Maybe God Doesn't play dice with the universe?

[Michael]

Bohmian Mechanics
First published Fri Oct 26, 2001; substantive revision Tue Jul 3, 2012

Bohmian mechanics, which is also called the de Broglie-Bohm theory, the pilot-wave model, and the causal interpretation of quantum mechanics, is a version of quantum theory discovered by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952. It is the simplest example of what is often called a hidden variables interpretation of quantum mechanics. In Bohmian mechanics a system of particles is described in part by its wave function, evolving, as usual, according to Schrödinger’s equation. However, the wave function provides only a partial description of the system. This description is completed by the specification of the actual positions of the particles. The latter evolve according to the “guiding equation,” which expresses the velocities of the particles in terms of the wave function. Thus, in Bohmian mechanics the configuration of a system of particles evolves via a deterministic motion choreographed by the wave function. In particular, when a particle is sent into a two-slit apparatus, the slit through which it passes and its location upon arrival on the photographic plate are completely determined by its initial position and wave function.

Bohmian mechanics inherits and makes explicit the nonlocality implicit in the notion, common to just about all formulations and interpretations of quantum theory, of a wave function on the configuration space of a many-particle system. It accounts for all of the phenomena governed by nonrelativistic quantum mechanics, from spectral lines and scattering theory to superconductivity, the quantum Hall effect and quantum computing. In particular, the usual measurement postulates of quantum theory, including collapse of the wave function and probabilities given by the absolute square of probability amplitudes, emerge from an analysis of the two equations of motion: Schrödinger’s equation and the guiding equation. No invocation of a special, and somewhat obscure, status for observation is required.

1. The Completeness of the Quantum Mechanical Description
2. The Impossibility of Hidden Variables ... or the Inevitability of Nonlocality?
3. History
4. The Defining Equations of Bohmian Mechanics
5. The Quantum Potential
6. The Two-Slit Experiment
7. The Measurement Problem
8. The Collapse of the Wave Function
9. Quantum Randomness
10. Quantum Observables
11. Spin
12. Contextuality
13. Nonlocality
14. Lorentz Invariance
15. Objections and Responses
Bibliography
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1. The Completeness of the Quantum Mechanical Description

Conceptual difficulties have plagued quantum mechanics since its inception, despite its extraordinary predictive successes. The basic problem, plainly put, is this: It is not at all clear what quantum mechanics is about. What, in fact, does quantum mechanics describe?

It might seem, since it is widely agreed that any quantum mechanical system is completely described by its wave function, that quantum mechanics is fundamentally about the behavior of wave functions. Quite naturally, no physicist wanted this to be true more than did Erwin Schrödinger, the father of the wave function. Nonetheless, Schrödinger ultimately found this impossible to believe. His difficulty had little to do with the novelty of the wave function (Schrödinger 1935): “That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message.” Rather, it was that the “blurring” that the spread out character of the wave function suggests “affects macroscopically tangible and visible things, for which the term ‘blurring’ seems simply wrong.”

For example, in the same paper Schrödinger noted that it may happen in radioactive decay that

the emerging particle is described … as a spherical wave … that impinges continuously on a surrounding luminescent screen over its full expanse. The screen however does not show a more or less constant uniform surface glow, but rather lights up at one instant at one spot ….

And he observed that one can easily arrange, for example by including a cat in the system, “quite ridiculous cases” with

the ψ-function of the entire system having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.

It is thus because of the “measurement problem,” of macroscopic superpositions, that Schrödinger found it difficult to regard the wave function as “representing reality.” But then what does? With evident disapproval, Schrödinger observes that

the reigning doctrine rescues itself or us by having recourse to epistemology. We are told that no distinction is to be made between the state of a natural object and what I know about it, or perhaps better, what I can know about it if I go to some trouble. Actually — so they say — there is intrinsically only awareness, observation, measurement.

Many physicists pay lip service to the Copenhagen interpretation — that quantum mechanics is fundamentally about observation or results of measurement. But it is becoming increasingly difficult to find any who, when pressed, will defend this interpretation. It seems clear that quantum mechanics is fundamentally about atoms and electrons, quarks and strings, not those particular macroscopic regularities associated with what we call measurements of the properties of these things. But if these entities are not somehow identified with the wave function itself — and if talk of them is not merely shorthand for elaborate statements about measurements — then where are they to be found in the quantum description?

There is, perhaps, a very simple reason why it is so difficult to discern in the quantum description the objects we believe quantum mechanics ought to describe. Perhaps the quantum mechanical description is not the whole story, a possibility most prominently associated with Albert Einstein. (For a general discussion of Einstein’s scientific philosophy, and in particular of his approach to the conflicting positions of realism and positivism, see the entry on Einstein’s philosophy of science.)

In 1935 Einstein, Boris Podolsky and Nathan Rosen defended this possibility in their famous EPR paper (Einstein et al. 1935). They concluded with this observation:

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.

The argument that the EPR paper advances to support this conclusion invokes quantum correlations and an assumption of locality. (See the entries on the Einstein-Podolsky-Rosen argument in quantum theory and on quantum entanglement and information.)

Later, on the basis of more or less the same considerations as those of Schrödinger quoted above, Einstein again concluded that the wave function does not provide a complete description of individual systems, an idea he called “this most nearly obvious interpretation” (Einstein 1949, p. 672). In relation to a theory incorporating a more complete description, Einstein remarked that “the statistical quantum theory would … take an approximately analogous position to the statistical mechanics within the framework of classical mechanics.” We note here, and show below, that Bohmian mechanics exactly fits this description.
2. The Impossibility of Hidden Variables … or the Inevitability of Nonlocality?

John von Neumann, one of the greatest mathematicians of the twentieth century, claimed that he had proven that Einstein’s dream of a deterministic completion or reinterpretation of quantum theory was mathematically impossible. He concluded that (von Neumann 1932, p. 325 of the English translation)

It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics — the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.

Physicists and philosophers of science almost universally accepted von Neumann’s claim. For example, Max Born, who formulated the statistical interpretation of the wave function, assured us that (Born 1949, p. 109)

No concealed parameters can be introduced with the help of which the indeterministic description could be transformed into a deterministic one. Hence if a future theory should be deterministic, it cannot be a modification of the present one but must be essentially different.

Bohmian mechanics is a counterexample to the claims of von Neumann. Thus von Neumann’s argument must be wrong. In fact, according to John Bell (Mermin 1993, p. 805), von Neumann’s assumptions (about the relationships among the values of quantum observables that must be satisfied in a hidden-variables theory) are so unreasonable that the “the proof of von Neumann is not merely false but foolish!” Nonetheless, some physicists continue to rely on von Neumann’s proof.

Recently, however, physicists more commonly cite the Kochen-Specker Theorem and, more frequently, Bell’s inequality in support of the contention that a deterministic completion of quantum theory is impossible. We still find, a quarter of a century after the rediscovery of Bohmian mechanics in 1952, statements such as these (Wigner 1976):

The proof he [von Neumann] published …, though it was made much more convincing later on by Kochen and Specker, still uses assumptions which, in my opinion, can quite reasonably be questioned. … In my opinion, the most convincing argument against the theory of hidden variables was presented by J. S. Bell (1964).

Now there are many more statements of a similar character that we could cite. This quotation is significant because Wigner was one of the leading physicists of his generation. Unlike most of his contemporaries, moreover, he was also profoundly concerned about the conceptual foundations of quantum mechanics and wrote on the subject with great clarity and insight.

There was, however, one physicist who wrote on this subject with even greater clarity and insight than Wigner himself: the very J. S. Bell whom Wigner praises for demonstrating the impossibility of a deterministic completion of quantum theory such as Bohmian mechanics. Here’s how Bell himself reacted to Bohm’s discovery (Bell 1987, p. 160):

But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the ‘observer,’ could be eliminated. …

But why then had Born not told me of this ‘pilot wave’? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing ‘‘impossibility’’ proofs, after 1952, and as recently as 1978? … Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

Wigner to the contrary notwithstanding, Bell did not establish the impossibility of a deterministic reformulation of quantum theory, nor did he ever claim to have done so. On the contrary, until his untimely death in 1990, Bell was the prime proponent, and for much of this period almost the sole proponent, of the very theory, Bohmian mechanics, that he supposedly demolished.

Bohmian mechanics is of course as much a counterexample to the Kochen-Specker argument for the impossibility of hidden variables as it is to the one of von Neumann. It is obviously a counterexample to any such argument. However reasonable the assumptions of such an argument, some of them must fail for Bohmian mechanics.

Wigner was quite right to suggest that the assumptions of Kochen and Specker are more convincing than those of von Neumann. They appear, in fact, to be quite reasonable indeed. However, they are not. The impression that they are arises from a pervasive error, an uncritical realism about operators, that we discuss below in the sections on quantum observables, spin, and contextuality.

John Bell replaced the “arbitrary axioms” (Bell 1987, page 11) of Kochen-Specker and others by an assumption of locality, of no action-at-a-distance. It would be hard to argue against the reasonableness of such an assumption, even if one were so bold as to doubt its inevitability. Bell showed that any hidden-variables formulation of quantum mechanics must be nonlocal, as, indeed, Bohmian mechanics is. But he showed much much more. (For more detail on Bell’s locality assumption, see Bell’s theorem in Scholarpedia.)

In a celebrated paper he published in 1964, Bell showed that quantum theory itself is irreducibly nonlocal. (More precisely, Bell’s analysis applies to any single-world version of quantum theory, i.e., any version for which measurements have outcomes that, while they may be random, are nonetheless unambiguous and definite, in contrast to the situation with Everett’s many-worlds version of quantum theory.) This fact about quantum mechanics, based as it is on a short and mathematically simple analysis, could have been recognized soon after the discovery of quantum theory in the 1920’s. That this did not happen is no doubt due in part to the obscurity of orthodox quantum theory and to the ambiguity of its commitments. It was, in fact, his examination of Bohmian mechanics that led Bell to his nonlocality analysis. In the course of investigating Bohmian mechanics, he observed that (Bell 1987, p. 11):

in this theory an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece.

Bohm of course was well aware of these features of his scheme, and has given them much attention. However, it must be stressed that, to the present writer's knowledge, there is no proof that any hidden variable account of quantum mechanics must have this extraordinary character. It would therefore be interesting, perhaps, to pursue some further “impossibility proofs,” replacing the arbitrary axioms objected to above by some condition of locality, or of separability of distant systems.

In a footnote, Bell added that “Since the completion of this paper such a proof has been found.” He published it in his 1964 paper, “On the Einstein-Podolsky-Rosen Paradox.” In this paper he derives Bell’s inequality, the basis of his conclusion of quantum nonlocality. (See the entry on Bell’s Theorem. For a discussion of how nonlocality emerges in Bohmian mechanics, see Section 13.)

It is worth stressing that Bell’s analysis indeed shows that any (single-world) account of quantum phenomena must be nonlocal, not just any hidden variables account. Bell showed that the predictions of standard quantum theory itself imply nonlocality. Thus if these predictions govern nature, then nature is nonlocal. [That nature is so governed, even in the crucial EPR-correlation experiments, has by now been established by a great many experiments, the most conclusive of which is perhaps that of Aspect (Aspect et al., 1982).]

Bell, too, stressed this point (by determinism Bell here means hidden variables):

It is important to note that to the limited degree to which determinism plays a role in the EPR argument, it is not assumed but inferred. What is held sacred is the principle of ‘local causality’ — or ‘no action at a distance’…

It is remarkably difficult to get this point across, that determinism is not a presupposition of the analysis. (Bell 1987, p. 143)

Despite my insistence that the determinism was inferred rather than assumed, you might still suspect somehow that it is a preoccupation with determinism that creates the problem. Note well then that the following argument makes no mention whatever of determinism. … Finally you might suspect that the very notion of particle, and particle orbit … has somehow led us astray. … So the following argument will not mention particles, nor indeed fields, nor any other particular picture of what goes on at the microscopic level. Nor will it involve any use of the words ‘quantum mechanical system’, which can have an unfortunate effect on the discussion. The difficulty is not created by any such picture or any such terminology. It is created by the predictions about the correlations in the visible outputs of certain conceivable experimental set-ups. (Bell 1987, p. 150)

The “problem” and “difficulty” to which Bell refers above is the conflict between the predictions of quantum theory and what can be inferred, call it C, from an assumption of locality in Bohm’s version of the EPR argument, a conflict established by Bell’s inequality. C happens to concern the existence of a certain kind of hidden variables, what might be called local hidden variables, but this fact is of little substantive importance. What is important is not so much the identity of C as the fact that C is incompatible with the predictions of quantum theory. The identity of C is, however, of great historical significance: it is responsible for the misconception that Bell proved that hidden variables are impossible, a belief that physicists until recently almost universally shared, as well as for the view, even now almost universally held, that what Bell’s result does is to rule out local hidden variables, a view that is misleading.

Here again is Bell, expressing the logic of his two-part demonstration of quantum nonlocality, the first part of which is Bohm’s version of the EPR argument:

Let me summarize once again the logic that leads to the impasse. The EPRB correlations are such that the result of the experiment on one side immediately foretells that on the other, whenever the analyzers happen to be parallel. If we do not accept the intervention on one side as a causal influence on the other, we seem obliged to admit that the results on both sides are determined in advance anyway, independently of the intervention on the other side, by signals from the source and by the local magnet setting. But this has implications for non-parallel settings which conflict with those of quantum mechanics. So we cannot dismiss intervention on one side as a causal influence on the other. (Bell 1987, p. 149)

As with just about everything else in the foundations of quantum mechanics, there remains considerable controversy about what exactly Bell’s analyis demonstrates. Nonetheless, the opinion of Bell himself about what he showed is perfectly clear.
3. History

The pilot-wave approach to quantum theory was initiated by Einstein, even before the discovery of quantum mechanics itself. Einstein hoped that interference phenomena involving particle-like photons could be explained if the motion of the photons was somehow guided by the electromagnetic field — which would thus play the role of what he called a Führungsfeld or guiding field (see Wigner 1976, p. 262 and Bacciagaluppi and Valentini 2009, Ch. 9). While the notion of the electromagnetic field as guiding field turned out to be rather problematical, Max Born explored the possibility that the wave function could play this role, of guiding field or pilot wave, for a system of electrons in his early paper founding quantum scattering theory (Born 1926). Heisenberg was profoundly unsympathetic.

Not long after Schrödinger’s discovery of wave mechanics in 1926, i.e., of Schrödinger’s equation, Louis de Broglie in effect discovered Bohmian mechanics: In 1927, de Broglie found an equation of particle motion equivalent to the guiding equation for a scalar wave function (de Broglie 1928, p. 119), and he explained at the 1927 Solvay Congress how this motion could account for quantum interference phenomena. However, despite what is suggested by Bacciagaluppi and Valentini (2009), de Broglie responded very poorly to an objection of Wolfgang Pauli (Pauli 1928) concerning inelastic scattering, no doubt making a rather bad impression on the illustrious audience at the congress.

Born and de Broglie very quickly abandoned the pilot-wave approach and became enthusiastic supporters of the rapidly developing consensus in favor of the Copenhagen interpretation. David Bohm (Bohm 1952) rediscovered de Broglie’s pilot-wave theory in 1952. He was the first person to genuinely understand its significance and implications. John Bell became its principal proponent during the sixties, seventies and eighties.
4. The Defining Equations of Bohmian Mechanics

In Bohmian mechanics the wave function, obeying Schrödinger’s equation, does not provide a complete description or representation of a quantum system. Rather, it governs the motion of the fundamental variables, the positions of the particles: In the Bohmian mechanical version of nonrelativistic quantum theory, quantum mechanics is fundamentally about the behavior of particles; the particles are described by their positions, and Bohmian mechanics prescribes how these change with time. In this sense, for Bohmian mechanics the particles are primary, or primitive, while the wave function is secondary, or derivative.

Warning: It is the positions of the particles in Bohmian mechanics that are its “hidden variables,” an unfortunate bit of terminology. As Bell (1987, page 201) writes, referring to Bohmian mechanics and similar theories,

Absurdly, such theories are known as ‘hidden variable’ theories. Absurdly, for there it is not in the wavefunction that one finds an image of the visible world, and the results of experiments, but in the complementary ‘hidden’(!) variables. Of course the extra variables are not confined to the visible ‘macroscopic’ scale. For no sharp definition of such a scale could be made. The ‘microscopic’ aspect of the complementary variables is indeed hidden from us. But to admit things not visible to the gross creatures that we are is, in my opinion, to show a decent humility, and not just a lamentable addiction to metaphysics. In any case, the most hidden of all variables, in the pilot wave picture, is the wavefunction, which manifests itself to us only by its influence on the complementary variables.

Bohmian mechanics is the minimal completion of Schrödinger’s equation, for a nonrelativistic system of particles, to a theory describing a genuine motion of particles. For Bohmian mechanics the state of a system of N particles is described by its wave function ψ = ψ(q1,…,qN) = ψ(q), a complex (or spinor-valued) function on the space of possible configurations q of the system, together with its actual configuration Q defined by the actual positions Q1,…,QN of its particles. (The word ‘spinor’ refers to a suitable array of complex numbers in place of a single one. Spinor-valued wave functions are used in quantum mechanics to describe electrons and other quantum particles that ‘have spin’.) The theory is then defined by two evolution equations: Schrödinger’s equation

iℏ(∂ψ/∂t) = Hψ

for ψ(t), where H is the nonrelativistic (Schrödinger) Hamiltonian, containing the masses of the particles and a potential energy term, and (writing Im[z] for the imaginary part b of a complex number z = a +ib) a first-order evolution equation,

The Guiding Equation:
dQk/dt = (ℏ/mk) Im ψ*∂kψ/ ψ*ψ] (Q1,…,QN)

for Q(t), the simplest first-order evolution equation for the positions of the particles that is compatible with the Galilean (and time-reversal) covariance of the Schrödinger evolution (Dürr et al. 1992, pp. 852–854). Here ℏ is Planck’s constant divided by 2π, mk is the mass of the k-th particle, and ∂k = (∂/∂xk,∂/∂yk,∂/∂zk) is the gradient with respect to the generic coordinates qk = (xk,yk,zk) of the k-th particle. If ψ is spinor-valued, the two products involving ψ in the equation should be understood as scalar products (involving sums of products of spinor components). When external magnetic fields are present, the gradient should be understood as the covariant derivative, involving the vector potential. (Since the denominator on the right hand side of the guiding equation vanishes at the nodes of ψ, global existence and uniqueness for the Bohmian dynamics is a nontrivial matter. It is proven in Berndl, Dürr, et al. 1995 and in Teufel and Tumulka 2005.)

For an N-particle system these two equations (together with the detailed specification of the Hamiltonian, including all interactions contributing to the potential energy) completely define Bohmian mechanics. This deterministic theory of particles in motion accounts for all the phenomena of nonrelativistic quantum mechanics, from interference effects to spectral lines (Bohm 1952, pp. 175–178) to spin (Bell 1964, p. 10). It does so in an entirely ordinary manner, as we explain in the following sections.

For a scalar wave function, describing particles without spin, the form of the guiding equation above is a little more complicated than necessary, since the complex conjugate of the wave function, which appears in the numerator and the denominator, cancels out. If one looks for an evolution equation for the configuration compatible with the space-time symmetries of Schrödinger’s equation, one almost immediately arrives at the guiding equation in this simpler form as the simplest possibility.

However, the form above has two advantages: First, it makes sense for particles with spin — and, in fact, Bohmian mechanics without further ado accounts for all the apparently paradoxical quantum phenomena associated with spin. Secondly, and this is crucial to the fact that Bohmian mechanics is empirically equivalent to orthodox quantum theory, the right hand side of the guiding equation is J/ρ, the ratio of the quantum probability current to the quantum probability density. This shows that it should require no imagination whatsoever to guess the guiding equation from Schrödinger’s equation, provided one is looking for one, since the classical formula for current is density times velocity. Moreover, it follows from the quantum continuity equation ∂ρ/∂t + div J = 0, an immediate consequence of Schrödinger’s equation, that if at some time (say the initial time) the configuration Q of our system is random, with distribution given by |ψ|2 = ψ*ψ, this will always be true (provided the system does not interact with its environment).

This demonstrates that it is wrong to claim that the predictions of quantum theory are incompatible with the existence of hidden variables, with an underlying deterministic model in which quantum randomness arises from averaging over ignorance. Bohmian mechanics provides us with just such a model: For any quantum experiment we merely take as the relevant Bohmian system the combined system, including the system upon which the experiment is performed as well as all the measuring instruments and other devices used to perform the experiment (together with all other systems with which these have significant interaction over the course of the experiment). We then obtain the “hidden variables” model by regarding the initial configuration of this big system as random in the usual quantum mechanical way, with distribution given by |ψ|2. The guiding equation for the big system then transforms the initial configuration into the final configuration at the conclusion of the experiment. It then follows that this final configuration of the big system, including in particular the orientation of instrument pointers, will also be distributed in the quantum mechanical way. Thus our deterministic Bohmian model yields the usual quantum predictions for the results of the experiment.

As the preceding paragraph suggests, and as we discuss in more detail later, Bohmian mechanics does not need any “measurement postulates” or axioms governing the behavior of other “observables”. Any such axioms would be at best redundant and could be inconsistent.

Besides the guiding equation, there are other velocity formulas with nice properties, including Galilean symmetry, and yielding theories that are empirically equivalent to orthodox quantum theory — and to Bohmian mechanics (Deotto and Ghirardi, 1998). The Bohmian choice is arguably the simplest. Moreover, Wiseman (2007) has shown that it is the Bohmian velocity formula, given by the guiding equation, that, according to orthodox quantum theory, would be found in a “weak measurement” of the velocity of a particle. And, somewhat paradoxically, it can be shown (Dürr et al., 2009) that according to Bohmian mechanics such a measurement is indeed a genuine measurement of the particle’s velocity — despite the existence of empirically equivalent velocity formulas! Similarly, weak measurements could be used to measure trajectories. In fact, quite recently Kocsis et al. (2011) have used weak measurements to reconstruct the trajectories for single photons “as they undergo two-slit interference,” finding “those predicted in the Bohm-de Broglie interpretation of quantum mechanics.”
5. The Quantum Potential

Bohmian mechanics as presented here is a first-order theory, in which it is the velocity, the rate of change of position, that is fundamental. It is this quantity, given by the guiding equation, that the theory specifies directly and simply. The second-order (Newtonian) concepts of acceleration and force, work and energy do not play any fundamental role. Bohm, however, did not regard his theory in this way. He regarded it, fundamentally, as a second-order theory, describing particles moving under the influence of forces, among which, however, is a force stemming from a “quantum potential.”
In his 1952 hidden-variables paper (Bohm 1952), Bohm arrived at his theory by writing the wave function in the polar form ψ = Rexp(iS/ℏ), where S and R are real, with R nonnegative, and rewriting Schrödinger’s equation in terms of these new variables to obtain a pair of coupled evolution equations: the continuity equation for ρ = R2 and a modified Hamilton-Jacobi equation for S. This differs from the usual classical Hamilton-Jacobi equation only by the appearance of an extra term, the quantum potential

U  =  −∑k(ℏ2/2mk) (∂k2 R / R ),

alongside the classical potential energy term.

Bohm then used the modified Hamilton-Jacobi equation to define particle trajectories just as one does for the classical Hamilton-Jacobi equation, that is, by identifying ∂kS with mkvk, i.e., by setting

dQk/dt = ∂kS / mk.

This is equivalent to the guiding equation for particles without spin. [In this form the (pre-Schrödinger equation) de Broglie relation p = ℏk, as well as by the eikonal equation of classical optics, already suggest the guiding equation.] The resulting motion is precisely what would be obtained classically if the particles were acted upon by the force generated by the quantum potential, in addition to the usual forces.

The quantum potential formulation of the de Broglie-Bohm theory is still fairly widely used. For example, the monographs by Bohm and Hiley and by Holland present the theory in this way. And regardless of whether or not we regard the quantum potential as fundamental, it can in fact be quite useful. In order to see most clearly that Newtonian mechanics should be expected to emerge from Bohmian mechanics in the classical limit, it is convenient to transform the theory into Bohm’s Hamilton-Jacobi form. Then the (size of the) quantum potential provides a measure of the deviation of Bohmian mechanics from its classical approximation. Moreover, the quantum potential is also useful for developing approximation schemes for solutions to Schrödinger’s equation (Nerukh and Frederick 2000).

However, Bohm’s rewriting of Schrödinger’s equation in terms of variables that seem interpretable in classical terms is not without a cost. The most obvious is an increase in complexity: Schrödinger’s equation is rather simple, and it is linear, whereas the modified Hamilton-Jacobi equation is somewhat complicated, and highly nonlinear. Moreover the latter, since it involves R, requires the continuity equation for its closure. The quantum potential itself is neither simple nor natural. Even to Bohm it seemed “rather strange and arbitrary” (Bohm 1980, p. 80). And it is not very satisfying to think of the quantum revolution as amounting to the insight that nature is classical after all, except that there is in nature what appears to be a rather ad hoc additional force term, the one arising from the quantum potential. The artificiality that the quantum potential suggests is the price one pays for casting a highly nonclassical theory into a classical mold.

Moreover, the connection between classical mechanics and Bohmian mechanics that the quantum potential suggests is rather misleading. Bohmian mechanics is not simply classical mechanics with an additional force term. In Bohmian mechanics the velocities are not independent of positions, as they are classically, but are constrained by the guiding equation. (In classical Hamilton-Jacobi theory we also have this equation for the velocity, but there the Hamilton-Jacobi function S can be entirely eliminated and the description in terms of S simplified and reduced to a finite-dimensional description, with basic variables the positions and the (unconstrained) momenta of all the particles, given by Hamilton’s or Newton’s equations.)

Arguably, the most serious flaw in the quantum potential formulation of Bohmian mechanics is that it gives a completely false impression of the lengths to which we must go in order to convert orthodox quantum theory into something more rational. The quantum potential suggests, as has often been stated, that transforming Schrödinger’s equation into a theory that can account in “realistic” terms for quantum phenomena, many of which are dramatically nonlocal, requires adding to the theory a complicated quantum potential of a grossly nonlocal character. It should be clear that this view is inappropriate. After all, the quantum potential need not even be mentioned in the formulation of Bohmian mechanics, and it in any case merely reflects the wave function, which Bohmian mechanics shares with orthodox quantum theory.
6. The Two-Slit Experiment

According to Richard Feynman, the two-slit experiment for electrons is (Feynman et al. 1963, p. 37–2) “a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality it contains the only mystery.” This experiment (Feynman 1967, p. 130) “has been designed to contain all of the mystery of quantum mechanics, to put you up against the paradoxes and mysteries and peculiarities of nature one hundred per cent.” As to the question (Feynman 1967, p. 145), “How does it really work? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it.”

But Bohmian mechanics is just such a deeper explanation. It resolves in a rather straightforward manner the dilemma of the appearance of both particle and wave properties in one and the same phenomenon: Bohmian mechanics is a theory of motion describing a particle (or particles) guided by a wave. Here we have a family of Bohmian trajectories for the two-slit experiment.

figure 1

Figure 1: An ensemble of trajectories for the two-slit experiment, uniform in the slits.
(Adapted by Gernot Bauer from Philippidis et al. 1979.)

While each trajectory passes through only one slit, the wave passes through both; the interference profile that therefore develops in the wave generates a similar pattern in the trajectories guided by the wave.

Compare Feynman’s presentation with Bell’s (Bell 1987, p. 191):

Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave? De Broglie showed in detail how the motion of a particle, passing through just one of two holes in screen, could be influenced by waves propagating through both holes. And so influenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.

Perhaps the most puzzling aspect of the two-slit experiment is the following: If, by any means whatsoever, it is possible to determine the slit through which the particle passes, the interference pattern will be destroyed. This dramatic effect of observation is, in fact, a simple consequence of Bohmian mechanics. To see this, one must consider the meaning of determining the slit through which the particle passes. This must involve interaction with another system that the Bohmian mechanical analysis must include.

The destruction of interference is related, naturally enough, to the Bohmian mechanical analysis of quantum measurement (Bohm 1952). It occurs via the mechanism that in Bohmian mechanics leads to the “collapse of the wave function.”
7. The Measurement Problem

The measurement problem is most the commonly cited of the conceptual difficulties that plague quantum mechanics. (It amounts, more or less, to the paradox of Schrödinger’s cat.) Indeed, for many physicists the measurement problem is not merely one conceptual difficulty of quantum mechanics; it is the conceptual difficulty.

The problem is as follows. Suppose that the wave function of any individual system provides a complete description of that system. When we analyze the process of measurement in quantum mechanical terms, we find that the after-measurement wave function for system and apparatus that arises from Schrödinger’s equation for the composite system typically involves a superposition over terms corresponding to what we would like to regard as the various possible results of the measurement — e.g., different pointer orientations. In this description of the after-measurement situation it is difficult to discern the actual result of the measurement — e.g., some specific pointer orientation. But the whole point of quantum theory, and the reason we should believe in it, is that it is supposed to provide a compelling, or at least an efficient, account of our observations, that is, of the outcomes of measurements. In short, the measurement problem is this: Quantum theory implies that measurements typically fail to have outcomes of the sort the theory was created to explain.

In contrast, if we, like Einstein, regard the description provided by the wave function as incomplete, the measurement problem vanishes: There is no measurement problem with a theory or interpretation in which, as in Bohmian mechanics, the description of the after-measurement situation includes, in addition to the wave function, at least the values of the variables that register the result. In Bohmian mechanics pointers always point.

Often, the measurement problem is expressed a little differently. Textbook quantum theory provides two rules for the evolution of the wave function of a quantum system: A deterministic dynamics given by Schrödinger’s equation when the system is not being “measured” or observed, and a random collapse of the wave function to an eigenstate of the “measured observable” when it is. However, the objection continues, textbook quantum theory does not explain how to reconcile these two apparently incompatible rules.

That this formulation of the measurement problem and the preceding one are more or less equivalent should be reasonably clear: If a wave function provides a complete description of the after-measurement situation, the outcome of the measurement must correspond to a wave function that describes the actual result, that is, a “collapsed” wave function. Hence the collapse rule. But it is difficult to take seriously the idea that different laws than those governing all other interactions should govern those interactions between system and apparatus that we happen to call measurements. Hence the apparent incompatibility of the two rules.

The second formulation of the measurement problem, though basically equivalent to the first, raises an important question: Can Bohmian mechanics itself reconcile these two dynamical rules? How does Bohmian mechanics justify the use of the “collapsed” wave function instead of the original one? This question was answered in Bohm’s first papers on Bohmian mechanics (Bohm 1952, Part I, Section 7, and Part II, Section 2). What would nowadays be called effects of decoherence, which interaction with the environment (air molecules, cosmic rays, internal microscopic degrees of freedom, etc.) produces, make difficult the development of significant overlap between the component of the after-measurement wave function corresponding to the actual result of the measurement and the other components of the after-measurement wave function. (This overlap refers to the configuration space of the very large system that includes all systems with which the original system and apparatus come into interaction.) But without such overlap that component all by itself generates to a high degree of accuracy the future evolution of the configuration of the system and apparatus. The replacement is thus justified as a practical matter. (See also Dürr et al. 1992, Section 5.)

Many proponents of orthodox quantum theory believe that decoherence somehow resolves the measurement problem itself. It is not easy to understand this belief. In the first formulation of the measurement problem, nothing prevents us from including in the apparatus all sources of decoherence. But then decoherence can no longer be in any way relevant to the argument. Be that as it may, Bohm (Bohm 1952) gave one of the best descriptions of the mechanisms of decoherence, though he did not use the word itself. He recognized its importance several decades before it became fashionable. (See also the encyclopedia entry on The Role of Decoherence in Quantum Mechanics.)
8. The Collapse of the Wave Function

In the previous section we indicated that collapse of the wave function can be regarded in Bohmian mechanics as a pragmatic affair. However, there is a sense in which the collapse of the wave function in Bohmian mechanics is more than a matter of convenience. If we focus on the appropriate notion of the wave function, not of the composite of system and apparatus — which strictly speaking remains a superposition if the composite is treated as closed during the measurement process — but of the system itself, we find that for Bohmian mechanics this does indeed collapse, precisely as the quantum formalism says. The key element here is the notion of the conditional wave function of a subsystem of a larger system, which we describe briefly in this section and that Dürr et al. 1992, Section 5, discuss in some detail, together with the related notion of the effective wave function.

For the evolution of the wave function, Bohmian mechanics is formulated in terms of Schrödinger’s equation alone. Nonetheless the textbook collapse rule is a consequence of the Bohmian dynamics. To appreciate this one should first note that, since observation implies interaction, a system under observation cannot be a closed system but rather must be a subsystem of a larger closed system, which we may take to be the entire universe, or any smaller more or less closed system that contains the system to be observed, the subsystem. The configuration Q of this larger system naturally splits into X, the configuration of the subsystem, and Y, the configuration of the environment of the subsystem.

Suppose the larger system has wave function Ψ = Ψ(q) = Ψ(x, y). According to Bohmian mechanics, the larger system is then completely described by Ψ, evolving according to Schrödinger’s equation, together with X and Y. The question then arises — and it is a critical question — as to what should be meant by the wave function of the subsystem.

There is a rather obvious answer for this, a natural function of x that suitably incorporates the objective structure at hand, namely the conditional wave function

ψ(x)  =  Ψ(x, Y)

obtained by plugging the actual configuration of the environment into the wave function of the larger system. (This definition is appropriate only for scalar wave functions; for particles with spin the situation would be a little more complicated.) It then follows immediately that the configuration of the subsystem obeys the guiding equation with the conditional wave function on its right-hand side.

Moreover, taking into account the way that the conditional wave function depends upon time t

ψt(x) = Ψt(x, Yt)

via the time dependence of Y as well as that of Ψ, it is not difficult to see (Dürr et al. 1992) the following two things about the evolution of the conditional wave: First, that it obeys Schrödinger’s equation for the subsystem when that system is suitably decoupled from its environment. Part of what is meant by this decoupling is that Ψ has a special form, what might be called an effective product form (similar to but more general than the superposition produced in an “ideal quantum measurement”), in which case the conditional wave function of the subsystem is also called its effective wave function. Second, using the quantum equilibrium hypothesis, that it randomly collapses according to the usual quantum mechanical rules under precisely those conditions on the interaction between the subsystem and its environment that define an ideal quantum measurement.

It is perhaps worth noting that orthodox quantum theory lacks the resources that make it possible to define the conditional wave function, namely, the actual configuration Y of the environment. Indeed, from an orthodox point of view what should be meant by the wave function of a subsystem is entirely obscure.
9. Quantum Randomness

According to the quantum formalism, for a system with wave function ψ the probability density for finding its configuration to be q is |ψ(q)|2. To the extent that the results of measurement are registered configurationally, at least potentially, it follows that the predictions of Bohmian mechanics for the results of measurement must agree with those of orthodox quantum theory (assuming the same Schrödinger equation for both) provided that it is somehow true for Bohmian mechanics that configurations are random, with distribution given by the quantum equilibrium distribution |ψ(q)|2. Now the status and justification of this quantum equilibrium hypothesis is a rather delicate matter, one that has been explored in considerable detail (Dürr et al. 1992). Here are a few relevant points.

It is nowadays a rather familiar fact that dynamical systems quite generally give rise to behavior of a statistical character, with the statistics given by the (or a) stationary probability distribution for the dynamics. So it is with Bohmian mechanics, except that for the Bohmian system stationarity is not quite the right concept. Rather it is the notion of equivariance that is relevant. A probability distribution on configuration space ρψ, depending upon the wave function ψ, is equivariant if

(ρψ)t = ρψt

where the dependence on t on the right arises from Schrödinger’s equation and on the left from the evolution on probability distributions arising from the flow that the guiding equation induces. Thus equivariance expresses the mutual compatibility, relative to ρψ, of the Schrödinger evolution of the wave function and the Bohmian motion of the configuration. It is an immediate consequence of the guiding equation and the quantum continuity equation that ρψ = |ψ(q)|2 is equivariant. (It can be shown in fact that this is more or less the only equivariant possibility that is suitably local (Goldstein and Struyve 2007).)

In trying to understand the status in Bohmian mechanics of the quantum equilibrium distribution, it is perhaps helpful to think of

quantum equilibrium, ρ  =  |ψ|2

as roughly analogous to (classical)

thermodynamic equilibrium, ρ  =  exp(-H/kT) /Z,

the probability distribution of the phase-space point of a system in equilibrium at temperature T. (Z is a normalization constant called the partition function and k is Boltzmann’s constant.) This analogy has several facets: In both cases the probability distributions are naturally associated with their respective dynamical systems. In particular, these distributions are stationary or, what amounts to the same thing within the framework of Bohmian mechanics, equivariant. In both cases it seems natural to try to justify these equilibrium distributions by means of mixing-type, convergence-to-equilibrium arguments (Bohm 1953, Valentini and Westman 2005). It has been argued, however, that in both cases the ultimate justification for these probability distributions must be in terms of statistical patterns that ensembles of actual subsystems within a typical individual universe exhibit (Bell 1987, page 129, Dürr et al. 1992). In both cases the status of, and justification for, equilibrium distributions is still controversial. It is also perhaps worth noting that the typicality-grounded account of quantum randomness in Bohmian mechanics is extremely similar to Everett’s account (Everett III 1957) of quantum randomness for “many worlds,” despite the huge metaphysical differences that exist between these two versions of quantum theory. It can be shown (Dürr et al. 1992) that probabilities for positions given by the quantum equilibrium distribution emerge naturally from an analysis of “equilibrium” for the deterministic dynamical system that Bohmian mechanics defines, much as the Maxwellian velocity distribution emerges from an analysis of classical thermodynamic equilibrium. (For more on the thermodynamic side of the analogy see Goldstein 2001.) Thus with Bohmian mechanics the statistical description in quantum theory indeed takes, as Einstein anticipated, “an approximately analogous position to the statistical mechanics within the framework of classical mechanics.”
10. Quantum Observables

Orthodox quantum theory supplies us with probabilities not merely for positions but for a huge class of quantum observables. It might thus appear that it is a much richer theory than Bohmian mechanics, which seems exclusively concerned with positions. Appearances are, however, misleading. In this regard, as with so much else in the foundations of quantum mechanics, Bell made the crucial observation (Bell 1987, p. 166):

*n physics the only observations we must consider are position observations, if only the positions of instrument pointers. It is a great merit of the de Broglie-Bohm picture to force us to consider this fact. If you make axioms, rather than definitions and theorems, about the “measurement” of anything else, then you commit redundancy and risk inconsistency.

Consider classical mechanics first. The observables are functions on phase space, functions of the positions and momenta of the particles. The axioms governing the behavior of the basic observables — Newton’s equations for the positions or Hamilton’s for positions and momenta — define the theory. What would be the point of making additional axioms, for other observables? After all, the behavior of the basic observables entirely determines the behavior of any observable. For example, for classical mechanics, the principle of the conservation of energy is a theorem, not an axiom.

The situation might seem to differ in quantum mechanics, as usually construed. Here there is no small set of basic observables having the property that all other observables are functions of them. Moreover, no observables at all are taken seriously as describing objective properties, as actually having values whether or not they are or have been measured. Rather, all talk of observables in quantum mechanics is supposed to be understood as talk about the measurement of the observables.

But if this is so, the situation with regard to other observables in quantum mechanics is not really that different from that in classical mechanics. Whatever quantum mechanics means by the measurement of (the values of) observables — that, we are urged to believe, don’t actually have values — must at least refer to some experiment involving interaction between the “measured” system and a “measuring” apparatus leading to a recognizable result, as given potentially by, say, a pointer orientation. But then if some axioms suffice for the behavior of pointer orientations (at least when they are observed), rules about the measurement of other observables must be theorems, following from those axioms, not additional axioms.

It should be clear from the discussion towards the end of Section 4 and at the beginning of Section 9 that, assuming the quantum equilibrium hypothesis, any analysis of the measurement of a quantum observable for orthodox quantum theory — whatever it is taken to mean and however the corresponding experiment is performed — provides ipso facto at least as adequate an account for Bohmian mechanics. The only part of orthodox quantum theory relevant to the analysis is the Schrödinger evolution, and it shares this with Bohmian mechanics. The main difference between them is that orthodox quantum theory encounters the measurement problem before it reaches a satisfactory conclusion while Bohmian mechanics does not. This difference stems of course from what Bohmian mechanics adds to orthodox quantum theory: actual configurations.

The rest of this section will discuss the significance of quantum observables for Bohmian mechanics. (It follows from what has been said in the three preceding paragraphs that what we conclude here about quantum observables for Bohmian mechanics holds for orthodox quantum theory as well.)

Bohmian mechanics yields a natural association between experiments and so-called generalized observables, given by positive-operator-valued measures (Davies 1976), or POVM’s, O(dz), on the value spaces for the results of the experiments (Berndl, Daumer, et al. 1995). This association is such that the probability distribution of the result Z of an experiment, when performed upon a system with wave function ψ, is given by <ψ | O(dz)ψ> (where < | > is the usual inner product between quantum state vectors).

Moreover, this conclusion is follows immediately from the very meaning of an experiment from a Bohmian perspective: a coupling of system to apparatus leading to a result Z that is a function of the final configuration of the total system, e.g., the orientation of a pointer. Analyzed in Bohmian mechanical terms, the experiment defines a map from the initial wave function of the system to the distribution of the result. It follows directly from the structure of Bohmian mechanics, and from the fact that the quantum equilibrium distribution is quadratic in the wave function, that this map is bilinear (or, more precisely, sesquilinear, in that its dependence on one factor of the wave function is antilinear, involving complex conjugation, rather than linear). Such a map is equivalent to a POVM.

The simplest example of a POVM is a standard quantum observable, corresponding to a self-adjoint operator A on the Hilbert space of quantum states (i.e., wave functions). For Bohmian mechanics, more or less every “measurement-like” experiment is associated with this special kind of POVM. The familiar quantum measurement axiom that the distribution of the result of the “measurement of the observable A” is given by the spectral measure for A relative to the wave function (in the very simplest cases just the absolute squares of the so-called probability amplitudes) is thus obtained.

For various reasons, after the discovery of quantum mechanics it quickly became almost universal to speak of an experiment associated with an operator A in the manner just sketched as a measurement of the observable A — as if the operator somehow corresponded to a property of the system that the experiment in some sense measures. It has been argued that this assumption, which has been called naive realism about operators, has been a source of considerable confusion about the meaning and implications of quantum theory (Daumer et al., 1997).
11. Spin

The case of spin illustrates nicely both the way Bohmian mechanics treats non-configurational quantum observables, and some of the difficulties that the naive realism about operators mentioned above causes.

Spin is the canonical quantum observable that has no classical counterpart, reputedly impossible to grasp in a nonquantum way. The difficulty is not quite that spin is quantized in the sense that its allowable values form a discrete set (for a spin-1/2 particle, ±ℏ/2). Energy too may be quantized in this sense. Nor is it precisely that the components of spin in the different directions fail to commute — and so cannot be simultaneously discussed, measured, imagined, or whatever it is that we are advised not to do with noncommuting observables. Rather the problem is that there is no ordinary (nonquantum) quantity which, like the spin observable, is a 3-vector and which also is such that its components in all possible directions belong to the same discrete set. The problem, in other words, is that the usual vector relationships among the various components of the spin vector are incompatible with the quantization conditions on the values of these components.

For a particle of spin-1 the problem is even more severe. The components of spin in different directions aren’t simultaneously measurable. Thus, the impossible vector relationships for the spin components of a quantum particle are not observable. Bell (1966), and, independently, Simon Kochen and Ernst Specker (Kochen and Specker 1967) showed that for a spin-1 particle the squares of the spin components in the various directions satisfy, according to quantum theory, a collection of relationships, each individually observable, that taken together are impossible: the relationships are incompatible with the idea that measurements of these observables merely reveal their preexisting values rather than creating them, as quantum theory urges us to believe. Many physicists and philosophers of physics continue to regard the Kochen-Specker Theorem as precluding the possibility of hidden variables.

We thus might naturally wonder how Bohmian mechanics copes with spin. But we have already answered this question. Bohmian mechanics makes sense for particles with spin, i.e., for particles whose wave functions are spinor-valued. When such particles are suitably directed toward Stern-Gerlach magnets, they emerge moving in more or less a discrete set of directions — 2 possible directions for a spin-1/2 particle, having 2 spin components, 3 for spin-1 with 3 spin components, and so on. This occurs because the Stern-Gerlach magnets are so designed and oriented that a wave packet (a localized wave function with reasonably well defined velocity) directed towards the magnet will, by virtue of the Schrödinger evolution, separate into distinct packets — corresponding to the spin components of the wave function and moving in the discrete set of directions. The particle itself, depending upon its initial position, ends up in one of the packets moving in one of the directions.

The probability distribution for the result of such a Stern-Gerlach experiment can be conveniently expressed in terms of the quantum mechanical spin operators — for a spin-1/2 particle given by certain 2 by 2 matrices called the Pauli spin matrices — in the manner alluded to above. From a Bohmian perspective there is no hint of paradox in any of this — unless we assume that the spin operators correspond to genuine properties of the particles.
12. Contextuality

The Kochen-Specker Theorem, the earlier theorem of Gleason (Gleason 1957 and Bell 1966), and other no-hidden-variables results, including Bell’s inequality (Bell 1964), show that any hidden-variables formulation of quantum mechanics must be contextual. It must violate the noncontextuality assumption “that measurement of an observable must yield the same value independently of what other measurements may be made simultaneously” (Bell 1987, p. 9). To many physicists and philosophers of science contextuality seems too great a price to pay for the rather modest benefits — largely psychological, so they would say — that hidden variables provide.

Even many Bohmians suggest that contextuality departs significantly from classical principles. For example, Bohm and Hiley (1993) write that “The context dependence of results of measurements is a further indication of how our interpretation does not imply a simple return to the basic principles of classical physics.”

However, to understand contextuality in Bohmian mechanics almost nothing needs to be explained. Consider an operator A that commutes with operators B and C (which however don’t commute with each other). What is often called the “result for A” in an experiment for “measuring A together with B” usually disagrees with the “result for A” in an experiment for “measuring A together with C.” This is because these experiments differ and different experiments usually have different results. The misleading reference to measurement, which suggests that a pre-existing value of A is being revealed, makes contextuality seem more than it is.

Seen properly, contextuality amounts to little more than the rather unremarkable observation that results of experiments should depend upon how they are performed, even when the experiments are associated with the same operator in the manner alluded to above. David Albert (Albert 1992, p. 153) has given a particularly simple and striking example of this dependence for Stern-Gerlach experiments “measuring” the z-component of spin. Reversing the polarity in a magnet for “measuring” the z-component of spin while keeping the same geometry yields another magnet for “measuring” the z-component of spin. The use of one or the other of these two magnets will often lead to opposite conclusions about the “value of the z-component of spin” prior to the “measurement” (for the same initial value of the position of the particle).

As Bell insists (Bell 1987, p. 166):

A final moral concerns terminology. Why did such serious people take so seriously axioms which now seem so arbitrary? I suspect that they were misled by the pernicious misuse of the word ‘measurement’ in contemporary theory. This word very strongly suggests the ascertaining of some preexisting property of some thing, any instrument involved playing a purely passive role. Quantum experiments are just not like that, as we learned especially from Bohr. The results have to be regarded as the joint product of ‘system’ and ‘apparatus,’ the complete experimental set-up. But the misuse of the word ‘measurement’ makes it easy to forget this and then to expect that the ‘results of measurements’ should obey some simple logic in which the apparatus is not mentioned. The resulting difficulties soon show that any such logic is not ordinary logic. It is my impression that the whole vast subject of ‘Quantum Logic’ has arisen in this way from the misuse of a word. I am convinced that the word ‘measurement’ has now been so abused that the field would be significantly advanced by banning its use altogether, in favour for example of the word ‘experiment.’

13. Nonlocality

Bohmian mechanics is manifestly nonlocal. The velocity, as expressed in the guiding equation, of any particle of a many-particle system will typically depend upon the positions of the other, possibly distant, particles whenever the wave function of the system is entangled, i.e., not a product of single-particle wave functions. This is true, for example, for the EPR-Bohm wave function, describing a pair of spin-1/2 particles in the singlet state, that Bell and many others analyzed. Thus Bohmian mechanics makes explicit the most dramatic feature of quantum theory: quantum nonlocality, as discussed in Section 2.

It should be emphasized that the nonlocality of Bohmian mechanics derives solely from the nonlocality, discussed in Section 2, built into the structure of standard quantum theory. This nonlocality originates from a wave function on configuration space, an abstraction which, roughly speaking, combines — or binds — distant particles into a single irreducible reality. As Bell (1987, p. 115) has stressed,

That the guiding wave, in the general case, propagates not in ordinary three-space but in a multidimensional-configuration space is the origin of the notorious ‘nonlocality’ of quantum mechanics. It is a merit of the de Broglie-Bohm version to bring this out so explicitly that it cannot be ignored.

Thus the nonlocal velocity relation in the guiding equation is but one aspect of the nonlocality of Bohmian mechanics. There is also the nonlocality, or nonseparability, implicit in the wave function itself, which is present even without the structure — actual configurations — that Bohmian mechanics adds to orthodox quantum theory. As Bell has shown, using the connection between the wave function and the predictions of quantum theory about experimental results, this nonlocality cannot easily be eliminated (see Section 2).

The nonlocality of Bohmian mechanics can be appreciated perhaps most efficiently, in all its aspects, by focusing on the conditional wave function. Suppose, for example, that in an EPR-Bohm experiment particle 1 passes through its Stern-Gerlach magnet before particle 2 arrives at its magnet. Then the orientation of the Stern-Gerlach magnet for particle 1 will significantly affect the conditional wave function of particle 2: If the Stern-Gerlach magnet for particle 1 is oriented so as to “measure the z-component of spin,” then after particle 1 has passed through its magnet the conditional wave function of particle 2 will be an eigenvector (or eigenstate) of the z-component of spin (in fact, belonging to the eigenvalue that is the negative of the one “measured” for particle 1), and the same thing is true for any other component of spin. You can dictate the kind of spin eigenstate produced for particle 2 by appropriately choosing the orientation of an arbitrarily distant magnet. As to the future behavior of particle 2, in particular how its magnet affects it, this of course depends very much on the character of its conditional wave function and hence the choice of orientation of the distant magnet strongly influences it.

This nonlocal effect upon the conditional wave function of particle 2 follows from combining the standard analysis of the evolution of the wave function in the EPR-Bohm experiment with the definition of the conditional wave function. (For simplicity, we ignore permutation symmetry.) Before reaching any magnets the EPR-Bohm wave function is a sum of two terms, corresponding to nonvanishing values for two of the four possible joint spin components for the two particles. Each term is a product of an eigenstate for a component of spin in a given direction for particle 1 with the opposite eigenstate (i.e., belonging to the eigenvalue that is the negative of the eigenvalue for particle 1) for the component of spin in the same direction for particle 2. Moreover, by virtue of its symmetry under rotations, the EPR-Bohm wave function has the property that any component of spin, i.e., any direction, can be used in this decomposition. (This property is very interesting.)

Decomposing the EPR-Bohm wave function using the component of spin in the direction associated with the magnet for particle 1, the evolution of the wave function as particle 1 passes its magnet is easily grasped: The evolution of the sum is determined (using the linearity of Schrödinger’s equation) by that of its individual terms, and the evolution of each term by that of each of its factors. The evolution of the particle-1 factor leads to a displacement along the magnetic axis in the direction determined by the (sign of the) spin component (i.e., the eigenvalue), as described in the fourth paragraph of Section 11. Once this displacement has occurred (and is large enough) the conditional wave function for particle 2 will correspond to the term in the sum selected by the actual position of particle 1. In particular, it will be an eigenstate of the component of spin “measured by” the magnet for particle 1.

The nonlocality of Bohmian mechanics has a remarkable feature: it is screened by quantum equilibrium. It is a consequence of the quantum equilibrium hypothesis that the nonlocal effects in Bohmian mechanics don’t yield observable consequences that can be controlled — we can’t use them to send instantaneous messages. This follows from the fact that, given the quantum equilibrium hypothesis, the observable consequences of Bohmian mechanics are the same as those of orthodox quantum theory, for which instantaneous communication based on quantum nonlocality is impossible (see Eberhard 1978). Valentini (1991) emphasizes the importance of quantum equilibrium for obscuring the nonlocality of Bohmian mechanics. (Valentini (2010a) has also suggested the possibility of searching for and exploiting quantum non-equilibrium. However, in contrast with thermodynamic non-equilibrium, we have at present no idea what quantum non-equilibrium, should it exist, would look like, despite claims and arguments to the contrary.)
14. Lorentz Invariance

Like nonrelativistic quantum theory, of which it is a version, Bohmian mechanics and special relativity, a central principle of physics, are not compatible: Bohmian mechanics is not Lorentz invariant. Nor can it easily be modified to accommodate Lorentz invariance. Configurations, defined by the simultaneous positions of all particles, play too crucial a role in its formulation, with the guiding equation defining an evolution on configuration space. (Lorentz invariant extensions of Bohmian mechanics for a single particle, described by the Dirac equation (Bohm and Hiley 1993, Dürr et al. 1999) or the Klein-Gordon equation (Berndl et al. 1996, Nikolic 2005), can easily be achieved, though for a Klein-Gordon particle there are some interesting subtleties, corresponding to what might seem to be a particle traveling backwards in time.)

This difficulty with Lorentz invariance and the nonlocality in Bohmian mechanics are closely related. Since quantum theory itself, by virtue merely of the character of its predictions concerning EPR-Bohm correlations, is irreducibly nonlocal (see Section 2), one might expect considerable difficulty with the Lorentz invariance of orthodox quantum theory as well with Bohmian mechanics. For example, the collapse rule of textbook quantum theory blatantly violates Lorentz invariance. As a matter of fact, the intrinsic nonlocality of quantum theory presents formidable difficulties for the development of any (many-particle) Lorentz invariant formulation that avoids the vagueness of orthodox quantum theory (see Maudlin 1994).

Bell made a somewhat surprising evaluation of the importance of the problem of Lorentz invariance. In an interview with the philosopher Renée Weber, not long before he died, he referred to the paradoxes of quantum mechanics and observed that “Those paradoxes are simply disposed of by the 1952 theory of Bohm, leaving as the question, the question of Lorentz invariance. So one of my missions in life is to get people to see that if they want to talk about the problems of quantum mechanics — the real problems of quantum mechanics — they must be talking about Lorentz invariance.”

The most common view on this issue is that a detailed description of microscopic quantum processes, such as would be provided by a putative extension of Bohmian mechanics to the relativistic domain, must violate Lorentz invariance. In this view Lorentz invariance is an emergent symmetry obeyed by our observations — for Bohmian mechanics a statistical consequence of quantum equilibrium that governs the results of quantum experiments. This is the opinion of Bohm and Hiley (1993), of Holland (1993), and of Valentini (1997).

However — unlike nonlocality — violating Lorentz invariance is not inevitable. It should be possible, it seems, to construct a fully Lorentz invariant theory that provides a detailed description of microscopic quantum processes. One way to do this is by using an additional Lorentz invariant dynamical structure, for example a suitable time-like 4-vector field, that permits the definition of a foliation of space-time into space-like hypersurfaces providing a Lorentz invariant notion of “evolving configuration” and along which nonlocal effects are transmitted. See Dürr et al. 1999 for a toy model. Another possibility is that a fully Lorentz invariant account of quantum nonlocality can be achieved without the invocation of additional structure, exploiting only what is already at hand, for example, the wave function of the universe (see Dürr et al. 1999) or light-cone structure. For a step in this direction, see Goldstein and Tumulka 2003’s model in which they reconcile relativity and nonlocality through the interplay of opposite arrows of time.

Be that as it may, Lorentz invariant nonlocality remains somewhat enigmatic. The issues are extremely subtle. For example, Bell (1987, page 155) rightly would find “disturbing … the impossibility of ‘messages’ faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we [emphasis added] can actually perform. The exact elucidation of concepts like ‘message’ and ‘we’, would be a formidable challenge.” While quantum equilibrium and the absolute uncertainty that it entails (Dürr et al. 1992) may be of some help here, the situation remains puzzling.
15. Objections and Responses

Bohmian mechanics has never been widely accepted in the mainstream of the physics community. Since it is not part of the standard physics curriculum, many physicists—probably the majority—are simply unfamiliar with the theory and how it works. Sometimes the theory is rejected without explicit discussion of reasons for rejection. One also finds objections that are based on simple misunderstandings; among these are claims that some no-go theorem, such as von Neumann’s theorem, the Kochen-Specker theorem, or Bell’s theorem, shows that the theory cannot work. Such objections will not be dealt with here, as the reply to them will be obvious to those who understand the theory. In what follows only objections that are not based on elementary misunderstandings will be discussed.

A common objection is that Bohmian mechanics is too complicated or inelegant. To evaluate this objection one must compare the axioms of Bohmian mechanics with those of standard quantum mechanics. To Schrödinger’s equation, Bohmian mechanics adds the guiding equation; standard quantum mechanics instead requires postulates about experimental outcomes that can only be formulated in terms of a distinction between a quantum system and the experimental apparatus. And, as noted by Hilary Putnam (2005),

In Putnam ([1965]), I rejected Bohm's interpretation for several reasons which no longer seem good to me. Even today, if you look at the Wikipedia encyclopaedia on the Web, you will find it said that Bohm's theory is mathematically inelegant. Happily, I did not give that reason in Putnam ([1965]), but in any case it is not true. The formula for the velocity field is extremely simple: you have the probability current in the theory anyway, and you take the velocity vector to be proportional to the current. There is nothing particularly inelegant about that; if anything, it is remarkably elegant!

One frequent objection is that Bohmian mechanics, since it makes precisely the same predictions as standard quantum mechanics (insofar as the predictions of standard quantum mechanics are unambiguous), is not a distinct theory but merely a reformulation of standard quantum theory. In this vein, Heisenberg wrote,

Bohm's interpretation cannot be refuted by experiment, and this is true of all the counter-proposals in the first group. From the fundamentally “positivistic” (it would perhaps be better to say “purely physical”) standpoint, we are thus concerned not with counter-proposals to the Copenhagen interpretation, but with its exact repetition in a different language (Heisenberg 1955, p. 18). 

More recently, Sir Anthony Leggett has echoed this charge. Referring to the measurement problem, he says (Leggett 2005) that Bohmian mechanics provides “little more than verbal window dressing of the basic paradox.” And in connection with the double-slit experiment, he writes,

No experimental consequences are drawn from [the assumption of definite particle trajectories] other than the standard predictions of the QM formalism, so whether one regards it as a substantive resolution of the apparent paradox or as little more than a reformulation of it is no doubt a matter of personal taste (the present author inclines towards the latter point of view) (Leggett 2002, p. R419). 

Now Bohmian mechanics and standard quantum mechanics provide clearly different descriptions of what is happening on the microscopic quantum level. So it is only with a purely instrumental attitude towards scientific theories that Bohmian mechanics and standard quantum mechanics can possibly be regarded as different formulations of exactly the same theory. But even if they were, why would this be an objection to Bohmian mechanics? Even if they were, we should still ask which of the two formulations is superior. Those impressed by the “not-a-distinct-theory” objection presumably give considerable weight to the fact that standard quantum mechanics came first. Supporters of Bohmian mechanics give more weight to its greater simplicity and clarity.

The position of Leggett, however, is very difficult to understand. There should be no measurement problem for a physicist with a purely instrumentalist understanding of quantum mechanics. But for more than thirty years Leggett has forcefully argued that quantum mechanics indeed suffers from the measurement problem. For Leggett the problem is so serious that it has led him to suggest that quantum mechanics might fail on the macroscopic level. Thus Leggett is no instrumentalist, and it is hard to understand why he so cavalierly dismisses a theory like Bohmian mechanics that obviously doesn’t suffer from the measurement problem, with which he has been so long concerned.

Sir Roger Penrose (2005, page 811) also seems to have doubts as to whether Bohmian mechanics indeed resolves the measurement problem. He writes that

it seems to me that some measure of scale is indeed needed, for defining when classical-like behaviour begins to take over from small-scale quantum activity. In common with the other quantum ontologies in which no measurable deviations from standard quantum mechanics is expected, the point of view (e) [Bohmian mechanics] does not possess such a scale measure, so I do not see that it can adequately address the paradox of Schrödinger's cat. 

But contrary to what he writes, his real concern seems to be with the emergence of classical behavior, and not with the measurement problem per se. With regard to this, we note that the Bohmian evolution of particles, which is always governed by the wave function and is always fundamentally quantum, turns out to be approximately classical when the relevant de Broglie wave length, determined in part by the wave function, is much smaller than the scale on which the potential energy term in Schrödinger’s equation varies (see Allori et al., 2002). Under normal circumstances this condition will be satisfied for the center of mass motion of a macroscopic object.

It is perhaps worth mentioning that despite the empirical equivalence between Bohmian mechanics and orthodox quantum theory, there are a variety of experiments and experimental issues that don’t fit comfortably within the standard quantum formalism but are easily handled by Bohmian mechanics. Among these are dwell and tunneling times (Leavens 1996), escape times and escape positions (Daumer et al. 1997a), scattering theory (Dürr et al., 2000), and quantum chaos (Cushing 1994, Dürr et al., 1992a).

Another claim that has become popular in recent years is that Bohmian mechanics is an Everettian, or “many worlds,” interpretation in disguise (see entry on the many worlds interpretation of quantum mechanics for an overview of such interpretations). The idea is that Bohmians, like Everettians, must take the wave-function as physically real. Moreover, since Bohmian mechanics involves no wave-function collapse (for the wave function of the universe), all of the branches of the wave function, and not just the one that happens to be occupied by the actual particle configuration, persist. These branches are those that Everettians regard as representing parallel worlds. As David Deutsch expresses the charge,

the ‘unoccupied grooves’ must be physically real. Moreover they obey the same laws of physics as the ‘occupied groove’ that is supposed to be ‘the’ universe. But that is just another way of saying that they are universes too. … In short, pilot-wave theories are parallel-universes theories in a state of chronic denial (Deutsch 1996, p. 225). 

See Brown and Wallace (2005) for an extended version of this argument. Not surprisingly, Bohmians do not agree that the branches of the wave function should be construed as representing worlds. For one Bohmian response, see Maudlin (2010). Other Bohmian responses have been given by Lewis (2007) and Valentini (2010b).

The claim of Deutsch, Brown, and Wallace is of a novel character that we should perhaps pause to examine. On the one hand, for anyone who, like Wallace, accepts the viability of a functionalist many-worlds understanding of quantum mechanics — and in particular accepts that it follows as a matter of functional and structural analysis that when the wave function develops suitable complex patterns these ipso facto describe what we should regard as worlds — the claim should be compelling. On the other hand, for those who reject the functional analysis and regard many worlds as ontologically inadequate (see Maudlin 2010), or who, like Vaidman (see the SEP entry on the many-worlds interpretation of quantum mechanics), accepts many worlds on non-functionalist grounds, the claim should seem empty. In other words, one has basically to have already accepted a strong version of many worlds and already rejected Bohm in order to feel the force of the claim.

Another interesting aspect of the claim is this: It seems that one could consider, at least as a logical possibility, a world consisting of particles moving according to some well-defined equations of motion, and in particular according to the equations of Bohmian mechanics. It seems entirely implausible that there should be a logical problem with doing so. We should be extremely sceptical of any argument, like the claim of Deutsch, Brown, and Wallace, that suggests that there is. Thus what, in defense of many worlds, Deutsch, Brown, and Wallace present as an objection to Bohmian mechanics should perhaps be regarded instead as an objection to many worlds itself.

There is one striking feature of Bohmian mechanics that is often presented as an objection: in Bohmian mechanics the wave function acts upon the positions of the particles but, evolving as it does autonomously via Schrödinger’s equation, it is not acted upon by the particles. This is regarded by some Bohmians, not as an objectionable feature of the theory, but as an important clue about the meaning of the quantum-mechanical wave function. Dürr et al. 1997 and Goldstein and Teufel 2001 discuss this point and suggest that from a deeper perspective than afforded by standard Bohmian mechanics or quantum theory, the wave function should be regarded as nomological, as an object for conveniently expressing the law of motion somewhat analogous to the Hamiltonian in classical mechanics, and that a time-dependent Schrödinger-type equation, from this deeper (cosmological) perspective, is merely phenomenological.

Bohmian mechanics does not account for phenomena such as particle creation and annihilation characteristic of quantum field theory. This is not an objection to Bohmian mechanics but merely a recognition that quantum field theory explains a great deal more than does nonrelativistic quantum mechanics, whether in orthodox or Bohmian form. It does, however, underline the need to find an adequate, if not compelling, Bohmian version of quantum field theory, and of gauge theories in particular. Some rather tentative steps in this direction can be found in Bohm and Hiley 1993, Holland 1993, Bell 1987 (p. 173), and in some of the articles in Cushing et al. 1996. A crucial issue is whether a quantum field theory is fundamentally about fields or particles — or something else entirely. While the most common choice is fields (see Struyve 2010 for an assessment of a variety of possibilities), Bell’s is particles. His proposal is in fact the basis of a canonical extension of Bohmian mechanics to general quantum field theories, and these “Bell-type quantum field theories” (Dürr et al., 2004 and 2005) describe a stochastic evolution of particles that involves particle creation and annihilation. (For a general discussion of this issue, and of the point and value of Bohmian mechanics, see the exchange of letters between Goldstein and Weinberg by following the link provided in the Other Internet Resources section below.)*
plato.stanford.edu/entries/qm-bohm/

[Paidion]

The details of physical phenomena have no bearing whatever on “God playing dice with the universe.”

One cannot conclude determinism from physics unless one assumes that the universe contains ONLY physical particles. But what if God is a metaphysical Self, and in addition to physical particles, He created other metaphysical selves, including man?
And what if these metaphysical selves (hereafter called “persons”) have libertarian free will so that they can cause the various parts of their physical bodies to move as they wish, and that there is not prior conditions which cause that movement, though such conditions may influence these persons?

“Playing dice” or “random event” is not part of the equation. Indeed, what IS a random event? I think it is none other than an event which we cannot explain. Every event has a cause. Sometimes the cause is a prior event, and sometimes the cause is a person or some other rational self.

[Michael]

The details of physical phenomena have no bearing whatever on “God playing dice with the universe.”

Yes it does.

What you mean is that “the details of physical phenomena have no bearing whatever” on questions regarding creaturely free will, but pog seemed to think that they do have some bearing, and kept bring quantum theory up on another thread (and suggesting C.S. Lewis’s comments on the subject were somehow out of date.)

This thread was intended more as an answer to him (and those who share his views), than an invitation to debate libertarian free will.

But since you brought the subject up here, I’ve been reading a paper by our own Prof. Thom Talbott, and his thoughts are very interesting (and if he ever drops in here, I’d be very interested in anything he might say relative to my questions.)

The paper can be found here.

willamette.edu/~ttalbott/God-Freedom.pdf

…what IS a random event? I think it is none other than an event which we cannot explain.

I think that’s what Lewis was saying.

P.S. After reading this, would you consider Prof. Talbott a libertarian or a compatibilist?

[pog]

Sorry Michael, but I think you may have misunderstood me. All that quantum stuff in the previous thread was mostly about whether there is an uncaused and truly random non-free agent dimension to the physical universe, and although some theologians and scientists speculate that the mind-body problem can be solved by appeal to quantum phenomena, I am no where wedded to it nor used it as an explanation for free willed agency.

I basically agree with Paidon when it comes to free agents as self-determining causal agents, and I thought I’d made that clear in the previous thread - where I also noted the problems with determinism.

Regarding this research and quantum indeterminacy all I can say is what I previously posted: the jury’s still out on this, the scientists really are arguing about whether randomness is ontic (Lewis was wrong on that), and I have no problem with it if it is. Or if it isn’t, for that matter, so long as free will exists.

[Michael]

Sorry Michael, but I think you may have misunderstood me. All that quantum stuff in the previous thread was mostly about whether there is an uncaused and truly random non-free agent dimension to the physical universe, and although some theologians and scientists speculate that the mind-body problem can be solved by appeal to quantum phenomena, I am no where wedded to it nor used it as an explanation for free willed agency…

So “all that quantum stuff” was off topic on a thread that had to do with “Libertarian Free Will and the Existence of God,” wasn’t it?

Especially when I started the thread to explore whether it was possible to explain all the contingency in the universe without positing a God who has libertarian free will (whether or not man does.)

You’re saying you successfully derailed the thread, got us talking about “whether there is an uncaused and truly random non-free agent dimension to the physical universe” (which you now say is irrelevant), and caused me to question libertarian free will (when I started out questioning the compatibilist position I had always more or less taken for granted.)

Good job.

Now if there is “an uncaused and truly random non-free agent dimension to the physical universe,” I would once again ask you where it came from?

That position seems fundamentally atheistic (or dualistic) to me.

How does the one uncaused Cause (Who created and maintains all things, visible and invisible ex nihilo according to clasic Christian Theism) turn things over to “Chance” whenever He wants to?

Where did this “Chance” come from?

Is it uncreated and co-eternal?

If it’s created, is it logically coherent to say that God created and maintains some law or force that somehow makes 50% of photons shot at a mirror go thru the mirror, and 50% go around it, but neither God or this thing He created determines which 50% do what (they just kinda do this on their own, even though they have no minds or wills of their own)?

And if this law or force (that I’ve called "Chance’) actually does determine which 50% do what (however unpredictable that might be to human observers), how is it that God isn’t aware of how this thing He created and maintains operates?

How is it that He doesn’t in effect decide which 50% do what by creating and maintaining this hypothetical law?

Or put metaphorically, how is it that He doesn’t (in effect) decide which way the coin lands before He even tosses it?

Since you kept bringing up and defending the theory that there is some “uncaused and truly random non-free agent dimension to the physical universe,” could you please explain that?

Are you suggesting that it is uncreated?

[Sobornost]

Hi Michael –
I don’t think that Pog tried to derail the discussion – he was simply introducing a matter of speculative interest that some up in all current debates about determinism and freedom.
I personally think that the relationship between determinism and freedom is a matter that we can attempt to clarify as best we can – but I don’t think there will ever be a fully cogent water tight description of the relationship (at least in my lifetime).

What I can see is that the matter is one of personal anguish for you – whereas Pog is seeking as much clarity as he can on the matter through discussion but is a bit more detached from the anxieties adn therofore takes the subject more discursively (as I would actually).

Some sage once said that all the biggest problems in life are such that they are not completely solvable but have to be lived with creatively. And knowing that the greatest minds over centuries have pondered this one – and never agreed completely – I just think that, yes it’s great to talk about but when we come up against paradoxes that test the limits of our knowing, it’s at that very point we have to start trusting in God as Abba (something that Cindy said to you in another conversation on his thread. So rest you merry and be not so anxious about this – ‘Consider the lilies , consider the birds of the air’.

All the best

Dick

[Michael]

…Some sage once said that all the biggest problems in life are such that they are not completely solvable but have to be lived with creatively. And knowing that the greatest minds over centuries have pondered this one – and never agreed completely – I just think that, yes it’s great to talk about but when we come up against paradoxes that test the limits of our knowing, it’s at that very point we have to start trusting in God as Abba…

Which God?

The God of Classic Christian Theism (Who created all things, visible and invisible ex nihilo)?

The dualistic god of the Parsees (who’s kingdom of light is co-eternal with a kingdom of darkness, and who didn’t create matter at all)?

The God of Joseph Smith Jr., who organized the world out of pre-existing matter (and who didn’t “create” us at all, according to the most common interpretation of certain Mormon texts, which takes individual human “intelligences” to be as co-eternal with God as physical “element”)?

Or the God of Process Theology (which seems similar to the Mormon God, in that He didn’t create everything ex nihilo)?

[Sobornost]

Hi Michael - I’ll answer as best as I can -

Which God?

The God of Classic Christian Theism (Who created all things, visible and invisible ex nihilo)?

Many modern theologians have objections to the God of Classical Christian theism as found in Augustine, Aquinas, Anselm etc, I understand. The God of classical Christian theism is spoken of in terms of his mighty attributes – Aseity, Omniscience, Omnipotence and Impassibility for example. In all of these God stands outside and apart from God’s creation, and the creation ‘ex-nihilo’ is an act of transcendent power. I understand that the problem with this picture is that it downplays the central tenet of orthodox Christianity that God is relational. There is relationship within God in the Trinity community of loving persons. God in the Holy Spirit and as Logos are intimately related to the universe as a work in progress. God becomes incarnate in Jesus and lives a fully human life taking our subjectivity into Godself etc. So the picture of God in Classical theism as wholly transcendent and in control doesn’t do our faith or human experience of God full justice.

Process theology is one attempt to speak of God in a more relational way – but it is criticised for sacrificing the transcendence of God in favour of a completely immanent theology (at least in some of its variations). I understand that

Boyd’s Open theism – that Pog finds compelling – is an attempt to recombine the insights of process theology with Christian orthodoxy.

But when I answer the question Which God? I have to say that it is God who is both the creator of all things, and God who also limits himself in the Incarnation and makes Himself vulnerable in relationship with us. Otherwise the incarnation would just be a puppet show in an almost docetic way.

The dualistic god of the Parsees (who’s kingdom of light is co-eternal with a kingdom of darkness, and who didn’t create matter at all)?

No not this God – and I’m not sure that this is a correct description of the Zoroastrian worldview – as I understand classical Zoroastrians they believe in an ethical dualism that is played out in the human heart. However, The Manicheans and other later Parsee sects misunderstood this and instead believed in a metaphysical dualism between matter(evil) and spirit (good). So if we are talking about Manicheans and by extensions the Hellenistic Gnostics – no I don’t believe in their god or their Christ. Their high god is pure spirit; the world was created by an inferior godling/demiurge that got above himself and created the world, accidently trapping human spirit in the prison of the flesh; and their Christ came to remind us of our spiritual nature – and only seemed to be a human being and to suffer and die. No I don’t believe in any of this – I affirm the Incarnation.

The God of Joseph Smith Jr., who organized the world out of pre-existing matter (and who didn’t “create” us at all, according to the most common interpretation of certain Mormon texts, which takes individual human “intelligences” to be as co-eternal with God as physical “element”)?

I know very little about Mormonism, but I understand it is a form of American Gnosticism – a bit like the offshoots of New Thought. I understand their God is the craftsman of pre-existing stuff rather than the creator. I didn’t know that they believe human intelligences to be co-eternal with God. I can’t say that any of their shtick resonates with me.

Or the God of Process Theology (which seems similar to the Mormon God, in that He didn’t create everything ex nihilo)?

I’m not sure it has to be similar to the Mormon idea of God. The case against creation ex nihilo is argued by the Christina philosopher Thomas Oord on the following grounds (cut and pasted from Wiki I’m afraid since I’m no expert; i understand that Oord is a Weslyan evangelical who has tried to integrate the best of process thought into orthodox channels)

*Thomas Jay Oord (born 1965), a Christian philosopher and theologian, argues that Christians should abandon the doctrine of creation ex nihilo. Oord points to the work of biblical scholars, such as Jon D. Levenson, who point out that the doctrine of creatio ex nihilo does not appear in Genesis. Oord speculates that God created our particular universe billions of years ago from primordial chaos. This chaos did not predate God, however, for God would have created the chaotic elements as well. Oord suggests that God can create all things without creating from absolute nothingness.

Oord offers nine objections to creatio ex nihilo:
1.Theoretical problem: One cannot conceive absolute nothingness.
2.Biblical problem: Scripture – in Genesis, 2 Peter, and elsewhere – suggests creation from something (water, deep, chaos, etc.), not creation from absolutely nothing.
3.Historical problem: The Gnostics Basilides and Valentinus first proposed creatio ex nihilo on the basis of assuming the inherently evil nature of creation, and in the belief that God does not act in history (interesting). Early Christian theologians adopted the idea to affirm the kind of absolute divine power that many Christians now reject.
5.Creation-at-an-instant problem: We have no evidence in the history of the universe after the big bang that entities can emerge instantaneously from absolute nothingness. As the earliest philosophers noted, out of nothing comes nothing (ex nihilo, nihil fit).
6.Solitary power problem: Creatio ex nihilo assumes that a powerful God once acted alone. But power, as a social concept, only becomes meaningful in relation to others.
7.Errant revelation problem: The God with the capacity to create something from absolutely nothing would apparently have the power to guarantee an unambiguous and inerrant message of salvation (for example: inerrant Bible). An unambiguously clear and inerrant divine revelation does not exist.
8.Problem of Evil: If God once had the power to create from absolutely nothing, God essentially retains that power. But a God of love with this capacity appears culpable for failing to prevent evil.
9.Empire Problem: The kind of divine power implied in creatio ex nihilo supports a theology of empire, based upon unilateral force and control of others. Is this so? *

Some additional arguments I’ve found against creation ex nihilo on Patheos are -

*Early Church Fathers didn’t sweat Creation out of ‘something.’ Both the Hellenistic tradition via Platonism and Judaism assumed that God created out of some unoriginate matter. Justin Martyr, Athenagoras, and Clement of Alexandria all explicitly affirm the doctrine. In one of his apologies to Greek philosophers Justin martyr insists that Plato stole the idea from Genesis! If Creation out of Nothing was necessary to preserve Monotheism or the Biblical doctrine of Creation then someone needs to call Justin.

Creation Out of Nothing isn’t Biblical, as in it isn’t in the Bible. If you read through the Bible you will not find the affirmation that God created the world out of nothing. It’s just not in there. In fact, even Biblical scholars who in the end want to affirm the doctrine for theological reasons will not point to the idea being present in the Bible. Just re-read Genesis 1 and ask yourself ‘where did the darkness and waters come from?’ They weren’t created but were there when God began to create.

Creation Out of Nothing isn’t a part of the Biblical Imagination. Not only is the doctrine absent in scripture but in the quite robust doctrine of Creation in both the Hebrew and Christian scriptures you don’t even see an interest in the question itself. There is plenty of interest in the goodness of Creation, God’s on-going relationship with Creation, Creation’s role in God’s on-going mission, the Cosmic Christ’s relationship to Creation, Creation’s groaning and it’s worship of God but not an affirmation that it came from nothing. It seems odd to me to insist on a doctrinal nuance that isn’t in scripture or even asked. Sure you can hold it but if no author of scripture thought about asking, relax with the dogmatism.*

A theologian who has tried to chart a middle path between process theology and classical theism is Jurgen Moltmann. Here I quote another post from Patheos:

*The central reason why Moltmann wants to preserve creatio ex nihilo is that, following Barth, he sees the emphasis on the divine freedom as critical. - but it should be remembered that as with Barth and the Confessing Church there are also definite political commitments in his emphasis on divine transcendence (as a guarantee that God stands in judgment over all idolatrous political systems, i.e. Nazism).

So much for Moltmann when he’s doing straight ‘confessional’ theology. He’s going for creatio ex nihilo, no arguments. HOWEVER, because he is such a consistently creative thinker, he doesn’t want to leave things there (and is surprisingly charitable towards Tillich in the discussion that follows). So, putting his ‘radical’ hat on and drawing from the Jewish kabbalistic notion of ‘zimsum’, he tweaks creatio ex nihilo in a way that nobody here has discussed as far as I can see:
‘In order to create a world ‘outside’ himself, the infinite God must have made room beforehand for a finitude in himself. It is only a withdrawal by God into himself that can free the space into which God can act creatively. The nihil for his creatio ex nihilo only comes into being because - and in as far as - the omnipotent and omnipresent God withdraws his presence and restricts his power.’ (God in Creation, 86-87)

I’m not necessarily saying that Moltmann is correct here. For all I know, the universe may have arisen from fluctuations in a quantum vacuum (whether you call that a ‘nihil’ is a question of definitions). But for sheer theological and speculative creativity, you have to admit that Moltmann’s idea is top-drawer stuff. And the notion that God’s relationship to creation is characterized by SELF-limitation of divine power - a self-limitation without which there could be no world at all, represents a middle way between classical theism and process theology which still seems attractive to many of us.*

OK I may be a heretic – but I’m a Jurgen Moltmann type of heretic. I find his ideas on God’s self limitation in creation suggestive too and consonant with a theology of kenosis.

All the best

Dick

[Michael]

Oord speculates that God created our particular universe billions of years ago from primordial chaos. This chaos did not predate God, however, for God would have created the chaotic elements as well.

But if nothing predated God, and He created the chaotic elements as well, you’re talking about creation ex nihilo.

All creation ex nihilo (as it is used in creedal statements and by Church Fathers) means is that God didn’t create the universe out of pre-existing matter.

I don’t know what you (or Oord) mean by “absolute nothing,” but you seem to be thinking of “nothing” as some kind of material that God created the universe out of, and that’s not what the doctrine of creation ex nihilo means.

And as far as impassibility is concerned, I was looking for an Orthodox explanation of the true Patristic understanding of this doctrine that I bookmarked a long time ago.

Unfortunately, I have so many book marks that I was unable to find it, but I believe it was in essential agreement with the explanation offered here.

What about the charge that impassibility turns God into an iceberg? The complaint turns out to be bogus. In truth, mainstream classic theism has always denied that God is cold and remote from his creation. One of the earliest Church Fathers, Justin Martyr, said any view of God that sees Him as apathetic amounts to a kind of atheistic nominalism:

If any one disbelieves that God cares for [His creation], he will thereby either insinuate that God does not exist, or he will assert that though He exists He delights in vice, or exists like a stone, and that neither virtue nor vice are anything, but only in the opinion of men these things are reckoned good or evil. And this is the greatest profanity and wickedness.[17]

God isn’t like a stone or an iceberg. His immutability is not inertia. The fact that He doesn’t change His mind certainly doesn’t mean He is devoid of thought. Likewise, the fact that He isn’t subject to involuntary passions doesn’t mean He is devoid of true affections. What it does mean is that God’s mind and God’s affections are not like human thoughts and passions. There’s never anything involuntary, irrational, or out of control about the divine affections. Here’s how J. I. Packer describes the doctrine of impassibility:

This means, not that God is impassive and unfeeling (a frequent misunderstanding), but that no created beings can inflict pain, suffering and distress on him at their own will. In so far as God enters into suffering and grief (which Scripture's many anthropopathisms, plus the fact of the cross, show that he does), it is by his own deliberate decision; he is never his creatures' hapless victim. The Christian mainstream has construed impassibility as meaning not that God is a stranger to joy and delight, but rather that his joy is permanent, clouded by no involuntary pain.

spurgeon.org/~phil/articles/impassib.htm

[pog]

So “all that quantum stuff” was off topic on a thread that had to do with “Libertarian Free Will and the Existence of God,” wasn’t it?
Especially when I started the thread to explore whether it was possible to explain all the contingency in the universe without positing a God who has libertarian free will (whether or not man does.)
You’re saying you successfully derailed the thread, got us talking about “whether there is an uncaused and truly random non-free agent dimension to the physical universe” (which you now say is irrelevant), and caused me to question libertarian free will (when I started out questioning the compatibilist position I had always more or less taken for granted.)
Good job.

Michael, that was not a very nice thing to say. It was rude and incorrect. I’m not sure exactly what I’ve done to upset you (other than disagree with you). Consider:

On your Libertarian Freewill and the Existence of God thread started Sat Nov 24, 2012 9:43 am, you posted a long cut-and-paste text from another forum to which no one responded. In that post you (not me) first introduced talk of quantum randomness:

‘It seems to me to suggest effects without causes, and to equate choices made because of some random quantum flux within the brain with “freedom.”’

You then began trying to initiate a dialogue by goading/insulting the forum members here:

‘The folks over at William Lane Craig’s Reasonable Faith were a little more responsive.’

I saw that, and I tried to offer my thoughts so it wouldn’t become a dead thread (like many other of the threads you start on exactly the same topic), and I noted from the outset that:

‘I’m no philosopher, and I can’t say I fully understand what your query is getting at (sorry), so be patient with me. But I’m not sure what the problem is with arbitrariness in the universe (and its relation to free-will)?’

I did not mention anything quantum at all in that opening post, although following your lead I mooted that the universe could have ‘randomness’ in it as well as freewill:

‘This naescent universe has self-unfolding and self-determining potential generated from the interplay of law-like boundaries, randomness and the creative participation of free-willed beings, and has a general (but not meticuously) determined end/goal.’

You responded to the ‘randomness’ and the ‘free will’ parts separately, thus starting two tangents. You directly asked me with regard to ‘randomness’ (not freewill):

‘If there’s a cause for every effect that requires a cause, and if God is the only uncaused Cause, where would the randomness come from, and how would God introduce it? How can He play dice with the universe when there’s no one and nothing except Him and His will to cause the dice to roll, or to stop them rolling at a particular point, and He must know how they will land?’

And:

‘I don’t see where the randomness comes from, or what you mean by freewill in relation to God or creatures?’

You had by this stage *already given *three astronomical examples that were clearly related to randomness in the universe more than freewilled agency. Again you cut-and-pasted a long block of text from another forum which this time discussed libertarian freewill in terms of casual indeterminacy.

Again I apologised for not being able to deal with your queries:

‘I’m not sure I can engage your points on the philosophical level your looking for, Micahel You may need to speak to some experts. I’ll try my best to address your points, but I’m way out of my depth …’

But tried to deal with freewill and randomness separately (as you had initiated) in relation to your comments, and very cautiously tried to say that I thought it was possible that there could be randomness in the physical universe:

‘Let’s say He created the universe with a set of physical laws and that some of these laws were indeterminate and probablisitc in nature. Like say, God made a spinning coin and made the laws of the universe such that there was nothing (outside the working of the system) that would determine the coin falling heads or tails but rather there was an irreducible and unchangeable probablility that it would land either heads or tails - a true 50/50 chance. Even with all the knowledge of all other contingent things one couldn’t accurately predict the outcome - it would always be either heads or tails. God doesn’t know; no one knows. Maybe quantum stuff works like that, maybe there is a true indeterministic element to reality. If so, then I don’t see why God couldn’t have caused the system or willed those probabilistic laws. To be really speculative, it might be that it’s through those probabilistic laws that God works His providential care without revealing his background tinkering (we can never know if it was chance or if God skewed the maths in that particular quantum event), or even that this system is how spirit interacts with matter or how freewill minds interact with material brains. No idea, but fun to consider! There’s a good chance I’m missing something here’

At this point you said:

‘I’m not sure where or who the experts are?’

Which is why I started to give names and quotes (which you later had a go at me for!)

And you also then *furthered our tangential discussion on randomness *(note: *you were driving *this discussion, not me) saying:

‘I’m not sure what “indeterminate” or “probabilistic” mean here.’

You introduced science:

‘When computer programers try to introduce randomness into a system, they do so in one of two ways.’

And asked me a direct question regarding how I would explain randomness in the physical universe:

‘I don’t see how this randomness could be introduced into the system–do you?’

Again, I apologised saying:

‘It seems like we’re coming at this from quite widely different angles, Michael. And judging by the number of threads you’ve started, this topic is one you’re seriously wrestling with. So again I apologise for not being much help.’

And then gave you specific books to read. I continued to discuss your primary topic of freewill and God’s free will, but on the tangent of randomness (which you introduced and you furthered and you asked directly about) I tentatively stated:

‘Aren’t things truely (according to some interpretations) random, indeterminate and probabilistic in the quantum world? Note, this isn’t just our ignorance - it’s not that we don’t know what determines the quantum events, or that they are simply unmeasurable, but rather that randomness and chaos are part of the true and deep sturcture of nature. I have no idea whether this is an accurate reflection of reality or not, and I have no idea if this chaos was originated in God or not, but I am perfectly happy to say that’s possibly true and that possibly God wanted it that way.’

8 very long posts in and quantum randomness (specifically) has hardly been mentioned … then in post 9 you say (with a quote):

‘pog wrote: Aren’t things truely (according to some interpretations) random, indeterminate and probabilistic in the quantum world? I don’t think there’s any evidence of that. I think something C.S. Lewis said about that a long time ago is still valid today.
pog wrote:Note, this isn’t just our ignorance - it’s not that we don’t know what determines the quantum events, or that they are simply unmeasurable, but rather that randomness and chaos are part of the true and deep sturcture of nature.
Why do you say “this isn’t just our ignorance - it’s not that we don’t know what determines the quantum events, or that they are simply unmeasurable”?

And:

What has changed since Lewis wrote “Those who like myself have had a philosophical rather than a scientific education find it almost impossible to believe that the (new) scientists (in the field of quantum mechanics) really mean what they seem to be saying. I cannot help thinking they mean no more than that the movements of individual units (sub-atomic particles) are permanently incalculable to us, not that they are in themselves random and lawless.”
(Miracles, page 20, parenthesis mine.)’

You clearly escalated all the quantum talk to specifics, not only questioning my tentative comments about the possibility of such a thing being ontic, but also specifically attacking the scientists own views on their own theory!
In response I was again restrained and tentative:

‘As much as I respect C. S. Lewis I don’t consider him much of an authority of quantum physics, and it’s prefectly possible that he’s wrong here. Note: I’m not arguing that the universe does have a probabilistic indeterminate side, rather I’m simply arguing that it might have such a side. I have no philosophical issue with randomness … Even a quick glance at wikipedia: en.wikipedia.org/wiki/Interpreta … _mechanics shows the wide range of differing interpretations of quantum theory, including the possibility that the universe really is indeterminate at the quantum level. I think the jury’s still out on this one - and I see no reason to decide the matter ahead of time.’

I then gave evidence/quotes to back up my point contra CS Lewis.
**You then responded massively **to my post, spending two-thirds of your time writing about what I’d said about quantum theory rather than on what I had said about freewill. **You asked me oddly irrelevant direct questions **(‘So is chad Ozel a Theist or an Atheist?’/ ‘Does Polkinghorne -as a Theist, not a scientist- explain where this randomness, chance, and chaos come from?’), and **asked me to expand my view on quantum theory **(‘Do you?’).

At this point your questions started to become a little too direct and **somewhat adversarial **in tone:

‘If you can, I wish you (or Professor Talbott) would explain it to me.’

Again,* I was tentative and apologetic *as I tried my best to answer your very hard questions:

‘By doing this I wanted to show that it is possible to believe that the universe does have randomness in it; that many highly intelligent and informed people accept this; that both Christians and non-Christians can hold this view (and Einstein was a determinist, but also an atheist). **Note again, I have no idea whether it is true or not, but I accept that a number of key thinkers have no issue with it scientifically, philosophically or theologically **… They really did mean what they said - and time has shown this. **They might be wrong (how would I know?), **… I think the Christian philosopher Keith Ward has some stuff to say about freedom and quantum physics - but again, I’m not well read enough to offer more … I agree that your argument is strong, that it is dificult, and that I really cant give you the kind of clear answer you’re looking for. Remember, I’m just a nobody.’

You then continued to further this debate on quantum science and *even introduced a new sub-topic *in our discussion (using bolded text no less)! ‘or particles that slip in and out of existence without His involvement’

You also then started getting snarky, making *unprovoked rude little asides *like:

‘Not according to some of the quantum physicists you’re fond of quoting.’

And:

‘In other words, why did you never answer this question?’

And:

‘Those are the questions that neither you or your sources have answered’

In response, *I admitted when I was wrong *(‘You are correct. I was wrong to label Einstein an atheist’); and I tried to prevent another tangential sub-topic from opening up (‘Besides, I was really talking about indeterminancy about particle behaviour, not really expanding it to cover fluctutions in the quantum vacuum - whatever that is all about!’).
I spent most of my post trying to discuss issues directly related to determinism, freedom and God (rather than science) and always in a tentative manner:

‘I don’t think anyone’s been able to solve the micro- macro- scale problem of quantum stuff. But I’m no quantum physicist! Remember, all I’m arguing for here is the possibility that the scientists might have got it right - that’s all (I’m not wedded to quantum indeterminism - indeed it is very much an ongoing debate and Einstein might be proved right after all).’ It’s not that we simply cammot measure - it’s that it is always undetermined - it’s probabilistic. Do I understand this? No, I’m not a quantum physicist.’

And:

‘They might be wrong, but they (the scientists, philosophers and theologians) are not being nonsensical.’
‘I thought I had, sorry. Let’s just pretend that the mainstream scientific view is right (**again, I’m not arging for it **- I just think it’s a reasnoble possibility).’

I (not you!) even tried to get the thread back on the OP topic:

‘Regardless,** I’m pretty sure we’ve wandered far from your OP!** We’ve got side-tracked onto quantum randomness, but **that seems somewhat tangential to the main point **which seems to be freewill! I’m not sure that I care too much whether the universe has a non-deterministic level or not, so long as beings have freewill.’

Even going so far as to compromise my own view, and *agree with you *to bring the debate back onto freedom and away from randomness:

‘So, for the sake of argument, let’s say that the universe is strictly deterministic in the physical. Are we agreed that beings (like humans) have freewill? If so, I’m not quite sure what the problem is.’

But in your next post **you ignored **all the stuff about freedom, and fixated on the quantum sub-topic I had tried twice to avoid:

‘I’ll post more later, but (for now) I’d like to ask just one question. Are we agreed that it’s Philosophically nonsensical to speak of particles popping in and out of existence of their own accord?’

I again responded tentatively:

‘My gut reaction would be yes, … Though, as far as I understand it, when physicists talk of sub-atomic particles appearing out of nothing they don’t, by their own detailed explanations, mean literally nothing in a philosophical or common-sensical manner’

Your next post was then again* fixated *on quantum stuff – entirely driven by you! You also started using an inordinate about of bolded text (shouting at me, basically) and your comments became even ruder:

‘Can you put aside your respect for these scientists long enough to consider whether it’s as philosophically nonsensical to speak of mindless particles moving’

And:

‘You’re good at quoting scientists who are trying to make philosophical statements…’

You also added to your already lengthy post a huge range of direct questions and another cut-and-paste (this time from another discussion you were having with Paidon).

At this point, growing exasperated with your rudeness, I tried to cool off for a while:

‘Hi Michael, this’ll be my last post for a while, what with Christmas looming and all. And I apologise if my posts have made you angry (or if I’ve misread your tone). I think we have reached an impasse and are now arguing against or past each other rather than trying to understand each other, so it’s probably best if I have a break after this anyway.’

Again, I was always polite and tentative:

‘I tried to communicate that they were using ‘nothing’ in a different way but obviously I failed to make myself clear to you.’

And:

‘No. God made a chaotic system 'cause He wanted one (if there is one - again, I’m just arguing that it’s possible - trying to keep an open mind etc).’ and ‘I’ll leave it there. There’s much more you said that I could respond to, but I have limited time and it seems like it’s getting us nowhere now. I’ve learned a bit from our chat, and you’ve helped me sharpen up my views. Thank you. I hope that I too may have been of some use - I did warn you to start with that I wasn’t an expert and that I couldn’t engage you on the level you wanted.’

But you wouldn’t let it go at that (**you even asked for a direct quote **even though you’d already told me that you didn’t like me quoting???):

‘Are you serious?’
‘But unless you don’t really mean what you seem to be saying’
‘if you really mean what you seem to be saying’
‘I find some of your posts frustrating … Talking to you is like talking to someone who agrees that an explosion at a typewriter factory couldn’t produce a typewriter’
‘What’s more frustrating is that all you’re doing is defending the theories of these quantum physicists you admire, without even trying to make sense (or help anyone else make sense) of what they’re saying.’
‘You may be up on the science, but you seem to be intentionally avoiding the real philosophical difficulty here (and I find it almost impossible to believe you don’t see it.)’
‘Would you please try to say something (or quote someone who has said something) that’s relevant to the question?’

Again *I tried to respond sensitively *to your rudeness and fixation:

‘I notice that you’ve started at least four threads on this website that pertain to this issue and that you are also discussing the same topic on other websites. You have made many posts, often quite complex ones, and posted even on Christmas Day. Clearly, this issue is one that you find deeply troubling, that you have no clear answer to, and which, I suspect, you have a deep personal motivation for pressing on with. I wish I could help, but I really don’t think that either I, or anyone else, will be able to give you the kind of clarity and perfect answers you seek.’

And again I was tentative:

‘And I am saying that I think such a view is possibly right. I guess you think that makes me mad I don’t find the view nonsense, merely confusing’

You were then rude and dismissive to Paidon (‘I’m more interested in logic here than I am in Patristic quotes that simply support your pre-stated position.’) before again focussing your attention on me:

‘So let’s try to focus on the philosophical issues this time.’
‘If I do, will you try to put the same thought and effort into answering my questions?’
‘BTW: If you had really given any thought to what Padre Pio said…’
‘it seems to me, that I’m doing a better job of answering your questions here than you are of answering mine.)’
‘I hope you (or someone here) will kindly take the time to give as much thought to my questions (and perhaps try to say something that might at least be helpful in trying to answer them.)’
‘I’ve given some thought to the philosophical questions you asked, and I’ve tried to answer them (using reason and logic–not the language of scientific theories nether of us understand). Could you please try to address my questions in the same way?’

I posted a reply saying:
‘O.K I’m sorry. I am of no use to you. I hope you find the answers you’re looking for’
‘Why the snarkiness? You seem to have adopted an adversarial tone in this post - is this how you want our discussion to progress? I will try my best.’
‘Quantum theory would be the way I view probabilistic causal relations as possible - I can see no other way. Note, this regards probabilisitc relations - I firmly hold to the choice of agents which is also non-deterministic but not proabilisitic.’

You made no apology for your rudeness, but said:

’Why the snarkiness? Because you don’t really seem to be addressing the questions I’m asking.’

Nice

You followed that up with:

‘That’s the question I’d like some thoughts on (and you keep wanting to talk about quantum physics, or Calvinism vs. Arminianism.)’

And more back-handed compliments to make me sound like some sophomoric imbecile:

‘see you’ve read “Candide,” by Voltaire. So did I, in Philosophy 101-where the Professor told us we wouldn’t get any answers, but would be able to ask better questions.’

So I made my final post in that thread and left the discussion.
But even then you responded to me, referred to me (disparagingly and inaccurately) in posts with other people (and even whole other threads!) and tried to get me back into the discussion (why?) – and you even acted paranoid and defensive with Sobornost (why?) and got annoyed with Puddy (why?):

‘perhaps it’s not the total distraction I believe all the talk about quantum theory is.’
‘I think most of them would admit they don’t really believe in anything like what Pog means by libertarian free will’
‘This is from a blog I quoted in my last post to pog.’
‘If so, many would argue (as I think pog would) that the randomness is based on some connection between the human mind (or brain) and quantum flux, but I see that as irrelevant.’ (note: I had made no clear connection and it was your topic)
‘If you, or Dick, or Pog (or anyone else) can point them out to me, I’d be truly grateful.’
‘That’s my thoughts for starters. I’ve plenty more to say. I’d like t have a look at libertarianism next – including a consideration of Pog’s and Don’s arguments.’
‘Here’s the problem I have with pog’s quantum mechanics based speculation.’ (what speculation is that??)
‘I’ve come to think of it somewhat like this hypothetical quantum chaos field that pog kept wanting to talk about here.’ (where did I keep wanting to talk about anything?)
‘questions regarding creaturely free will, but pog seemed to think that they do have some bearing, and kept bring quantum theory up on another thread’ (on yet another thread about the same topic with a huge cut-and-paste of a whole, possibly copyrighted, article!)
‘This thread was intended more as an answer to him (and those who share his views), than an invitation to debate libertarian free will.’ (You even started a whole thread just to refute some speculative and tentative comments I made on another old thread I’d stopped responding on … it’s like having a stalker!)

So, Michael, I am angry that you started this thread to hound me, and that you never apologised for being rude, and that you still refer to me on other threads inaccurately and that you have now responded to my measured post here in such a rude manner. It’s also irritating that you clutter up the Philosophy board with loads of threads on the same topic.

[Sobornost]

Hi Michael –

I think you have been rude to Pog actually. Also I’m uncomfortable with your idea that creating a thread means that you should be absolutely in control of it - to focus as you please, or to diverge as you please. If you wan to take on the informal role of thread faciltator, yes you have an informal function to guide and to clarify (and to see the thread out without creating another to qiockly) - but not to dictate, coerce and dismiss in this way.

Anway I post below what I intended before Pog posted.

You must note that I am not a metaphysician; I take a general interest in a lot of things, but I quickly get lost when things get very abstract (I find mathematics very difficult – and perhaps this is why metaphysics is not my metier, neither are thought experiments about randomness and probability). A lot of the contradictions expressed in the debates alluded to here and on related threads – objectivity/subjectivity, determinism/freedom, transcendence/immanence – are held in a completely liveable with tension for me just by meditating on the Incarnation as mystery. This will more than suffice as a gift of food for my mind that is quite simple when it comes to these matters.

Bearing this in mind, I will now have a look at your questions about my last slap dash cut and paste post.

First, I must say that I don’t necessarily agree with Oord. I’ve only read about him ever in passing. I have read Levenson’s Book on creation themes in the Hebrew Bible which, according to the Wiki article, Oord has been influenced by. I enjoyed the book – but haven’t followed it up with rigorous reflection. I mentioned Oord only to suggest that theologians who want to qualify the Classical notion of ‘creation ex nihilo’ – probably influenced in some way by Process Theism – are not necessarily in any way analogous to the Mormons in this aspect of their theological reflections.

Oord speculates that God created our particular universe billions of
years ago from primordial chaos. This chaos did not predate God, however,
for God would have created the chaotic elements as well.

But if nothing predated God, and He created the chaotic elements as well,
you’re talking about creation ex nihilo.

That’s a very fair point. Technically this is creation by ex nihilo – and since I’ve not read Oord I can only give my own answer here. My problems with classical ‘ex nihilo’ accounts is that they tell us that the creation was an act of Divine fiat/power. I have problems with pictures of God that focus too much on power. All of the critics of the Process Theist, the Open Theist, and the Personalist Theist critics of Classical Theism would agree that God is not omnipotent in the sense of being coercive. The divine has a power of persuasion rather than coercion. Process theologians interpret the classical doctrine of omnipotence as involving force, and suggest instead a forbearance in divine power. “Persuasion” in the causal sense means that God does not exert unilateral control.

So yes God does create the chaotic elements (and I’m at the limits of my thoughts here). However this creation takes place in a way nearer to what Moltmann speculates on so richly (outlined in my previous post through a cut and paste job). The chaotic elements come into being by God’s act of self limitation of omnipotence which God does in order to allow the universe the space to be in freedom – that is to be other than God. It is this chaotic freedom that God creates and then shapes with persuasion and forbearance in an ongoing process. This picture satisfies me because it suggests that the creation and the kenosis of incarnation are intimately related. And it all I say is very different from Mormon ideas about the eternity of matter, God as craftsman and not creator, and the coeternity of human beings with God. Perhaps Oord argues along these lines – I dunno.

All creation ex nihilo (as it is used in creedal statements and by Church
Fathers) means is that God didn’t create the universe out of pre-existing
matter.

Just a quibble here – some of the Church Fathers didn’t argue for a creation ex nihilo doctrine apparently (see above concerning Clement of Alexandria, Athenagoras, and Justin Martyr).

I don’t know what you (or Oord) mean by “absolute nothing,” but you seem to
be thinking of “nothing” as some kind of material that God created the
universe out of, and that’s not what the doctrine of creation ex nihilo
means.

I dunno what Oord means – the position that chimes with me is Motlmann’s where the idea of nothing as some kind of material is not an issue.

And as far as impassibility is concerned, I was looking for an Orthodox
explanation of the true Patristic understanding of this doctrine that I
bookmarked a long time ago.

Unfortunately, I have so many book marks that I was unable to find it, but
I believe it was in essential agreement with the explanation offered here.

What about the charge that impassibility turns God into an iceberg?
The complaint turns out to be bogus. In truth, mainstream classic theism
has always denied that God is cold and remote from his creation. One of the
earliest Church Fathers, Justin Martyr, said any view of God that sees Him
as apathetic amounts to a kind of atheistic nominalism:

If any one disbelieves that God cares for [His creation], he will
thereby either insinuate that God does not exist, or he will assert that
though He exists He delights in vice, or exists like a stone, and that
neither virtue nor vice are anything, but only in the opinion of men these
things are reckoned good or evil. And this is the greatest profanity and
wickedness.[17]

God isn’t like a stone or an iceberg. His immutability is not inertia. The
fact that He doesn’t change His mind certainly doesn’t mean He is devoid of
thought. Likewise, the fact that He isn’t subject to involuntary passions
doesn’t mean He is devoid of true affections. What it does mean is that
God’s mind and God’s affections are not like human thoughts and passions.
There’s never anything involuntary, irrational, or out of control about the
divine affections. Here’s how J. I. Packer describes the doctrine of
impassibility:

This means, not that God is impassive and unfeeling (a frequent
misunderstanding), but that no created beings can inflict pain, suffering
and distress on him at their own will. In so far as God enters into
suffering and grief (which Scripture’s many anthropopathisms, plus the fact
of the cross, show that he does), it is by his own deliberate decision; he
is never his creatures’ hapless victim. The Christian mainstream has
construed impassibility as meaning not that God is a stranger to joy and
delight, but rather that his joy is permanent, clouded by no involuntary
pain.

My only beef with what Packer says is his emphasis on God’s absolute control – God is in relationship with us. God cares about us. God make Himself vulnerable to us – I’m not sure that Packer’s description of God does full justice to God’s vulnerability and forbearance for me. But who knows?

One last thing I’d say is that Classical Theism and Christian Orthodoxy are not the same thing. The metaphysics of Classical Theism are derived from Plato and Aristotle as well as being formative on the theologies of Augustine, Anselm and Aquinas, were also formative on the Classical Theists of Judaism (Maimonides) and Islam (Averroes). So Classical Theism is a current within Christian Orthodoxy but there are perfectly orthodox Christian thinkers who disagree with aspects of the Classical synthesis – for example Alvin Plantinga ( rejects divine simplicity), Richard Swinburne (rejects divine timelessness) and William Lane Craig (who reject both divine simplicity and timelessness).

Well that’s the best I can do. As I said my spirituality focused on Incarnation is not horribly troubled by the issues discussed here. I thank God for this – because I’m no metaphysician.

All the best

Dick

[Michael]

and William Lane Craig (who reject both divine simplicity and timelessness).

I don’t know if Craig rejects simplicity, but it’s an oversimplification of his position to say he rejects timelessness.

His position is that God is timeless sans creation, and temporal with creation (and whether you say God is dipolar, or omni temporal, some such position seems absolutely necessary to me–otherwise you’re left with questions like how God could have transversed an actual infinite of successive moments to arrive at the moment of creation, and how He can be relational–it seems to me that God has to be both in time and out of time.)

Perhaps you owe Craig an apology?

And BTW, I believe you referred to Aaron as a friend of yours?

I received a PM from him last month thanking me for these threads, and telling me he failed to see how I had been at all “snarky,” so I don’t really think I owe pog any apology.

(but if it makes you happy, I will apologize for being somewhat annoyed at being forced into a debate on quantum physics.)

While he appears to be a compatibilist in regards to human free will, Aaron had some interesting thoughts regarding God’s free will that might answer some of my questions, and when not distracted by the QM debate that pog forced on me, I tried to explore that issue.

I didn’t actually post any of Aaron’s thoughts at the time because I wanted to see what others thought (and because I thought he might want to.)

But since no one else (except maybe Chrisguy90) has really said anything, and Aaron still hasn’t posted his thoughts, I don’t think he’d mind if I share them here.

Here are his exact words (emphasis mine.)

God and LFW

Sent: Fri Dec 28, 2012 7:15 pm
From: Aaron
To: Michael
Hi Michael,

While I’m not ready to begin contributing to the forum again, I just wanted to say that I’ve enjoyed your recent posts, and especially your discussion with pog (I should also add that I haven’t found your responses “snarky” at all!). I also share some of your frustrations. I, too, have wrestled with the kinds of questions you’re asking, and have experienced similar evasion when seeking opinions/answers from others. While I still consider determinism in some form far superior to its alternative (as have many Christian Universalists, such as the highly influential Hosea Ballou, Sylvanus Cobb, and A.E. Knoch), it’s certainly not without its difficulties.

Having said that, here’s my attempt at answering your question regarding God and LFW: I believe God necessarily transcends the circumstances that shape and define the choices of all finite persons he brings into existence. While every choice created beings make is elicited and influenced by circumstances and factors outside of our control (many of which we are completely unaware), I don’t see this as being true of God. It seems to me that the only constraint on God’s choices is his nature (which scripture seems to teach is supremely benevolent). Thus, if there is such a thing as libertarian freedom, then I’m inclined to believe that God is the only being who possesses it. And even then, the exercise of his LFW - whatever the choices may be - would be consistent with his nature (and thus consistent with the best interests/final happiness of any intelligent beings he chooses to bring into existence). IOW, God’s LFW would have certain “boundaries,” being constrained by his benevolent nature. If God is love, then any randomness or arbitrariness in God’s choices/decrees would still have to be such that they are consistent with the best interests/final happiness of whatever number of intelligent beings he chooses to bring into existence. God would thus not be free (in a libertarian sense) to be malevolent rather than benevolent, although he would be free with regards to any choices that are consistent with his benevolent nature (e.g., how many intelligent beings with which he chooses to populate a universe).

Another possibility (to which I admittedly have not given much thought, and am not sure how coherent it is) is that God is actualizing (or has actualized, or has always been actualizing) EVERY state of affairs that is consistent with his benevolent nature. This would mean that God is choosing - or has always chosen - to do literally EVERYTHING that his nature allows him to do, and that every state of affairs that is consistent with God’s nature has been, or is being, actualized by God. This might also mean that our universe is but one of a vast multitude of different universes, with each being unique in some way but all being consistent with, and an expression of, God’s benevolent nature (such that every intelligent being he brings into existence in every universe would have its best interests promoted by God). But again, I’m not sure of the coherence of this view.

The following is an essay on determinism that I thought you might find helpful: anthonyflood.com/blansharddeterminism.htm.

See also chapter 13 of Sylvanus Cobb’s “A Compend of Christian Divinity,” where he attempts to reconcile man’s accountability with divine determinism (books.google.com/books?id=_z4RAA … CDoQ6AEwAA).

In Christ,
Aaron

Aaron at least attempted to answer my questions.

We were actually talking about the same issues (not Quantum Mechanics.)

And his suggested answers actually seemed like they might make some sense.

I’d like to take this opportunity to publicly thank him, and to ask the rest of you what you think of his thoughts?

[Sobornost]

Dear Michael -
I’m really sorry about the intro to my last email - must have been feeling snarky myself after a difficult day. None of us are perfect – so please let it go.
Of course I think your ideas are really interesting and well worth the effort you put in posting here – and you stimulate good conversation too.
Obviously I am sad that the questions of metaphysics are a source of great anguish to you rather than something that fills you with wonder at mystery. Perhaps my willingness to embrace ‘mystery’ is a copout sometimes. As I said – I’m just not very good at abstract thinking. I’m very interested in people that think in the abstract – I had a good friend once who was a brilliant mathematician and talking to him filled me with wonder at how different our ways of looking at the world can be. But as for me – I do my best when it comes to metaphysics; but I’m soon out of my depth and just trying to get a cut and paste general knowledge idea of what’s going on when I’m trying to answer or questions. Also I only have a dilettante knowledge of thinkers in this field (so I don’t think Craig needs an apology from me if I have misrepresented him – which is very likely because I was only quoting someone else’s view and haven’t checked this stuff out; he would need an apology from a peer, but not from a bungling amateur like me).

Anyway friend, my apologies again. And please understand that while I can take a very general interest in metaphysics, when we get down to the fine detail I do start to get lost. Getting lost doesn’t bother me – but I can only concentrate on abstract conversations and contribute to these positively in very small spurts.

Bless you Michael –

Dick

P.S. I don’t know Aaron – but that’s a fine letter. You obviously can communicate well together, so he can speak to your condition as a fellow metaphysician.

[Michael]

if I have misrepresented him – which is very likely because I was only quoting someone else’s view

Then perhaps your source of information owes him (and you) an apology.

I don’t know Aaron

My apologies.

I see now that it was Chisguy90 who referred to Aaron as a friend.

Or, as a friend of mine on this board Aaron would say, “a being which must undergo pain/be saved from its own fallibility/delivered from evil in order to be maximally happy.”

Of course I think your ideas are really interesting and well worth the effort you put in posting here – and you stimulate good conversation too.

I don’t know about that, but I thank you.

Anyway friend, my apologies again

I never said you or anyone else owed me an apology, but I thank you again for your kind words (and for sharing your thoughts on these threads.)

P.S. If your source of information on Craig’s views is what you’d call a metaphysician, it would be nice if he actually took the time to share some of his thoughts on these threads.

[Michael]

P.S. I just received an interesting reply to some of my questions on Craig’s Forum (and I’ll post it here for those who might be interested in these questions.

Basically, your question supposes that ‘East’ and ‘counter-cockwise’ are meaningful without implicit reference to the rest of the universe. The term ‘East’ does seem arbitrary if we consider it alone, as in such particular questions you raised. However, when we give the meaning of those terms, we give them as the objects referred relate to other objects, which are defined in terms relative to other objects and so on. Eventually, a complete denotation will be wholistic in the sense that the whole universe will be referred to (atleast implicitly), and so the underlyng question is:

Why is the universe the way that it is?

This however still separates the universe from it being the effect of which God is the cause, and when we look at the picture wholistically, ie., God and creation, our question has to do with asking:

Why is God the way that He is?

At this point, one assumes that the ontological ground for logical truths are more fundamental than the nature of God. What I mean is – when we ask what the sufficient reason there is for God being the way He is, we are presupposing that the reason why the laws of Logic are the way they are, precede the ontology of God.

We can only ask for the reason why God has the ontology that He has, at the same level as asking for why the laws of logic are the way they are.

Because the PSR can only be meaningfully applied to truths which are logically posterior to the ontology of logic, it is inapplicable in the case of God. The reason why the laws of logic hold the way they do, is because the negation of such is logically impossible, which in this case is to say is ontologically impossible (logicality being the ontology of logic).
Similarly, the reason why God is the way He is, is because the negation of such is ontologically impossible.

[Michael]

So “all that quantum stuff” was off topic on a thread that had to do with “Libertarian Free Will and the Existence of God,” wasn’t it?
Especially when I started the thread to explore whether it was possible to explain all the contingency in the universe without positing a God who has libertarian free will (whether or not man does.)
You’re saying you successfully derailed the thread, got us talking about “whether there is an uncaused and truly random non-free agent dimension to the physical universe” (which you now say is irrelevant), and caused me to question libertarian free will (when I started out questioning the compatibilist position I had always more or less taken for granted.)
Good job.

Michael, that was not a very nice thing to say. It was rude and incorrect. I’m not sure exactly what I’ve done to upset you

You haven’t upset me, but your back quoting here was selective and misleading.

Did you forget this?

I find some of your posts frustrating. Talking to you is like talking to someone who agrees that an explosion at a typewriter factory couldn’t produce a typewriter, but doesn’t agree that a typewriter couldn’t unpredictably start typing and produce random works of literature without any human input

.
Yes, it is frustrating And, actually, I think you’ve represented me (or at least the science I’m saying I find possible) accurately - and I know that it goes totally against our common sense and intuitions - but that is what the majority of quantum scientists do seem to be saying - *and meaning *

When you wrote this, you seemed to understand my philosophical objections to the interpretations of QM you kept wanting to argue about, and even seem to see how illogical it is (though how you could go on to say you find it “possible” I don’t know.)

But you still bring it up (post after post, page after page) and insist that it must somehow be possible because it’s the opinion held by the scientists you quote.

That’s the only defense you ever offered!

You never offered any logical reason particles could move, move in this direction instead of that, or change direction without being caused to by God, or some deterministic force created and controlled by God, or offered any philosophical explanation of how such a thing could be.

You just tried to wear me down by quoting your quantum scientists, and endlessly insisting that their view was the majority view.

And please note, you agreed that what you were saying was analogous to the keys of a typewriter moving by themselves and randomly typing out works of human literature, and then said “actually, I think you’ve represented me (or at least the science I’m saying I find possible) accurately.”

That’s the kind of irrationality I was subjected to on that thread.

“No, an explosion at a typewriter factory couldn’t produce a typewriter; but, yes, a typewriter could mysteriously start typing on it’s own and produce the complete works of William Shakespeare.”

“Yes, that’s an accurate representation of what I’m saying; and no, I can’t offer any logical explanation of how it could be possible–but look at the scientific credentials of the men I’m quoting. If they believe it’s possible, I find it possible (and so should you.)”

That was your argument, all through that thread (and then you accuse me of being snarkey.)

On your Libertarian Freewill and the Existence of God thread started Sat Nov 24, 2012 9:43 am, you posted a long cut-and-paste text from another forum to which no one responded. In that post you (not me) first introduced talk of quantum randomness:

‘It seems to me to suggest effects without causes, and to equate choices made because of some random quantum flux within the brain with “freedom.”’

The argument that random quantum fluctuations somehow explain human free will has been made by some of the people you’re fond of quoting, and from your comments here it seems that we both agree that that argument is bogus (because it would wrongly equate random effects upon the brain with freedom.)

Strange that you managed to argue about human free will with me so long, on a thread where I wasn’t even interested in discussing that subject, when we both seem to agree that it has nothing to do with QM.

You then began trying to initiate a dialogue by goading/insulting the forum members here:

‘The folks over at William Lane Craig’s Reasonable Faith were a little more responsive.’

I saw that, and I tried to offer my thoughts so it wouldn’t become a dead thread

Which would have been nice if you actually had something to say.

Instead you chose to argue about QM.

and I noted from the outset that:

‘I’m no philosopher, and I can’t say I fully understand what your query is getting at (sorry), so be patient with me. But I’m not sure what the problem is with arbitrariness in the universe (and its relation to free-will)?’

If I owe you any apology, it’s for not picking up on this at the time.

You really didn’t understand the opening post from the beginning.

The only free will that I was interested in was God’s (not man’s.)

The question was whether evidence of randomness in creation pointed to a God who had to have some kind of libertarian free will (and Aaron, a determinist, seemed to understand that, and replied in the affirmative.)

While I still consider determinism in some form far superior to its alternative (as have many Christian Universalists, such as the highly influential Hosea Ballou, Sylvanus Cobb, and A.E. Knoch), it’s certainly not without its difficulties.

Having said that, here’s my attempt at answering your question regarding God and LFW: I believe God necessarily transcends the circumstances that shape and define the choices of all finite persons he brings into existence. While every choice created beings make is elicited and influenced by circumstances and factors outside of our control (many of which we are completely unaware), I don’t see this as being true of God. It seems to me that the only constraint on God’s choices is his nature (which scripture seems to teach is supremely benevolent). Thus, if there is such a thing as libertarian freedom, then I’m inclined to believe that God is the only being who possesses it. And even then, the exercise of his LFW - whatever the choices may be - would be consistent with his nature (and thus consistent with the best interests/final happiness of any intelligent beings he chooses to bring into existence). IOW, God’s LFW would have certain “boundaries,” being constrained by his benevolent nature.

Now while I’m sure you’d like to have a long argument with Aaron about humans having free will, my point here is that he understood the OP far better than you did, and if it makes you feel any better, I apologize for not realizing sooner that you really had no idea of what we were talking about.

You responded to the ‘randomness’ and the ‘free will’ parts separately

Because, to me, they were always two separate issues (and I wasn’t interested in discussing human free will on that thread at all), but you kept conflating the two.

You kept going off on a QM tangent, and arguing that mindless particles could conceivably do things on their own (without being caused to do what they do by God or any determining force created, sustained, and understood by God.)

I kept telling you that I found that logically incoherent, and you kept bringing it up, repeating it, and going on your QM tangents.

Now you say you don’t know what you did to upset me, unless it was to disagree with me, but when I look at what you said here, I don’t see where we really disagreed?

I find some of your posts frustrating. Talking to you is like talking to someone who agrees that an explosion at a typewriter factory couldn’t produce a typewriter, but doesn’t agree that a typewriter couldn’t unpredictably start typing and produce random works of literature without any human input.

Yes, it is frustrating And, actually, I think you’ve represented me (or at least the science I’m saying I find possible) accurately - and I know that it goes totally against our common sense and intuitions - but that is what the majority of quantum scientists do seem to be saying - *and meaning *

The one thing that is annoying is that you somehow kept the argument going anyway.

Think about that pog.

Now back to the topic here (where free will isn’t even in the topic heading.)

You quoted scientists who supported your indeterminist interpretation of QM ad nauseam on the other thread, and thanks to you I’m somewhat interested in the de Broglie–Bohm theory (a determinist interpretation of QM.)

If you’re at all interested, here’s a little more on it for you (and if you’re not, here’s a little more for anyone else who might be.)

**Bohmian trajectories are no longer “hidden variables”
Posted Jun6-11 at 10:18 AM by Demystifier
The recent weakly measured photon trajectories

sciencemag.org/content/332/6034/1170.full [Science 3 June 2011: Vol. 332 no. 6034 pp. 1170-1173 DOI: 10.1126/science.1202218]

scienceblogs.com/principles/2…erfere_obs.php

coincide with the Bohmian trajectories. Hence, the Bohmian trajectories should no longer be considered “hidden variables”. Now they are no longer more “hidden” than the wave function.

To avoid confusion, this experiment does not prove that the Bohmian INTERPRETATION is right. After all, this experiment does not reveal a Bohmian trajectory of a single particle. Instead, this trajectory can only be measured through a large number of measurements on an ensemble of equally prepared systems. But this is fully analogous to the wave function. A single measurement cannot measure the wave function either. Only a large number of measurements on an ensemble of equally prepared systems can determine the wave function of the system. Thus, from an experimental point of view, Bohmian trajectories are neither more nor less real than the wave function. Hence, if the wave function is not thought of as a hidden variable, then the Bohmian trajectory should not be thought of as a hidden variable either.

Of course, we still have different interpretations alive. For example, the Bohmian interpretation claims that such Bohmian trajectories exist even at the level of single systems. The Copenhagen interpretations denies it. Likewise, many-world interpretation and some variants of the “Copenhagen” interpreation claim that a wave function exists even at the level of single systems. The statistical ensemble interpretation denies it. Yet, no direct measurement is able to say which interpretation is correct. Or at least, not yet.**

physicsforums.com/blog.php?b=3077

And for those who missed it.

Bohmian Mechanics
First published Fri Oct 26, 2001; substantive revision Tue Jul 3, 2012

Bohmian mechanics, which is also called the de Broglie-Bohm theory, the pilot-wave model, and the causal interpretation of quantum mechanics

…in Bohmian mechanics the configuration of a system of particles evolves via a deterministic motion choreographed by the wave function. In particular, when a particle is sent into a two-slit apparatus, the slit through which it passes and its location upon arrival on the photographic plate are completely determined by its initial position and wave function…

plato.stanford.edu/entries/qm-bohm/