

Is de Broglie-Bohm Theory Specially
Equipped to Recover Classical Behavior?

Joshua Rosaler

Abstract

Supporters of the de Broglie-Bohm (dBB) interpretation of quantum the-
ory argue that because the theory, like classical mechanics, concerns the mo-
tions of point particles in 3D space, it is specially suited to recover classical
behavior. I offer a novel account of classicality in dBB theory, if only to
show that such an account falls out almost trivially from results developed
in the context of decoherence theory. I then argue that this undermines any
special claim that dBB theory is purported to have on the unification of the
quantum and classical realms.

1 Introduction

Several advocates of the de Broglie-Bohm (dBB) interpretation of quantum
theory hold that because, like classical mechanics, it concerns the motions of
point particles in 3D space, it is specially suited to recover classical behavior.
1 2 They note that in dBB theory, we can ask simply: under what circum-
stances do the additional particle configurations posited by the theory follow
approximately Newtonian trajectories? Moreover, the equations of motion
for the additional “hidden” variables in dBB theory take the form of classical
equations of motion, but with an additional “quantum potential” or “quan-
tum force” term that produces deviations of the trajectories from classicality.
On this basis, a number of authors have suggested that dBB theory furnishes
special tools to recover classical behavior that other interpretations do not,
via the requirement that the quantum potential or force be negligibly small
(Allori et al., 2002), (Bohm and Hiley, 1995).

1By“classical,” I mean having sharply defined values for properties such as position
and momentum and conforming approximately to Newtonian equations of motion.

2For brevity, I refer to the de Broglie-Bohm interpretation of quantum theory as “dBB
theory.”

1


Here, I provide an alternative to existing accounts of classicality in dBB
theory, if only to show that such an account falls out almost trivially from
results developed outside the context of dBB theory, in the literature on deco-
herence. Formal tools specific to dBB theory, such as the quantum potential
or quantum force, turn out to be neither necessary nor helpful to the anal-
ysis. Here, I regard decoherence theory as an interpretation-neutral body of
results that follow when both a system and its environment (including any
observers, measuring apparati, and residual microscopic degrees of freedom)
are modeled as a closed system that always obeys the unitary Schrodinger
evolution. On my account of classicality in dBB theory, the Bohmian con-
figuration evolves classically only because it succeeds in tracking one among
the many branches of the total quantum state defined by decoherence, and
the approximate classicality of such a branch in turn is a consequence of the
unitary Schrodinger evolution, on which dBB theory has no special claim.

Since the dBB account of classicality is entirely parasitic on the branch-
ing structure defined by decoherence, the claim that dBB theory is uniquely
suited to recover classical behavior must already presuppose what is at is-
sue in debates about the interpretation of quantum mechanics: namely, that
it provides the one true account of how one among the many potentiali-
ties/branches contained in a decohered quantum superposition gets selected
as “the outcome” (necessarily, in accordance with Born Rule). Insofar as
other interpretations furnish viable resolutions to this issue, they should be
able to make similar use of decoherence-based results to recover classical be-
havior on their own terms. For example, (Wallace, 2012), Ch. 3 provides a
decoherence-based account of macroscopic classical behavior on the Everett
Interpretation. At most, the fact that dBB theory can recover classical be-
havior on its own terms can be regarded as an internal test of the theory’s
adequacy; it should not be regarded as a point in favor of dBB theory over
other interpretations.

In Section 2, I review the major existing lines of approach to modelling
classical behavior in dBB theory and highlight a number of gaps in these
accounts. In Section 3, I explain the basic mechanism whereby environ-
mental decoherence helps to recover macroscopic classical behavior in dBB
theory. In Section 4, I summarize the analysis of macroscopic classical be-
havior furnished by decoherence theory on the bare formalism of quantum
mechanics (that is, Schrodinger evolution without collapse) . In Section 5,
I show how the dBB account of classical behavior follows straightforwardly
from the decoherence-based analysis of the previous section.

2


2 Existing Accounts of Classicality in dBB

Theory

Bohm’s theory posits that the state of any closed system is given by a wave
function Ψ(X,t) that always evolves according to the Schrodinger equation
(with the same Hamiltonian as in conventional quantum mechanics), and a

configuration q that evolves according to the equation dq
dt

=
∇S(X,t)|q

m
, where S

is the phase of the wave function; it also posits that, at some initial time t0,
our knowledge of the particle configuration is characterized by the probability
distribution |Ψ(X,t0)|2 (Bohm, 1952a), (Bohm, 1952b). Efforts to model
classical behavior in Bohm’s theory fall into two broad categories: 1) what
I call “quantum potential” approaches, which rely on the vanishing of the
quantum potential and/or force, and in which the Bohmian configuration
occupies center stage; 2) what I call “narrow wave packet” approaches, which
take an analysis of the wave function as their starting point and treat the
Bohmian configuration as being in some sense simply “along for the ride.”

2.1 Quantum Potential Approaches

If one plugs the polar decomposition of the wave function, Ψ(X,t) = R(X,t)ei
S(X,t)

~ ,
into Schrodinger’s equation, one obtains the following relations as the real
and imaginary parts, respectively, of the resulting relation:

∂S

∂t
=

(∇S)2

2M
+ V −

~2

2M

∇2R
R

(1)

∂R2

∂t
+ ∇· (

∇S
M

R2) = 0. (2)

The first equation takes the form of the classical Hamilton-Jacobi equation,
except for the additional term Q ≡ − ~

2

2M
∇2R
R

, known as the “quantum po-
tential.” The second equation takes the form of a continuity equation for the
probability distribution ρ ≡ R2. Together with the evolution equation for
the dBB configuration, dq

dt
=
∇S(X,t)|q

M
, the first of these equations implies that

this configuration obeys the relation,

M
d2q

dt2
= −∇V |q −∇Q|q, (3)

3


which mathematically resembles Newton’s Second Law, but with an addi-
tional “quantum force” term −∇Q. When ∇Q ≈ 0, q follows an approxi-
mately Newtonian trajectory; some authors have also suggested Q ≈ 0 as a
requirement for classicality, though the sufficiency of this condition for clas-
sicality presupposes implicitly that ∇Q ≈ 0 is also satisfied Many supporters
of dBB theory feel that the quantum potential/force’s being approximately
equal to zero furnishes a simple, transparent condition for classicality, and
moreover, one that is unique to dBB theory. Bohm and Hiley, Holland, and
Allori, Durr, Goldstein and Zanghi are among the authors who offer analyses
of classicality rooted in this supposition (Bohm, 1952a), (Bohm and Hiley,
1995), (Holland, 1995), (Allori et al., 2002), (Durr and Teufel, 2009). Some of
these authors also suggest ways of generalizing this approach to incorporate
environmental decoherence, but do not explain in detail how the conditions
Q ≈ 0, ∇Q ≈ 0 are to be extended to the case where the system is open,
or why this would be useful given that the results (1) and (3) were derived
under the now-abandoned assumption that the system is closed and in a pure
state.

2.2 Narrow Wave Packet Approaches

A second approach that is sometimes adopted in the effort to model classical
behavior in dBB theory makes use of Ehrenfest’s Theorem,

M
d〈P̂〉
dt

= −〈
∂̂V

∂X
〉, (4)

which applies generally to wave functions evolving under the Schrodinger

equation with a Hamiltonian of the form Ĥ = P̂
2

2M
+ V (X̂). The crucial

point in this approach is that for wave packets narrowly peaked in position
(narrowly, that is, relative to some characteristic length scale on which the
potential V changes ), one has approximately that

M
d〈P̂〉
dt
≈−

∂V (〈X̂〉〉)
∂〈X̂〉

, (5)

which entails that the expectation value of position 〈X̂〉 evolves approxi-
mately along a Newtonian trajectory. In dBB theory, the property of equiv-
ariance, whereby the Born Rule probability distribution |Ψ(X,t)|2 is pre-
served by the flow of the configuration variables, ensures that the Bohmian

4


trajectory will follow the wave packet and therefore traverse the same New-
tonian trajectory as the expectation value 〈X̂〉. The primary advocate of
the narrow wave packet approach in the literature has been Bowman, who
has also actively criticized approaches based on the quantum potential and
quantum force (Bowman, 2005). Bowman notes that the narrow wave packet
approach, as applied to isolated systems, does not explain why the state of
S should be a narrow wave packet to begin with; however, he argues, cor-
rectly on my view, that this can be corrected by incorporating environmental
decoherence into the analysis. However, Bowman’s account does not recog-
nize the need, not just for decoherence, but for a special type of decoherence
that ensures disjointness of the branches of the total quantum state in the
combined configuration space of the system and environment. Moreover,
Bowman confines his attention to the reduced dynamics of the open system
whose classicality we seek to model (say, the center of mass of the moon)
and does not consider the structure of the overall pure state of the system
and environment; for reasons that will become clear in the next section, this
restriction obscures the true mechanism whereby the Bohmian configuration
of the system comes to be guided by just one of the narrow wave packets
present in the overall superposition.

3 The Role of the Environment

To illustrate the role of the environment in recovering classicality from dBB
theory, and why it is not sufficient to model a macroscopic system like the
moon’s center of mass as an isolated system, it is instructive to consider a
simple example. Let S be the center of mass of some macroscopic body, and
assume to begin with that the system is always closed and in a pure state.
Let the pure state be a superposition of narrow wave packets with opposite
average momenta, initially separated across a macroscopically large distance
in space; in addition, let the time evolution of the state be such that the
trajectories of the packets - for simplicity, assume they are straight lines and
that the system is free - overlap at some point in S’s configuration space
and then pass through each other (so that the trajectories of the two packets
over time make the shape of an “X”). Then the quantum state at every time
takes the form,

5


|Ψ〉 =
1
√

2
[|q1,p〉 + |q2,−p〉], (6)

where |q,p〉 designates the quantum state of a wave packet simultaneously
peaked about postion q and momentum p (to within the restrictions of the
uncertainty principle) and q1 and q2 change with time in the manner specified.
Assuming the mass M to be macroscopically large, we can neglect spreading
of these wave packets. 3 Now consider an ensemble of initial Bohmian config-
urations associated with this pure state. Those trajectories associated with
initial conditions in the first packet initially will follow the classical straight-
line trajectory of that packet, and likewise for the second packet. However,
Bohmian trajectories of a closed pure-state system cannot cross, so when the
packets overlap in configuration space, trajectories initially associated with
one wave packet will exit the overlap region in the wave packet in which they
did not begin, rather than proceeding in a straight line with their original
wave packet. Thus, the trajectories will exhibit highly non-classical kink as
a consequence of the overlap. In dBB theory, such non-classicalities in the
Bohmian trajectory are a generic consequence of wave packets overlapping
in configuration space, even in cases where the mass M is macroscopically
large (M & 1kg.) and wave packet spreading can be neglected; also, they are
not restricted to the simple case of a free particle discussed here.

Let us now abandon the assumption that the system S is isolated and
allow it to interact with and become entangled with its environment E. Now
it is the combined system SE, rather than S, that is closed and in a pure
state, though it is still S’s classicality that we wish to recover. Let us assume
that at every time the wave function of the closed system SE consisting
of the center of mass and its environment (which may consist of photons,
neutrinos, or other particles of matter) takes the form

|Ψ〉 =
1
√

2
[|q1,p〉⊗ |φ1〉 + |q2,−p〉⊗ |φ2〉], (7)

for some states |φ1〉 and |φ2〉 in E’s Hilbert space HE, where |q1,p〉 and
|q2,p〉 follow the same trajectories through S’s configuration space as in the

3Using the formula σ(t) =
√
σ20 + (

~t
Mσ0

)2 for the time dependence of the width of

an initial Gaussian under free evolution, one can show that for a free particle of mass
M ∼ 1kg., the time it takes for a wave packet initially localized on the scale of an Angstrom
to spread to a centimeter is longer than the age of the universe.

6


isolated case just considered. Moreover, assume that |φ1〉 and |φ2〉 have
disjoint supports in E’s configuration space QE - that is, that they are “su-
perorthogonal”: 4

〈φ1|y〉〈y|φ2〉≈ 0 for all y ∈ QE (8)

where |y〉 is a position (or more accurately, configuration) eigenstate of the
environment. Because of the disjointness of the supports of |φ1〉 and |φ2〉 in
QE, the packets |q1,p〉⊗|φ1〉 and |q2,−p〉⊗|φ2〉 will remain disjoint in the total
configuration space QSE, even when |q1,p〉 and |q2,−p〉 overlap in QS. The
non-overlap condition for Bohmian trajectories applies only to trajectories
in QSE since the state is pure only relative to SE, so the configuration
qSE = (QS,qE) of the whole system will forever remain in one wave packet
or the other - say, the first one. As a consequence, the configuration QS
associated with S always follows the classical trajectory of the wave packet
|q1,p〉 in QS. There is no “kink” as in the isolated case. If E contains on
the order of 1023 microscopic degrees of freedom, as it typically will, we can
expect the relation (8) to hold irreversibly, and for this reason can effectively
ignore the second wave packet since it will have no influence on the motion of
the total configuration. Moreover, the configuration qE of the environment
will be irreversibly correlated to the wave packet |q1,p〉, since it is bound lie
in the support of 〈y|φ1〉 (if qSE had started in the second packet, qE would
instead be in the disjoint region associated with the support of 〈y|φ2〉 and be
correlated with the wave packet |q2,−p〉).

4 Decoherence-Based Models of Classical Be-

havior

In this section, I describe the evolution of the pure state of a closed system
SE consisting of the center of mass S of some macroscopic body and its
environment E, on the assumption that this evolution is always governed
by the Schrodinger equation. This analysis draws most directly from (Joos

4By “support” of a configuration space function, I mean the region of configuration
space in which the function’s value is not negligibly small - so, greater than some arbitrarily
chosen small �. Note that superorthogonality implies orthogonality but is not implied by it;
the term “superorthogonal” can be traced back to (Bohm and Hiley, 1995) and (Maroney,
2005) .

7


et al., 2003), Ch.’s 3 and 5, (Hartle, 2011), and (Wallace, 2012), Ch.3. In
the next section, I show that an account of classicality at the level of the
Bohmian configuration follows straightforwardly from this analysis.

The quantum description of the closed system in question takes as its
Hilbert space H = HS ⊗HE, the tensor product of the Hilbert space HS
associated with the center of mass of the body in question and the Hilbert
space HE associated with the residual microscopic degrees of freedom in the
environment (which includes degrees of freedom both internal and external
to the body in question). The dynamics with respect to this set of variables
are given by a Schrodinger equation of the form

i~
∂|Ψ〉
∂t

=
(
ĤS ⊗ ÎE + ÎS ⊗ ĤE + ĤI

)
|Ψ〉, (9)

where |Ψ〉 ∈ HS ⊗HE, ÎE is the identity operator on HE, ÎS the identity
operator on HS, and ĤI is an interaction Hamiltonian acting on HS⊗HE. In
the models of interest here, ĤS =

P̂2

2M
+ V (X̂), and ĤI is a function of only

of center-of-mass position X̂ and the positions of environmental particles,
represented collectively by ŷ. At a more coarse-grained level, we can examine
the evolution of the reduced density matrix ρ̂S ≡ TrE|Ψ〉〈Ψ| of S. For a
wide variety of models, in which environmental decoherence is significant
but dissipative effects can be ignored, the evolution of ρ̂S is governed by the
equation,

i~
∂ρ̂S
∂t

=
[
ĤS, ρ̂S

]
− iΛ

[
X̂,
[
X̂, ρ̂S

]]
, (10)

where the first term generates unitary evolution prescribed by ĤS and the
second represents the effect of decoherence from the environment; the sec-
ond term suppresses the off-diagonal elements of 〈X′|ρ̂S|X〉 throughout its
evolution, and Λ is a constant derived from the parameters in the closed
system Hamiltonian in (9) (This is an important special case of the well-
known Caldeira-Leggett equation; for further discussion and derivation of
this equation, see (Joos et al., 2003) and (Schlosshauer, 2008)). From (10),

one can show that M
d〈P̂〉
dt

= −〈 ∂̂V
∂X
〉, where 〈Ô〉≡ Tr[ρ̂SÔ] for any Hermitian

operator Ô on HS; this constitutes a generalization of Ehrenfest’s Theorem
to open, decohering quantum systems (Joos et al., 2003). By analogy with
the case of closed systems, one can show that when the width of the dis-
tribution ρS(X) ≡ 〈X|ρ̂S|X〉, known as the ensemble width of S, is narrow

8


by comparison with the characteristic length scales on which V varies, we

have M
d〈P̂〉
dt
≈−∂V (〈X̂〉〉)

∂〈X̂〉
, which entails that the expectation value of position

〈X̂〉 = TrS(ρ̂SX̂) follows an approximately Newtonian trajectory as long as
the width of the distribution ρS(X), also known as the ensemble width of ρ̂S,
remains narrowly peaked relative to the characteristic length scales on which
V varies. The timescales on which the ensemble width of an initially narrow
ρS(X) remains narrowly peaked will depend both on the value of the mass
M and on the strength of chaotic effects in the Hamiltonian ĤS (for further
discussion of the role of chaos in wave packet spreading in open systems, see
(Zurek and Paz, 1995) ).

Let us now consider what constraints this analysis of ρ̂S places on the
the evolution of the pure state |Ψ〉 of the total system SE, recalling that
ρ̂S ≡ TrE|Ψ〉〈Ψ|. The decoherent or consistent histories formalism will prove
especially useful for this purpose. 5 Consider a partition {Σα} of the classical
phase space associated with the system S such that the cells Σα all have equal
phase space volume. Using this partition, we can define the positive operator-
valued measure (POVM) given by the operators Π̂α ≡

∫
Σα
dz |z〉〈z|, where

z ≡ (q,p) is a notational shorthand for a point in phase space, and |z〉 is
a minimum-uncertainty coherent state centered on the phase space point z.
6 7 If the cells Σα are significantly larger than the volume in phase space
over which coherent states have strong support (i.e., ~), then the operators
Π̂α constitute an approximate PVM since in this case Π̂αΠ̂β ≈ δαβΠ̂α. We
can extend this approximate PVM on HS to an approximate PVM {P̂α}
on HS ⊗HE by defining P̂α = Π̂α ⊗ ÎE. Inserting factors of the identity
ÎSE =

∑
αi
P̂αi at regular time intervals of the unitary evolution, we can

then write the state evolution at successive time intervals N∆t as follows:

5For an introduction to the decoherent histories formalism, see for example (Gell-Mann
and Hartle, 1993), (Griffiths, 1984), (Halliwell, 1995).

6For my purposes, it is sufficient for the reader to think of a coherent state state simply
as a Gaussian wave packet narrowly peaked both in position and momentum.

7A positive-operator-valued measure (POVM) on H is a set {Π̂α} of positive opera-
tors such that

∑
α Π̂α = Î; recall that an operator Ô is positive if it is self-adjoint and

〈Ψ|Ô|Ψ〉 ≥ 0 for every |Ψ〉 ∈ H. A projection-valued measure (PVM) {P̂α} on Hilbert
space H is a POVM such that P̂αP̂β = δαβP̂α (no summation over repeated indices).

9


|Ψ(N∆t)〉 = e−
i
~ĤN∆t|Ψ0〉 (11)

= (
∑
αN

P̂αN )e
− i~Ĥ∆t(

∑
αN−1

P̂αN−1 )...(
∑
α1

P̂α1 )e
− i~Ĥ∆t(

∑
α0

P̂α0 )|Ψ0〉 (12)

=
∑

α0,...,αN

Ĉα0,...,αN|Ψ0〉 (13)

where the components Ĉα0,...,αN|Ψ0〉 are defined by

Ĉα0,...,αN|Ψ0〉≡ P̂αNe
− i~Ĥ∆tP̂αN−1...P̂α1e

− i~Ĥ∆tP̂α0, |Ψ0〉. (14)
The reason for using this particular approximate PVM will be made clear
shortly. Each component Ĉα0,...,αN|Ψ0〉, corresponds to a particular “history”
or sequence (Σα0, ..., ΣαN ) of regions through phase space. Let us examine in
more detail the structure of one of these components. Using the definition
of the operators P̂αi we can write,

Ĉα0,...,αN|Ψ0〉 =
∫

Σi0

...

∫
ΣiN

dz0...dzN |zN〉⊗ |φ̃(z0, ...,zN )〉 (15)

=

∫
Σi0

...

∫
ΣiN

dz1...dzN w(z0, ...,zN ) |zN〉⊗ |φ(z0, ...,zN )〉

(16)

with |φ̃(z0, ...,zN )〉 ≡
∑

i |ei〉〈zN,ei|Ĉα0,...,αN|Ψ0〉 ∈ HE for {|ei〉} any basis
of HE, w(z0, ...,zN ) ≡

√
〈φ̃(z0, ...,zN )|φ̃(z0, ...,zN )〉, and |φ(z0, ...,zN )〉 ≡

|φ̃(z0,...,zN )〉
w(z0,...,zN )

. As Zurek has shown, the coherent states |z〉 for systems like S
are special in that under fairly generic conditions, they become entangled
with the environment only on much longer timescales than other states in
HS; he calls such states “pointer states” (Zurek et al., 1993). As Wallace
demonstrates in detail in (Wallace, 2012), Ch.3, continuous monitoring of
the center of mass position by the environment (usually via scattering of
photons, air molecules, etc. by the center of mass) enforces the relation:

〈φ(z′0, ...,z
′
N )|φ(z0, ...,zN )〉≈ 0 (17)

for zi and z
′
i differing by more than the width of a coherent wave packet, for

any 0 ≤ i ≤ N. From (15) and (17) it follows immediately that

10


〈Ψ0|Ĉ
†
α′0,...,α

′
N
Ĉα0,...,αN|Ψ0〉≈ 0 (18)

if αi 6= αi for any 0 ≤ i ≤ N. When this condition holds, each component
Ĉα0,...,αN|Ψ0〉 of the total superposition is said to constitute a “branch” of the
quantum state, or simply a branch state (note that, as written, they are not
normalized). Thus, we can see the reason for the choice of the approximate
coherent state PVM: the histories defined in terms of this PVM are mutually
decoherent, which follows as a consequence of the fact that the states |z〉
are generically pointer states for systems like S. In turn, satisfaction of (18)
for each N ensures that the only allowable transitions from branch states
Ĉα0,...,αN|Ψ0〉 at an earlier time to branch states Ĉβ0,...,βM|Ψ0〉 at a later time
(with N < M), are those for which (β0, ...,βN ) = (α0, ...,αN ) - that is, such
that the history associated with the earlier state is an initial segment of the
history associated with the later state. This is part of what is meant when
decoherence is said to generate a branching structure for the quantum state.

As a consequence of the open systems version of Ehrenfest’s Theorem, on
time scales where ensemble spreading can be ignored, Ĉα0,...,αN|Ψ0〉 ≈ 0 for
all histories (α0, ...,αN ) that are not approximately classical. Thus, we can
restrict the sum (11) to the subset Hc of histories that are approximately
classical:

|Ψ(N∆t)〉≈
∑

(α0,...,αN )∈Hc

Ĉα0,...,αN|Ψ0〉 (19)

From this we can see that relative to a single branch Ĉα0,...,αN|Ψ0〉, the
mean values of S’s position and momentum at each time step i∆t (with
1 ≤ i ≤ N) lie along an approximately classical trajectory, and the en-
semble distributions in position and momentum relative to this branch re-
main tightly peaked around these values. 8 Moreover, it follows from (17)
that the reduced density matrix of E relative to branch α ≡ (α0, ...,αN ),
ρ̂αE ≡

1
|w(α)|2 TrS[Ĉα|Ψ0〉〈Ψ0|Ĉ

†
α], exhibits a strong correlation to this trajec-

tory in that it is orthogonal to the reduced density matrix ρ̂α
′
E associated with

any other trajectory/branch α′ - i.e., TrE(ρ̂
α
Eρ̂

α′
E ) ≈ δαα′.

8Relative to the branch Ĉα0,...,αN |Ψ0〉, the expectaiton values of position and momen-
tum at times i earlier than N are given respectively by 1|w(α0,...,αi)|2〈Ψ0|Ĉ

†
α0,...,αi

(X̂ ⊗
ÎE)Ĉα0,...,αi|Ψ0〉 and

1
|w(α0,...,αi)|2

〈Ψ0|Ĉ†α0,...,αi (P̂ ⊗ ÎE)Ĉα0,...,αi|Ψ0〉.

11


5 The dBB Model of Macroscopic Newtonian

Systems

Since the wave function in dBB theory obeys the same Schrodinger dynam-
ics as was assumed in the analysis of the previous section, the quantum
state in the corresponding dBB model also takes the form (19). However,
we saw in Section (8) that classicality in Bohm’s theory requires not just
the orthogonality of environmental states associated with different branches,
represented in (17), but the stronger condition of superorthogonality, which
ensures disjointness of these states in QE:

〈φ(z′0, ...,z
′
N )|y〉〈y|φ(z0, ...,zN )〉≈ 0 ∀y ∈ QE, (20)

for zi and z
′
i sufficiently different for any 0 ≤ i ≤ N. Typically, the unitary

Schrodinger evolution will also enforce this stronger condition. It follows im-
mediately from (20) that the branch states associated with different histories
are disjoint in the full configuration space QSE:

〈Ψ0|Ĉ
†
α′0,...,α

′
N
|X,y〉〈X,y|Ĉα0,...,αN|Ψ0〉≈ 0 ∀(X,y) ∈ QSE (21)

if αi 6= α′i for any 0 ≤ i ≤ N. As a consequence of this disjointness,
the Bohmian configuration qSE will lie in the support of just one branch
Ĉβ0,...,βN|Ψ0〉, and the influence of all other branches on its evolution, and all
future sub-branches of those other branches, can be neglected.

Let us now examine what this implies about the evolution of the system
configuration QS and the environmental configuration qE. Let SEβ0,...,βN
designate the support of Ĉβ0,...,βN|Ψ0〉 in QSE. This region will be contained
in the region Sβ0,...,βN ×Eβ0,...,βN , the direct product of the regions in which
the marginal distributions over QS and QE have support, with Sβ0,...,βN ≡
supp

[∫
dy|〈X,y|Ĉβ0,...,βN|Ψ0〉|

2
]
⊂ QS and Eβ0,...,βN ≡ supp

[∫
dX|〈X,y|Ĉβ0,...,βN|Ψ0〉|

2
]
⊂

QE. Now the region Sβ0,...,βN should roughly coincide with the range of posi-
tions associated with the phase space region ΣβN ; thus, Sβ0,...,βN for each N
should lie close to the Newtonian configuration space trajectory Xcl(N∆t)
associated with the sequence (Σβ0, ..., ΣβN ). Moreover, because of (20), the
regions Eα0,...,αN corresponding to the environmental support of each distinct
branch will be disjoint, so that

Eα′0,...,α′N ∩Eα0,...,αN = ∅ (22)

12


if αi 6= α′i for any 0 ≤ i ≤ N.
Since qSE = (QS,qE), and qSE ∈ SEβ0,...,βN ⊂ Sβ0,...,βN × Eβ0,...,βN , it

follows that QS ∈ Sβ0,...,βN and qE ∈ Eβ0,...,βN . Thus, the Bohmian config-
uration QS of the system S follows an approximately Newtonian trajectory
Xcl(N∆t) near to that associated with the sequence (Σβ0, ..., ΣβN ), while the
configuration of the environment E becomes correlated to this trajectory and
thereby serves as a record of it. So, at last, we have that

|QS(N∆t) −Xcl(N∆t)| < δ,
qE(N∆t) ∈ Eβ0,...,βN

(23a)

(23b)

for all N such that N∆t is less than the time when ensemble spreading of
ρS(X) becomes appreciable, and for δ some suitably chosen small margin
of error. Moreover, if qSE = (QS,qE) lies in the support of a single branch
Ĉα0,...,αN|Ψ0〉, at later times it may be found in the support only of branches
Ĉβ0,...,βM|Ψ0〉 such that (α0, ...,αN ) are the first N indices in (β0, ...,βM ),
where N < M. If M∆t is less than the timescale on which ensemble spread-
ing of ρS(X) becomes appreciable, (β0, ...,βM ) will represent the continu-
ation up to M∆t of the classical trajectory approximated by (α0, ...,αN ).
This follows from (21) and the equivariance of the Bohmian configuration’s
dynamics.

6 Conclusions

The analysis of classicality advanced in the previous section extends the effec-
tive collapse mechanism originally developed by (Bohm, 1952b), as applied
to the context of a laboratory quantum measurement, to the context of a
classically evolving macroscopic body interacting with some environment. In
both cases, decoherence renders the total state a superposition of disjoint
packets, so that the configuration comes to be guided by only one of these
packets. 9

9Some have suggested that dBB theory does not require decoherence to solve the mea-
surement problem because the theory is already about objects with determinate positions
and momenta. There are two problems with this line of thinking. First, it presupposes
that decoherence is somehow optional in dBB theory. It isn’t: decoherence is a generic
consequence of unitary evolution, and thus happens in dBB theory whether or not the
empirical adequacy of dBB theory requires it. Second, the empirical adequacy of dBB
theory does rely on decoherence, since as Bohm showed in (Bohm, 1952b), decoherence

13


Given this analysis, the position that dBB theory is specially equipped
to recover classical behavior must presuppose the very point that is at is-
sue in debates about the measurement problem: namely, that the Bohmian
mechanism for effectively collapsing a decohered superposition onto a single
component is the true mechanism employed in nature. Because advocates of
other interpretations provide their own mechanisms for the collapse or effec-
tive collapse of a decohered superposition, those who do not already submit
to the dBB interpretation are unlikely to be impressed by its account of
classical behavior. In particular, advocates of the Everett interpretation are
likely to regard the analysis given in Section 5, concerning the evolution of
the Bohmian configuration, as utterly superfluous to a quantum description
of classical behavior since they regard the structure of a unitarily evolving
quantum state as sufficient for this purpose; see (Brown and Wallace, 2005).
However, many are also hesitant to accept that the structure associated with
a unitarily evolving quantum state on its own is sufficient to save the ap-
pearances, not least because this supposition entails the existence of a vast,
ever-growing proliferation of worlds associated with the different branches of
the quantum state. Barring the objection that the Bohmian configuration
is superfluous, the fact that dBB theory is able to support an account of
classical behavior on its own terms should at least provide some reassurance
of its continuing viability in the nonrelativistic domain. However, it should
not be counted as an advantage of dBB theory over other interpretations.

Acknowledgments: Thanks to David Wallace, Simon Saunders and Harvey
Brown for comments on earlier drafts of this work, and to Cian Dorr for
helpful discussions of de Broglie-Bohm theory. Thanks also to audiences in
Oxford, Sussex and Vallico Soto, Tuscany.

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