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• Pt. 3 of 4

[Full Table of Contents]
8. Bohm’s Spin State Formulation of the EPR Problem
    8.1 Bohm’s Early Concepts of Quantum Theory
    8.2 Entangled Spin Thought Problem
    8.3 Bohm’s Hidden Variables Interpretation of Quantum Mechanics
        8.3.1 Establishing the Viability of a Hidden Variables Interpretation
        8.3.2 Implications for the Theory of Measurement and the EPR Paradox
    8.4 Entangled-Polarization Photons as a Test of the EPR Paradox
9. Bell’s Theorem

8. Bohm’s Spin State Formulation of the EPR Problem

8.1 Bohm’s Early Concepts of Quantum Theory

The position-momentum thought experiments considered by Bohr and Einstein are not realizable in practice, for they require extraordinary instrument sensitivity, approaching the Heisenberg limit. David Bohm presented a more practicable thought experiment in his textbook Quantum Theory (1951), using a pair of particles with entangled spin states.

Bohm at the time upheld a Copenhagen-style interpretation of quantum theory as requiring the following conceptual changes:

1. Replacement of the notion of continuous trajectory by that of indivisible transitions.
2. Replacement of the concept of complete determinism by that of causality as a statistical trend.
3. Replacement of the assumption that the world can be analyzed correctly into distinct parts, each having a fixed intrinsic nature (for instance, wave or particle), by the idea that the world is an indivisible whole in which parts appear as abstractions or approximations, valid only in the classical limit.[36]

This last idea echoes Bohr’s denial of separability, and indeed Bohm acknowledges that much of his discussion elaborates Bohr’s famous discourses on Atomic Theory and Description of Nature (1934). Bohm, however, extends inseparability beyond the apparatus and object to imply a fundamentally holistic universe, an insight that eventually would culminate in his famous non-local interpretation of quantum theory.

Under classical mechanics, Bohm remarks, the causal concepts of momentum and energy were mathematically superfluous. If you knew the equations of motion and the positions and velocities of system components at a given point in time, you could deduce all future (and past) states of the system without recourse to any other properties. Under quantum mechanics, however, momentum and energy are indispensable causal factors. Momentum causes displacement insofar as it determines the mean distance covered by a particle in a given time. You can no longer predict future (or past) spatiotemporal information from that of the present, without explicitly introducing momentum or energy. With the aid of momentum or energy, you can predict future spatiotemporal information, but only statistically, not with deterministic certainty. To those who say that mere statistical causality is not causality at all, he retorts that it is precisely this non-determinism that makes energy and momentum indispensable as causal factors. After all, if a system’s spatiotemporal values were fully deterministic, we would have no need of energy or momentum at all, as in classical physics.[37]

Bohm interprets complementarity to mean that the properties of matter are opposing potentialities, more precisely defined only at each other’s expense and in interaction with an appropriate environment. This is seen formally by the fact that the spatial wave function contains momentum information in its phase factor, and the converse is true in the momentum representation. This motivates him to interpret the uncertainty principle for position and momentum to mean that, within the range of indefiniteness of each, exist the factors responsible for the definition of the other, so the very existence of either requires the indefiniteness of the other. Thus Bohm considers them interwoven potentialities, a notion calculated to evade EPR’s insistence on definite elements of physical reality, i.e., properties considered only as sharply defined actualities.[38]

Potential properties are essential to descriptions of the behavior of matter. The wave function alone will not suffice, because it is not in one-to-one correspondence with the actual behavior of matter, only in a statistical correspondence.[39] The wave function gives the probability a system will develop a certain definite value of a complementary (i.e., non-commuting) property after an interaction.

Since a complementary property of matter can become definite only by interacting with another system, that property belongs not only to the material object, but also to the system with which it interacts. An object has no intrinsic properties of its own; instead, it shares all its properties mutually and indivisibly with the systems with which it interacts.[40] This implies a breakdown of the principle of separability.

Classically, even when an object interacts with its environment, we can distinguish object from environment by spatial separation, analyzing the interaction in terms of the changes that happen in each region of space. We can still identify an object even as it changes by the continuity of such changes over time. If the object were to change abruptly in an instant, or if it changed in a way that followed no causal laws, i.e., in an unsystematic or unpredictable way, we could have no guarantee that we were still looking at the same object. We need continuity and causality in order to attribute definite and characteristic effects to each part. These criteria were used to identify fundamental particles, i.e., by the appearance of continuous tracks in a cloud chamber, and their predictable behavior under electric and magnetic forces. The process of identification [of parts], as carried out in practice, always involves the tacit assumptions of continuity and causality.[41] Bohm has not proved that separability logically requires determinism or even causality, only that separability as such cannot be useful for identifying objects with intrinsic physical properties, unless we also suppose some regularity of reaction to external factors, and some continuity in changes to its state.

Unlike some other Copenhagen interpreters, Bohm recognizes that there is no limit to how accurately an observation can be made for any particular property. Analysis of the world into separable parts breaks down only when objects of interest depend critically on the transfer of a few quanta. In the first place, we cannot have a deterministic causal account, for the momentum of each part of the system (e.g., each particle of a multi-particle system) cannot be precisely defined to the extent that the system is localized at all (per Heisenberg’s principle). So we cannot identify the properties of each part by means of reaction to external forces, as they reflect light quanta erratically and discontinuously, and fluctuate in their own properties discontinuously. Probing their properties by applying force would give erratic reactions due to uncontrollable exchanges of quanta between object and probe.

The lack of continuity of motion, coupled with the rapidly and uncontrollably changing nature of all of the parts, would make it difficult for us to continue to identify each part with the passage of time, since between observations a part might change in a very fundamental way. For example, it might turn from something resembling a wave to something resembling a particle, but it would be impossible to follow the transition between the two in any detail… If there were many similar interacting parts…, it would soon become impossible to make certain we were following the same part that we had started with.[42]

Bohm here seems to be looking at limits on knowledge, rather than denying absolutely that the world is composed of parts. He continues to the measurement problem:

…the quanta connecting object and environment constitute irreducible links that belong, at all times, as much to one part as to the other. Since the behavior of each part depends as much on these quanta as on its ‘own’ properties, it is clear that no part of the system can be thought of as separate… these quanta do not constitute separate objects, but are only a way of talking about indivisible transitions of the objects already in existence. The fact that quanta are unpredictable and uncontrollable would, in any case, prevent their introduction as a third object from being of any use…[43]

Again, this might be taken as a purely epistemological limitation.

In a system whose behavior depends critically on the transfer of a few quanta, however, the separation of the world into parts is a non-permissible abstraction because the very nature of the parts (for instance, wave or particle) depends on factors that cannot be ascribed uniquely to either part, and are not even subject to complete control or prediction.[44]

Here we have no merely epistemological basis for denying separability, but an appeal to a real interdependence of parts for the definition of their nature, which affects their localizability, if we treat the wave nature as a definite state rather than a device for predicting outcomes based on competing potentialities. If we do not do so, Bohm’s argument here depends entirely on his claim that separability requires regularity of reactions and continuous states for each part in order to be useful for predictions. This does not disprove the reality of separability, only its usefulness.

Bohm makes a sweeping inference from non-separability: The entire universe must, on a very accurate level, be regarded as a single indivisible unit in which separate parts appear as idealizations permissible only on a classical level of accuracy of description.[45] This would seem to make the classical realm just an approximation of reality, but on the contrary we need classical reality in order for quantum theory to say anything at all.

For the physical interpretation of the wave function is always in terms of the probability that when a system interacts with a suitable measuring apparatus, it will develop a definite value of the variable that is measured. But, as we have seen, the last stages of a measuring apparatus are always classically describable. In fact, it is only at the classical level that definite results for an experiment can be obtained, in the form of distinct events which are associated in a one-to-one correspondence with the possible values of the physical quantity that is being measured. This means that without an appeal to a classical level, quantum theory would have no meaning.[46]

Classical reality is not reducible to quantum reality; it is not just some large scale limit of quantum reality. The semiclassical WKB approximation does not establish such a reduction, for:

…even when a wave packet is defined to only a classical order of accuracy, it will eventually spread over tremendous distances. Yet the object in question (for example, an electron) can always be found within an arbitrarily small region of space when its position is measured. We conclude that a description at the quantum level (i.e., in terms of the wave function alone) does not, in general, adequately represent the definiteness of physical properties that the electron is capable of manifesting when it interacts with suitable measuring devices. In order to obtain a means of interpreting the wave function, we must therefore at the outset, postulate a classical level in terms of which the definite results of a measurement can be realized.[47]

This contradicts some popular interpretations of quantum mechanics, including those used by physicists for pedagogy, where we act as if classical mechanics were just some large scale limit of quantum reality. Bohm reminds us that such a reduction is incomplete, for we need to add to quantum mechanics the definiteness of classical concepts: the large scale behavior of a system is not completely expressible in terms of concepts that are appropriate at the small scale level… Classical properties are not mere useful fictions, as if only quantum concepts truly described reality, for it is only in terms of well-defined classical events that quantum-mechanical potentialities can be realized… We need both small-scale properties, i.e., the quantum potentialities that constitute the wave function, and large-scale properties, i.e., the definite-valued properties of classical mechanics, to fully characterize a system.

Modern theorists who like to say that the world is fundamentally quantum mechanical should keep in mind this fact. It is something more than the historical fact that we could not have discovered quantum theory without first establishing the classical. Rather, the content of quantum theory itself presupposes a purely classical level in order for its concepts of measurement and statistical prediction to have any meaning. Take away the classical apparatus from a thought experiment, and quantum theory is devoid of content.

When we apply the denial of separability to the EPR problem, we find that EPR’s search for elements of reality in each subsystem is misguided. Bohm understands the uncertainty principle to mean that one non-commuting observable can become better defined only at the expense of the precision of the other, if the system (e.g., an electron) should interact with a suitable measurement apparatus.

We see then that position and momentum are not only incompletely defined and opposing potentialities, but also that in a very accurate description, they cannot be regarded as belonging to the electron alone; for the realization of these potentialities depends just as much on the systems with which it interacts as on the electron itself. This means that there are no precisely defined elements of reality belonging to the electron.[48]

Bohm recognizes, however, that science would be impossible without the principle of separability. Fortunately, all real observations are, in their last stages, classically describable. It is only thanks to the fact that a measurement apparatus can be treated classically that we can apply separability, making science possible, for only because of this classical treatment can the observer (scientist) be treated as separate from the apparatus (though there is unavoidable entanglement of the apparatus with the object). Even if the scientist has significant interaction with his apparatus, there is no problem as long as we can correct for this interaction by known causal laws.[49]

The classical nature of scientist-apparatus interaction makes it possible to disentangle what comes from the scientist from what comes from the object of study, and also for various scientists to examine the same apparatus and get the same results. While this avoids paradox and gives a much more lucid account of the measurement problem than Bohr’s, it would seem that the validity of the scientific method is purely a fortuitous accident falling out of the large-scale limit of classical interactions. How, then, can this method be used validly to make inferences about the non-classical world? Bohm does not address this directly, though we should recall that he considers classical concepts to stand on their own, not as something derivable from quantum mechanics: quantum theory presupposes the classical level and the general correctness of classical concepts in describing this level; it does not deduce classical concepts as limiting cases of quantum concepts. [50]

8.2 Entangled Spin Thought Problem

With this conceptual toolset, Bohm interprets the following thought problem. Suppose a diatomic molecule has a total angular momentum or spin[51] of zero, and it then splits into two atoms. Before the split, the total angular momentum of the combined system has a definite value (zero), but each atom has only an indefinite z-spin (likewise for x and y components). After the split, if you measure the spin of one atom with respect to the z axis, you know the other atom must have an equal and opposite value, so you have a definite z-spin for each. The same would be true if you had chosen to measure spin along any other axis. Thus the total spin along any component is a conserved quantity of zero, which means the two-particle system wavefunction can be represented as:

ψ₀ = (1/√2)(ψ_c - ψ_d) = (1/√2)[u₊(1)u_-(2) - u_-(1)u₊(2)]

ψ_c = u₊(1)u_-(2) is the state of the first atom having z-spin +1/2 (in units of ℏ) and the second atom having z-spin -1/2. ψ_d = u_-(1)u₊(2) is the state of the first atom having z-spin -1/2 and the second atom having z-spin +1/2. The state ψ₀ for the two-atom system with total angular momentum of zero (vector sum of x, y, and z spins) is a superposition of these two possibilities for the z-spin states of the atoms. (The state ψ₀ is known as the singlet state, because the spin magnetic number can only have one possible value, m = 0. We do not deal explicitly with m in this problem.)

The formula above has ψ_c and ψ_d with equal probabilities of 1/2. If we were to replace the minus with a plus, this would change the phase relations between ψ_c and ψ_d, but leave their probabilities unchanged. The resulting wavefunction

ψ₁ = (1/√2)(ψ_c + ψ_d)

would correspond to a system with total angular momentum of 1, though the z-component, of course, would still be zero, the sum of the two opposing z-spins.[52] If we knew only that the atomic z-spins were opposite, we could not know whether the total angular momentum was 1 or 0. ψ₀ and ψ₁ are eigenstates of the total spin operator S². Thus we cannot have a definite value of total spin while having definite z-spin values for each of the two atoms. One or the other property can be well-defined at the expense of the other. Keep in mind that total spin considers the vector sum of all three coordinates, so it is possible to know the sum of the z-spins of the two atoms without knowing the total spin. Under Bohm’s current interpretation, this is not a mere limitation of knowledge, but a lack of definition for one or the other property.

The definite phase relations between ψ_c and ψ_d implies correlation between measurements of spin of each particle. Returning to our example where the initial total spin is zero, suppose both spins are measured at the same time (assuming a space-time scale where relativistic effects are negligible). The wavefunction during measurement may be expressed as:

ψ = f_cψ_c + f_dψ_d

where the coefficients f_c and f_d evolve over time. Solving the Schrödinger equation to yield the wave function after measurement, i.e., after interacting with a magnetic field gradient for a sufficient duration Δt, the coefficients have the same magnitude, signifying equal probability for the two outcomes. More importantly, they differ by a minus sign in their phase factor, so they have the forms:

f_c = (1/√2)e^-iμH'⁽^{z₁ - z₂}⁾^Δt/ℏ

f_d = -(1/√2)e^iμH'⁽^{z₁ - z₂}⁾^Δt/ℏ

where μ is the magnetic moment, H' is the magnetic field gradient (change in magnetic field strength with respect to the z-coordinate, evaluated at z = 0), and z_i are the z coordinates of the two particles in the apparatus frame. The opposite signs in the phase factor or exponent signifies the opposite momentum obtained by each atom (depending on which has positive spin), yielding an opposite time evolution of states. Otherwise, the coefficients retain their initial magnitudes from the formula for ψ₀.

We can obtain a good classical measurement only if our apparatus (magnetic field gradient) can provide sufficient separation between the positive and negative spin states, so that each of the atoms ends up with a well-defined z-spin, and the apparatus does not alter the value of the z-spin. For the particles to be distinguished from the initial beam (recall that a quantum measurement requires repeated iterations, and from the statistical results we infer the probabilities), the momentum gained by interaction with the apparatus, μH'Δt, where Δt is the duration of interaction, must be greater than the initial momentum uncertainty Δp₀. To have a classical measurement, we must be well above the Heisenberg limit, so Δz ≫ ℏ/Δp, where Δz is the range of z-coordinates where the wavefunction is appreciable(i.e., where the particles have a non-negligible probability of appearing). Combining these two inequalities, we infer that μH'ΔtΔz/ℏ ≫ 1. Compare this quantity to the exponent or phase factor of f_c or f_d. Δz is the uncertainty in the z-coordinate of each particle in the apparatus frame. Since the quantity μH'ΔtΔz/ℏ ≫ 1, this means the uncertainty in the phase factor is much greater than one, or 2π for that matter, so the phase factor effectively fluctuates randomly, as each particle may take a different value of z within Δz from one moment to the next. Thus, the precise conditions necessary to make a classical measurement also introduce the randomization of each phase factor, breaking the fixed phase relation between ψ_c and ψ_d. We can now give the wavefunction as:

ψ = (1/√2)(ψ_ce^iα_c + ψ_de^iα_d)

where α_c and α_d are separate and uncontrollable phase factors.[53] Thus the two possible results (with either atom 1 or atom 2 having positive spin, and the other negative) do not interfere with each other after the measurement is complete. This is how quantum theory maintains logical consistency when showing that the two possible states interfere with each other prior to measurement but not after, and thus the spins of the two particles are (anti-)correlated to each other, without compromising the assumption that they cease to interact after the measurement is complete.

The EPR problem arises from supposing that we could have instead have measured some other variable, say the x-spin. The initial two-particle system wavefunction ψ₀ may be expressed in terms of the eigenfunctions of x-spin, which we denote by v instead of u:

ψ₀ = (1/√2)(ψ'_c - ψ'_d) = (1/√2)[v₊(1)v_-(2) - v_-(1)v₊(2)]

ψ'_c = v₊(1)v_-(2) is the state of the first atom having x-spin +1/2 (in units of ℏ) and the second atom having x-spin -1/2. ψ'_d = v_-(1)v₊(2) is the state of the first atom having x-spin -1/2 and the second atom having x-spin +1/2.

This has the same form as when it was expressed in eigenfunctions of z-spin, so the same results ensue. The issue is that, by choosing to measure x-spin, we cannot have z-spin eigenstates, and vice versa, since v₊(1) = (1/√2)(u₊(1) + u_-(1)) and v_-(1) = (1/√2)(u₊(1) - u_-(1)). Since potentialities are not realized irrevocably until interaction with the apparatus actually takes place, there is no inconsistency in the statement that while the atoms are still in flight, one can rotate the apparatus into an arbitrary direction, and thus choose to develop definite and correlated values for any desired spin component of each atom.[54]

If you measure z-spin, the system will subsequently behave as if it is in either ψ_c or ψ_d, depending on outcome. If you measure x-spin, it will instead behave as if it is in either ψ'_c or ψ'_d, depending on outcome. This is despite the fact that ψ₀ is the same in both cases. There is no one-to-one correspondence between a system’s wavefunction and its behavior, for the wavefunction is just a probabilistic expression of how a system may behave depending on which observable is measured, i.e., which configuration of an apparatus should interact with it.

Einstein had considered it problematic that the wave function of the second subsystem should change instantly upon measurement of the first, even if they are distant and non-interacting. Bohm’s explanation that phase relations are broken by measurement of one of the atoms, so that the system is subsequently characterized entirely by ψ_c (or by ψ_d, depending on outcome) is insufficient. He must also invoke Bohr’s denial of separability, denying that the two-atom system is analyzable into parts prior to this measurement, even after the atoms have ceased to interact.

The measurement itself, Bohm notes, effectively multiplies the spin wave function by an uncontrollable phase factor. The correlation of the spins of the two particles does not imply non-local action, for the existence of correlations does not imply that the behavior of either atom is affected in any way at all by what happens to the other, after the two have ceased to interact.[55] This is proved by showing that multiplication by random phase factors does not change the mean values of any function of the spin variables for the second particle. That is to say, the mean value of any function of the coordinate spin of particle 2 will be the same both before and after the measurement of the first particle. Thus the correlation exists both before and after the measurement. The fact that the wave function changes from ψ₀ to ψ_c or ψ_d or ψ'_c or ψ'_d does not imply that the behavior of a particle changes at that instant, for there is no one-to-one correspondence between the wave function and a particle’s behavior. Thus there is no action at a distance.

…contrary to general opinion, quantum theory is less mathematical in its philosophical basis than is classical theory, for, as we have seen, it does not assume that the world is constructed according to a precisely defined mathematical plan. Instead we have come to the point of view that the wave function is an abstraction, providing a mathematical reflection of certain aspects of reality, but not a one-to-one mapping. To obtain a description of all aspects of the world, one must, in fact, supplement the mathematical description with a physical interpretation in terms of incompletely defined potentialities.[56]

Here Bohm effectively concedes that the wavefunction is an incomplete description of the world, but also holds that this description is made complete by the notion of incompletely defined potentialities. Elsewhere in his book,[57] he showed that the supposition that a particle’s behavior is always determined by some definite, though imprecisely knowable, position and momentum, cannot yield the same predictions as quantum wave mechanics, at least not without some highly unparsimonious assumptions, perhaps not at all. The wave nature of fundamental particles, with a wavevector indicating a range or cone of momenta rather than a single value, is essential to any account of empirically observed phenomena, so fundamental particles cannot be in particle states, with sharply defined positions and momenta, at all times, even in principle, since this would actually contradict observed experimental results.

Rather, properties exist only in an imprecisely defined form… they are not really well-defined properties at all, but instead only potentialities, which are more definitely realized in interaction with an appropriate classical system, such as a measuring apparatus. This contradicts EPR’s implicit assumptions that the world can correctly be analyzed into elements of reality, each of which is a counterpart of a precisely defined mathematical quantity appearing in a complete theory, and the hypothesis that reality is built upon a mathematical plan.[58] There are no such hidden quantities waiting to be discovered, at least not as actualities. Since the realization of potentialities depends on interaction with apparatus, they are as much properties of the apparatus as of the particle. So there are no precisely defined elements of reality belonging to the particle alone. This denies realism in the narrow sense of continuously existing sharply defined properties.

Applying this to Bohm’s spin problem, we may say that all three coordinate components of spin exist only in roughly defined forms, with each component having the potential to become better defined (at the expense of the others) if it is measured, i.e., if it interacts with an apparatus for sufficient duration to distinguish the plus and minus states of that spin component with full predictability.

If we measure the z-spin of one particle, that value (say +1/2) will thenceforth be fully predictable for future z-spin measurements of that particle (if no x or y-spin measurements intervene). Thus it would meet EPR’s criterion of an element of reality. Since the opposite value of z-spin (-1/2) would likewise be fully predictable for the second particle, they would argue that particle’s z-spin is also an element of reality. If both of these elements of reality, which are clearly correlated to each other, are developed only upon measurement of the first particle, it remains inexplicable how the measurement of the first particle could cause the distant second particle to have an opposite value.

Bohm acknowledges that correlated values are developed upon measurement. Our choice of which property to measure (e.g., which coordinate spin) determines which pair of correlated values may be realized (if we subsequently measure the same coordinate spin for the other particle). Bohm does not consider this paradoxical, for he does not regard the wavefunction as a system’s equation of motion. It is not in one-to-one correspondence with behavior, but a probability function contingent on the choice of property to be measured. This may be seen more clearly in an earlier illustration. The wavefunction of a free electron can be expressed in two forms, as a probability function of positions and as a probability function of momenta. But the wave function itself does not tell us which of these two mutually incompatible probability functions is the appropriate one.[59] Only one or the other of these forms of the wave function can predict statistical outcomes, depending on whether we measure position or momentum. If we choose to measure position, the momentum wavefunction will not give an accurate statistical prediction. The two forms are mutually incompatible, as far as physical realization is concerned. This abolishes Einstein’s paradox about two wavefunctions describing the same reality. Rather, the two different wavefunctions describe different sets of probabilities depending on distinct contingencies, i.e., the type of measurement interaction that will occur, resolving one or another property.

When a measurement brings a property into sharp realization, it puts an end to any interference with the other potential values that might have been realized. This affects the subsequent range of potentiality for complementary variables. If we measure a particle’s x-position exactly, we destroy interference with all other possible x values. This in turn prevents the formation of wave packets with a sharply defined momentum. Analogously, in our problem, measuring z-spin destroys interference between ψ_c and ψ_d, and prevents the states ψ'_c or ψ'_d from characterizing subsequent behavior.

When one of the particles interacts with a magnetic field gradient in the z-direction, the two-particle system undergoes an irreversible process in which phase relations are destroyed between ψ_c and ψ_d. This occurs even before we check which of the two possibilities is realized. If we find, for example, that ψ_c is realized, we may instantly replace the wavefunction with ψ_c to predict subsequent behavior. This sudden change in wavefunction is allowable, because there are no phase relations with ψ_d, i.e., no more interference, and because ψ_c is analyzable into substates, i.e., the z-spin state of each particle, as they are no longer interacting.

The wavefunction is the most complete possible description of a particle using variables belong to that particle alone, yet it is insufficient for determining which of two complementary variables (e.g., x-position and x-momentum, z-spin and x-spin) will be manifested with sharp definition, since that depends on what kind or interaction the particle actually undergoes. Thus complementary variables are latent, incompletely defined potentialities prior to interaction. The wavefunction is indeed incomplete, but it is made complete by an actual, irreversible interaction, not by any hidden variables.

In our initial two-particle system, there is no precisely defined spin variable associated with each atom, contrary to the supposition by EPR that this must be an element of reality. It is true, as EPR say, that after measurement, the definitely realized value becomes fully predictable (as long as no measurement of a complementary variable intervenes), so it then meets their criterion of an element of reality. This, however, is insufficient cause for inferring the definite reality of that variable prior to measurement. Without the latter supposition, there is no strong basis for claiming that what we choose to measure on the first particle causes a change in the second particle. Rather, the states of the particles are only correlated when they are still potentialities. If we measure the z-spin of one particle, then we know that the z-spin of the other particle, if measured after the particles were separated yet before any measurement of a complementary variable, must have the opposite value. The same pair of values may have occurred if we measured the other particle first; indeed, if the particle measurements are spatially separated in the relativistic sense, there is no unequivocal way to determine which occurred first, so the values must be independent of this priority, if physics is to be coherent and self-consistent.

Note that Bohm is able to uphold a broadly conventional interpretation of quantum mechanics against EPR’s criticism only by emphasizing the distinction between potentiality and actual existence, perhaps even more forcefully than Heisenberg. He only needs to deny the principle of separability as applied to incompletely defined potentialities. He does not need to deny the applicability of separability and locality to actual, definitely realized existents. Indeed, the case of spatially separated measurements of the particles would render incoherent any attempt to make the measurement of one cause a non-local change in the other, since we could not unequivocally define which occurred first. We might resort to some kind of holistic fatalism, but this is to abandon the principle that the measurement is an irreversible intervention, completing the incomplete description provided by the wavefunction. If the outcomes of such interventions were predetermined, we would essentially be appealing to hidden variables.

Bohm dismisses the possibility of mechanically determined hidden variables on the grounds that this would require a violation of the uncertainty principle, which thus far has been confirmed by a wide range of experiments not correctly treated by any other known theory. He draws this inference (of a violation of Heisenberg’s principle), however, by reading EPR as concluding that two noncommuting variables… correspond to simultaneously existing elements of reality.[60] As I have shown in section 4.6, EPR did not argue for the simultaneous reality of non-commuting variables, but only showed that this followed from the supposition that the wave function is complete, thereby contradicting that premise, which even Bohm concedes, in a way, to be incorrect. Thus EPR do not seem to require a violation of the uncertainty principle.

Nonetheless, Bohm’s appeal to the uncertainty principle’s agreement with experiment raises serious problems for any would-be hidden variables account of quantum phenomena. At the very least, it strongly implies that the values of quantum observables themselves are not sharply defined between measurements. This is not merely a limit on knowledge, for the contrary assumption leads to conclusions that contradict experiment (unless some implausible suppositions are made[61]). Further, as Bohm noted previously (Sec. 8.1), the existence of separate elements of reality (i.e., the joint supposition of narrow-sense realism and separability) requires assumptions of continuity and causality in order to be applied consistently. Without such assumptions, we cannot say which effect arises from which part, so we cannot even be sure about which part we observing. [T]he assumption that there are separately existing and precisely defined elements of reality would be at the base of any precise causal description in terms of hidden variables; for without such elements there would be nothing to which a precise causal description could apply. In other words, determinism presupposes narrow-sense realism. Yet the uncertainty principle requires that these variables should be truly hidden, i.e., they cannot directly correspond to the observable variables bound by that principle.

Einstein himself was aware of Bohm’s two-particle spin system example, but his notes on the subject show that he misunderstood it, even in terms of quantitative predictions,[62] so we pass over his response.

8.3 Bohm’s Hidden Variables Interpretation of Quantum Mechanics

8.3.1 Establishing the Viability of a Hidden Variables Interpretation

Only a year later, in 1952, Bohm reversed his position, not only admitting that a hidden variables interpretation of quantum mechanical phenomena was possible, but explicitly formulating such a possibility. In the first of two papers, he showed that all the standard predictions of quantum theory could be obtained equivalently on the assumption that a particle’s wavefunction is a mathematical representation of an objectively real field.[63] This field exerts a real force on the particle, altering its trajectory in complicated ways so that, at observable scales, its future position and momentum can be known only probabilistically. This account is truly causal, deterministic, and mechanical, at least in principle, if the hidden variables, i.e. the precise values of position and momentum, could be known.

The one-particle wavefunction can be expressed in polar form: ψ = R exp(iS/ℏ), where R and S are real-valued functions of position (x), R being the amplitude of the wavefunction and S being the phase. Using the Schrödinger equation to derive formulas for the partial derivatives ∂R/∂t and ∂S/∂t, we consequently find that:

(∂S/∂t) + ((∇S)²/2m) + V(x) - (ℏ²/2m)(∇²R/R)

This has the same form as the Hamilton-Jacobi equation of classical mechanics, except for the last term, which Bohm chooses to interpret as a quantum-mechanical potential added to the classical potential V(x).[64] This enables us to treat the above equation as truly mechanical, where S becomes Hamilton’s principal function, and each particle has a well-defined trajectory with definite position and momentum at all times. As in classical mechanics, the particle velocity may be given by v = ∇S(x)/m if the trajectory is orthogonal to at least one surface of constant S. The relation |R(x)|² = P(x) guarantees conservation of probability, but under the new interpretation, R is more than a mere probability amplitude for an ensemble of trajectories, for it appears directly in the formula for quantum potential: - (ℏ²/2m)(∇²R/R). This means that the force on a particle depends on the wavefunction magnitude R, so the wavefunction may now be interpreted as a real force field.

In order to secure all the same predictions as conventional quantum theory, we need only stipulate that the initial momentum of a particle is p = ∇S(x), in which case this relation holds for all time. We note this preserves the classical form of linear momentum: p = mv. As Bohm notes, however, this restriction is not inherent in the conceptual structure of the new interpretation. It is conceivable that this form of the momentum breaks down at subatomic scales (< 10^-13 meters, then thought to be the fundamental length), in which case the new interpretation would permit genuinely new physics not allowed under the conventional interpretation. Otherwise, without any predictions to distinguish them, the new and old interpretations are equally valid.

Under the new interpretation, the use of probability and statistics in quantum mechanics arises from a purely practical limitation on our knowledge of the precise values of position and velocity, due to the uncontrollable disturbances caused by the measuring apparatus. The form of the quantum potential U(x) = -(ℏ²/2m)(∇²R(x)/R(x)), with its factor of ℏ² (10^-68 in SI units) and extreme sensitivity to the precise value of x (due to R(x) in the denominator) and the rate at which R varies over x (as expressed by the Laplacian ∇²), requires impossibly precise knowledge of position and velocity in order to be useful for the prediction of a specific particle’s trajectory. Without such knowledge, the quantum potential makes deterministic prediction possible only in principle, not in practice.

The uncertainty principle expresses the practical limitation on measurement precision imposed by the uncontrollable interaction of an apparatus with the observed system. This limitation holds as long as Schrödinger’s equation holds and the particle velocities have the form v = v = ∇S(x)/m. If the velocity condition breaks down at subatomic levels, then it should be possible in principle to make some measurements with greater precision than permitted by the uncertainty principle. Unless and until such a subatomic deviation from the conventional theory is discovered, the precise positions and momenta of particles may be regarded as hidden variables, since we cannot localize them to a degree that would enable us to make deterministic calculations using the quantum potential.

The probability density P(x) is now understood to apply to an ensemble of trajectories within the measurement limits set by the uncertainty principle. We resort to this device because of our ignorance of precise initial conditions. Under the new interpretation, P^1/2 = R of the quantum potential, and R is more fundamentally representative of this field for each particle, while only coincidentally corresponding to the probability density for an ensemble. If either Schrödinger’ equation or the velocity condition breaks down at subatomic levels, the wavefunction magnitude squared |ψ|² = R² would no longer equal the probability density P.

Bohm’s new interpretation has the advantage of perfect philosophical lucidity, enabling us to maintain realism (in the narrow sense), causality, and even determinism (at least in principle). It also provides a clearly physical explanation of individual outcomes, appealing to a real physical force instead of a mystical dependence on an ensemble of unrealized possibilities. Its principal drawback is that it may seem overly contrived, as it seems far too convenient that the field variable R should coincide with a probability density, and that the quantum potential should have the form that it does.[65] Of course, this is no coincidence, as Bohm just reverse-engineered the quantum potential and velocity formulae from the existing mathematics of quantum theory. Whether the reality of this potential is credible or not really boils down to judgments of aesthetics and parsimony.

We are not principally concerned with whether Bohm’s causal interpretation is correct, but with its implications regarding (1) the non-necessity of the conventional interpretation and (2) the principle of locality. Both of these factors have important bearing on the EPR paradox.

The conventional interpretation, Bohm says, rests on two assumptions:

(1) The wave function with its probability interpretation determines the most complete possible specification of the state of an individual system.
(2) The process of transfer of a single quantum from observed system to measuring apparatus is inherently unpredictable, uncontrollable, and unanalyzable.

This accurately recapitulates the thought of Bohr, though with the usual ambiguity in the notion of something being inherently unpredictable, uncontrollable and unanalyzable. Arguably, the term inherently is ill-matched with the subsequent adjectives, all of which have implied reference to an extrinsic measuring apparatus and observer.

The uncertainty principle may be derived in two ways. First, one may assume (1) the probabilistic interpretation of the wave function and use the de Broglie relation p = ℏk (where k gives the number of waves per meter in each direction) to deduce the principle. Alternatively, one may use a theoretical analysis of the measurement process, finding that the measurement apparatus interacts with the observed system by means of indivisible quanta, so there is always an irreducible disturbance of some observed property of the system. If, in fact, the probability interpretation of the wave function gives the most complete possible specification of a state—and this is an assumption, not something proved—then it must be impossible even in principle to predict or control this disturbance. Thus we must introduce assumption (2) in order to maintain the assumption of completeness.

The latter approach, expressed by Bohr in terms of complementarity, requires us to renounce our hitherto successful practice of conceiving an individual system as a unified and precisely definable whole, all of whose aspects are, in a manner of speaking, simultaneously and unambiguously accessible to our conceptual gaze. Up until now, the scientific method had required us to be able to define isolatable features of a system in order for us to analyze it. Bohr suggested that there was a point where physical reality was beyond analysis. As David Bohm remarks, this denial implies the impossibility of any model, mathematical or otherwise, that is in a precise one-to-one correspondence with the behavior of a system. For Bohr, the wave function is not a model of a system, but has only a statistical correspondence with its behavior.

Under the principle of complementarity, we have pairs of complementary properties, each of which can be more precisely defined only to the proportion that the other is less precisely defined. There are two ways to interpret this limit on precision. We could attribute this limit to the measuring apparatus, the action of which precludes simultaneous and precise measurement of the complementary property. Alternatively, we may take the assumption that a state is completely specified by the probabilistic interpretation of the wavefunction, understanding this to mean that there is an unavoidable lack of precision in the very conceptual structure that we use to describe the behavior of the system. This latter interpretation would make reality beyond the Heisenberg limit to be literally inconceivable, leaving no basis for a theoretical description even in principle. Although Bohm describes this as a limit in our conceptual structure, it necessarily implies a lack of definition in reality itself, which is philosophically problematic.

Rather than criticize the standard interpretation of quantum theory on philosophical grounds, Bohm notes only that

…it requires us to give up the possibility of even conceiving precisely what might determine the behavior of an individual system at the quantum level, without providing adequate proof that such a renunciation is necessary. The usual interpretation is admittedly consistent; but the mere demonstration of such consistency does not exclude the possibility of other equally consistent interpretations, which would involve additional elements or parameters permitting a detailed causal and continuous description of all processes…

These additional parameters may be called hidden variables; i.e., they are not among the variables in standard quantum theory. All previous statistical theories in physics ultimately appealed to hidden variables; e.g., thermodynamic properties result from statistical averages of atomic positions and momenta.

In Bohm’s new interpretation, the hidden variables are the precise values of particle position and momenta, localized to a region smaller than that in which the intensity of the ψ-field is appreciable. As long as we are unable to measure position and momentum with such precision, it cannot be proved that these hidden variables are necessary.

Why should such variables, if they exist, remain hidden? If their macro-effects are adequately described by the statistics of the wave function, we have no practical need for the precise values of the hidden variables. Possibly at subatomic scales we may find effects that depend on these precise values, enabling us to make predictions that would not be possible under standard quantum theory, even disagreeing with standard quantum theory at that scale. Even if there is no divergence from the standard theory at smaller scales, the reality of hidden variables may be sustained as long as we accept that any measurement of dynamic variables (e.g., position, momentum) causes an uncontrollable disturbance of these variables, so that their precise values are irretrievably lost.

Can the two assumptions of the standard interpretation be tested? Mere consistency with experiment is not enough, because the standard interpretation does not make any unique predictions that could not be obtained under contrary assumptions. Worse, the standard interpretation could be made unfalsifiable by repeatedly postulating further operators to account for any future disagreement with experiment:

…it is always possible, and, in fact, usually quite natural, to assume that the theory can be made to agree with experiment by some as yet unknown change in the mathematical formulation alone, not requiring any fundamental changes in the physical interpretation. This means that as long as we accept the usual physical interpretation of the quantum theory, we cannot be led by any conceivable experiment to give up this interpretation, even if it should happen to be wrong. The usual physical interpretation therefore presents us with a considerable danger of falling into a trap, consisting of a self-closing chain of circular hypotheses, which are in principle unverifiable if true.

To avoid such unscientific a priorism, we must study the consequences of contradicting the assumptions of the standard theory. We must try to design experiments whose outcomes depend uniquely on the hidden variables, enabling us to make different predictions. If these predictions are verified, then the hidden variables exist. Even if they are not verified, the standard interpretation is not proved, because there could be another hidden variables theory.

We conclude then that a choice of the present interpretation of the quantum theory involves a real physical limitation on the kinds of theories that we wish to take into consideration. From the arguments given here, however, it would seem that there are no secure experimental or theoretical grounds on which we can base such a choice because this choice follows from hypotheses that cannot conceivably be subjected to an experimental test and because we now have an alternative interpretation.

The new interpretation has conceptual advantages over the standard interpretation, as it can at least attempt to answer questions that the standard interpretation regards as unaddressable in principle. For one thing, the dependence of an individual outcome on the statistical ensemble is rationally explained by the supposition that the wave function represents a force, not just a probability density. In the standard interpretation, such dependence is utterly mysterious.

If the polar wave function components P(x) = R²(x) and S(x) are taken to represent a field coordinate and conjugate canonical momentum respectively, we can derive Hamiltonian equations of motion in the P-S phase space, which match the formulae for ∂P/∂t and ∂S/∂t derived from the wave equation. This mechanical interpretation is also consistent with an intuitive notion of a stationary state, where the energy is a constant of the motion and the newly posited quantum mechanical potential remains time-independent, as one would expect.

Applying this mechanical interpretation to the double slit problem, Bohm asks:

How can the opening of a second slit prevent the electron from reaching certain points that it could reach if this slit were closed? If the electron acted completely like a classical particle, this phenomenon could not be explained at all. Clearly, then the wave aspects of the electron must have something to do with the production of the interference pattern. Yet, the electron cannot be identical with its associated wave, because the latter spreads out over a wide region. On the other hand, when the electron’s position is measured, it always appears at the detector as if it were a localized particle.

The standard interpretation requires us to oscillate between two incompatible conceptual models, those of wave and particle. The alternative interpretation, by contrast, has a single, consistent conceptual model: a real ψ-field acting on a particle. R (and therefore P) is constant prior to the electron reaching the diaphragm, so the quantum potential is zero. After the electron passes through the diaphragm, R varies by position, so you have a non-zero quantum potential U, and the motion of the particle becomes complicated, but the probability of it entering a region is given by ∫P(x)dx, as in the standard interpretation.

In particular, the particle is never found where R (and therefore ψ) is zero. The quantum potential U is infinite wherever R is zero.

If the approach to infinity happens to be through positive values of U, there will be an infinite force repelling the particle away from the origin. If the approach is through negative values of U, the particle will go through this point with infinite speed, and thus spend no time there.

Since closing a slit alters the ψ field, it alters the range of places where the particle may be found. This interpretation is perfectly perspicuous, rational and causal, but in some conditions it relies on non-locality (infinite speed), which is inconsistent with relativity. Bohm will attempt to harmonize his hidden variables interpretation with the principle of locality in a companion paper.

8.3.2 Implications for the Theory of Measurement and the EPR Paradox

In his second paper,[66] Bohm proceeds to give an alternative account of the measurement problem, one which relies on a single physical conceptualization instead of arbitrary distinctions among apparatus, system and measurement. He also invokes his quantum potential as a means of defending (in Appendix B) de Broglie’s pilot wave interpretation, of which he learned only after completing his first paper. Most pertinently for us, he discusses the implications of the new interpretation for the EPR paradox and the relativistic principle of locality.

The observed particle and measuring apparatus may be treated as a single system described by a four-dimensional wave function, whose variables are the particle’s three position coordinates x and an apparatus coordinate y. Under the new interpretation, the combined wavefunction Ψ represents a real field acting on the physical variables x and y. Since the apparatus and the observed particle are independent at first, we can express the initial wavefunction as a product of two wavefunctions in the variables x and y respectively.

Ψ₀(x, y) = ψ₀(x)g₀(y)

where ψ₀ may be expressed as a linear sum of eigenfunctions: Σ_qψ_q(x)

More generally, however, when the particle interacts with the apparatus, solving Schrödinger’s equation for the time-dependent combined wavefunction Ψ(x, y, t) yields a correlation between the eigenvalues q of ψ(x) and the apparatus coordinate y. R, S, and U fluctuate violently during this interaction. Eventually, the wave packets for different values of q cease to overlap in y space; i.e., distinct values of the apparatus variable correspond to distinct eigenstates of the measured particle. This non-overlap requires sufficient duration and strength of interaction so that the separation of wave packets δy is much larger than the width Δy of the initial packet g₀(y), which must be broad enough to be limited by the uncertainty principle, for the apparatus to behave classically.

Note that this analysis does not require us to make any arbitrary cut between system and apparatus, nor is there anything mysterious about the act of measurement (aside from the complexity of movement during interaction, making prediction of outcome impossible without precise knowledge of initial conditions), nor is there anything privileged about the role of an observer. We treat the observed particle and the apparatus as a single combined system, and the ability to obtain definite measurements of eigenvalues, the so-called collapse of the wavefunction, is simply a result of sustained physical action by the Ψ-field to a degree that breaks the correlation between eigenvalues of ψ(x) and the variable y. This break is explained mechanically, but using the standard quantum formalism, indeed borrowing arguments from Bohm’s 1951 textbook.

The causal interpretation does introduce a new idea, however, since we interpret the probability density |Ψ(x, y)|² as representing the strength of a real field. Accordingly, the wave packet of the apparatus variable y must, after measurement, enter one of the eigenstate packets and remain there, not in one of the spaces where Ψ = 0. The packet entered determines the result of the measurement, and the other packets can be ignored henceforth, as they are no longer correlated to the variables x and y. Thus we have a truly physical explanation for the collapse of the wavefunction, though we may question the parsimony of what happens to all the unused packets.

The impossibility of having simultaneous eigenstates of non-commuting observables is explained physically by noting that the measurement of each observable disturbs the system in a way that is incompatible with carrying out the process necessary for the measurement of the other.

The hidden variables are the precise values of position and momentum. In practice, we only know these within some range of uncertainty signified by Δ. Since the postulated hidden variables depend both on the state of the apparatus and on the observed system, von Neumann’s argument that quantum theory is inconsistent with hidden variables does not apply.

The new interpretation is indistinguishable in its predictions from the standard theory, in all possible experiments, if we make these suppositions:

1. ψ satisfies Schrödinger’s equation.
2. Momentum is confined to p = ∇ S(x), where S is the phase factor of ψ in polar form.
3. The statistical ensemble of possible states has a probability density of |ψ(x)|². This is the result of our ignorance of precise initial conditions.

If ψ should cease to be described by Schrödinger’s equation at small distances (i.e., if there are inhomogeneities), then the magnitude of ψ would cease to equal a probability density. In such cases we could violate uncertainty principle at small distance scales (less than the fundamental length). This is conceivably testable, and would allow one to contradict the conventional interpretation.

The statistical results of quantum theory come from the interaction of a particle with a classical apparatus, yielding an uncontrollable fluctuation in quantum potential, which is highly sensitive to precise values of position and momentum. Even if it were possible to know the precise initial values, we would still have statistical results due to interaction with the classical system.

Coming to the EPR paradox, suppose you have two particles with coupled position and momentum. If you measure the position of one, you know the position of other, but not the momentum of the other, and so forth. The paradoxical question posed by EPR is: how can this correlation be transmitted across a distance? Bohr had answered enigmatically that we should not even look for a model of such transmission, but should just accept that the correlation somehow occurs. Bohm, with his new interpretation, posits that we can model this as a trajectory in six-dimensional space (x, y, z for each of the two particles). Measuring one or another coordinate observable forces the ψ-field into one or another eigenstate. Thus force is transmitted an infinite speed! This non-locality is not problematic for a classical system, but it seems to contradict relativity.

Bohm remarks that if we are operating under the three assumptions cited above, then we will have the same predictions as conventional quantum theory, which do not contradict relativity, as there is no faster-than-light signaling. To constitute a signal, you would have to know what a particle would have done if you had measured the other variable, but that is unknowable. Suppose, going beyond the three assumptions, that you could know precise position and momentum simultaneously. How can this be reconciled with relativity? Bohm suggests that the ψ-field alters the spacetime metric, or that Lorentz invariance does not apply to small scales. This last claim seems inadequate, for the EPR paradox can be applied to any distance in principle. Even Bohm’s first retort seems evasive, as though our nescience of alternative outcomes were sufficient grounds for contradicting the relativistic structure of spacetime. If Bohm is to be taken seriously that something truly physical and causal is going on even where we are ignorant, then he cannot, in consistency, abandon this notion when trying to circumvent the non-local implications of his interpretation.

Later theorists have upheld the consistency of Bohmian mechanics with relativity by postulating that there is a preferred foliation of space-time. In fact, proofs have been offered that such a supposition is necessary. We need not concern ourselves with the validity of these arguments. Bohmian mechanics (as refined by Bohm & Hiley 1993) is indistinguishable from conventionally interpreted quantum mechanics in its predictions above the Planck scale (i.e., for any observable phenomena), and is still a viable interpretation in the face of all experimental results,[67] though it has the problem of reconciling non-locality with relativity.

The relevant conclusion is that a hidden variables interpretation of quantum theory in general, and the EPR paradox in particular, is indeed possible, but Bohm’s interpretation is non-local if it is taken physically. The defense of saying that there is no non-local signaling is an appeal to a limitation on knowledge. If Bohm purports to give an account of what is really going on, regardless of our knowledge or ignorance, and ψ is a physical field, then we can hardly avoid the conclusion that this is non-local action.

8.4 Entangled-Polarization Photons as a Test of the EPR Paradox

In 1957, Bohm and his student Yakir Aharanov (1932-) proposed another test of the EPR paradox, namely a pair of photons produced by positron-electron annihilation.[68] Due to conservation of angular momentum, such a pair will invariably be polarized orthogonally with respect to each other. If one photon is left circularly polarized, the other will be right circularly polarized. If one is linearly polarized along a certain axis, the other will be polarized along an axis perpendicular to that, while still in the same plane perpendicular to the direction of propagation. The mathematics of the possible wavefunctions is fully analogous to the entangled spin problem. The advantage is that this experiment was already physically realizable to a degree sufficient to provide a test that the EPR paradox really occurs.

To date it had only been shown that angular momentum was conserved for the statistical ensemble, not for each individual pair. Direct measurement of the polarization states for each individual pair was still not possible, but in 1949 Wu and Shaknov at Columbia University had measured the relative rate in which such photons are scattered in different planes.[69] Bohm and Aharonov showed that the result of this experiment was consistent with the supposition of antisymmetric correlation of polarization of each individual pair, and inconsistent with the supposition that the pair were already in definite polarization states (whether circular, linear, or some intermediate elliptical state) prior to measurement along a definite axis.

Supposing that a photon propagates along the z-axis, its wave function may be expressed as:

ψ = rC_k^xψ₀ + sC_k^yψ₀

where C_k represents a creation operator, exciting the above the ground state ψ₀ so that it is polarized along the x or y axis. The squared magnitudes of the coefficients r and s are proportionate to the probability of excitation in the x and y directions respectively. For circular polarization, r and s are equal in magnitude. For linear polarization along some angle α with the x axis: r = cos α; s = sin α.

If, as quantum theory predicts, each photon is always polarized orthogonally to its counterpart, regardless of the choice of axes for measurement, then for some choice of x and y, the wavefunction of the combined system will be the antisymmetric combination:

Ψ = 2^-1/2(C₁^xC₂^y - C₁^yC₂^x)ψ₀

This should hold for every individual pair, not just the statistical ensemble. It will hold no matter what our choice of orientation for the x and y axes of measurement, but, once chosen, there will not in general be similar correlations for polarization along other directions.

Suppose, on the other hand, that this definite phase relation breaks down for photons that are no longer interacting, and instead each photon goes into some definite (though perhaps random) polarization state, related to the other’s state only in a way that gives the correct distribution for the statistical ensemble. We could have each member of a pair circularly polarized in opposite directions (left-handed versus right-handed), or we could have pairs of mutually orthogonal linearly polarized photons, but not each such pair is necessarily produced by the same process. Rather a uniform distribution among possible directions accounts for the results of the ensemble.

This contrary hypothesis would give a range of possible values for the relative probability of scattering in different planes of 1 to 1.5. The hypothesis that quantum theory is correct even in individual cases gives a predicted ratio of 2.0, in agreement with the observed value of 2.04 ± 0.08.

Experiment therefore eliminates the possibility that quantum theory breaks down when particles are far apart with non-overlapping wavefunctions. On the contrary, the predicted correlation persists even for individual pairs that are no longer interacting. Further, as this correlation is dependent on the choice of axes, and does not exist among other non-chosen axes, the EPR paradox remains in its fullness. Available explanations include that of Bohr, which is merely a declaration that the observed system and measurement apparatus are an unanalyzable whole. Bohm and Aharonov agree that the object and apparatus are a single combined system, but hold that this could be analyzable into components, either via Bohm’s causal interpretation, or by the supposition that quantum mechanics is an approximation of a deeper, subquantum-mechanical physics.

9. Bell’s Theorem

In 1964, John S. Bell presented as a thought experiment the singlet spin state discussed by Bohm (1951), but considering the possibility of measuring the spin along different directions varying by some angle, as in Bohm and Aharanov’s (1957) example of entangled-polarization photons. Bell would use this thought experiment to test Einstein’s contention that the principles of determinism and locality could both be upheld and yield statistical predictions identical with those of quantum theory. Stunningly, Bell would prove mathematically, using only the axioms of probability theory, that any deterministic system of hidden variables that obeys the principle of locality (i.e., no instantaneous action-at-a-distance) is inconsistent with the distribution of outcomes predicted by quantum theory. The subsequent empirical verification of this distribution therefore constitutes a strong refutation of local determinism, which no appeal to ignorance can overcome.

Bell understood EPR as arguing that quantum theory, to be complete, should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. EPR mentioned the possibility of a complete theory only as a closing remark, but Bell was well aware of Einstein’s long-standing advocacy that there truly were missing variables, representing unaccounted elements of reality, that would restore local determinism. Moreover, Bell astutely perceived that the core issue is not reality, as EPR claimed, but causality and locality, where causality is understood in the sense of classical mechanics, a deterministic causality. Although Einstein had tried to make his argument rest on more fundamental principles, such as reality and separability, in fact he never successfully removed the implicit assumptions that causality is local and deterministic. Bell chose to test Einstein’s claims rather than Bohr’s, if only because the latter was unintelligible, while Einstein at least expressed himself clearly.

Aware of Bohm’s causal interpretation of quantum mechanics, Bell did not deny that a hidden variables theory was possible, but would prove that all such theories must have a grossly non-local structure. This is a stunning result, because it implies (if we accept basic probability theory) that there can be no local hidden variables theory (i.e., a theory that preserves both determinism and locality) even in principle, and no appeal to the ignorance or limitations of human knowledge can circumvent it. This is a testable theorem, as it yields statistical distributions utterly contrary to the assumption that any hidden variables, even if known only to God or Nature, are actually determining definite outcomes locally. For those who accept locality as an axiom, this would mean that reality is fundamentally non-deterministic to some extent. If we insist on determinism, whether on the level of physics or metaphysics, then the determining agent(s) must be able to act non-locally, notwithstanding the temporal paradoxes this might create under relativity. Neither prospect is especially pleasing to those who would like a clear, rationalistic, intelligible notion of causality, free of paradox.

If we have a distant pair of spin 1/2 particles (e.g., electrons) in a singlet state, so that spin measurements are anti-correlated, and we use a magnet to measure the spin σ₁ of one particle in an arbitrary direction given by unit vector a, then a measurement of the second particle’s spin σ₂ along the same direction a must have the opposite value.

Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.

Since we can predict in advance the result of measuring any chosen component of σ₂, by previously measuring the same component of σ₁, it follows that the result of any such measurement must actually be predetermined.

The first statement posits Einstein’s principle of locality as a hypothesis to be tested. The second statement supposes that the system is deterministic. The logical relation between the two statements is unclear. Most commentators have supposed that determinism is a second hypothesis, while a few have considered it to be consequent to the supposition of locality. On the latter interpretation, violations of Bell’s theorem would refute all local theories, not just deterministic models.

The first clause of the second statement is true regardless of the principle of locality, for it is a consequence of quantum theory. We could certainly conduct an entangled spin experiment where the first measurement is in the absolute past of the second, so that we would have perfect predictability of the second measurement, if made along the same axis as the first. Does the conclusion follow only from this predictability, or is the supposition of locality or some other supposition also necessary?

As we have noted (Sec. 5.3), perfect predictability need not imply physical determinism, unless the effects are in fact generated by antecedent physical conditions. The anti-correlation of singlet spins may admit other explanations than physical determinism. Simply because we observe one particle’s spin first, it does not follow that that spin state causes the anti-parallel state of the other particle. Indeed, the hypothesis of locality makes that seem unlikely. Does the perfect predictability at least imply that there must be some common deterministic causation underlying both results? Even that need not follow, for the predictability is only in terms of anticorrelation, not an antecedent determination of which particle has which value (i.e., +1 or -1, in units of ℏ/2), so there could be a less than fully deterministic process that gives anticorrelated results. Nonetheless, the anticorrelation itself indicates a common causality, though not necessarily a deterministic one, unless of course the anticorrelation is the result of one measurement being antecedent to the other. Yet this last supposition is expressly denied by the hypothesis of locality.

In short, the supposition of determinism is a second hypothesis, not something logically consequent to the principle of locality and the anticorrelated values of the EPR experiment. A common causality might be inferred from anticorrelation (with or without locality), but this need not be physical determinism, where effects are fully determined by antecedent causes. Of course, the principle of locality is only intelligible in terms of causation, though in the form of a denial that there can be causation between events that are spatially separated (in the relativistic sense).

Bell, following EPR, uses the supposition of determinism to imply the incompleteness of quantum theory, since the two-particle wavefunction does not determine the result of either individual spin measurement. Bell clearly regards the supposition of determinism to entail determination of which particle has which value, not mere anticorrelation. Whether he understood determinism to be a consequence of locality or not, he clearly takes determinism in this stronger sense, so it is in fact a second hypothesis.

The more complete specification of state may be represented by a collection of a parameters designated as λ. Bell’s proof works regardless of whether λ is one or many variables or functions. We may further note that it does not matter even if λ itself is a random variable. What is essential is that the outcome of an individual spin measurement is fully determined by the axis of measurement and λ. So even if λ itself is nondeterministic in origin, we are assuming a determinism on the level of quantum phenomena. Having used the anticorrelation between σ₁ ⋅ a and σ₂ ⋅ a as a basis for supposing determinism, Bell now makes the stronger supposition that there would be determinism even if the two particles had spin measurements on different axes. The result A of measuring σ₁ ⋅ a is then determined by a and λ, and the result B of measuring σ₂ ⋅ b in the same instance is determined by b and λ. The key phrase in the same instance makes this a stronger supposition, as it posits determinism even beyond the anticorrelated circumstances from which it was ostensibly inferred.

It does not matter whether only some of the variables λ determine A and others determine B, or if the same variables λ jointly determine (in combination with the axis of measurement) both A and B. Thus we theoretically could allow for non-locality in λ itself. What we disallow is any causality between the orientation a and the result B, or between B and a. That is, we disallow non-local causality between the act of orienting the magnet that measures particle 1 and the measured spin value of distant particle 2. We hypothesize that the observed anticorrelations might instead be accounted by some additional variables or functions λ, which could have any form.

Let ρ(λ) be the probability distribution of values for λ. The expectation value of the product of the spins of each particle (for particular choices of a and b) is then given by the sum of the product of A and B for each value λ multiplied by the probability of that value of λ. Again, λ could be one or many parameters, and it could take discrete or continuous values. Bell supposes it to be continuous, without loss of generality, so the sum has the integral form:

E(a, b) = ∫dλρ(λ)A(a, λ)B(b, λ)

This is the probability-weighted sum of the product A(a)B(b) over all possible values of λ. Implicit assumptions include the axioms of probability theory and that the range of possibilities for λ is no greater than the infinity of the continuum (real numbers).

To agree with the predictions of quantum theory, this integral must equal the expectation value of the singlet state, which is: - a ⋅ b. Since a and b are unit vectors, their dot product is just cos α, where α is the angle from a to b. If they are parallel (α = 0), the expectation value is -1, since the two particles invariably have anticorrelated values of 1 and -1. If they are perpendicular (α = π/2, the two values are completely uncorrelated, and the expectation value is 0. These results can be duplicated by the supposition of hidden variables, but the same can not be said in general for values of α between 0 and π/2, at least not without introducing a dependence of A on b or B on a for a given value of λ.

When a = b, E(a, a) = -1, which implies that A(a) = - B(a) for a given value of λ. Thus the integral form of E is also:

E(a, b) = -∫dλρ(λ)A(a, λ)A(b, λ)

A contradiction with the expectation value of quantum theory arises, however, if we consider that the same experiment can be conducted with a different orientation c instead of b. Bell proves that:

|E(a, b) - E(a, c)| ≤ 1 + E(b, c)

where each of the expectation values is integrated over values of λ as above. The right side of the equation is obtained by exploiting the fact that A and B can only take the values of 1 or -1, so they are equal to their reciprocal. This inequality holds no matter what λ is, as long as we add the singlet constraint that E(a, a) = -1.

Bell then shows that E(b, c) cannot equal the quantum mechanical value, i.e., the cosine of the angle from b to c, nor even made to approximate it arbitrarily closely. Thus we have a contradiction with the predictions of quantum theory. No set of hidden variables whatsoever can generate all the expectation values predicted by quantum theory (and later confirmed by experiments) while upholding the principle of locality.

Continue to Part IV

Notes

[36] Bohm, D. Quantum Mechanics (New York: Prentice-Hall, 1951), p.144.

[37] Ibid., pp.151-57. Energy, not discussed here, is indispensable to mechanics since, in the form of the Hamiltonian operator, it determines the time evolution of the wave function, in both the position and momentum representations.

[38] Ibid., pp.157-59.

[39] Ibid., p.159. This denial of one-to-one correspondence differs from that of Einstein. Bohm is merely pointing out that the wavefunction, though it uniquely and completely characterizes the state and thus the statistical distribution of the ensemble, it only probabilistically describes behavior in individual instances of measured objects, which may vary from each other. Einstein, on the other hand, was claiming existence of a special scenario where two non-trivially distinct forms of the wavefunction for system II (depending on the choice of measurement on system I) ostensibly corresponded to the same state of system II (assuming that the choice of measurement on system I could not affect the state of system II).

[40] Ibid., p.161.

[41] Ibid., p.164.

[42] Ibid., p.166.

[43] Loc cit..

[44] Ibid., p.167.

[45] Loc. cit.

[46] Ibid., p.625.

[47] Ibid., p.626.

[48] Ibid., p.633.

[49] Ibid., pp.585-86.

[50] Ibid., p.624.

[51] For this problem, we can speak of spin and angular momentum interchangeably, but angular momentum is a broader concept, applicable also to orbital angular momentum, not just the intrinsic spin of fundamental particles and their composites.

[52] ψ₁ is just one of three possible states for total spin of 1, namely the m = 0 state of the triplet. The m = 1 and m = -1 states are ψ_a = u₊(1)u₊(2) and ψ_b = u_-(1)u_-(2), i.e., when both atoms have parallel z-spin, both positive or both negative.

[53] Bohm, op. cit., p.617.

[54] Ibid., pp.621-22.

[55] Ibid., p.618.

[56] Ibid., p.622.

[57] Ibid., pp.135-136. Bohm proposes a hypothetical experiment where we consider an electron as having definite position and momentum, but these cannot be measured simultaneously with greater accuracy than allowed by the uncertainty principle. Particle diffraction phenomena may be explained, as in William Duane’s hypothesis, by quantized momentum transfers to the grating, without recourse to wave interference. The proposed experiment uses a proton microscope, in which a beam of protons with well-defined momentum p is directed at a lens. This proton beam intercepts some electrons having well-defined momentum. The position of an electron (i.e., its distance from the lens) may be inferred from how protons are deflected into different parts of the lens. Under a wave model of the proton, we expect the proton wave to diffract at the edge of the lens, and interpret outcomes accordingly. Under Duane’s hypothesis, the same outcomes may be explained by uncontrollable transfers of momenta between the protons and the edge of the lens. We know from other particle diffraction experiments that the range of possible momentum transfers is independent the direction or point of origin of the particles, so diffraction does not depend critically on the position or velocity of the particle. Thus the range of uncontrollable deflections should be determined mainly by the size and shape of the lens.

To give the same results as the wave theory, the resolving power of the lens ought to be about λ / sin φ₀, where φ₀ is the aperture of the lens, and λ is the de Broglie wavelength of the proton. (This assumes that the pencil of rays covers the whole lens.) Thus the uncertainty Δx in the electron’s position will be on the order of λ ≅ h/p.

However, if momentum is conserved in each transfer, the electron can gain no more momentum than (m/M)p, where m is the electron mass and M is the proton mass. Since the initial uncertainty in electron momentum is negligible, we have a final momentum uncertainty of Δp ≅ (m/M)p. This makes ΔxΔp ≅ (m/M)h, much less than what is allowed by the uncertainty principle.

Thus the supposition that the electron always has a definite position and momentum, even if the momentum abruptly changes in an unpredictable manner, must lead to the possibility of simultaneously measuring its position and momentum to an accuracy that contradicts Heisenberg’s principle. Such a contradiction is incompatible with the wave-like behavior of matter under the de Broglie hypothesis, which is experimentally confirmed.

This difficulty might be avoided by supposing that the proton behaved like a wave originating at a point from where it was scattered by the electron, up until it arrived at the detecting plate. Its small range of momenta that could be received from the electron makes for a narrow pencil of rays, narrower than the lens. Thus resolving power is determined not by aperture size, but instead by the angular width of the pencil of rays. We regain consistency with the uncertainty principle, but can keep our model of definite yet uncertain particle properties (position and momentum) only by assuming that the range of permissible momentum transfers between the particle and lens is somehow determined by the range of permissible transfers between the electron and proton. This would require the proton to retain a memory of its previous interaction; not just the determinate outcome of that interaction (i.e., its definite momentum received), but also the range of possible outcomes, or the fact that it had interacted with an electron. This is a highly contrived assumption, to say the least, and it is unclear if it could be developed in a way to be consistent with all experimental facts.

[58] Ibid., pp.619-620.

[59] Ibid., p.146.

[60] Ibid., p.623.

[61] See note 57 above.

[62] Einstein asserts: If the spin of subsystem I is measured along the x axis, it is found to be either 1 or -1 in that direction. It then follows that the spin of the subsystem II equals 0 along the y-direction. This is incorrect under quantum theory (and confirmed by experiment). Sauer, Tilman. An Einstein manuscript on the EPR paradox for spin observables. Studies in the History and Philosophy of Modern Physics (2007) 38: 879-887. p.882.

[63] Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables. I Phys. Rev. (1952), 85:166-179.

[64] Since the quantum potential is fully defined by the wave function ψ, now interpreted as a field, and is not dependent on an arbitrary constant, the addition of this potential to the classical potential is not a mere gauge transformation. The quantum potential should indeed have real physical significance, if it is assumed that ψ represents a physical field.

[65] The presence of R in both the numerator and the denominator means that the force applied by the quantum potential is independent of field strength. The homogeneity of Schrödinger’s equation implies that the ψ-field is not radiated or absorbed, at least not at observable scales. These features break analogy with other known physical fields.

[66] Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables. II Phys. Rev. (1952), 85:180-193.

[67] In 2005, Yves Couder and Emmanuel Fort found that droplets may self-propel through resonant interaction with their own wave field, giving a real world classical fluid mechanical example of pilot wave behavior, analogous to de Broglie’s quantum theory. In 2017, however, Giuseppe Pucci and John W.M. Bush showed that this pilot wave phenomenon could not replicate the quantum mechanical double slit interaction. [Pucci, G.; Harris, D.M.; Faria, L.M.; Bush, J.W.M. Walking droplets interacting with single and double slits. J. Fluid Mech. (2018) 835:1136-1156.] This was widely reported in the media as refuting pilot wave interpretations of quantum mechanics, even though such interpretations do not depend on or imply the existence of a particular classical mechanical analog. Bohmian mechanics in particular is unaffected by this result.

[68] Bohm, D. and Aharonov, Y. Discussion of Experimental Proof for the Paradox of Einstein, Rosen and Podolsky. Phys. Rev. (1957) 108: 1070-1076.

[69] Wu, C.S. and Shaknov, I. The Angular Correlation of Scattered Annihilation Radiation. Phys. Rev. (1950) 77:136.

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