces no further SV collapse because the knowledge gained is redundant with that already obtained by M1.

On the other hand, relativity tells us that since the two detection events are separated by a spacelike interval, either detection event can be made to precede the other in time sequence by an appropriate choice of reference frames. Therefore, suppose that we describe the same experiment from some second reference frame F2 in which measurement event M1 occurs after M2 in time sequence. Now CI4a tells us that event M2 alters the SV describing the entire system. The SV collapses to a state that is consistent with the results of measurement M2, and this collapse is a local event occurring at the location and time of M2. When the other photon reaches the left polarimeter and measurement M1 is made, the system is already in a definite quantum mechanical state, determined by the result of M2. Therefore, measurement M1 produces no further SV collapse because the knowledge gained is redundant with that already obtained by M2.

2.4.4 Two Histories?

Clearly, these two histories of the collapse of the overall SV are mutually exclusive and contradictory. Moreover, they conflict with the principle of relativistic invariance because the collapse event is not a phenomenon that is independent of the reference frame in which it is viewed. Thus relativity is inconsistent with CI4a.

One might try to avoid this conflict with relativity by making the ad hoc assumption that one of these descriptions (say that describing M1 as producing the SV collapse) is the correct one independent of reference frame. This assumption becomes troublesome because it favors one measurement and one observer over the other for no apparent reason. But it does reduce the level of conflict with relativity. However, it has another problem. In reference frame F2 it permits a cause, the collapse event at M1, to occur after its effect, the arrival of the other photon at M2 in a definite quantum mechanical state.

Thus, CI4a leads to conflicts at the interpretational level with either special relativity or the principle of causality. There has been some recognition of this dilemma among the founders of quantum mechanics. For example, Dirac (Hiley, 1981) said with reference to this problem: "It is against the spirit of relativity, but it is the best we can do ... We cannot be content with such a theory."

We should emphasize that the contradictions discussed above do not apply to CI4b, which uses a different SV for each observer. In any case, these contradictions do not have consequences at the observational level because state vector collapse is not an observable event. Collapse is a construct perceived in the formalism (von Neumann, 1932), a pseudo-event that is asserted by the CI to occur when the state of knowledge changes. It is only when we require that the CI give an account of the collapse of some unique overall state vector and require that this account is interpretationally consistent with other established laws of physics, that we reveal an interpretational paradox. The paradox is not a new one. It is the Einstein-Podolsky-Rosen paradox, but it is here restated in the language of the CI itself.

If CI4a is to be rejected because it leads to interpretational paradoxes, is CI4b an acceptable alternative? In our opinion CI4b is acceptable in the sense that it successfully dodges nonlocality problems of the second kind. But it does this ostrich-fashion, retreating behind a solipsistic blindfold of local knowledge and positivism. The most serious criticism of CI4b, in the view of the author, is that the account of the SV given by CI4b bears little resemblance to the SV most physicists think they are calculating (Weisskopf, 1980) when they perform quantum mechanical calculations implicitly involving SV collapse. Nonlocality is dealt with by CI4b in an air-tight but counter-intuitive way.

2.5 Completeness: Do Canonically Conjugate Variables have Simultaneous Reality?

Another problem raised in the Einstein-Podolsky-Rosen (1935) paper is that of the correspondence between the QM formalism and reality for the case of pairs of canonically conjugate variables, i.e., pairs of variables like position and momentum having QM operators which do not commute. The EPR paper argues that "every element of the physical reality must have a counterpart in the physical theory" and pointed out that in terms of the QM formalism "when the operators corresponding to physical quantities do not commute, they cannot have simultaneous reality". Thus (goes the argument) there is a lack of correspondence between quantum mechanics and reality, and the former must be "incomplete". This is this part of the EPR criticism of quantum mechanics which received the most subsequent discussion in the literature. It became the central focus of the debate over quantum mechanics and its interpretation for a long period thereafter.

And yet, from one point of view the QM formalism contains the solution to the completeness problem. The SV of the formalism which describes a particle (say an electron) is clearly complete in the sense that it contains components or projections which can localize either of a pair of conjugate variables. For the case of position and momentum the SV contains projections which localize the position of an electron to arbitrary precision and other components which will similarly localize its momentum. When a measurement is made which collapses the SV only one of these two kinds of components can be projected out by the collapse, so the simultaneous measurement of both variables can only be made to the precision specified by the uncertainty principle. Thus, the variables do have "simultaneous reality" in the uncollapsed SV but can never have "simultaneous reality" in a single component of the SV which results from the collapse. This should satisfy the EPR criterion of completeness.

The above resolution of the EPR completeness criticism is, however, demolished by the CI itself, since CI4b denies the objective reality of the SV and associates it instead with the "knowledge" of an observer. If the SV is not a physical entity, but rather an ephemeral construction existing only as "knowledge" in the mind of one observer (as beauty in the eye of the beholder) then the "reality" of the conjugate variables becomes only a subjective one arising from the observer's lack of information, in support of the EPR criticism.

This leads us to the conclusion that there is indeed a completeness problem associated with quantum mechanics as the EPR paper asserted. It is not, as was supposed however, a problem with the QM formalism, but with the interpretation of the formalism. An interpretation which gives physical reality to the SV of the formalism provides a de facto solution to the problem of completeness.

2.6 Predictivity: Why Cannot the Outcome of an Individual Quantum Event Be Predicted?

The third criticism of QM by the EPR paper was that a proper theory should enable the user to, "without in any way disturbing the system, ... predict with certainty ... the value of a physical quantity". Quantum mechanics, on the other hand, provides the user with a way of predicting only average behavior of an ensemble of quantum events but not the behavior of a particular particle in a particular event {footnote 13}. This is the problem of predictivity.

Born's statistical interpretation as embodied in CI2 meets the problem of predictivity head-on. It asserts that there is an intrinsic randomness in the microcosm which precludes the kind of predictivity we have come to expect in classical physics, and that the QM formalism provides the only predictivity which is possible, the prediction of average behavior and of probabilities as obtained from Born's probability law (P = Psi Psi*).

While this element of the CI may not satisfy the desires of some physicists for a completely predictive and deterministic theory, it must be considered as at least an adequate solution to the problem unless a better alternative can be found. Perhaps the greatest weakness of CI2 in this context is not that it asserts an intrinsic randomness but that it supplies no insight into the nature or origin of this randomness. If "God plays dice", as Einstein (1932) has declined to believe, one would at least like a glimpse of the gaming apparatus which is in use.

2.7 The Copenhagen Interpretation and the Uncertainty Principle

Element CI1, the uncertainty principle of Heisenberg (1927), is one of the most important aspects of the CI. It is also an interpretational aspect of quantum mechanics which has received a large amount of attention in the literature. It has been the subject books and symposia, and it was the focus of the famous Bohr-Einstein debate.

And yet, it is the aspect of quantum mechanical interpretation which has perhaps the best grounding in analogous classical phenomena and which is easiest to understand from the viewpoint of classical physics. Heisenberg's uncertainty relations are a direct consequence of the character of the solutions of the Schrödinger equation and its relativistic equivalents, solutions which are functions of products of conjugate variables such as k.r and Et. In fact, Heisenberg's original derivation of the uncertainty principle dealt directly with this property of the wave equation solutions by showing that the Fourier transform of a localized Gaussian position wave function is a localized Gaussian momentum-space wave function, with the momentum width of the latter Gaussian proportional to the reciprocal of the position width of the former Gaussian. This property of Gaussian distributions under Fourier transformations is well known. And perhaps more important, it has many analogs in classical physics.

As an example, consider the representations of fast electrical pulses in the time and the frequency domains. Such a pulse can be represented either as in the time domain as a set of voltages varying continuously as a function of time, or in the frequency domain as a continuous set of Fourier components, i.e., a set of voltages varying continuously as a function of frequency. These representations of such pulses have exactly the Bohr-Heisenberg complementary relationship and exhibit their own "uncertainty principle". The localization of a fast pulse in the time domain (by making it extremely short in duration) requires a corresponding de-localization in the frequency domain, since the Fourier frequency spectrum of such a pulse must include a broader range of frequencies including very high ones. Conversely, one can increase the localization of the pulse in the frequency domain by passing the pulse through an electrical "band pass filter", which eliminates the Fourier components which do not fall within the frequency "window" of the filter. The observable result of this frequency localization is a corresponding broadening of the pulse in the time domain. Here then is a purely classical phenomenon which exhibits an "uncertainty principle". This fast pulse uncertainty principle can be observed directly on an oscilloscope screen in any well-equipped electronics laboratory.

However, CI4b asserts that the SV which is the carrier of these canonically conjugate quantities is not real wave. Rather, according to CI4b the SV is a mathematical representation of the knowledge of some observer. This renders more questionable any association of the uncertainty principle of quantum mechanics with similar phenomena of classical physics. If it is not a physical wave but the observer's knowledge which is being localized or delocalized, one is less secure in associating that behavior with classical analogs which show an "uncertainty principle" directly in the object itself.

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