[Full Table of Contents]
5. Relevant Philosophical Concepts
5.1 Realism
5.2 Causality
5.3 Determinism
5.4 Non-Determinism and Randomness
5.5 Locality
5.6 Priority of Principles
6. Conceptual Problem as Understood by EPR
7. Early Responses by Bohr and Einstein
7.1 Bohr and Complementarity
7.2 Einstein and Separability
Discussion of the EPR paradox invokes concepts of natural philosophy such as realism, determinism, causality and locality. We have already seen that EPR conflated the concepts of realism and determinism, and similar confusions have plagued subsequent commentary. Although the definitions of terms may be purely conventional, the same is not true of the underlying concepts as such, so it is advantageous to define terms in a way that allows us to represent as many relevant conceptual distinctions as possible. The definitions presented here have been chosen with that aim in mind, and most of them are fairly common usages among philosophers, so they are not idiosyncratic or unfamiliar.
Realism, in the generic sense, is any philosophy in which it is admitted that there can be one or more entities that exist independently of anyone’s beliefs, thoughts or modes of conceptualizing or denoting ideas. In short, it is the premise that there exists an extramental reality that is to some degree independent of what we think of it. Epistemological realism is the premise that it is possible in principle for minds to obtain some knowledge of those aspects of extramental reality that do not depend entirely on our thoughts. Epistemological realism is a presupposition of the scientific method.
Realism in the context of the EPR problem, however, means something more specific, namely that physical properties have definite values in objective reality, independent of our state of knowledge. These values need not be predictable; it is only required that, at a given time, the property for that system should have only one value to the exclusion of all others, and that this value is independent of our state of knowledge. To avoid confusion, I will always speak of definite values
rather than determinate values,
since determinate
may also suggest determinism and the notion of predictability.
This more specific realist supposition, that every physical property should have definite values at all times, is not a strict necessity of the scientific method, unlike generic realism and epistemological realism. It could be that certain properties resolve themselves into definite values only upon interacting with some object, an interaction which we call a measurement. Perhaps there is no such thing as what the z-spin really is
before interacting with a magnetic field gradient in the z-direction. Prior to interaction, there may be potentials or propensities toward the z-spin up
or z-spin down
states, but neither of these potentials is realized until the interaction or measurement occurs.
Some early interpreters of quantum mechanics, such as von Neumann, thought that the act of measurement had to do with the subjective knowledge of an observer, rather than an objective physical interaction. On such an interpretation, even the definite value resulting from a measurement might be considered to fail the criterion of objective reality, since it is conditioned by the observer’s state of knowledge.
Scientific theories are supposed to do more than merely describe reality; they should also explain phenomena or give an account of them in terms of more fundamental universal principles.[13] Such explanatory principles are called causes. There are at least two modes of explaining phenomena, enabling a distinction in kinds of causality.
First, we may explain a phenomenon formally, by giving a mathematical formula that generalizes a class of phenomena. We call this explanation formal
because it gives the form or structure of a phenomenon in quantitative or relational terms. We are not necessarily claiming that this formal cause
is a real physical agent that generates physical effects, nor that the form itself is a principle of action (as in some metaphysical systems). Nonetheless, the fact that phenomena may share a common formal structure is at least suggestive of a real agent that is responsible for the uniformity indicated by the formal explanation. In many cases, the terms of mathematical expressions may in fact correspond to properties of real physical agents.
Formal explanations alone cannot give us a clear order of causality in the sense of agency, especially when these explanations are mathematical. In mathematics, a logically consistent theory can be formalized equivalently, making one or another set of premises the axioms from which all other propositions in the theory can be proven. Thus it is not always possible to define unequivocally which general principle is more fundamental.
A second, more ordinary use of the term causality
refers not to formality, but to a real physical existent acting in a way that produces or generates a phenomenon, in which case we call the phenomenon the effect and the actor or agent the efficient cause. In ordinary usage by scientists and laymen, when we simply say a cause,
we mean an efficient cause. A paradigmatic example of efficient causality is found in billiards, where a moving cue ball strikes an object ball, causing it to move, and that moving object ball in turn strikes a second object ball, causing it to move. Thus we have a clear chain of cause and effect, with each effect making it possible for an agent to produce the next effect.[14]
Efficient causes explain
phenomena in the sense of giving an account of a phenomenon’s concrete existence. The existence and action of the causal agent are considered to be a sufficient explanation of why the effect has actually come into being. This explanation is not merely for our formal understanding, but ostensibly gives an account of how things work in reality, namely that each cause brings into being the corresponding effect.
Note that I consistently refer to agents, not events, as causes. This avoids a conceptual confusion common in philosophy of science. Without pretending to resolve the status of event
as a metaphysical category, we may acknowledge that an event would be empty without actors. If there are no agents doing anything, then nothing happens and there is no event.[15] The agent remains essential, even if we choose to reify its action as an event.
Without the supposition of an agent, we are effectively denying efficient causation, and our explanations of events are confined to formalism.
When we understand that agents rather than events are efficient causes in the primary sense, we cannot fall into the error of thinking that an effect must occur some time after the cause. In fact, the cause must be simultaneous with the effect. The cause, after all, is what produces the effect, so if it were not present, the effect could not be produced. Confusion arises from the fact that the effect may be a long-term phenomenon, i.e., a motion that continues in time. A struck object ball may continue rolling long after it has ceased to interact with the cue ball. Yet it received this motion at the event of collision, and the presence of the cue ball was certainly required at this time for the event to occur and the object ball’s motion to be altered.
The idea that a cause must be prior to its effect arises when we have a succession of two or more causal acts in time. The cue ball alters the motion of the first object ball at some time t1, and as a result of this altered motion, the first object ball is able to alter the motion of a second object ball at some later time t2. The altered motion of the second object ball is an indirect effect of the cue ball’s action. Granting that some time t2 - t1 has elapsed between collisions, we may say that the action of the initial cause (the cue ball) is certainly earlier in time than the production of its indirect effect (altered motion of the second object ball), due to the fact that the agent bearing its immediate effect (the first object ball with altered motion) retains its power to act for some period of time after the first interaction.
Since the second effect is produced by an act of the agent bearing the first effect and the first effect is produced by the initial cause, it would be paradoxical for the second effect to be produced at a point in time before the initial cause acted. If the initial cause has not yet acted, its immediate effect cannot yet exist, yet the presence of the agent bearing that first effect is absolutely indispensable at the time when the second effect is produced, for this agent is the immediate cause of the second effect.[16]
Both formal and efficient causes are indispensable modes of scientific explanation. If we were to abolish either of them, we would thereby overthrow our understanding of what we mean by science and scientific theories, and have to start over on new foundations.
Determinism is the premise that an effect or outcome is fully determined by some antecedent conditions with complete certainty. The claim that determinism admits no exceptions whatsoever, not even human free will, may be called strong determinism, hard determinism or superdeterminism.
For the mechanistic physicists of early modernity, determinism meant that a physical effect or outcome, i.e., some sensible datum, is fully determined by antecedent physical conditions. If a physical phenomenon is deterministic, then full knowledge of antecedent physical conditions enables us to predict with unerring certainty what the effect or outcome will be.
This perfect predictability, by itself, does not give a physical account of the connections between the determining antecedent conditions and the predicted effects. It is only when we take determining
to entail generating or causing an effect that determinism attains the character of a physical explanation. Physically interpreted, determinism is a particular mode of causality, where an effect is fully determined in all its particulars by its causes and the circumstances under which the causes act. Strong physical determinism would make possible, in principle, a complete explanation of the entirety of concrete physical reality, a prospect that has held much appeal to those who spend their lives trying to understand nature. It is no accident that those, like Einstein, who are most concerned with theoretical completeness also tend toward strong determinism.
If we were to deny the principle of efficient causality in physics, we could not have a truly physical determinism. The perfect predictability of effects from antecedent conditions would indicate a purely formal correspondence between effects and conditions, unmediated by any physical agency.
Although physical determinism requires (efficient) causality, the converse is not true. It is at least conceivable for there to be non-deterministic causality, in which a cause and the circumstances under which it acts do not determine the effect with perfect certainty. Some philosophers argue that determinism is the only viable mode of causality, but this requires appeal to theses beyond those directly required by empirical science.
Similarly, determinism requires realism, but realism does not require determinism. This is obviously true when speaking of realism in a general sense, i.e., that extramental reality exists and is knowable. Yet it is also true when treating realism in the specific sense of physical properties having definite values at all times. The values of physical properties cannot be fully predictable at all times if these properties do not even have definite values at all times. Still, as noted when discussing the EPR paper, it is conceivable for properties to have definite values at all times without being fully predictable.
Even on the assumption of denying realism (in the narrower sense), we might still admit a sort of determinism, if we stipulated that the only truly physical outcomes or events are those instances when a property does have a definite value. Then all physical effects could, at least in principle, be fully predictable even if properties did not have definite values at times between interaction events.
Many scientists and philosophers hold, at least implicitly, that determinism is the only viable form of realism, but this requires metaphysical argument. Determinism and realism are conceptually distinct, and we should emphasize this so we do not mistake a refutation of one for a refutation of the other.
The negation of determinism is commonly called indeterminism,
but I will instead use the term non-determinism.
This is to avoid confusing a denial of determinism with a denial of realism (in the narrow sense), since indeterminism
may suggest the notion of having indeterminate
or indefinite values. Non-determinism
means that an effect or outcome is not fully determined by antecedent conditions with complete certainty. This need not entail total unpredictability, only less than perfect predictability.
Non-determinism in physics means that there exist events or outcomes, i.e., sensible data, that are not fully determined by antecedent physical conditions. Even full knowledge of all physical conditions in the universe antecedent to the event in question would not enable us to predict the outcome with complete certainty. This makes it impossible for there to be any physical theory that can predict all physical outcomes, so all physical theories would be incomplete
in that sense. Still, a non-determinist universe could conceivably admit a complete theory
as defined by EPR. Although particular outcomes are not predictable with certainty, a physical theory may nonetheless contain a counterpart to every element of physical reality. EPR never showed that an element of physical reality must be predictable with certainty.
Just as determinism is distinct from causality, so is non-determinism distinct from acausality. Physical determinism may presuppose causality, but it does not logically follow that physical non-determinism is acausal. Non-determinism does not require us to abandon causality, but only to accept that the same cause may produce different effects under the same conditions. Admittedly, the question of why this outcome rather than another occurred in a particular instance would remain unanswered. The cause itself does not explain this, so we might say the cause is insufficient for the effect. Indeed, the cause and other antecedent conditions do not constitute a logically sufficient condition
from which the particular outcome necessarily follows. Nonetheless, the cause is sufficient
in the sense that it has the requisite power and inclination to produce that outcome, notwithstanding the fact that it also has the power and inclination to produce other outcomes. When it actually does produce a particular outcome, it is truly the efficient cause of that effect.
There are philosophical objections to this account, among which one may argue that powers, inclinations or dispositions really belong to the material aspect of being, not agency. Another objection is that a cause that can produce more than one effect cannot be a proper cause for a particular effect, since an outcome’s proper cause makes that fact true and renders it impossible to be otherwise. It is not our task here to resolve the philosophical questions of whether or not all efficient causality must be deterministic, or if there can be free agents, or if free agents must have intelligence. We have only to describe the concepts of causation, determinism and non-determinism, so that we may distinguish them and later identify which, if any, are directly affected by the findings of physics.
Randomness is closely related to non-determinism, yet distinct from it. Randomness, considered as a statistical concept, characterizes a set of trials whose outcomes cannot be predicted with certainty. This does not imply total unpredictability, for some outcomes may occur with greater frequency than others, enabling us to determine their probability. We can at least predict the frequency distribution of outcomes for a large set of trials, with generally improving accuracy as the number of trials increases. The non-certainty of predictability implies that the sequence of outcomes cannot be generated by a computable function.[17] The successive outcomes of these random trials are represented by numerical values of a random variable, with a distinct value assigned to each distinct possible outcome. The successive values of a random variable are not correlated to each other. Note that it is unintelligible to speak of an individual outcome as random,
for randomness is definable only correlatively, or more exactly, by the absence of computable correlations between successive outcome values.
To determine that a set of outcomes is random, we must order them in some sequence. If we were free to order them however we pleased, we might place all the trials with similar outcomes consecutively, breaking the random structure. It is only when the sequence of outcomes is treated as an external given, rather than something we can define arbitrarily, that it becomes meaningful to test the data for computability. Thus the notion of randomness is not purely mathematical, but requires at least the hypothetical supposition of outcomes given to us in a sequence we do not control. This sequence need not be temporal.
Randomness is evidence of non-determinism only when it is applied to physical (or metaphysical) reality. The notion of antecedent conditions is essential to defining determinism and non-determinism. Thus we must have some sort of temporal (or meta-temporal) order in order to interpret a random sequence as representing non-deterministic outcomes. The sequence itself need not be temporal, for it may represent simultaneous trials in identically prepared conditions. Still, for the sequence to represent non-deterministic effects, it must be held that the outcome values are not fully predictable or computable from any antecedent conditions whatsoever. This is true if all trials have identically prepared conditions.
In reality, however, it is difficult to be sure if trial conditions are truly identical. Perhaps some subtle variation in initial conditions accounts for the difference in outcomes. Since we prepare our trials in ignorance of these variations, which is to say independently of them, the distribution of outcomes may be modeled with a random variable. In classical probability theory, conceived in the age of mechanistic natural philosophy, unpredictability of outcomes is a consequence of our partial or total ignorance of the determining antecedent conditions. We may have random variables even under a paradigm of (strongly) deterministic physics, but here the randomness is only contextual, showing a lack of correlation between our preparation of trials and the subtle antecedent conditions that determine outcomes. Similarly, Aristotle’s classic example of chance, where I happen to meet someone who owes me money at the marketplace, is consistent with the supposition that each of us behaved in a fully deterministic manner. We each went to the marketplace for definite reasons, but neither of these had anything to do with one of us owing money to the other. The events are uncorrelated to each other except accidentally, but they may nonetheless be causally, even deterministically, linked to other antecedent conditions. Likewise, we may model a tossed coin or die with a random discrete variable, even though each outcome is the result of fully deterministic mechanics, as long as the initial conditions of the toss are prepared with great variability and without regard for any preferred outcome.
If the randomness of quantum mechanical phenomena is not the result of mere ignorance or imprecision in our preparation of ostensibly identical trials, then the sequence of outcomes is not just contextually random, but absolutely random. That is to say, it is not merely uncorrelated to a particular set of conditions, but is not computable from any antecedent physical conditions whatsoever.
Supposing one could demonstrate that some event outcomes are absolutely random
and therefore evidence of non-determinism, it would not follow that these outcomes are acausal.
Full predictability is not essential to the notion of causation. When an unpolarized photon is absorbed by a linear polarizer, this outcome is not predictable, as there was a 50% chance it would have passed through as linearly polarized light. Nonetheless, it would be rash to conclude that its absorption was not caused by anything. One might say its absorption was caused by the linear polarizer, which has the power to do such a thing, and without which the absorption would not have occurred. Likewise, if the photon had passed through the filter as linearly polarized light, one may say this polarization would have been caused by the linear polarizer, which had the power to produce this effect as well, and without which the light could not have become polarized.
Since randomness means an absence of correlation among trial outcomes, it deals only with an ontology of events, and does not directly treat causal agents or their absence. A demonstration of absolute randomness would imply that there are no deterministic causes that can generate the effect, but it would remain consistent with the presence of free agents acting as causes.[18]
The principle of locality is a constraint upon physical causation where an agent can act only upon what is in its immediate vicinity. Some version of locality has been imposed by physicists since the age of mechanism, when Newtonians and Cartesians insisted that all action is by direct contact of bodies. There is no logical necessity for this principle; rather it is an expression of metaphysical and epistemological preference. If one allowed that terrestrial events could be instantaneously effected by remote causes in the heavens (as in the correspondences
of medieval physics), the search for physical causes could not be constrained to empirical verification. After all, one could always posit remote cosmic influences without limit to explain anything. Locality removed the need for occult causes, and made spatial continuity a visible expression of how an effect grows directly out of its cause.
The negation of locality would be instantaneous action at a distance, where one body, acting at one moment in time, alters the behavior of some distant body at that exact same moment. One might say that its action propagates at infinite speed. Under locality, action can propagate only by intermediate agents moving at finite speeds. In our billiards example, the cue ball can alter the motion of the second object ball only by the mediation of the first object ball, which requires some amount of time to move from its point of collision with the cue ball to its second point of collision. Although causation at each point of collision remains instantaneous, locality implies that a chain of cause and effect mediated by physical agents must follow a temporal sequence. The generation of each effect must be temporally prior to the generation of all physically consequent effects.
In classical mechanics, gravitation seemed to be a perplexing exception to the principle of locality. The same later seemed to be the case for electricity and magnetism. Einstein’s theory of special relativity, however, appeared to shut the door on non-local action, for it constructed space and time in such a way that infinite speed was geometrically impossible. Worse, it was no longer possible to speak unequivocally of distant events occurring at the same moment.
Under general relativity, gravitation was explained by curvature of the spacetime metric, thereby subjecting it to the principle of locality. Electromagnetic fields have been conceived as qualitative alterations of empty space, so that the field itself is an agent that acts locally, though this is an interpretation, not something empirically demonstrated.
Under relativity, the principle of locality must be reformulated to account for our new understanding of the geometry of space and time. The essence of the principle remains the same, for an agent can act only upon what is in its immediate vicinity. What changes are the constraints defining which events may be consequent to a causal action. Let O be the event of a cause generating its immediate effect. Then locality requires that any consequent effects, i.e., effects generated by the agent that was altered at event O, must be in the future light cone of O. This excludes not only the absolute past of O, but also any event that is spacelike separated from O.
The exclusion of the past is nothing new or even unique to locality, for any logically coherent account of causality must prohibit the present from altering the past, unless we allow that time is an absolutely fatalistic loop. Even on the latter assumption, the past would be altered not qua past, but qua future. By alteration, of course, we mean a real physical change by efficient causation, not a mere revision of our state of knowledge of the past.
The exclusion of spacelike separated events takes a distinctively relativistic view of so-called instantaneous action at a distance. Events that are spacelike separated from O are simultaneous
with O only in certain reference frames. In other frames they would be in the future or the past of O. Unless we want to insist that there are privileged frames, it seems clear that all spacelike separated events are inadmissible as effects of the agent altered at O, since we would thereby be admitting that the present could alter the past. On the other hand, as we have noted elsewhere, the temporal order of spacelike separated events is purely constructed, having no physical implications. Yet that lack of physical implications depends on excluding the possibility of causality between such events, so once again, under relativity, a principle of locality seems essential to maintaining a logically coherent account of physical causality.
If we instead take O as representing some secondary effect, then its remote cause must have acted somewhere in the past light cone of O, and nowhere else. Moreover, if we take a deterministic account of causality, then all antecedent conditions
must likewise be in the past light cone of O. If we admitted that an antecedent condition
could occur in a spacelike separated event, then we could avoid causal paradox only if we either (a) denied that the antecedent condition is a physical cause or (b) held that a frame in which this condition is in the past of O is physically privileged.
It is worth noting that all Bell’s experiments and Wheeler’s delayed choice experiments purporting to demonstrate non-locality deal with spacelike separated events. There has never been an experiment where a measurement event altered something in its absolute past. Indeed, it is unintelligible how we would even notice such a result, supposing it were not absurd. Thus all supposed demonstrations of the present altering the past
deal with the constructed or frame-dependent chronologies of spacelike separated events.
The principle of locality should not be conflated with that of realism, in the narrow sense of properties having definite values at all times. Such confusion can arise when the property in question is position, for then we say that a particle is localized
if its position has a definite value. Conceivably, if a particle were delocalized,
its range of immediate action might allow it to circumvent the principle of locality, yet delocalization
itself would have paradoxical implications under relativity, making something at once in its own past, present and future. Nonetheless, the question of localization
is distinct from the principle of locality.
Locality as such is not a requirement of the principles of causality or realism, but, under the relativistic account of space and time, locality becomes essential to maintaining a coherent account of causality. Still, locality and causality are distinct principles. The current version of locality is a consequence of the peculiar structure of spacetime, such that we cannot define a unique temporal order for spacelike separated events. A refutation of locality as currently understood need not contradict causality, but it would contradict the well-established relativistic structure of spacetime, or at least the interpretation that there are no privileged frames of reference.
Although the principle of locality has long been a mainstay of modern physics, it has been identified with Einstein because his theory of relativity defined the current constraints of this principle, and because he famously insisted on it as a necessary axiom in the EPR paper and in later publications. Ironically, his articulations of this principle often used non-relativistic examples, making its necessity less evident.
Certainly, under a relativistic paradigm, Einstein was quite correct to make locality a necessary axiom. If we recognize, however, that locality as such is not required by causality, there may be ways to allow for non-local formal influences
distinct from efficient causation, or non-local conditions that affect information states, distinct from physical states.
Lastly, although locality is a constraint on causal actions, the relativistic version of the principle is usually formulated in terms of an event ontology. As we have noted, speaking of events without agents is not properly a causal account of reality. If locality really tells us anything about the structure of causal relations, we must always refer the events back to the agents acting at those spatiotemporal points.
How one analyzes the EPR paradox depends on how one ranks the aforementioned principles in terms of logical or metaphysical priority. For EPR, reality
was a priority, but this was understood in a way that entailed determinism, not just realism. It was also taken for granted that causation must act locally, even in a non-relativistic context:
…since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system.[19]
Here the time of measurement
is mentioned as though it unequivocally applied also to the second system, i.e., as if relativistic effects were negligible.
Thus EPR assumed realism, determinism, causality, and locality, so it is not immediately clear which of these principles may be contradicted if their thought experiment should be realized. It is helpful to structure these principles in some order of metaphysical priority or logical dependence.
Realism in the sense that extramental reality exists and is knowable is a presupposition of the scientific method, but this generic realism need not entail that every variable should have definite values at all times. Even if we do accept that more specific form of realism, it need not entail strong determinism.
Causality is another presupposition of science, but causality need not imply strong determinism, except according to the opinions of some philosophers and scientists. Conversely, non-determinism does not imply acausality. On the contrary, quantum randomness is circumscribed by definite causal factors, such as the time evolution of the wavefunction, and the measurement interactions that result in definite values.
Locality is not necessary to causality, except under the relativistic model of spacetime. Even here, it is necessary only under the assumption that there can be no privileged reference frame, which, strictly speaking, is unproven, though its contradiction ought to be empirically testable.
We may rank the principles schematically, with arrows such that A → B signifies A is a necessary condition of B.
Generic realism and causality are bedrock assumptions of science, but the specific realism where properties have definite values at all times is less essential. Deterministic predictability is a sufficient condition for the realism where properties have definite values at all physical events. A refutation of determinism need not refute this specific form of realism, nor the more generic forms, nor causality. In general, contradicting a principle in the diagram would consequently refute any of the principles downstream
of it, but would leave those upstream
unaffected. Although a contradiction of locality would not affect realism, causality or determinism, it would refute special relativity as we know it, creating a genuine crisis in physics. This is why historically most quantum theorists have chosen to abandon determinism or realism (in the narrow sense of definite values) rather than locality. Those who opt for non-local interpretations face the burden of reconciling this with the relativistic account of spacetime, by invoking non-causal influences or conditions (e.g., formal or logical necessity, changes in information).
EPR defined a physical theory to be complete when every element of reality is represented by some concept in the theory. It was then averred that the perfect predictability of a physical quantity is a sufficient condition for the existence of an element of reality corresponding to that quantity. In our terms, determinism is a sufficient condition for generic realism.
We have seen that, later in the paper, EPR apparently conflated generic realism with determinism and with a more specific sense of realism where properties have definite values at all times. Yet it is possible to have realism, and even determinism, without insisting that properties have definite values at all times.
Nonetheless, in the entangled system proposed by EPR, we do indeed have quantities with definite values and fully predictable states, at least after measuring system I at some time tm, though not prior to this. Once system I has been measured for property A, it is certain that system II is in an eigenstate for the property P, which means it has a definite value, and that value will remain the same in subsequent measurements with perfect predictability. Thus P is an element of physical reality. If instead we measured B in system I, then system II would be in an eigenstate of Q, and thus Q would be an element of reality. Since it is presumed that system I cannot affect system II, i.e., that causality is limited by the principle of locality, then we infer that both P and Q are elements of physical reality. Since P and Q might be non-commuting operators, they cannot both be simultaneously definite in value under quantum mechanics, so this is not a complete theory.
One may retort that there is no contradiction, since we can only measure A or B, not both. The state of having the definite value pk or qr becomes a reality only after the measurement of system I, not prior to this.
EPR anticipated this criticism that our criterion of reality is not sufficiently restrictive.
One might hold that two physical quantities are simultaneous elements of reality only when they can be simultaneously measured or predicted.
Thus the quantities P and Q are not simultaneously real. EPR respond:
This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.
It is not problematic per se to make the reality of a quantity depend on a measurement. What is problematic is making it depend on the measurement of another system, which is not interacting. Yet the supposition of non-interaction depends on a principle of locality, so EPR have found it necessary to incorporate locality into their definition of reality.
We have found, on the contrary, that their criterion of reality is too restrictive, as it mistakes perfect predictability for a necessary condition of reality. Thus EPR incorporated determinism into their definition of reality, just as they have included locality. The first premise of EPR’s argument is: Either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) the two physical quantities corresponding to non-commuting operators cannot have simultaneous definite reality. The proof of this premise relied on an implicit assumption that a complete theory must render all real values predictable, i.e., that a complete theory must be deterministic. Thus one must assume determinism in order for the overall argument to hold and yield a contradiction of (1).
While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.
This is the only oblique reference to hidden variables, i.e., that there could be other variables besides xI and xII that may complete what quantum mechanics leaves unexplained. In keeping with EPR’s assumption of strong determinism, a theory that somehow included these hidden variables would in principle render the whole system deterministic.
Einstein was dissatisfied with Podolsky’s revisions to the EPR paper, finding that it lost sight of the main problem while getting bogged down in mathematical details. As he confided to Schrödinger in a letter, he considered the essential point to be that there are two different wavefunctions for the second system, ξk(xII) and φr(xII) in our notation (Sec. 4.5). I claim that this being different is incompatible with the hypothesis that the Ψ description is coordinated in a one-to-one way with the physical reality (the real physical state of affairs).
[20]
Einstein truly believed that the problem was a question where realism was at stake. Yet why should we suppose that both wave functions ought to describe the exact same physical state of affairs? After all, the form ξk for system II arises if property A is measured in system I and the form φr arises if property B is measured in system I. Since A and B are non-commuting, we can only measure one or the other at a given time, not both, so there are two different states of affairs, depending on which property we actually measure. This is problematic only if one also introduces a principle of locality. In Einstein’s words, The real state of affairs of [the second system] now cannot depend on what kind of measurement I perform [on the first system].
[21]
Even if we admit that the only way to preserve locality is to allow there to be two different wave functions for the same physical reality, would it thereby follow that the quantum mechanical description is incomplete? As Tilman Sauer (2013) notes, this seems to be a problem of overdetermination rather than incompleteness.[22] All completeness requires is that it accounts for every element of reality. Evidence of incompleteness should come in the form of failure to distinguish between two distinct physical realities.
Before returning to Einstein, let us examine Niels Bohr’s immediate response to the EPR paper, published five months later in October 1935.[23] Following their lead, he construed the proposed problem as having to do with physical reality,
rather than locality or determinism. Subsequent commentators were influenced by Bohr in this construction of the problem.
In his typical style, Bohr makes extreme-sounding philosophical claims to account for an apparent quantum paradox. He considers that the impossibility of controlling an object’s reaction to a measurement implies the necessity of a final renunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality.
Do we really need to renounce realism or causality, as Bohr appears to suggest? How can we do this without invalidating the scientific method, upon which the theory of quantum mechanics depends?
Rather, for Bohr, the classical ideal of causality
refers specifically to mechanistic determinism, where spatiotemporal information, i.e., the future trajectory of a body, is fully determined by definite quantities of momentum and energy. The problem of physical reality,
likewise, refers to a specific notion of reality, namely whether both spatiotemporal position and conserved dynamical quantities (momentum and energy) have definite values at all times. EPR attempted to define the problem with their criterion of reality,
but Bohr finds this suffers from an ambiguity. He recounts their criterion:
If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.
As we have noted, this establishes that determinism is a sufficient condition of reality, though elsewhere EPR, without justification, treat it as a necessary condition or criterion
of reality. Bohr’s response takes a different approach, finding ambiguity in the phrase: without in any way disturbing a system.
He uses the thought experiment of a particle passing through a single-slit and multiple-slit diaphragms to construct his own version of the EPR problem, reducing it to a canonical uncertainty relation, to be explained by his principle of complementarity.
We will amplify his discussion with a description of the assumed wave mechanics.
Bohr calls the object of study a particle,
and considers that, initially, the symbolic representation of its state
is given by a plane wave. Notably, Bohr here regards the wave function as an abstract representation of an object’s state, not as another version of the object. In this paper, complementarity
is applied primarily to non-commuting observables rather than wave-particle duality.
Bohr explains the uncertainties in momentum and position as unavoidable consequences of the experimental setup. First, he considers interaction with a single slit.
Even if the momentum of this particle is completely known before it impinges on the diaphragm, the diffraction by the slit of the plane wave giving the symbolic representation of its state will imply an uncertainty in the momentum of the particle, after it has passed the diaphragm, which is greater the narrower the slit.
Although the plane wave is a symbolic representation of the particle’s state, Bohr speaks of it being diffracted by the slit, as though it were a physical object. This refers to the fact that the distribution of outcomes (when the experiment is repeated for many particles) follows a diffraction pattern, as though the symbolic plane wave were diffracted, altering the relative probabilities accordingly.[24]
Since each particle ends up deflected up or down at some angle, its momentum varies (for even a mere change in direction involves momentum change). Thus the symbolic plane wave representation of the state implies an uncertainty in the momentum after passing through the slit. For Bohr, uncertainty
refers not to our subjective knowledge or belief, but to an indefiniteness in the value of a physical property, agreeing with Heisenberg’s understanding.[25]
Likewise, the uncertainty in position Δx is a consequence of the physical setup. When the slit is much wider than the wavelength, the uncertainty is simply the width of the slit. We could reduce position uncertainty by making the slit narrower, but when this width approaches the scale of the wavelength, we have noticeable diffraction, increasing momentum uncertainty. Any further reduction in position uncertainty by making the slit narrower has a tradeoff of making the momentum after interaction less predictable.
This tradeoff in uncertainties is just a specific case of Heisenberg’s uncertainty principle: ΔpΔx ~ h, where x is the transverse spatial coordinate, p is the momentum in the direction of that same coordinate, and h is Planck’s constant.[26] This principle is a mathematical consequence of the non-commutation of the position and momentum operators, so it holds regardless of the particular experimental setup.
Bohr considers that the momentum’s uncertainty Δp is inseparably connected with the possibility of an exchange of momentum between the particle and the diaphragm….
He then inquires to what extent this momentum exchange can be taken into account in a description of the phenomenon. Evidently, the intent is to see whether a complete description, free of uncertainty, is possible even in principle.
To explore this possibility, he considers two versions of the same experiment. The basic setup is to have two diaphragms, the first with a single slit and the second with multiple slits (parallel to each other and to the first slit), followed by a photographic plate to detect particles. As before, we begin with particles whose states are characterized by a plane wave. Bohr supposes that the first diaphragm causes diffraction, which means the slit width must be comparable to the wavelength (though larger than it). The output state will be symbolically represented by a diffracted wave pattern, which, per Huygen’s principle in classical wave mechanics, may be treated as a superposition of waves emitted from different points within the width of the slit, additively forming a cylindrical wave propagating from the slit. As this wave in turn encounters the second diaphragm, each of its slits likewise emits cylindrical waves. These waves, however, interfere with each other constructively and destructively at overlap points, resulting in an interference pattern of peaks and troughs of greater or lesser intensity (i.e., more or fewer particles at each point) on the photographic plate. The function of the first diaphragm is to confine the initial position and momentum of the particles within some range, which we characterize as the uncertainty.
First, we assume that the first diaphragm (with a single slit) is rigidly attached to the second diaphragm (with multiple slits) and the photographic plate. This fixes the position of the first slit with respect to the slits of the second diaphragm and the photographic plate. As always, we are concerned with position in the transverse direction, i.e., along the width of the slit. So we can now know the initial position of the first slit exactly, confining the initial position uncertainty of the particle to the slit width. There is a tradeoff, however, since this rigid connection of the apparatus implies that any momentum exchanged between the particle and the first diaphragm (or the second for that matter) will be absorbed by the entire apparatus, including the detecting plate. This makes it impossible to take such momentum exchange, in each individual instance, into account for our experimental result, i.e., the position of the particle on the photographic plate.
This might seem to be a purely practical limitation on our knowledge, owing to the unavoidable uncertainties generated by physical interactions. Bohr, on the other hand, recognizing that this situation would be generated by any measurement apparatus with a fixed position, points elsewhere to the critical factor: where we have to do with a feature of individuality completely foreign to classical physics.
It is impossible for some definite amount of momentum exchange by an individual particle to be a determining factor in the outcome for that particle. This does not preclude momentum from being a determining factor in a non-individualistic sense, namely in terms of the momentum of the wave characterizing the state, which may be constructed approximately by repeated measurements of particles in that state. This would seem to situate causality in the realm of this collective or wave state. At any rate, it is excluded from the realm of individual particles, as Bohr makes clear:
In fact, any possibility of taking into account the momentum exchanged between the particle and separate parts of the apparatus would at once permit us to draw conclusions regarding thecourseof such phenomena,—say through what particular slit of the second diaphragm the particle passes on its way to the photographic plate—which would be quite incompatible with the fact that the probability of the particle reaching a given element of area on this plate is determined not by the presence of any particular slit, but by the positions of all the slits of the second diaphragm within reach of the associated wave diffracted from the slit of the first diaphragm.
If we could know exactly how much momentum is exchanged between a particle and each diaphragm, we would be able to calculate exactly which slit in the second diaphragm the particle transited. This is contradicted, Bohr claims, by the fact that the outcome probability for an individual particle is affected by the presence of all slits that are within range of the wave representing the state of the collective. Note that Bohr is not discarding the principle of causality altogether. On the contrary, he explicitly relies on a kind of physical causality to make his argument. An account where we have a well-defined trajectory for an individual particle is unacceptable precisely because it fails to take into account some determining factors, namely the presence of the other slits. To regard these as determining outcomes (albeit probabilistically) is to regard them as physical causes (though not with the full predictability of determinism). The supposition of a definite trajectory with well-defined momentum at all points would fully predict outcomes without resort to these known causal factors, so it cannot possibly yield the correct statistical result, which is known to vary depending on the presence or absence of other slits.
In this first scenario, then, it is impossible even in principle (not just with respect to our ignorance) for outcomes to be fully determined by the particular trajectory of an individual particle, obtaining some definite amount of momentum change at each interaction point. Yet why is this impossible? Precisely because the quantity of this momentum exchange is supposed to be independent of the presence or absence of other slits. In other words, it is only when we include the supposition of locality that we contradict a deterministic account where the outcome results from the definite reality of an amount of momentum exchanged. Yet locality is violated only if we consider the individual particle as an agent, but not if we regard the wave state of the collective as the agent, for, as Bohr notes, the wave state indeed reaches
other slits; i.e., the state includes tendencies toward these other possibilities. The fact that only the slits reached by the wave are determining factors indicates that locality may be upheld if the wave state itself could be regarded as an agent.
Strictly speaking, Bohr has not directly refuted the idea that a definite amount of momentum is exchanged (even under the supposition of locality) for each particle. He has shown only that this definite amount, if it exists, plays no decisive role in determining the probabilities of outcomes. Yet a quantity of momentum that does not affect spatiotemporal outcomes can hardly be considered momentum in any physically meaningful sense. Bohr therefore seems justified in his denial of definite reality to the quantity of momentum of the individual particle.
Bohr proceeds to a second scenario, where the first diaphragm is not rigidly connected to the rest of the apparatus, and it is possible in principle to measure the diaphragm’s momentum with any desired accuracy before and after the passage of the particle, and thus to predict the momentum of the latter after it has passed through the slit.
(As always, we are concerned with momentum only in the direction of the width of the slit.) The first diaphragm is made to collide with a test body, whose momentum before and after collision is controlled. Then the diaphragm’s momentum can be calculated just as exactly, using the law of momentum conservation. We can repeat this with another test body after the diaphragm has interacted with the particle in its slit. We then take the difference of the calculated momentum for the diaphragm before and after interaction in the slit, thereby learning exactly how much momentum it imparted to the particle.
Application of the law of momentum conservation at two successive time points requires us to be operating on a space-time scale sufficiently large for classical mechanical ideas to be applied. If we assume a large enough scale (i.e., much larger than the Planck scale), then momentum can be measured without any limitation on accuracy. On that supposition, however, there is a necessary lack of exact control on the space-time coordinates of the diaphragm and the test bodies.
Moreover, fixing the momentum of the first diaphragm made it necessary for it to be not attached rigidly to the rest of the apparatus. As we saw in the first scenario, such rigid attachment would fix its position in the experimental frame (i.e., that of the plate) with unlimited accuracy, but at the expense of not being able to apply momentum conservation to learn the exact amount of momentum imparted to the particle. The diaphragm’s unattached state in the second scenario means that we cannot use it to fix the position of particles passing through its slit. Like the particles themselves, the diaphragm moves somewhat freely with respect to the rest of the apparatus, and its collision with the test body alters its position unpredictably. The first diaphragm’s momentum may be fixed with sufficient accuracy so that we can know exactly which slit in the second diaphragm a particle will enter, but…
…then even the minimum uncertainty of the position of the first diaphragm compatible with such a knowledge will imply the total wiping out of any interference effect—regarding the zones of permitted impact of the particle on the photographic plate—to which the presence of more than one slit in the second diaphragm would give rise in case the positions of all apparatus are fixed relative to each other.
The reason for this wiping out of the interference pattern, roughly, is as follows. In order for us to know which slit in the second diaphragm the particle transits, we must know the particle’s momentum with enough accuracy so that the uncertainty Δp is less than the difference in momenta between particles passing through adjacent slits, |p1 - p2|, which nonetheless end up on the same point on the photographic plate. In this setup, the particle’s momentum uncertainty Δp equals that of the first diaphragm. Yet, per Heisenberg’s uncertainty principle, the first diaphragm necessarily also has a position uncertainty Δx ~ h/Δp in the lab frame. Since |p1 - p2| > Δp, it follows that Δx > h/|p1 - p2|. In the case of a photon, p = E/c = h/λ, where λ is the photon’s wavelength. Some trigonometry shows that Δx > λd/a, where d is the distance from the second diaphragm to the photographic plate, and a is the distance between slits.[27] Yet λd/a is also the distance between interference peaks on the photographic plate, so the position uncertainty is such that the interference pattern is wiped out.
In short, fixing the momentum rather than position alters the nature of the experiment, so that it is no longer suited for measuring the interference phenomenon of the first setup. The first diaphragm is no longer a measuring instrument of position, but becomes an object of observation due to its independence of motion with respect to the experimental frame.
Although Bohr does not spell this out explicitly here, the complementarity of the two scenarios is not just between well-defined position and momentum (having one or the other, but not both), but relates to wave-like and particle-like behavior. Diffraction with interference, after all, is distinctively wave-like behavior, which we can observe only in the first setup. This comes at the cost of not being able to treat the object as a classical particle with a definite trajectory. In the second setup, we are able to treat the object as a classical particle with momentum conservation, but at the expense of wiping out the wave-like behavior. Whether the object behaves in a wave-like manner or a particle-like manner is determined by how we set up the experiment, fixing either position or momentum with respect to the measurement frame. The difference between the two scenarios is whether the first diaphragm is treated as part of the measurement instrument or part of the object being measured, i.e., whether or not it is fixed with respect to the measurement frame. The difference in perceived behavior owes to differences in relation to the measurement frame.
Relevant to the EPR paradox, Bohr notes that, in the second scenario, we could measure the momentum of the first diaphragm before the particle passes through it, and still be left with a free choice whether we wish to know the momentum of the particle or its initial position relative to the rest of the apparatus.
The first option, already discussed, results in a position uncertainty that wipes out the interference phenomenon. If we choose the second option, measuring the position of the diaphragm rather than its momentum after the particle passes (say, by making a rigid connection to the rest of the apparatus), then we can never know how much momentum it exchanged with the particle. In that case, the momentum uncertainty of the particle will remain sufficient to have the diffractive interference phenomenon. In other words, whether or not we observe the interference phenomenon after transit through the second diaphragm depends on our choice of measurement (position or momentum) for the first diaphragm after it has ceased interacting with the particle! This is the EPR paradox in its essentials, where the first diaphragm is system I and the particle is system II.
This freedom of choice after interaction is not paradoxical or unusual to Bohr, for it is merely our freedom of handling the measuring instruments, characteristic of the very idea of experiment.
Depending on which experimental arrangement we choose, we renounce one or another aspect of describing physical phenomena. In classical mechanics, two elements are required for a fully deterministic description: spatiotemporal coordinates and dynamical conservation laws (for energy and momentum). In quantum mechanics, however, we can only know one of these elements at the expense of the other, making the classical deterministic account of causality impossible. We could predict a trajectory perfectly if we knew the starting position and the amount of momentum received, yet fixing the starting position in some reference frame makes momentum uncertain in that same frame, and vice versa.
This uncertainty results not from a mere lack of knowledge, but from the impossibility… of accurately controlling the reaction of the object on the measuring instruments.
This has not merely to do with an ignorance of the value of certain physical quantities, but with the impossibility of defining these quantities in an unambiguous way.
In earlier writings, Bohr described this uncontrollability in terms of Planck’s quantum of action,
h.[28] The mere existence of this quantum of action shows that there are limits beyond which it is impossible to do classical mechanics. That is to say, mechanics has predictable, deterministic causality only on scales larger than h. The quantum of action itself is an element evading customary explanation,
i.e., it cannot be explained in the customary sense of physical causes.
If we fix the position of the particle, then an uncontrollable amount of momentum is exchanged with the apparatus. It is precisely because we cannot control this amount that we cannot use it to determine the outcome. This uncontrollability is not merely a question of our ignorance, but implies an intrinsic non-determinism, at least with respect to prior values of position and momentum. It is not acausal, for then we should not speak of momentum exchange at all, but it is not deterministic, precisely because the initial condition does not fully determine the final value of the momentum. What is left unclear and unknown is whether the momentum even has a definite value. At any rate, it is fundamentally unpredictable (from prior values of position and momentum) within a certain range.
If, on the other hand, we fix the momentum of the particle, then there is an uncontrollable displacement of the diaphragm interacting with a test body. The position is uncontrollable
in the same sense as described above, due to the irreducible quantum of action.
As noted by philosophers Bai & Stachel (2010)[29], these considerations provide a clear criterion for discerning whether an entity is part of the measuring instrument or not. It depends on whether the uncertainty resulting from the quantum of action must be applied to the entity in question. In the first setup, where the first diaphragm is rigidly attached to the rest of the apparatus, it may be considered part of the instrument. Even though the particle exchanges an uncontrollable amount of momentum with the diaphragm, the overall apparatus is so massive compared with the particle that its momentum change has negligible effect on its motion. (Nonetheless, the particle’s momentum uncertainty, equal to that of the diaphragm, is significant enough to affect the particle’s motion, resulting in the diffractive interference pattern.) In the second setup, by contrast, the first diaphragm has a significant uncontrollable displacement by virtue of its collisions with the test bodies, so it shares a position uncertainty with the particle entering its slit, since it moves freely with respect to the experiment frame. Thus the diaphragm is not part of the instrument, but another object of quantum measurement.
It is in the second setup where we have an EPR paradox, because the first diaphragm and the particle can be treated as an entangled pair. If we choose to measure the diaphragm’s momentum after interacting with the particle in its slit, then we can resolve the particle’s momentum with arbitrarily fine accuracy, but its position uncertainty is correspondingly greater, i.e., it is less controllable, due to the quantum of action h (dimensionally, a product of position and momentum). If we instead choose to measure the diaphragm’s position, by fixing it in relation to the rest of the apparatus, this can be accomplished only by an uncontrollable momentum exchange between the diaphragm and the apparatus (since we must stop the motion of the diaphragm with respect to the apparatus), thereby creating an equal uncertainty for the particle’s momentum in the experimental frame. Nothing has been done to the particle by connecting the diaphragm to the apparatus. Rather, we have done something to the measurement instrument so that its relation to the particle has changed. This does not involve any non-locality. Though Bohr does not spell this out, presumably the attachment must occur before each particle strikes the photographic plate, or we will not have the requisite uncertainty in momentum.
As Bai & Stachel remark, Bohr does not consider it meaningful to define the concepts used to describe a phenomenon without a full account of the measurement process.[30] You must define that entire process from start to finish in order to define what you mean by the position of X
or the momentum of X.
The physical phenomenon is the total process, from preparing a system, to interaction, to registering a quantity. The acts of preparation (fixing the initial state) and registering quantities (testing predictions) constitute measurement. By our choice of what to measure on the first diaphragm, we impose a condition on the phenomenon involving the particle, thereby affecting the kinds of predictions we can make. The first diaphragm and the particle are inseparable insofar as they are both essential to defining the conditions of the measurement process and therefore its results (tests of predictions). None of this, however, requires supposing that the diaphragm somehow acts on the particle after it has passed through its slit. Entanglement implies non-separability, but not non-locality.
It is notoriously difficult to interpret Bohr’s stance on the reality
of the unmeasurable. Like Heisenberg, Bohr did not believe that there were really
definite values of properties beyond the limits of measurement. On the other hand, he did not confine reality to measurable quantities, as shown by his endeavors to give a rational, causally local account of what was transpiring in the diaphragm example. He recognized, of course, that the notion of causation would have to be modified, replacing strong determinism with a degree of uncontrollability
(not just unknowability) in physical properties. Yet this uncontrollability made the properties in question, to that extent, useless for predicting outcomes, consequently losing their classical character. A momentum that cannot determine the future trajectory of a particle is arguably not momentum in any physically meaningful sense.
Bohr does not deny that either position or momentum can be fixed with arbitrarily fine accuracy at any time. He only acknowledges that, for some strange reason, namely the irreducibly inexplicable quantum of action, we cannot fix both at the same time, no matter how perfect our instruments. Therefore it is impossible, even in principle, to give a causal description in the sense of classical mechanics, for that requires both space-time coordination (having a fixed position at a given time in the frame in question) and a dynamical conservation law (having a conserved definite quantity of momentum or energy over time) to yield an equation of motion. At the quantum level, we can only have one or the other complementary aspect of ordinary causality.
Bohr’s notion of complementarity suggests that non-commuting properties such as momentum
and position
point to different aspects of physical reality, approaching the same object from different perspectives. He denies that they are inherent attributes of the object.[31] This seems to make Bohr an irrealist, but he does not deny physical reality altogether, only that perspective-dependent properties such as position and momentum inhere in objects with absolute or even definite quantities.
Bohr, unlike Heisenberg, does not attribute quantum uncertainty to the measurement disturbing
a physical phenomenon, for the phenomenon includes the measurement, and there is nothing other than the phenomenon. Again, this may sound irrealist, but Bohr is not denying that there is any physical reality besides the measurable; rather he affirms that the measurement is an inseparable condition of the physical reality being observed.
It is precisely because the measurement is part of the physical phenomenon that Bohr considers the EPR criterion of reality to be ambiguously defined. The ambiguity is in their expression: without in any way disturbing a system.
Bohr agrees with EPR that system I imposes no mechanical disturbance of system II after they have ceased interaction, and that the measurement chosen on system I after interaction does not constitute such a disturbance. Nonetheless, there is still an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system.
Our choice of measurement on system I does not involve some non-local mechanical alteration of system II, but it changes the conditions of measuring system II, restricting what we can possibly find out about it.
Critically, these conditions constitute an inherent element of the description of any phenomenon to which the term ‘physical reality’ can be properly attached.
EPR’s attempt to define physical reality without reference to these conditions makes their criterion of reality ambiguous, in Bohr’s view. Once again, it may seem that Bohr is taking an irrealist stance, denying that there is any reality beyond what is measured. On the contrary, he only affirms that a definition of physical reality must include the conditions of measurement, keeping in mind that measurement
is not about subjective human knowledge, but physical conditions imposed on a system by physical interactions. These conditions define bounds on the controllability of one or another complementary property in a given experimental frame.
The quantum mechanical description is complete in the sense that it is a rational utilization of all possibilities of unambiguous interpretation of measurements, compatible with the finite and uncontrollable interaction between the objects and the measuring instruments.
The EPR paradox does not contradict this notion of completeness, since its two different wave functions for system II (after measuring one or another property of system I) describe two distinct physical realities, in Bohr’s understanding of reality as phenomenon
inclusive of measurement conditions.
Bohr emphasizes the importance of distinguishing object from instrument in quantum mechanics, which has its root in the indispensable use of classical concepts in the interpretation of all proper measurements, even though the classical theories do not suffice in accounting for the new types of regularities with which we are concerned in atomic physics.
This seems to imply that classical properties such as position and momentum have no reality outside of measurement, but Bohr never says this, and elsewhere speaks hypothetically of uncontrollable changes in position and momentum. He only denies that these properties have definite values within their range of uncertainty, so that a causal description in the sense of classical mechanics is impossible even in principle. As we have noted, though it is conceivable that such definite values exist, they have no measurable physical consequences, and thus would not pertain to physical reality by the empiricist standard professed by EPR no less than Bohr.
It is unclear if the distinction between instrument and object has much relevance in the context of the EPR paradox. Although this distinction figures in Bohr’s diaphragm thought-experiment, it hardly seems germane to the measurement of entangled pairs of particles in a singlet state of correlated angular momentum or polarization. Our choice to measure along one or another axis for spin or polarization does not define a distinction between whether or not the particle in question is part of the instrument. Bohr himself elsewhere considered that quantum uncertainty is not to be attributed to the object or the instrument, but to the phenomenon as a whole.
According to Bohr, the only unambiguous interpretation of the symbols of quantum mechanics is embodied in the well-known rules which allow to predict the results to be obtained by a given experimental arrangement described in a totally classical way, and which have found their general expression through the transformation theorems….
The relevance of these transformation theorems,
whereby you can replace a pair of commuting observables with another pair obtained by a rotation
in the state space spanned by the first two, is not immediately obvious. Bohr discusses this in a footnote, the gist of which is as follows.
Suppose you have two pairs of commuting observables, q1, q2 and p1, p2, where q1, p1 are non-commuting, as are q2, p2. (For example, position q and momentum p observables along two Cartesian coordinates have such relations.) You could replace them with observables Q1, Q2, P1 P2, attained algebraically by rotating
by some angle θ in the planes defined by q1, q2 and p1, p2 respectively. Since Q1 and P2 do commute, you can measure both simultaneously. Since Q1 is expressible in terms of q1 and q2, and P2 is expressible in terms of p1 and p2, a subsequent measurement of q2 lets us predict the value of q1, and a subsequent measurement of p2 lets us predict the value of p1.
The unstated implication is that, by first making a measurement of the rotated
observables Q1 and P2, the observables q2 and p2 obtain predictive power for q1 and p1 respectively, which they may not have had otherwise (supposing q1 is orthogonal to q2 and p1 is orthogonal to p2). This is just a restatement of the problem in terms of commutation relations, though the use of rotations admirably prefigures the approach to be used by Bell. For Bohr, the apparent significance is that what we choose to measure (i.e., how we orient
the instrument at some angle θ in the operator space) impacts what kinds of predictions we can subsequently make, and even alters the dependence of one observable on another.
Bohr says that these formalistic rules regarding measurements exclude inconsistency in quantum mechanical descriptions related to a change in where we divide object from instrument, but the consistency of quantum mechanical formalism was never disputed by EPR, only its completeness. Bohr apparently considers the choice of measurement variables to be equivalent to the choice of where to divide the object from the instrument (though we have noted the latter is evidently inapplicable to entanglement of spin and polarization). Interestingly, he says we have the latter freedom of choice only within a region where the quantum-mechanical description of the process concerned is effectively equivalent with the classical description.
This is because, as stated earlier, the symbols of quantum mechanics can be applied unambiguously only to predict the results from an experimental arrangement described in a totally classical way.
What Bohr means by totally classical
is obscure,[32] but it would seem he means merely that we are operating in a domain where either complementary variable can be measured.
In his final remarks, Bohr returns to a theme that he had first broached in his 1927 Como lecture, namely that the quantum mechanical principle of complementarity ultimately comes from the complementarity of space and time discovered by relativity. In relativity, Bohr says, you need to maintain a sharp distinction between space and time coordinates to describe measuring processes, even though at a deeper level we discover laws that force us to renounce the customary separation of space and time ideas.
The relativistic invariance of quantum mechanical uncertainty relations, he adds, suffices to guarantee the compatibility of the principle of complementarity (which he identifies with said relations) with relativity theory.
By basing the completeness
of quantum mechanical descriptions on the formalism’s ability to make correct statistical predictions for all permutations of experimental setups, Bohr apparently has contented himself with a purely formalistic mode of physical explanation or causality. The fact that it gives us the correct mathematical correlations, quite apart from supposing any definite quantities of mechanical properties, suffices to account for the only sort of physical reality
Bohr considers meaningful. Yet he accomplishes this only by forsaking classical efficient causality, and it is not at all clear that the explanatory mode he proposes in its stead qualifies as efficient causality at all. If one really were to abolish efficient causality, then quantum mechanics (and science in general) does not really give accounts of how phenomena come into being, for such an account is impossible. Scientific explanations must be purely formal or mathematical, only allowing us to calculate the probabilities of outcomes, and showing mathematical correlations between events. This reduces scientific theory to a theory of knowledge about events, not a theory of how things really work.
EPR had attacked this purely formalistic description, arguing that it failed to provide a complete description of physical reality. Bohr’s retort is that quantum mechanical theory has no need of the classical notion of reality, apart from the interpretation of measurements. In this view, quantum mechanics is not even an attempt to be a partial description of classical reality; it is complete only with respect to descriptions of phenomena
as Bohr understands them, i.e., including the specific experimental measurement conditions. There is nothing else, at least nothing relevant to predicting measurement outcomes, that quantum mechanical theory omits. In a sense, Bohr and Einstein agree in substance on the limits of what quantum mechanical theory can do. Where they disagree is on what qualifies as physical reality or physical explanation.
Bohr’s thought-experiment used the position and momentum operators in part because these were also used as an example by EPR. This focus on the formalism of square-integrable functions displeased Einstein, as it distracted from the main problem, transcending the form of the operators, namely that there are two wave functions that describe the same physical reality. He took up this theme in his own terms in a lengthy discourse titled Physik und Realität
(1936). We will not attempt a comprehensive account of the discussions among Einstein, Schrödinger, Bohr, Heisenberg, von Neumann, Born and others, but we will review Einstein’s mature thought on the subject expressed in 1948, since his principle of separation
helps further define the conceptual problem.
In an article titled Quantum Mechanics and Reality,
[33] Einstein explains succinctly and clearly why he considers quantum mechanics to be an incomplete theory. He considers a free particle with a spatially limited (i.e., normalizable) ψ-function, with neither sharply defined momentum nor a sharply defined location, and asks: In what sense should I imagine that this description represents a true individual fact? Two conceptions seem to me possible and obvious, which we want to weigh against each other.
a) The (free) particle has in reality a definite place and a definite momentum, though not both at the same time can be determined by measurement in the same individual case. The ψ-function gives, according to this view, an incomplete description of the real state of affairs. This view is not the one accepted by physicists.…
b) The particle has in reality neither a definite momentum nor a definite place; the description by the ψ-function is a basically complete description. The sharp location of the particle that I get through a location measurement is not interpretable as the location of the particle. The sharp localization that emerges during the measurement is only produced by the unavoidable (not insignificant) measurement intervention. The measurement result depends not only on the real particle situation but also on the principle of the incompletely known nature of the measuring mechanism. It is analogous when the particle’s momentum or some other observable is measured. This is probably the interpretation currently favored by physicists; and one has to admit that it alone does justice to the empirical facts expressed in Heisenberg’s principle within the framework of quantum mechanics.
It is clear from (b) that Einstein did not fail to understand Bohr’s basic position, and indeed he expressed it more coherently than Bohr himself. Whereas Bohr denied that empirical physical reality was definable apart from the measurement process, he made a slippery argument from limitations of the measurement process to declarations about objective reality, or rather the absence of definite values therein. Einstein treats this denial as a statement about reality, so if Bohr were to reject this characterization, he would effectively renounce the claim that quantum mechanics completely describes reality, save by confining reality to measurements. Yet we have seen that Bohr admitted some objective reality prior to measurement, but only denied that this reality could be described classically with definite values of position and momentum.
Einstein continues:
According to this view, two (not only trivially)[34] different ψ-functions always describe two different real situations (e.g., the spatially sharp or the momentum-sharp particle). The above applies mutatis mutandi as well for the description of systems that consist of several mass points. Here, too, we assume (in the sense of interpretation I b) that the ψ-function completely describes a real state of affairs, and that two (essentially) different ψ-functions describe two different real facts, even though they, by performance of a complete measurement, can lead to consistent measurement results; the agreement of the measurement results is then attributed to the partially unknown influence of the measuring arrangement.
Here Einstein subjects Bohr’s interpretation to (generic) realism. If our choice of which variable to measure results in two different wave functions, it is because either measurement creates a distinct state of affairs in reality. Even retaining the sense of interpretation (b), namely that a property becomes sharply defined only after measurement, it is nonetheless really the case that one or the other property becomes sharply defined. Einstein admits with Bohr that there is no inconsistency between the two wave functions in terms of their respective measurement predictions, insofar as both sets of outcomes are consistent with the same initial (pre-measurement) form of the wave function. Nonetheless, he notes, this consistency is obtained only by invoking the unknown (or in Bohr’s term, uncontrollable
) interaction with the measuring instrument.
Einstein now invokes a modified version of the separation principle
[Trennungsprinzip] he had mentioned to Schrödinger in 1935, taking the latter’s critique into account. His original separation principle
had conflated ontological separability and causal non-locality, but now he is prepared to distinguish these.
If one asks what is characteristic of the world of physical ideas independent of the quantum theory, then the following is striking: the concepts of physics refer to a real external world; i.e., ideas are posited of things which claim areal existenceindependent of the perceiving subjects (bodies, fields, etc.), which ideas, on the other hand, are brought into sensible impressions as far as possible. Further, it is characteristic of these physical things that they are thought to be arranged in a space-time continuum. It is also essential for this classification of the things introduced in physics that at a certain time these things claim an independent existence, as far as these things liein different parts of space.Without the assumption of such an independence of the existence (of theso-being) of spatially distant things from each other, which originates first from everyday thinking, physical thinking would not be possible in the sense in which we are familiar. Without such a clean separation, one cannot see how physical laws could be formulated and tested.
Physics, by definition, deals with the sensible world, and this is arranged in a spatiotemporal continuum, which is necessary for the very notion of physical dynamics. The only way we can compartmentalize and analyze reality is by operating on the assumption that a physical object in this part of space is existentially independent of an object in another part of space. This does not mean that they cannot influence each other, but only that we can consider this object as being as it is
without reference to some spatially distant object. If we could not subdivide physical reality into this thing here
and that thing there
(in a given reference frame), we would not be able to even formulate coherent hypotheses about physical reality, much less test them.
How separate do things have to be to be existentially independent? Field theory has carried this principle to extremes by locating in the infinitesimal (four-dimensional) space-elements the elementary things on which it depends, existing independently of each other, and the elementary laws postulated for them.
As general relativity is a field theory par excellence, this remark counters Bohr’s use of that theory to support complementarity,
for it also affirms separability in the strongest terms.
For the relative independence of spatially distant things (A and B) the idea is characteristic: external influence on A has no immediate influence on B; this is known as thelocality principle[Prinzip der Nahewirkung], which is consistently applied only in field theory. Total abrogation of this principle would make the idea of the existence of (quasi-) closed systems and thus the establishment of empirically verifiable laws in the usual sense impossible.
Here locality
(more exactly, proximity
) is introduced as though it were a necessary consequence of separability, which it is not. Locality, rather, presupposes separability. We obviously could not speak of influences on A not immediately influencing some spatially distant object B unless we could first speak of A and B as spatially distant objects with independent existence. We could not have locality without separability, but we might, in principle, have separability without locality. Nonetheless, locality is practically necessary for physical science. If physical objects and phenomena were not isolatable, scientific inquiry would become deeply problematic, as we could have no adequately controlled experiments or closed systems in which to test hypotheses.
Einstein then proceeds to show that the standard interpretation (b) of quantum mechanics is inconsistent with the principle of separation, and is therefore incomplete. If this proof is valid, it would have a stronger conclusion than the EPR paper, since it relies on separability, which is more fundamental than locality.
As usual, we have a combined system S12 composed of subsystems S1 and S2, considered at some time t after their interaction has ceased. Due to prior interaction, the joint wavefunction ψ12 cannot be simply the product of subsystem wavefunctions ψ1(q1)ψ2(q2), but rather it is the sum of a product of orthogonal functions of q1 and q2.
Spatial separation of the two subsystems is defined to mean that, at time t, ψ12 is non-zero only for q1 confined to some volume of space R1, and only for q2 confined to some other space R2 that is not overlapping or contiguous with R1. If we take this as a two-particle system, it means that each particle is localized within R1 and R2 respectively.
Here Einstein evidently takes the qi to mean spatial coordinates, so the ψi are spatial wave functions. His definition of separation requires treating q1 and q2 (each consisting of three coordinates) as though they mapped to the same physical space (after transforming between coordinate systems). Yet we should recall that q1 and q2, considered as variables of ψi, inhabit entirely different state spaces.
Entanglement implies that there are no distinct wave functions for the two subsystems after interaction, but we can determine ψ2 of S2 from ψ12 if we also perform a complete measurement in the sense of quantum mechanics
on S1.[35] This measurement breaks the entanglement, so we henceforth have a definable wavefunction for each subsystem.
Depending on which observable we measure on S1, we will get different forms for ψ2, each of which gives different statistical predictions for subsequent measurements performed on S2, hence these different forms correspond to different quantum mechanical states of S2. Under interpretation (b), these different forms of ψ2 describe physically distinct states of affairs resulting from the measurement intervention. That implies a different state of affairs for S2 is generated depending on which observables we choose for our complete measurement of S1.
Under quantum mechanics, Einstein admits, this is not generally a problem, since we can only choose one or another set of observables for measurement of S1, and that specifies exactly one form of ψ2 for S2. We do not need to assign two incompatible forms of ψ2 to S2 simultaneously.
If, however, the two subsystems are separated in space as defined earlier, and we wish to uphold the principle of separability, we have a problem.
In our example, the complete measurement at S1 means a physical intervention that only affects the space part R1. However, such an intervention cannot directly influence the physical reality in a space part R2 remote from it. From this it would follow that any statement regarding S2, to which we can arrive on the basis of a complete measurement at S1, must also apply to the system S2, if no measurement at all takes place at S1. This would mean that all statements must be valid for S2 at the same time, which can be derived from the setting of ψ2 or ψ2 etc.
When dealing with a single particle, Bohr could say that it became position-sharp only by virtue of the uncontrollable momentum exchange with the instrument, or momentum-sharp by virtue of an uncontrollable displacement of the diaphragm with respect to the instrument; i.e., the difference in wave function form is attributable to the measurement act. In the present example, Einstein claims that it is impossible to attribute the specific form of the wave function ψ2 to the measurement act, since a measurement on S1 can only occur in R1, since the joint wavefunction ψ12 is non-zero only there for q1. Thus the measurement must be something occurrent only within that region of space.
This is well and good, but Einstein must still introduce the principle of locality as well. Only when we further assume that a measurement on S1 cannot immediately affect R2 can we conclude that ψ2 must describe S2 regardless of whether a measurement on S1 is performed! The same can be said of all other versions of ψ2, so all of them must be simultaneously attributable to S2, contradicting the interpretation (b) that they are descriptions of distinct physical states.
Einstein, apparently considering locality a necessary consequence of separability, thinks that his conclusion can be avoided only by denying the independent existence of the physical realities of different space parts,
i.e., the separation principle. He defends this principle by noting that there is no actually observed physical phenomenon, including those of quantum mechanics, that makes it likely for it to be violated. After all, every direct measurement, including those of quantum mechanics, situates an observed object here rather than there, and we can describe this object without reference to what is going on somewhere distant. Yet we have seen that it is locality, not separability, which is crucial to obtaining the contradiction.
We have noted that Bohr, in his holistic account of the measurement process, apparently denied separability, while still upholding locality insofar as causality can be applied. Some modern commentators have thought to solve
the entanglement paradox by reading Bohr as denying separability and admitting locality. This seems illogical, since locality presupposes separability. In fact, the separability denied by Bohr in 1935 is not the same as that affirmed by Einstein in 1948. Bohr considered both subsystems to be essential to defining the measurement process; it is only in that sense that they are inseparable. Indeed, as Max Born commented in response to Einstein, our measurement on system 1 can determine the quantum state of system 2 only in combination with knowledge of the joint wave function ψ12, so the measurement is conditioned in part by system 2. If we admit that our measurement on system 1 can be conditioned by its interaction with the long departed system 2, perhaps no greater difficulty is involved in supposing that a measurement on 1 can impose conditions on what we can subsequently measure on 2, at least as far as ontological separability is concerned. It is far more problematic, however, to reconcile this with locality, as J.S. Bell would bring into sharp relief.
[13] The principles need not be absolutely universal, i.e., applying to all entities under all conditions, but they should apply to at least some class of entities under some conditions if they are to have any explanatory power beyond a tautological re-labeling.
[14] There can be mutuality in causation. Under Newtonian principles, we understand that the cue ball is affected by the object ball with an equal and opposite force, so in reality both balls are simultaneously acting as causes, each producing the effect of altering the motion of the other.
Also, causes may be required to act in concert to produce an effect. In fluid mechanics, biology, and meteorology, the sheer multiplicity of agents and complexity of interactions can make it practically impossible to identify clear chains of cause and effect. Here we can only speak of causal factors contributing to the effect.
[15] This differs from the abstract Minkowski spacetime definition of an event as a spatiotemporal point. Yet if we treat special relativity as a physical theory rather than a mathematical formalism, we see that it is meaningless to speak of spatiotemporal points where nothing happens, since nothing can be observed there. A fundamental insight of relativity is that we cannot treat time as an absolute, abstracted from moving objects, so we should not speak of time where there is no motion. We must be mindful that special relativity, like all physical theories, is no stronger than its empirical basis.
[16] Only the immediate cause, not a more remote cause, needs to be present at the time of an effect’s production. It is not at all necessary for your grandparents to have remained alive at the time of your conception, but it would be quite miraculous if your parents (or at least their germ cells) were not.
[17] A computable function is one whose values can be calculated from inputs using an algorithm. Depending on what kinds of algorithms we allow, we can have broader or more restrictive definitions of randomness. Kolmogorov’s definition is most intuitive: a random string cannot be produced by any algorithm expressible in a string (which may include recursion) shorter than or equal to its length.
When one considers the infinite wealth of possible functions and algorithms, it becomes clear that random sequences, far from being simple, represent staggering complexity.
[18] Discussions of free will in the philosophy of science often conflate free agency with randomness. As we have shown, randomness denotes a lack of correlation among event outcomes or effects. Free agency deals with causes rather than effects, with subjects rather than events, so the notion of randomness is inapplicable. Free agency is, of course, non-deterministic in the sense of not being predictable with certainty from antecedent conditions. Nonetheless, the free agent also acts as the determiner of which inclination it will pursue, so it is determining,
but not in the determinist sense of relying entirely on antecedent conditions.
[19] Einstein, Podolsky and Rosen, op. cit.
[20] Einstein, A. Letter to Schrödinger, June 19, 1935. In: K. Von Meyenn, ed. Archive for the History of Quantum Physics,
microfilm 92, section 2-107; (2011), vol. 2, pp. 537–539. Translation by T. Sauer, note 22, p.109.
[21] Ibid.
[22] Sauer, T. (How) Did Einstein Understand the EPR Paradox?
In: T. Sauer and A. Wüthrich, eds. New Vistas on Old Problems: Recent Approaches to the Foundations of Quantum Mechanics (Berlin: Edition Open Access, 2013) pp.105-120.
[23] Bohr, N. Can Quantum-Mechanical Description of Physical Reality be Considered Complete?
Physical Review (1935), 48:696-702. All quotations in section 7.1 are from this paper, unless otherwise noted.
[24] Since a plane wave is not normalizable, we cannot compute absolute probabilities. The plane wave, however, is only an idealized example. In the real world, there are no wavefronts extending to infinity, so the wave function of an actual physical system will in fact be normalizable.
[25] It is now generally acknowledged that Heisenberg’s Unbestimmtheit is best translated as indeterminacy,
i.e., indefiniteness, rather than uncertainty.
Heisenberg attributed this indefiniteness to the measurement process, and though he recognized that this did not logically exclude the possibility of there always being definite values in an unobservable reality, he dismissed the idea, since physics describes only the observable.
[26] In this original form of the uncertainty principle, the Δ signifies the full spread of values, held by nearly 100% of instances. If we instead use it to signify the standard deviation from the expectation value, then the product of uncertainties must be greater than or equal to h/4π, or ℏ/2.
[27] See discussion in Cohen-Tannoudji et al., Quantum Mechanics (New York: John Wiley & Sons, 1977), I, pp.50-52. This example supposes a double-slit diaphragm (i.e. Bohr’s second diaphragm) is detached from the plate, but the position uncertainty is the same. For massive particles (not treated in the example), the position uncertainty will exceed λd/a, where λ is the particle’s de Broglie wavelength h/p.
[28] Bohr, N. Essays 1958-1962 on Atomic Physics and Human Knowledge (New York: John Wiley & Sons, 1963), p.12. h, being a product of position and momentum, may be thought of as a minimum allowable quantity, or quantum, of action. Classical action is the integral of momentum over a path, and we can predict that a particle will choose the path of stationary action (i.e., a local minimum) among the paths available to it under specified conditions. In quantum mechanics, by contrast, the behavior of a particle is influenced by all permitted paths for a specified value of the action. The fact that the action, at some point, can no longer vary continuously, but has a minimum discrete value, entails the replacement of a least-action principle with a multiple-paths
principle. This does not require us to hold that the particle literally follows all paths.
[29] Bai, Tongdong and Stachel, J. Bohr’s Diaphragms
(Cincinnati: Xavier University, 2010), pp.16-17.
[30] Ibid., p.20.
[31] Bohr, N. Causality and Complementarity,
Philosophy of Science (1937), 4(3):289-98, pp.292-3.
[32] Most interpreters take this to mean that the instrument or its action can be described classically. For a contrarian view, see: Plotnitsky, A. Reading Bohr: Physics and Philosophy (Dordrecht, Netherlands: Springer, 2006), pp.80-84.
A plausible alternative is that we must pretend
that instrument and object are unentangled in order to interpret results as measuring a classical property of an object, thereby supposing the object has such a property. Howard, Don. Revisiting the Einstein-Bohr Dialogue
(Caltech, October 2005; Jerusalem, Bar-Hillel Lecture, December 2005; Special issue of Iyyun in honor of Mara Beller), p.28.
[33] Einstein, A. Quanten-Mechanik und Wirklichkeit,
Dialectica (1948), 2, 320-324. Original translation from German.
[34] Non-trivial difference excludes mere addition or multiplication of an arbitrary constant, since this would not define a distinct quantum state (i.e., outcome predictions would be unchanged). This would also exclude any phase factor change that represents a mere shift in coordinates, as well as gauge transformations.
[35] A complete measurement
means measurements on a complete set of commuting observables. If each subsystem is a free particle, a complete measurement could consist of measurements of all three position components, or all three momentum components, or any permutation where we do not measure position and momentum along the same coordinate axis, since these observables are non-commuting.
© 2019 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org
| Home | Top |