Bell's Theorem and the Theory of Relativity

----An Interpretation of Quantum Correlation at a Distance based on the Philosophy of Organism------

Yutaka Tanaka

Summary

This paper starts with the observation that the combination of the so-calld EPR argument and Bell's theorem reveals one of the most paradoxical features of quantum reality, i.e. the non-separability of two contingent events. If we accept the conclusion of the revised EPR argument together with Bell's theorem, we are necessarily led to the denial of local causality which was presupposed by the original version of Einstein's criticism against quantum physics. As the concept of local causality is a cornerstone of Einstein's theory of relativity, we next consider the problem of compatibility between the theory of relativity and quantum physics Popper's proposal of going back to Lorentz's theory is examined and rejected because the quantum correlation of EPR is not to be interpreted as "an action at a distance' which we can control and use as the operational definition of absolute simultaneity. An inquiry into something like aether as hidden reality behind the theory of relativity is considered as retrogressive as the so-called hidden variable theory of quantum physics. Accepting the non-separability of local elements of reality as the undeniable fact, we discuss the possibility of a realistic interpretation of quantum physics which transcends scientific materialism and classical determinism. As an example of such projects, Stapp's theory is examined with respect to a Whiteheadian process philosophy which provides the metaphysical background for his realistic interpretation of quantum physics. Finally, we present another version of quantum metaphysics based on "the philosophy of organism" which is broad enough to include observer and observed, local causality and non-local correlation, space and time, and potentiality and actuality in the inseparable unity of physical reality.

I. Einstein's Criticism of Quantum Mechanics and Bell's Theorem

The original form of the EPR argument was summed up by Einstein and his coauthors as follows⁽⁴⁾:

In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

We can write the above argument in the form of syllogism which contains two propositions.

Proposition C (the completeness of quantum mechanics): Quantum mechanics is complete in the sense that there are no hidden parameters which explain the statistical data in a deterministic way.

Proposition S (the simultaneous reality of complementary physical quantities):

The complementary physical quantities, to which the canonical conjugate operators correspond in the standard formulation of quantum physics, have simultaneous reality in the sense that we can predict with certainty their values without disturbing the system.

The formal structure of the EPR argument is as follows:

The Major Premise
The Minor Premise
The Conclusion

This argument is sometimes called the EPR paradox, for it says that if we admit the completeness of quantum physics, then we are necessarily led into the contradiction (), for the major premise is equivalent to

It is noteworthy that the semantic structure of the EPR argument against the completeness of quantum mechanics is similar to Goedel's argument against the completeness of formalized arithmetic, for Goedel proved that if a formalized system of arithmetic is consistent, then it cannot be complete. As the criterions of completeness are different between formalized arithmetic and physics, this similarity only holds in an analogous sense, but it helps us to understand the meta-physical aspect of the EPR argument. Bohr seemed to understand this aspect of the argument, for he once said that he could see no reason why the prefix "meta"should be reserved for logic and mathematics and why it was anathema in physics.⁽⁵⁾ The Bohr-Einstein debate was essentially meta-physical in the sense that they tackled the aporias of quantum physics at and beyond the boundary of human observation.

The EPR argument was not generally accepted as valid by his contemporary physicists, because it was interpreted as an argument against the indeterminacy principle established by Heisenberg. Though Einstein's earlier arguments against the Copenhagen interpretation aimed at pointing out a possibility of measuring two complementary physical quantities beyond the limit of exactitude imposed by the indeterminacy principle, the purpose of the EPR argument was not the refutation of this principle, but essentially the semantic claim that if we accept the completeness of quantum physics, then we are, through considering a suitable imaginary experiment, necessarily led to the contradiction of both accepting and not accepting the indeterminacy principle.

The imaginary experiment in the EPR argument involved a system of two particles with the wave function

As and are commuting operators, the above wave function can have the determinate values of and p=0. So measuring enables the calculation of without in any way disturbing the system. There is an element of reality corresponding to this determinate value. If we measure instead of, then we can also calculate the determinate value which also corresponds to another element of reality. If quantum mechanics represents all elements of reality (the proposition C), then the position and momentum of the second particle have simultaneous reality, which contradicts the major premise, i.e. the principle of indeterminacy. This principle would not be violated if we insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted with certainty. So the EPR argument presupposes that the measurement of the second particle is independent of that of the first particle because the distance between two particles are so great that they may be considered as causally separable elements of reality. Making this assumption explicit, we can reformulate the EPR argument as follows:

The Assumption L (the separability of local elements of reality):

The physical system are separable into two or more parts which are causally independent of each other at an given instant. The observation of the one cannot causally influence that of the other in so far as the four-dimensional distance between them is space-like (dx²+dy²+dz²-c²dt^20).

The reformed EPR argument shows that the alleged incompleteness is proved only under the assumption of L.

The Major Premise
The Minor Premise
The Conclusion

Though Einstein mentioned this assumption toward the end of his paper, he took it for granted because the breakdown of L was so unreasonable for him. If the principle of local causality did not hold, then the partial description of the whole universe would be, strictly speaking, impossible on account of the dubious concept of a closed system in the level of quantum phenomena.

It was Bohm⁽⁶⁾who first explicitly stated that the assumption L was incompatible with the current theoretical structure of quantum mechanics. He even said that the name "quantum mechanics" might be a misnomer because "mechanics" is necessarily associated with L, i.e. the separability of local elements of reality.⁽⁷⁾

The local hidden variable theory is formally expressed as (), for it must presuppose the locality thesis L and the incompleteness of quantum mechanics. There have been many trials of hidden variable theories, and so many counter arguments which aimed at excluding such a possibility from the peculiarity of quantum statistics. The most famous criticism against the hidden variable theory was given by von Neumann who mathematically proved that the basic axioms of formalized quantum mechanics exclude hidden variables.⁽⁸⁾ His proof was, however, not conclusive in the case of the hidden variable theory which does not share the axioms of quantum mechanics.⁽⁹⁾

Bell proposed a crucial experiment between quantum mechanics and a local hidden variable theory.⁽¹⁰⁾He proved that there is a limit on the extent of correlation of statistical results that can be expected for any type of local hidden variable theory. The limit is expressed in the form of inequality which is now called the Bell inequality. Bell showed that quantum mechanics sometimes violates this inequality, especially in the correlation at a distance in the imaginary experiment of the EPR argument. This experiment became realizable when we replace the original version of the EPR experiment with the measurement of spin-components of two spin-1/2 particles or with the measurement of polarization of two photons. Real tests of the Bell inequality have been carried out by many groups of investigators.⁽¹¹⁾ The most conclusive was done by Aspect (1982) and the result was that the Bell inequality was really violated.

Aspect utilized the correlated photon pairs and which counter-propagate along Oz and impinge on the linear polarization analyzers I and II. The result +1 and -1 are assigned to linear polarizations parallel or perpendicular to the orientation of the polarizer, and this orientation is characterized by a unit vector a and b.

The quantum state of the whole system can be expressed as the following superposition:

Then we can calculate the probability of each photon's polarization along a given direction. P₊(a)=P_-(a)=P₊(b)=P_-(b)=1/2

P₊₊(a,b)=P_--(a,b)=(1/2)cos2(ab)

P₊-(a,b)=P_-+(a,b)=(1/2)sin2(ab)

Let EQM be the coefficient of correlation between two quantum events at the polarizers a and b:

EQM=P₊₊(a,b)+P_--(a,b)-P_+-(a,b)-P_-+(a,b)=cos2(ab)

If ab=0, then EQM=１, which means perfect positive correlation.

If ab=/2, then EQM= -1, which means perfect negative correlation.

This kind of perfect correlation seems miraculous if we assume the completeness of quantum mechanics and admit the coincidence between two contingent events, for we may wonder how “knows" which channel was chosen at the last moment for . This “miracle" would disappear if we succeeded in making a locally deterministic model for the above simultaneous perfect correlation. Such a model has to assume the hidden causal mechanism which predetermines both results of measurements. This causal mechanism can be represented by local hidden variable which has a probability distribution

For simplicity we assume one hidden variable , but we may use many hidden variables in the following consideration by using multiple integrals. The problem is whether or not such a hidden deterministic model can represent the strong correlation signified by EQM in the case that a and b are neither parallel nor perpendicular.

Let the function A(,a) and B(,b) determine the measured values of

polarization at the polarizera and b respectively:

A(,a)=1 or -1 ; B(,b)=1 or -1

Then the coefficient of correlation Ｅ(a,b) given by the statistical expectation value with respect to .

With respect to four different directions a,a',b,b', we define the quantity S=E(a,b)-E(a,b')+E(a',b)＋E(a',b').

Then we can prove the inequality -2S2
This is the Bell inequality which was tested by Aspect's experiment.⁽¹²⁾

Quantum physics shows that SQM based on EQM does not satisfy this inequality. In the experimental situation in which ab=ab'=∠a'b'=22.5⁰,

aa'=bb'=45⁰, ab'=67.5⁰, we get SQM=, which invalidates the Bell inequality.

As quantum mechanics and any kind of local hidden variable theory predict different statistical results, the above experiment may well be called a crucial experiment. As the result was for quantum mechanics, we must conclude that we cannot make quantum mechanics complete by introducing local hidden variables.

The violation of the Bell inequality means that the combined proposition ()is false,

for such a local hidden variable theory cannot explain the correlation at a distance in the system of two particles which have previously interacted with each other. If we admit both the validity of the EPR argument and the experimental test of Bell's theorem, we have to abandon L, i.e. the separability of local elements of reality.

The Conclusion of the Reformed EPR Argument	()
The Experimental Test of Bell's Theorem
The Final Conclusion	L

We must notice that the final conclusion is independent of our attitude toward the completeness versus incompleteness problem of quantum mechanics.

We cannot prove , as Einstein intended to do, the incompleteness of quantum mechanics as a result of the falsified premise L, but this falsification itself depends logically upon the validity of the EPR argument and empirically upon the violation of the Bell inequality, for the validity of the argument is one thing and the truth of its conclusion is quite another.

Moreover, the EPR argument makes us reconsider the nature of the indeterminacy principle, for there seems to exist no mechanical interference between the observer and the observed in the imaginary experiment concerned. This principle was originally interpreted by Heisenberg as the inevitable inexactitude of measurement due to uncontrollable mechanical interactions between the observer and the observed, but once we have verified the simultaneous correlation between distant events and admit the non-separability of local element of reality, we must amend Heisenberg's interpretation in such a way that the indeterminacy principle holds primarily on the level of the definition of quantum phenomena where the observer and the observed are not separable from each other. Bohr seemed to anticipate this view in his reply to the EPR argument⁽¹³⁾:

Of course there is in a case like that just considered no question of a mechanical disturbance of the system under investigation during the last critical stage of the measuring procedure. But even at this stage there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system.

Bohr, however, rejected the semantic criterion of completeness and reality in the EPR argument, and chose to talk only about quantum phenomena which we can define through the macroscopic apparatus of observation. Bohr's standpoint was that quantum mechanics did not require the depth structure under quantum phenomena , but certainly not that it was ontologically self-sufficient for the world-description. Rather, classical physics was to Bohr indispensable for the definition of quantum phenomena. It was important for Bohr to recognize that “however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms".⁽¹⁴⁾ So quantum phenomena need classical physics for their definition in terms of experimental apparatus, whereas any single classical model of reality cannot exhaust the varieties of quantum phenomena. Bohr's philosophy of complementarity was pragmatic and provisional in the sense that it sidestepped the difficult problem of quantum measurement, i.e. how to describe "the collapse of the wave function"within the framework of quantum mechanics.⁽¹⁵⁾ If we want to get a unified picture of macroscopic and microscopic reality, we must present a suitable framework of ontology which can assimilate the main characteristics of quantum physics, especially the non-separability of local elements of reality.

II. Quantum Correlation and the Theory of Relativity

Some philosophers and physicists, facing the breakdown of locality, proposed going back to the problem situation before Einstein. Bell himself suggested a possibility of the restoration of the absolute framework presupposed by Lorentz's theory of electrons and aether, because "behind the scenes something is going faster than light."⁽¹⁶⁾ Popper more explicitly stated this possibility⁽¹⁷⁾ :

It is only now, in the light of the new experiments stemming from Bell's work, that the suggestion of replacing Einstein's interpretation by Lorentz's can be made. If there is action at a distance, then there is something like absolute space. If we now have theoretical reasons from quantum theory for introducing absolute simultaneity, then we would have to go back to Lorentz's interpretation.

Popper's opinion, however, is very dubious when we reexamine Lorentz's own comparison of his and Einstein's interpretation of the "Lorentz" transformation in the supplements of his Theory of Electrons.⁽¹⁸⁾ Whereas Lorentz derived this transformation through considering the relation between the true and universal frame of reference (S) and an apparent and local one (S'), Einstein abolished the very distinction between true and apparent or between absolute and relative. The Lorentz transformation became the symmetric interrelation between two inertial systems in the theory of relativity. The crucial difference between the two interpretation is as follows :

(1) The contraction of a measuring rod and the delay of a clock was, according to Lorentz, caused by an electron's movement through aether absolutely at rest. Lorentz explained away these "weird" effects by appealing to aether as a hidden reality. The constant velocity of light was to Lorentz a paradoxical fact to be explained away on ad hoc hypotheses of the unknown causal mechanism of aether.

(2) Einstein considered the contraction of a measuring rod and the delay of a clock, not as causal effects of unknown reality, but as the symmetric effects between S and S' which should be interpreted to be derived from the definition of space-time metric. If we rely on S, then we must say the measuring rod of S' contracts and the clock of S' delays. Symmetrically, if we rely on S', then we must also say the measuring rod of S contracts and the clock of S delays. The hidden causal mechanism was, to Einstein, not only useless, but also contradictory because mathematical formulae of the Lorentz transformation exclude the non-symmetric interpretation. The constant velocity of light was, however paradoxical it might be seem, not to be explained away as exceptional phenomena, but to be accepted as the universal principle which made it possible to reconstruct Newtonian mechanics in combination with the principle of relativity.

It is noteworthy that the relation between Einstein's theory of relativity and Lorentz's theory of aether was similar to that between quantum mechanics and the hidden variable theory. This similarity suggests that the methodology of special relativity was more revolutionary and akin to quantum mechanics than Lorentz's correlations of quantum mechanics on the essentially classical and pre-relativistic model like Lorentz's seems fruitless and retrogressive.

Moreover, there are several arguments against the restoration of the absolute frame of reference. The simultaneous correlation in quantum physics is different from a Newtonian type of action at a distance. The former is probabilistic and non-controllable whereas the latter is deterministic and controllable. So we cannot send information with a superluminous speed on the basis of the distant simultaneous correlation in quantum physics. We cannot acquire information through the random sequence of measured values at one side without comparing them with the results of the other side. As the coincidence of two contingent events cannot be used for sending information with a superluminous speed for the purpose of synchronizing two clocks at a distance, the empirical test of Bell's theorem does not make Einstein's theory of relativity invalid through the alleged discovery of prohibited action. We may theoretically introduce absolute simultaneity, but we do not have any experimental arrangement to detect the existence of the absolute frame of reference.

Instead of the restoration of an abolished classical theory, Stapp made a radically progressive trial of introducing something like absolute time by supposing the deep structure below Lorentz invariant phenomena.⁽¹⁹⁾ This structure was described by him as that of events which have the absolutely linear order of "coming into existence". Stapp's theory had an ontological background provided by Hartshorne's version of process metaphysics, according to which the ultimate realities are events and the whole universe has a cumulative structure of creative advance with a cosmic simultaneous "front"of actuality. The purpose of Stapp's theory was to ensure both the macroscopic causality properties with Lorentz-invariance and all of quantum theory on the basis of his metaphysics of events. We may say that Stapp replaces the classical concept of aether with the absolute world of events which are logically prior to space-time. The main characteristic of Stapp's theory was that he adopted the absolute and universal concept of existence in which what comes into existence does not depend on a space-time standpoint, whereas Einstein's theory of relativity relied on the relative and local concept of existence in which what comes into existence depends on a space-time standpoint. As the breakdown of the Bell inequality requires some events to depend on other events whose positions lie outside their backward light-cones, Stapp postulated that the sequence of actualized events should be well-ordered even in the case of spatially distant events.

Though I agree with Stapp that the ontological framework of events is necessary for the unified picture of the world, I do not think he is justified in introducing the absolutely well-ordered structure of events. Einstein's theory of relativity which only admits the partially-ordered structure of events seems more plausible the consideration of the Bell-Aspect experiment.

In the simplest cases of Bell's phenomena there are four events E₀ E₁ E₂ , E₃ whose locations L_o ,L₁, L₂, and L₃, lie in four well-separated experimental areas A_o,A₁, A₂ and A_3. If all events lie in the well-ordered sequence of occurrence as Stapp assumed, there must be an unambiguous temporal order between E₁ and E₂ : one of the two events must be prior to the other. Suppose E₁ is the prior to E_2. Then E₂ depends on what the experimenter in A_l has decided to do whereas E_l is independent of what the experimenter in A₂ will decide to do. So he reduced the "simultaneous" correlation between E₁ and E₂ to the unilateral influence of one upon the other. The difficulty of the above picture is that there does not seem to be any experimental apparatus to determine which is prior, E_l or E_2. Though we guess that an influence or superluminous signal must have gone from L_l to L₂, or from L₂ to L₁, we do not know yet which one is the cause of the other. There is the remnant of classical causality in Stapp's model in which the mutuality or interdependence of quantum phenomena totally disappears. In other words, Stapp's model does not seem to consider the "individuality" of quantum system which Bohr emphasized in his doctrine of complementarity between space-time coordination and causality. This "individuality" can be expressed as the organic interdependence between parts of the quantum system : the whole may be in a definite state, i.e. may have as definite properties as quantum theory permits,' without its parts being in definite state. The two particles of the imaginary experiment in the EPR argument and the two photons of Aspect's experiment are examples of the inseparable parts of an "individual" organism. In this organic unity there cannot be a determinate causal order between all parts of the whole. In the above case there remains the essential ambiguity of causal order between E₁ and E₂ because their correlation is symmetrical and not detectable until we monitor and record it in L_3, i.e. the common causal future of L_l and L₂. This ambiguity is characteristic of the relativistic framework of space-time, and any attempt of restoring the absolute framework tends to violate not only the principle of relativity but also the principle of complementarity between space-time coordination and causality.

In the next section I will present another model which aims at synthesizing the principle of relativity and quantum correlation on the basis of the philosophy organism. In this model events are, as in Stapp's and Hartshorne's process me physics, basic ontological categories from which material objects and space-time are derived. The background philosophy of organism is more similar to Whitehead's own cosmology than Stapp's and Hartshorne's revised version, for the fundamental vision of Whitehead's philosophy is, as Nobo clearly explicated⁽²⁰⁾, the mutual immanence of discrete events regardless of their temporal relationship ⁽²⁰⁾, whereas "process" philosophers seem to stress only the immanence of earlier events in later ones. We will find that the immanence of later events in earlier ones and contemporaries in each other are indispensable for the understanding of quantum correlation. The "organic" model of quantum reality is also similar to the Hua-yen Buddhist doctrine of simultaneous interfusion and interpenetration signifying unity-in-multiplicity, for it rejects the notion of independent self-existence which Hua-yen Buddhists called svabhava in their doctrines of pratitya-samutpada (interdependent origination).⁽²¹⁾ The concept of the absolute frame of reference should replaced with the idea of thoroughgoing relativity : we need not postulate t absolutely unique temporal order. Even the absolute world of four-dimension space-time as prefixed reality in Einstein's theory of relativity should be abolished if we take into account the complementarity between space-time coordination a causality. If we are, as Bohr aptly stated, simultaneously actors as well as spectators on the great stage of life, the image of a scientist as an outside spectator should be replaced with that of a participating observer inseparably involved in the object to be observed.

III. Quantum Correlation viewed from the Philosophy of Organism

The peculiarity of quantum correlation is caused by the so-called "the collapse of the wave function". One of the unsolved problems of quantum mechanics is about the nature of this discontinuous phenomenon. The usual framework of quantum theory does not describe the process of collapse itself but simply accepted it as the result measurement in the statistical data of observation. In other words the collapse of the wave function belongs, not to the object language of quantum formulae, but the meta-language of quantum mechanics which correlates mathematical formula and experimental data. Many physicists tried to enlarge the framework of quantum mechanics enough to give a unified description of observer and observed, i.e. microscopic measured system and the macroscopic measuring apparatus, but there seems not to be an unanimous resolution of this conundrum.

d'Eespagnat pointed out the enigma of the "collapse of the wave function" follows : ⁽²²⁾

The puzzle with which we have to struggle is constituted by the fact that, since the wave function is a non-local entity, its collapse is a non-local phenomenon. According to the formalism, this phenomenon propagates instantaneously. In that sense we may say that the wave packet reduction is a non-covariant process. Again, this would create no difficulty if, like the reduction of probabilities in classical phenomena, this collapse were of a purely subjective nature. But we have seen quite strong arguments in favor of the thesis that it is not.

d'Espagnat's comment that the wave collapse is not to be solved by a subjective interpretation of probability is important, for it excludes an easy "solution" of the conundrum by appealing to our ignorance of initial conditions. Certainly, if we get a new information about the system, then the probability distribution of quantities which characterize the system changes discontinuously. The discontinuous change of quantum physics cannot be explained away by this kind of probabilistic arguments. Such general arguments are unsatisfactory because they do not take into consideration the peculiar characteristics of quantum mechanical algorithm of probability. The probability wave and the probability amplitude represented by a complex number were totally unknown before quantum physics. They behave in the very inconceivable way as if they violated classical logic.

For example, the famous double slit experiment shows that even in the case of only one particle, say a photon, the interference occurs between two mutually exclusive possibilities i.e. the possibility of the same particle's going through one slit A and the alternative possibility of its going through another slit B. So if we represent the third event, say the effect of the photon on the photographic plate with C, then has been experimentally confirmed, which violates the distributive law of classical logic.

Finkelstein stresses the need of quantum logic as a non-Aristotelian logic in the description of the microscopic world just as we need a non-Euclidean geometry in the theory of general relativity.⁽²³⁾ I prefer to say that if we need quantum logic, then it must be a kind of modal logic with the distinction of real (objective) possibility and actuality. In the above example of the double slit experiment, describes not an actuality but a real possibility whereas both and describe two actualities which are mutually exclusive. In the Whiteheadian terminology, the transition from the disjunctive many to the conjunctive one does not follow classical logic because the interference of alternative possibilities really occurs.

This phenomenon of probability interference shows that we have to face objective probability reflecting the experimental situation rather than subjective one reflecting only our ignorance of the determinate fact. In other words real possibility and actuality are inseparable with each other in quantum physics, and we must treat the collapse of the wave function as the objective transition from real possibility to actuality.

The next Problem is about the quantum transition itself. If the collapse of wave function is an objective phenomenon, then is it "an action at a distance" i.e. a non-covariant phenomenon which happens instantaneously This problem is crucial to our consideration of the Bell correlation and the theory of relativity. In the section II we confirmed the fact that quantum correlation and the principle of relativity are compatible, and we need not explain quantum correlation as the unilateral causal effect with the superluminous speed. Einstein's theory of relativity was more progressive than Lorentz's theory of aether in that Einstein introduced into physics a radically new perspective in which space and time are non-separable with each other.

It is regrettable that many discussions of physicists about the collapse of the wave function presuppose only non-relativistic framework. The "simultaneous" correlation would be meaningless in the relativistic framework, because such a terminology implicitly assumes that there exists only one time-system of classical physics. The non-relativistic quantum physics does not treat space and time in their non-separable unity. Time appears only in the form of a parameter and does not take the role of operator corresponding to an observable quantity whereas spatial coordinates are permitted status of operators which characterize the quantum system. So if we describe the collapse of the wave function in the non-relativistic framework, we must say that it happens instantaneously, i.e. non-locally with respect to space.

The dubious scenario roughly runs as follows : if the quantum system prepared at the time t₁ is measured at t₂ , it changes its states continuously and causally between t₁< t < t₂ according to Schroedinger's equation, but at the moment of t₂ the discontinuous irreversible event called "the collapse of the wave function" happens and its effects propagates instantaneously with the super-luminous speed. The above picture is not relevant to the relativistic concept of space-time, because the very concept of simultaneity and instantaneous transmission does not make sense. The non-separability of time from space means that non-locality of the collapse should be accepted, not only with respect to space but also with respect to time. The reason why temporal non-locality, more exactly spatio-temporal non-locality has been ignored may be simply that the collapse of the wave function has been discussed mainly in the non-relativistic framework. Einstein himself seemed to anticipate the problematic of spatio-temporal non-locality in his criticism of the indeterminacy principle, for he pointed out that "if we accept quantum physics, then it becomes impossible to restrict the indeterminacy principle to the future ; we must admit the indeterminacy of the past as well."⁽²⁴⁾

This criticism was not so famous as the EPR argument, but it is of decisive importance when we discuss the collapse of the wave function as a non-local phenomenon in space-time.

An example of the indeterminate past was given by Wheeler in his famous discussion of the "delayed choice" experiment.⁽²⁵⁾ We may use the same diagram to explain this experiment. In this diagram we assume that the present choice is made at A₃. The experimenter at A₃ can choose for one photon either the mode of non-interference or the mode of self-interference even after the photon has passed through A_l or A₂. In this experiment, whether the photon has passed through (either A_l or A₂) as a particle, or through (both A₁ and A₂) as a wave depends on the present choice made at A₃. Before the decision at A3 the location of the particle was essentially indeterminate. What we can say of past space-time and past events is decided by choices made in the near past and now. Wheeler discussed the possibility that the phenomena called into being by the present decision can reach backward in time, even to the earliest days of the universe. The above example shows that it makes sense to state that events occur in the four dimensional framework of space-time. This occurrence itself does not take place in time as the fourth coordinate of space-time.

In Einstein's theory of relativity the concept of events is static in the sense that an event simply is and occupies a determinate location without any regard to other regions of space-time sub specie aeternitatis. In the quantum indeterminism, on the other hand, the modified concept of events is dynamic in the sense that an event happens in the extensive continuum of space-time. We need not postulate, as Stapp did, that all events of the whole universe constitute the well-ordered sequence with respect to this kind of happening in space-time, because it would make geneses of events subordinate to space-time coordination to make becoming of events the fifth coordinate. The delayed choice experiment cannot be explained away by the introduction of anything like absolute time-order because any theory compatible with relativity must retain the order of causality within a light cone.

We cannot call the delayed-choice "retroactive causality" because it does not make sense to say that we can "change" the past if we mean by the past something determinate ; rather we should say that the past in the level of quantum description cannot be considered as totally determinate. The following analysis of quantum correlation is similar to that of Whitehead's analysis of "symbolic reference" though Whitehead seems to use this term to explicate the structure of perception only in the high-grade organisms such as a human being. As the wave function is an essentially non-local relational entity, the world itself has the structure of symbolism as well as that of causality.

The main difference of the proposed model from the Whiteheadian ontology is that this model does not take a single quantum event as the totally determinate individual. The specificity of any attribute of the quantum event is, as Shimony clearly showed,⁽²⁶⁾ always attained at the price of indefiniteness of other attributes on account of the indeterminacy principle. Every event is complementarily described as an entity with respect its actuality, and as a locus with respect to its potentiality. An event is a spatio-temporal entity, and a material body corresponds to the nexus of events (world-tube) which has various characteristics such as energy, momentum, and other observable physica quantities. It is essential in this organic model that observables are adjectives of event-nexus with alternative selective patterns of perspectives.

Two "elementary particles of the same kind are not two separate substances, but the same adjective which can have two contexts of actualization in different events. The fundamental relation of events is called "self-projection". This term i introduced for the purpose of explaining both objective and subjective aspects which necessarily emerge in quantum organism, but it should not be understood in psychological sense in which the self-projection of an observer has no objective correlate. The self-projection which I mean is a physical relation between events and signifies the organic unity between the observer and the observed. We cannot observe microscopic events without their self-projections in the macroscopic measuring apparatus. What we observe, however, is not a mere shadow of the separate self-existing substance, but in one sense a thing itself because every thing can exist only in the complex network of self-projections of events. What we observe depends on our choice of measuring apparatus which reciprocally projects itself in the microscopic events by influencing the possible pattern of actualized contingency.

First I will sketch the formal structure of self-projection ; a, b, c, signify events which, as loci, "mirror" the universe according to their own perspectives, and as entities, project themselves in every loci in the universe. There are two modes of self-projection : causal efficacy and mutual immanence.

a < b : a projects itself into b in the mode of causal efficacy

: a projects itself into b in the mode of mutual immanence

The mode of causal efficacy is cumulative ; it is non-reflexive, non-symmetrical and transitive :

(1) (a)(a<a)
(2) (a,b)(a<bb<a)
(3) (a, b, c) ((a<b)(b<c)(a<c))

The mode of mutual immanence is reflexive, symmetrical, and transitive ;
(4) (a)(aa) (5) (a,b)(abba) (6) (a, b, c) ((ab)(bc)(ac))

Let a, b, g signify classes of events. We can define the relativistic concept of the past, the future, and the contemporaries of an given event in terms of the self-projection in the mode of causal efficacy :

Def. P(a)={x | x<a}	P(a) is called the (causal) past of a
Def. F(a)={x | a<x}	F(a) is called the (causal) future of a
Def. aCb(a<b)(b<a)	a is contemporaneous with b

As the relation of contemporaneity is not always transitive, the existence of the uniquely-defined present of a given event is not guaranteed by the theory of relativity. We can introduce something like a cosmological "present" in terms of a maximum class of mutually contemporary events instead

Def. If a class d of events satisfies the following conditions, it is called a contemporary duration of the universe :

(1) (a, b) (adbdaCb (2) (a, b) (adaCbbd)

The relativity of simultaneity means that there are an infinite number of possibilities for a contemporary duration of the universe. As the arrow of local causality always passes from the past to the future in every frame of reference, it cannot explain the quantum correlation of the Bell experiment which holds between two contemporaries. The contemporaneity as defined above is essentially a negative (derivative) relation and also irrelevant to the explanation of positive correlation in quantum physics. The experimental test of Bell's theorem requires something positive to cover such a correlation. Self-projection in the mode of mutual immanence is introduced to satisfy this requirement. This mode should be non-causal in the sense that it does not pass immediately from the past to the future, but signifies a kind of mutual interpenetration among events in terms of which a composite system behaves as if it were one individual. Causal efficacy ranges from causal immanence to causal influence.

Causal immanence holds between two temporally separated events in the isolated microscopic system with a small number of degrees of freedom, when the causal influences from the outside are negligible. The relation of causal immanence is the basis of a deterministic description of the microscopic system before its interaction with the measuring apparatus.

The causal efficacy from the macroscopic system with a great number of degrees of freedom is called causal influence. It is practically impossible to give a deterministic description of the system on the basis of the exact control of causal influences, which only permit statistical treatment of complex thermo-dynamical processes with an increasing entropy. The irreversible process of quantum measurement, however, cannot be identified with the entropy-increasing process of thermodynamics, as Wigner showed in his argument against Daneri-Loinger-Prosperi's theory of measurement.⁽²⁷⁾ When a and b project themselves into each other in the mode of mutual immanence, they behave as if they were one individual on account of mutual immanence (the non-separability of quantum events). Even when the two loci of a and b are spatially separated, these two loci as potentialities have an internal relation with each other with regard to certain characteristics (e.g. polarization or spin).

The mutual immanence disappears when the system is causally influenced from the outside system. The collapse of the wave function of a composite system may give a distant simultaneous correlation when self-projections between contemporary parts of the system pass from the mode of mutual immanence to that of mutual transcendence (the disappearance of the term of phase interference between them). Every event is organically related with the whole universe by symbolic correlation which integrates the two modes of self-projection. The distant correlation h quantum physics holds between two contingent events with the same causal past immanent in both. This correlation does not mean the superluminous sending o information in terms of causality. but signifies the relation of mutually self projecting events which constitute the organic system. This system integrates two different modes of self-projection in the presented duration defined by the measuring apparatus. The whole setting of the measuring apparatus determines the kind o simultaneous correlation which holds between contingent patterns of physical value, measured in both parts. Each of two events with the same immanent causal past can be seen as the symbol of the other as if they were two sides of the same coin.

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