## 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
Schrodinger, the father of the wave function. Nonetheless,
Schrodinger ultimately found this impossible to believe. His
difficulty had little to do with the novelty of the wave function:




> 
> 
> 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. (Schrodinger [1935] 1980: 327)
> 
> 
> 
> 



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 Schrodinger, echoing Einstein in the 1927 Solvay discussions (pp. 440-41 in Bacciagaluppi & Valentini 2009), 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 .... (Schrodinger
> [1935] 1980: 327-328) 
> 
> 
> 



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



> 
> 
> the \(\psi\)-function of the entire system having in it the living and
> the dead cat (pardon the expression) mixed or smeared out in equal
> parts. (Schrodinger [1935] 1980: 328) 
> 
> 
> 



It is thus because of the "measurement problem", of
macroscopic superpositions, that Schrodinger found it difficult
to regard the wave function as "representing reality". But
then what does? With evident disapproval, Schrodinger 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. (Schrodinger [1935] 1980:
> 328) 
> 
> 
> 



Many physicists pay lip service to the Copenhagen interpretation--that quantum mechanics is fundamentally about
observation or results of measurement. But (with the exception of the proponents of QBism) 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. 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. (Einstein et al. 1935: 780)
> 
> 
> 
> 



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 Schrodinger 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: 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. (Einstein 1949: 672) 
> 
> 
> 



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 



> 
> 
> 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. (von Neumann [1932] 1955: 325) 
> 
> 
> 



Physicists and philosophers of science almost universally accepted von
Neumann's claim. (The philosopher Grete Hermann was a notable exception.) For example, Max Born, who formulated the
statistical interpretation of the wave function, assured us that 



> 
> 
> 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.
> (Born 1949: 109) 
> 
> 
> 



Bohmian mechanics is a counterexample to the claims of von Neumann and Born.
Thus von Neumann's argument must be wrong. In fact, according to
John Bell, von Neumann's assumptions (about the relationships
among the values of quantum observables that must be satisfied in a
hidden-variables theory, see Bell 1966) are so unreasonable that the "the proof
of von Neumann is not merely false but *foolish*!"
(Mermin 1993: 805, fn 8, quoting an interview in *Omni*, May,
1988: 88). Nonetheless, some physicists continue to rely on von
Neumann's proof. 



Recently, however, physicists more commonly cite the
 Kochen-Specker Theorem
 and, even 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:



> 
> 
> The proof he [von Neumann] published ..., though it was made much
> more convincing later on by Kochen and Specker (1967), 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). (Wigner
> [1976] 1983: 291) 
> 
> 
> 



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:



> 
> 
> 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? (Bell 1982, reprinted in 1987c: 160) 
> 
> 
> 



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 may be, 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 1966,
reprinted 1987c: 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 Goldstein et al.
2011 and Norsen 2011.) 



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 almost did happen:
Schrodinger 1935, in his famous cat paper, came remarkably close
to discovering a Bell-type argument for quantum nonlocality. For
details see Hemmick and Shakur 2012, chapter 4.) 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:



> 
> 
> 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.
> (Bell 1966: 452; reprinted 1987c: 11) 
> 
> 
> 



In a footnote, Bell added that "Since the completion of this
paper such a proof has been found" (1966: 452, fn. 19). 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 first rather
conclusive such experiment was that of Aspect (Aspect, Dalibard, &
Roger 1982). More conclusive still is the experiment of Weihs
et al. 1998. Very recently there have been several "loop-hole
free" tests of Bell's inequality (Giustina et al. 2015;
Hensen et al. 2015; and Shalm et al. 2015. See also the entry on
 Bell's Theorem.).]



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 1981a,
> reprinted 1987c: 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 1981a, reprinted 1987c: 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, concerning EPRB correlations:



> 
> 
> 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 1981a, reprinted 1987c: 149) 
> 
> 
> 



As with just about everything else in the foundations of quantum
mechanics, there remains considerable controversy about what exactly
Bell's analysis demonstrates. (For further insight into the
various controversies see Maudlin 2014 and Goldstein et al. 2011. See also the entry on
 Bell's Theorem.)
Nonetheless, the opinion of Bell himself about what he showed is
perfectly clear. See Norsen 2011 for a nice overview of Bell's
views on the matter.


## 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 *Fuhrungsfeld* or guiding field (see Wigner
[1976] 1983: 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 Schrodinger's discovery of wave mechanics in
1926, i.e., of Schrodinger's equation, a discovery inspired by Louis de Broglie's 1924 Ph.D. thesis, 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: 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 (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.



For a very good discussion of the history of quantum mechanics, the
debates about its foundations, and about the reception of Bohmian
mechanics in particular, see Bricmont 2016 and Becker 2018. See also Beller 1999.


## 4. The Defining Equations of Bohmian Mechanics



In Bohmian mechanics the wave function, obeying
Schrodinger'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. (Bohmian
mechanics, like classical mechanics, is thus a theory grounded in
a *primitive ontology*, for Bohmian mechanics one of particles
described by their positions. This terminology was introduced in
Durr et al. 1992a. For more on the notion of primitive ontology,
see Allori et al. 2008 and Allori 2015.)



Warning: It is the positions of the particles in Bohmian mechanics
that are its "hidden variables", an unfortunate bit of
terminology. As Bell 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. (1987a,
> reprinted 1987c: 201-202) 
> 
> 
> 



Bohmian mechanics is the minimal completion of
Schrodinger'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 \(\psi = \psi(\boldsymbol{q}\_1 ,\ldots
,\boldsymbol{q}\_N) = \psi(q)\), 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
\(\mathbf{Q}\_1 ,\ldots ,\mathbf{Q}\_N\) 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: Schrodinger's equation 

\[
i\hslash\frac{\partial \psi}{\partial t} = H\psi
\]

 for
\(\psi(t)\), where \(H\) is the nonrelativistic (Schrodinger)
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 \(z = a +\di b)\) a first-order evolution equation,





**The Guiding Equation** 

\[ \frac{d\mathbf{Q}\_k}{dt} = \frac{\hslash}{m\_k} \Im
{{\psi^\*\partial\_k\psi} \over{\psi^\*\psi}} (\mathbf{Q}\_1 ,\ldots
,\mathbf{Q}\_N) \]

 




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 Schrodinger evolution
(Durr, Goldstein, & Zanghi 1992a: 852-854). Here
\(\hslash\) is Planck's constant divided by \(2\pi\),
*m\(\_k\)* is the mass of the \(k\)-th particle, and
\(\partial\_k = (\partial /\partial x\_k,\partial /\partial y\_k,\partial
/\partial z\_k)\) is the gradient with respect to the generic
coordinates \(\boldsymbol{q}\_k = (x\_k,y\_k,z\_k)\) of the \(k\)-th
particle. If \(\psi\) is spinor-valued, the two products involving
\(\psi\) 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 \(\psi\), global existence and uniqueness for the Bohmian
dynamics is a nontrivial matter. It is proven in Berndl, Durr, 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: 175-178) to
spin (Bell 1964: 10). And 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
Schrodinger'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
\(J/\varrho\), 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
Schrodinger'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 \(\partial
\varrho /\partial t + \textrm{div} J = 0\), an immediate consequence
of Schrodinger's equation, that if at some time (say the
initial time) the configuration \(Q\) of our system is random, with
distribution given by \(\lvert \psi \rvert^2 = \psi \*\psi\), 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 \(\lvert \psi
\rvert^2\). 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 & 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. (Marian,
Zanghi, and Oriols 2016 have recently proposed an explicit
procedure for performing such a measurement.) And, somewhat
paradoxically, it can be shown (Durr, Goldstein, &
Zanghi 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". And Mahler et al. 2016 have experimentally
found, using weak measurements, "Bohmian trajectories" for
entangled photons, illustrating quantum nonlocality and the phenomenon
of "surreal Bohmian trajectories". 



For a single particle the guiding equation defines the motion of a
particle guided by a wave in physical 3-dimensional space. One might
expect that similar motions might arise classically. Couder & Fort
(2006) have shown, by investigating interference-like phenomena in the
motion of bouncing oil droplets in a fluid, that this is indeed so.
Bush (2015) has further explored this sort of possibility for the
emergence of a Bohmian version of quantum mechanics from something
like classical fluid dynamics. A serious obstacle to the success of
such a program is the quantum entanglement and nonlocality
characteristic of many-particle quantum systems.



The "many interacting worlds" approach of Hall, Deckert,
& Wiseman (2014), independently advanced by Sebens (2015), retains
particle trajectories but attempts to do away with a wave-function at
the level of fundamental ontology. What takes its place is a large
number of trajectories in configuration space, with each trajectory
regarded as actual and as corresponding to the motions of a finite
number of particles in physical space regarded as describing a world
in its own right. These interact in such a way as to mimic
trajectories guided by a wave function: a coarse-grained density of
worlds takes the place of \(\lvert \psi \rvert^2\). The velocities of
the world-particles are required to at least approximately be related
to their configurations via some smooth function on configuration
space. This condition would be a rather surprising one for the sorts
of large systems studied in statistical mechanics. Even more
unmotivated from a classical perspective, the velocity function must
be the gradient of a multivalued function that obeys a
Bohr-Sommerfeld-like quantum condition. 



We stress that Bohmian mechanics should be regarded as a theory in its
own right. Its viability does not depend on its being derivable from
some other theory, classical or otherwise.


## 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 arrived at his theory by
writing the wave function in the polar form \(\psi =
R\)exp\((iS/\hslash)\), where \(S\) and \(R\) are real, with \(R\)
nonnegative, and rewriting Schrodinger's equation in terms
of these new variables to obtain a pair of coupled evolution
equations: the continuity equation for \(\varrho = R^2\) 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* 

\[
U = {-}\sum\_k (\hslash^2 /2m\_k) (\partial\_{k}^2 R / R ),
\]

 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 \(\partial\_k S\)
with \(m\_k\boldsymbol{v}\_k\), i.e., by setting 

\[
d\mathbf{Q}\_k /dt = \partial\_k S / m\_k.
\]

 This is
equivalent to
 the guiding equation
 for particles without spin. [In this form the (pre-Schrodinger
equation) de Broglie relation \(\boldsymbol{p} = \hslash
\boldsymbol{k}\), 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
Schrodinger's equation (Nerukh & Frederick 2000, Wyatt 2006).



However, Bohm's rewriting of Schrodinger'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:
Schrodinger'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: 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 Schrodinger'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




> 
> 
> 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. (Feynman,
> Leighton, & Sands 1963: 37-2) 
> 
> 
> 



This experiment 



> 
> 
> 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. (Feynman 1967: 130) 
> 
> 
> 



As to the question, 



> 
> 
> 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. (Feynman 1967: 145) 
> 
> 
> 



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.



![[line drawing, a vertical bar on the left with two openings, multiple lines angle out to the right from each]](figure1.gif)

Figure 1: An ensemble of trajectories
for the two-slit experiment, uniform in the slits.
  

(adapted by Gernot Bauer from Philippidis, Dewdney, & Hiley 1979:
23, fig. 3)




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:



> 
> 
> 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. (Bell [1989] 1987c: 191) 
> 
> 
> 



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".



For an accessible presentation of the behavior of Bohmian trajectories
in scattering and tunneling phenomena, see Norsen 2013.


## 7. The Measurement Problem



The
 measurement problem
 is the most commonly cited of the conceptual difficulties that plague
quantum mechanics. (It amounts, more or less, to the paradox of
Schrodinger'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 Schrodinger'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
Schrodinger'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, produced by interaction with the environment
(air molecules, cosmic rays, internal microscopic degrees of freedom,
etc.), make it very difficult for significant overlap to develop
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 Durr, Goldstein &
Zanghi 1992a: 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 (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,
that the collapse of the wave function is an *effective*
collapse. However, there is a sense in which the collapse of the wave
function in Bohmian mechanics is more than a matter of convenience, a
sense in which the collapse is actual and not merely
effective. Indeed, 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 wave function does in
fact 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 Durr,
Goldstein, & Zanghi 1992a: Section 5, discuss in some
detail, together with the related notion of the effective wave
function. 



One of the defining equations of Bohmian mechanics is
Schrodinger's equation, governing the evolution of the wave
function. Among the defining equations there is no additional
evolution equation prescibing collapse of the wave function.
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 \(\Psi = \Psi(q) = \Psi(x,
y)\). According to Bohmian mechanics, the larger system is then
completely described by \(\Psi\), evolving according to
Schrodinger'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* 

\[\psi(x) = \Psi(x, Y)\]

 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 \(t\) 

\[\psi\_t (x) = \Psi\_t (x, Y\_t)\]

 via the time dependence
of \(Y\) as well as that of \(\Psi\), it is not difficult to see
(Durr, Goldstein, & Zanghi 1992a) the following two
things about the evolution of the conditional wave: First, that it
obeys Schrodinger's equation for the subsystem when that
system is suitably decoupled from its environment. Part of what is
meant by this decoupling is that \(\Psi\) 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
\(\psi\) the probability density for finding its configuration to be
\(q\) is \(\lvert\psi(q)\rvert^2\). 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 Schrodinger equation for both) provided that it is
somehow true for Bohmian mechanics that configurations are random,
with distribution given by the *quantum equilibrium*
distribution \(\lvert\psi(q)\rvert^2\). Now the status and
justification of this *quantum equilibrium hypothesis* is a
rather delicate matter, one that has been explored in considerable
detail (Durr, Goldstein, & Zanghi 1992a). 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 \(\varrho^{\psi}\), depending upon
the wave function \(\psi\), is *equivariant* if 

\[(\varrho^{\psi})\_t = \varrho^{\psi\_t}\]

where the dependence on \(t\) on the right arises from
Schrodinger'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 \(\varrho^{\psi}\), of the Schrodinger 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 \(\varrho^{\psi} =
\lvert\psi(q)\rvert^2\) is equivariant. (It can be shown in fact that
this is more or less the only equivariant possibility that is suitably
local (Goldstein & 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*, \(\varrho = \lvert\psi \rvert^2\)




as roughly analogous to (classical) 




*thermodynamic equilibrium*, \(\varrho = \textrm{exp} (-H/kT)
/Z\),




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 &
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 exhibited by ensembles of actual
subsystems within a *typical* individual universe (Bell 1981b,
reprinted 1987c: 129; Durr, Goldstein, & Zanghi
1992a). And in both cases the status of, and justification for,
equilibrium distributions remains controversial (Durr &
Struyve 2020; see also Goldstein 2012). 



It can nonetheless be shown (Durr, Goldstein, & Zanghi
1992a) 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--in much the same manner that 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".



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 1957) of quantum randomness for
"many worlds", despite the huge metaphysical differences
that exist between these two versions of quantum theory.


## 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: 



> 
> 
> [I]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. (Bell 1982, reprinted 1987c: 166)
> 
> 
> 
> 



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
Schrodinger 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 POVMs, \(O(dz)\),
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 \(\psi\), is given by \(\langle \psi |
O(dz)\psi \rangle\) (where \(\langle | \rangle\) is the usual
 inner product
 between quantum state vectors).



Moreover, this conclusion 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. 1997a).


## 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, \(\pm
\hslash/2\)). 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 (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 Schrodinger 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.



For further discussion and more detailed examples of the Bohmian
perspective on spin see Norsen 2014.


## 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" (1964, reprinted 1987c: 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 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. (1993: 100)
> 
> 
> 



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 (1992: 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:



> 
> 
> 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". (Bell 1982, reprinted 1987c: 166) 
> 
> 
> 


## 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 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. (Bell 1980, reprinted
> 1987c: 115) 
> 
> 
> 



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, the elimination of this nonlocality, if possible at all, is extremely difficult (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; hence the
choice of orientation of the distant magnet strongly influences
that behavior.



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 Schrodinger'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. (For a more
explicit and detailed discussion see Norsen 2014.)



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 is not compatible with special relativity, a central
principle of physics: 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 & Hiley 1993; Durr 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
Renee 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. (Interview with John Bell, in Weber 1989 [Other Internet
> Resources])
> 
> 
> 



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 in such a
theory would be 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 Durr 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 or light-cone
structure. For more on the latter possibility, see Goldstein and
Tumulka's model (2003), in which they reconcile relativity and
nonlocality through the interplay of opposite arrows of time. For a
discussion of the former possibility, see Durr et al. 2014. In
the sort of theory discussed there, the wave function of the universe
provides a covariant prescription for the desired foliation. Such a
theory would be clearly Lorentz invariant. But it is not so clear that
it should be regarded as relativistic.



Be that as it may, Lorentz invariant nonlocality remains somewhat
enigmatic. The issues are extremely subtle. For example, Bell 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. (1981a, reprinted 1987c: 155) 
> 
> 
> 



While quantum equilibrium and the absolute uncertainty that it entails
(Durr, Goldstein, & Zanghi 1992a) may be of some help
here, the situation remains puzzling.


## 15. Identical Particles



Fundamental particles of the same kind, for example electrons, are
treated in quantum mechanics as if they are somehow identical or
indistinguishable. This treatment is reflected in the symmetry
properties of the wave function of a many-particle system under
permutations of the coordinates of those particles. For electrons, and
for other *fermions*, the wave function must change its sign
when the coordinates of a pair of particles are exchanged; it must
be *antisymmetric*. For photons and other *bosons* it
must be symmetric, with no change at all.



The justification usually given for this is that the only way to keep
track of the individual particles and thereby retain their
individuality is by following their trajectories, which of course one
cannot and must not do, and in any case does not have, in standard
quantum mechanics. (Like so many arguments used to justify various
claims in quantum mechanics, this sort of argument, with its
positivistic slant, is rather weak. Its conclusion, however, is quite
solid.) On this basis one often hears it maintained that Bohmian
mechanics, because its particles have trajectories (deterministic ones
to boot), is unable to deal with identical particles.



There is, however, no problem whatsoever in incorporating bosons and
fermions into Bohmian mechanics. One simply stipulates that the wave
function in Bohmian mechanics has the same symmetry properties under
permutations as in standard quantum mechanics. Once again, in Bohmian
mechanics one does not change the wave function or its evolution
equation; one merely adds to it the actual configuration of the
particles and its guiding equation.



In fact, by taking the configuration seriously in Bohmian mechanics,
one perhaps arrives even more naturally than in standard quantum
mechanics at the classification of fundamental quantum particles as
bosons and fermions, with wave functions of the appropriate
symmetries. That's because the natural configuration space for a
system of \(N\) particles, say electrons, described by the same
dynamical parameters such as charge, consists of \(N\)-point subsets
of physical space, and not \(N\)-tuples of points in physical
space. It is natural, in other words, to regard the labelling we
assign to particles as an unphysical convenience, and to use on the
fundamental level unlabelled configurations rather than labelled
ones. Wave functions on this natural configuration space are in effect
symmetric, trivially so.



Moreover the natural configuration space has a non-trivial
topology. When this is taken into account the possibility of
antisymmetric wave functions, of fermions, naturally
emerges. (Durr et al. 2006, following Nelson 1985 for stochastic
mechanics. For a similar early analysis that is more traditionally
quantum, see Leinaas and Myrheim 1977.)


## 16. Quantum Motion on Shape Space



A natural thought that one sometimes hears expressed is that Bohmian
mechanics, for which particles with well-defined positions in space
play a central role, cannot easily accommodate a relational framework,
as most prominently advocated by Julian Barbour (1999). With such a
framework, location in space is not a fundamental physical
notion. Rather it is shapes formed by arrangements of particles,
shapes determined by relative positions, that are physical.



Bohmian mechanics can naturally be extended to a relational framework,
which also leads to a relational notion of time as well. Within this
framework, the more familiar descriptions in terms of absolute space
and time emerge merely as corresponding to a choice of gauge
(Durr et al. 2020a). Thus, rather than being a regression to
outdated modes of physics, a Bohmian perspective suggests the
possibility that much of what we regard as fundamental in physics
might in fact be imposed by us, through our choice of gauge.


## 17. 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
Schrodinger'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,



> 
> 
> 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 encyclopedia 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! (2005: 262) 
> 
> 
> 



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: 18) 
> 
> 
> 



More recently, Sir Anthony Leggett has echoed this charge. Referring
to the measurement problem, he says that Bohmian mechanics provides
"little more than verbal window dressing of the basic
paradox" (Leggett 2005: 871). 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: 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 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 Schrodinger's cat. (2005:
> 811) 
> 
> 
> 



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 Schrodinger'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.
1997b), scattering theory (Durr et al., 2000), and quantum chaos
(Cushing 1994; Durr, Goldstein, & Zanghi
1992b). Moreover, the additional resources present in Bohmian
mechanics, in particular the notion of the conditional wave function
defined in Section 8, have been useful for the development of
approximation schemes for dealing with practical quantum applications
(Oriols & Mompart 2012, Struyve 2020).



Especially problematical from an orthodox perspective is quantum
cosmology, for which the relevant quantum system is the entire
universe, and hence there is no observer outside the system to cause
collapse of the wave function upon measurement. In this setting
Bohmian models have clarified the issue of the inevitability of the
presence of singularities in theories of quantum gravity (Falciano,
Pinto-Neto, & Struyve 2015). For a discussion of how a Bohmian
perspective can eliminate some conceptual difficulties arising in
quantum gravity, such as the problem of time and the problem of the
absence of sufficiently many diffeomorphism-invariant observables, see
Goldstein & Teufel 2001. 



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: 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 Schrodinger'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. Durr,
Goldstein, & Zanghi (1997) and Goldstein & 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
Schrodinger-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 & Hiley 1993, Holland 1993, Bell 1987b), and
in some of the articles in Cushing, Fine, & Goldstein 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" (Durr 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.)



Inspired by the structure of Bell-type quantum field theories, Tumulka
has developed a novel approach to the problem of the ultraviolet
divergences of quantum field theory based on what he
calls *interior boundary conditions*. (See D. Durr et
al. 2020b and the references therein.) 



For a brief introduction to Bohmian mechanics see Tumulka 2021. For
longer accessible presentations see Bricmont 2016, Norsen 2017,
Bricmont 2018, and Maudlin 2019.