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Semi-classical approximations based on Bohmian mechanics
Ward Struyve∗
Abstract
Semi-classical theories are approximations to quantum theory that treat some de-
grees of freedom classically and others quantum mechanically. In the usual ap-
proach, the quantum degrees of freedom are described by a wave function which
evolves according to some Schrodinger equation with a Hamiltonian that depends
on the classical degrees of freedom. The classical degrees of freedom satisfy classical
equations that depend on the expectation values of quantum operators. In this pa-
per, we study an alternative approach based on Bohmian mechanics. In Bohmian
mechanics the quantum system is not only described by the wave function, but also
with additional variables such as particle positions or fields. By letting the classical
equations of motion depend on these variables, rather than the quantum expectation
values, a semi-classical approximation is obtained that is closer to the exact quantum
results than the usual approach. We discuss the Bohmian semi-classical approxi-
mation in various contexts, such as non-relativistic quantum mechanics, quantum
electrodynamics and quantum gravity. The main motivation comes from quantum
gravity. The quest for a quantum theory for gravity is still going on. Therefore
a semi-classical approach where gravity is treated classically may be an approxi-
mation that already captures some quantum gravitational aspects. The Bohmian
semi-classical theories will be derived from the full Bohmian theories. In the case
there are gauge symmetries, like in quantum electrodynamics or quantum gravity,
special care is required. In order to derive a consistent semi-classical theory it will
be necessary to isolate gauge-independent dependent degrees of freedom from gauge
degrees of freedom and consider the approximation where some of the former are
considered classical.
1 Introduction
Quantum gravity is often considered to be the holy grail of theoretical physics. One
approach is canonical quantum gravity, which concerns the Wheeler-DeWitt equation
and which is obtained by applying the usual quantization methods (which were so suc-
cessful in the case of high energy physics) to Einstein’s field equations. However, this
approach suffers from a host of problems, some of technical and some of conceptual
nature (such as finding solutions to the Wheeler-DeWitt equation, the problem of time,
∗Instituut voor Theoretische Fysica & Centrum voor Logica en Filosofie van de Wetenschappen, KU
Leuven, Belgium. E-mail: [email protected]
1
arX
iv:1
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0477
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. . . ). For this reason one often resorts to a semi-classical approximation where gravity
is treated classically and matter quantum mechanically [1, 2]. The hope is that such an
approximation is easier to analyze and yet reveals some effects of quantum gravitational
nature.
In the usual approach to semi-classical gravity, matter is described by quantum field
theory on curved space-time. For example, in the case the matter is described by a
quantized scalar field, the state vector can be considered a functional Ψ(ϕ) on the space
of fields, which satisfies a particular Schrodinger equation
i∂tΨ(ϕ, t) = H(ϕ, g)Ψ(ϕ, t) , (1)
where the Hamiltonian operator H depends on the classical space-time metric g. This
metric satisfies Einstein’s field equations
Gµν(g) = 8πG〈Ψ|Tµν(ϕ, g)|Ψ〉 , (2)
where the source term is given by the expectation value of the energy-momentum tensor
operator.
This semi-classical approximation is non-linear in Ψ. However, it is expected that
the full quantum theory for gravity will be linear. Consider for example a macroscopic
superposition of Ψ1 and Ψ2, where Ψ1 represents a lump of matter in one location and
Ψ2 represents that lump in another location. Then it is expected that the full quantum
theory for gravity will describe a superposition of the state Ψ1 with its gravitational
field and Ψ2 with its gravitational field. This is not the case for the semi-classical
approximation. Namely, for the macroscopic superposition Ψ = (Ψ1 + Ψ2)/√
2, we
have that 〈Ψ|Tµν |Ψ〉 ≈(〈Ψ1|Tµν |Ψ1〉+ 〈Ψ2|Tµν |Ψ2〉
)/2, so that the gravitational field
is affected by two matter sources, one coming from each term in the superposition. As
experimentally shown by Page and Geilker, the semi-classical theory becomes inadequate
in such situations [2, 3]. On the other hand it will form a good approximation when the
matter state approximately corresponds to a classical state (i.e., a coherent state).
Of course, as already noted by Page and Geilker, it could be that this problem is not
due to the fact that gravity is treated classically, but due to the choice of the version
of quantum theory. Namely, Page and Geilker adopted the Many Worlds point of view,
according to which the wave function never collapses. However, according to standard
quantum theory the wave function is supposed to collapse upon measurement. Which
physical processes act as measurements is of course rather vague and this is the source
of the measurement problem. But it could be that such collapses explain the outcome of
their experiment. If an explanation of this type is sought, one should consider so-called
spontaneous collapse theories, where collapses are objective, random processes that do
not in a fundamental way depend on the notion of measurement. (See [4] and [5, 6] for
actual proposals combining such a spontaneous collapse approach with respectively (2)
and its non-relativistic version.)
2
In this paper, we consider an alternative to standard quantum mechanics, called
Bohmian mechanics [7–10], and develop a semi-classical approximation which is expected
to improve upon the usual approach.
Bohmian mechanics solves the measurement problem by introducing an actual con-
figuration (particle positions in the non-relativistic domain, particle positions or fields
in the relativistic domain [11]) that evolves under the influence of the wave function.
Instead of coupling classical gravity to the wave function it is now natural to couple it
to the actual matter configuration. For example, in the case of a scalar field there is
an actual field ϕB whose time evolution is determined by the wave functional Ψ. There
is an energy-momentum tensor Tµν(ϕB, g) corresponding to this scalar field and this
tensor can be introduced as the source term in Einstein’s field equations:
Gµν(g) = 8πGTµν(ϕB, g) . (3)
While the resulting theory is still non-linear, it solves the problem with the macroscopic
superposition. Namely, even though the matter wave function is in a macroscopic su-
perposition of being at two locations, the Bohmian field ϕB and the energy-momentum
tensor Tµν(ϕB, g) will correspond to a matter configuration at one of the locations. This
means that according to eq. (3) the gravitational field will correspond to that of matter
localised at that location.
However, there is an immediate problem with this ansatz, namely that eq. (3) is
not consistent. The Einstein tensor Gµν is identically conserved, i.e., ∇µGµν ≡ 0. So
the Bohmian energy-momentum tensor Tµν(ϕB, g) must be conserved as well. However,
the equation of motion for the scalar field does not guarantee this. (Similarly, in the
Bohmian approach to non-relativistic systems, the energy is generically not conserved.
On the other hand, if gravity is also treated quantum mechanically, in the context
of the Wheeler-DeWitt theory, then the total energy-momentum tensor is covariantly
conserved [12, 13].)
We will explain that the root of the problem seems to be the gauge invariance, which
in this case is the invariance under spatial diffeomorphisms (i.e., spatial coordinate trans-
formations). The gauge invariance makes that the gauge invariant content is contained
in a combination of the metric and scalar field. Therefore, it seems that one can not
just assume the metric to be classical without also assuming the scalar field ϕB to be
classical (in which case the energy-momentum tensor is conserved).
We will see that a similar problem arises when we consider a Bohmian semi-classical
approach to scalar electrodynamics, which describes a scalar field interacting with an
electromagnetic field. In this case, the wave equation for the scalar field is of the form
i∂tΨ(ϕ, t) = H(ϕ,A)Ψ(ϕ, t) , (4)
where A is the vector potential. There is also a Bohmian scalar field ϕB and a charge
current jν(ϕB, A) that could act as the source term in Maxwell’s equations
∂µFµν(A) = jν(ϕB, A) , (5)
3
where Fµν is the electromagnetic field tensor. In this case, we have ∂ν∂µFµν ≡ 0 due to
the anti-symmetry of Fµν . As such, the charge current must be conserved. However, the
Bohmian equation of motion for the scalar field does not imply conservation. Hence, just
as in the case of gravity, a consistency problem arises. We will find that this problem
can be overcome by eliminating the gauge invariance, either by assuming some gauge
or (equivalently) by working with gauge-independent degrees of freedom. In this way,
we can straightforwardly derive a semi-classical approximation starting from the full
Bohmian approach to scalar electrodynamics (where all degrees of freedom are treated
quantum mechanically). For example, in the Coulomb gauge, the result is that there
is an extra current jνQ which appears in addition to the usual (classical) charge current
and which depends on the quantum potential, so that Maxwell’s equations read
∂µFµν(A) = jν(ϕB, A) + jνQ(ϕB, A) , (6)
which is consistent since the total current is conserved, because of the equation of motion
for the scalar field.
While it is easy to eliminate the gauge invariance in the case of electrodynamics, this
is notoriously difficult in the case of general relativity. One can formulate a Bohmian
theory for the Wheeler-DeWitt approach to quantum gravity, but the usual formulation
does not explicitly eliminate the gauge freedom arising from spatial diffeomorphism
invariance. Our expectation is that one could find a semi-classical approximation given
such a formulation. At least we find our expectation confirmed in simplified symmetry-
reduced models, called mini-superspace models, where this invariance is eliminated. We
will illustrate this for the model described by the homogeneous and isotropic Friedman-
Lemaıtre-Robertson-Walker metric and a uniform scalar field.
In this paper, we are merely concerned with the formulation of Bohmian semi-
classical approximations. Practical applications will be studied elsewhere. Such appli-
cations have already been studied for non-relativistic systems in the context of quantum
chemistry [14–19]. It appears that Bohmian semi-classical approximations yield better
or equivalent results compared to the usual semi-classical approximation. (They are
better in the sense that they are closer to the exact quantum results.) This provides
good hope that also in other contexts, such as quantum gravity, the Bohmian approach
also gives better results. Potential applications might be found in inflation theory, where
the back-reaction from the quantum fluctuations onto the classical background can be
studied, or in black hole physics, to study the back-reaction from the Hawking radiation
onto space-time.
Other semi-classical approximations have been proposed, see for example [20–23],
and in particular [24, 25] where also Bohmian ideas are used. We will not make a
comparison with these proposals here.
The semi-classical approximation is just one practical application of Bohmian me-
chanics. In recent years, others have been explored (see [26, 27] for overviews). For
example, one may use Bohmian approximation schemes to solve problems in many-
body systems [28, 29] or one may use Bohmian trajectories as a calculational tool to
4
simulate wave function evolution [30]. So even though Bohmian mechanics yields the
same predictions as standard quantum theory (insofar the latter are unambiguous), it
leads to new practical tools and ideas.
The outline of the paper is as follows. We start with an introduction to Bohmian
mechanics in section 2. In section 3, we present the Bohmian semi-classical approx-
imation to non-relativistic quantum theory and its derivation from the full Bohmian
theory. In section 4, we discuss the Bohmian semi-classical approximation to the quan-
tized Abraham model. This model describes extended rigid particles interacting with an
electromagnetic field. This is a gauge theory and we will consider two possible gauge fix-
ings, the Coulomb gauge and the temporal gauge. While the former completely fixes the
gauge, the latter does not and hence admits a residual gauge symmetry. No consistency
problems of the sort mentioned above arise. The reason seems to be that the in the full
Bohmian theory the gauge freedom only acts on the electromagnetic field and not on the
charges. So no consistency problems arise for the semi-classical approximation where
the electromagnetic field is treated classically. Nevertheless, in the case of the temporal
gauge, a separation of the remainig gauge freedom from the gauge-invariant degrees
of freedom will be important in order to find a good semi-classical approximation. In
section 5, we consider scalar electrodynamics, which describes a scalar field coupled to
an electromagnetic field. In this case, the gauge freedom acts on both the scalar and
electromagnetic field. This leads to consistency problems if the gauge is not completely
fixed. First we will consider two gauges, the Coulomb gauge and the unitary gauge,
which completely fix the gauge, and show that no consistency problems arise. Then we
will consider the temporal gauge which does not completely fix the gauge. In this case,
we encounter problems with consistency (due to non-conservedness of the charge cur-
rent). The temporal gauge is also interesting because it brings scalar electrodynamics
in a form which is very similar to that of canonical quantum gravity (described by the
Wheeler-DeWitt equation). We turn to gravity in section 6. We do not try to develop
a Bohmian semi-classical theory from full quantum gravity, but rather make a natural
guess for the Bohmian energy-momentum tensor that couples quantum matter to clas-
sical gravity. We consider both matter described by a scalar field and by relativistic
point-particles. In both cases, the natural guess for the energy-momentum tensor is not
conserved. A more careful analysis should start from full quantum gravity. However,
because of the spatial diffeomorphism invariance, which acts on both the matter and
gravitational field, such an analysis is not expected to help in the construction of a
consistent semi-classical approximation, unless a way can be found to gauge fix or to
isolate the gauge-independent degrees of freedom. In the case of Newtonian gravity cou-
pled to non-relativistic particles, we can formulate a consistent Bohmian semi-classical
approximation, which can be seen as a Bohmian analogue of the Schrodinger-Newton
equation. Finally, we also consider mini-superspace models for quantum gravity in sec-
tion 6. The model we consider is homogeneous and isotropic and therefore the spatial
diffeomorphism invariance is eliminated. In this case, we can derive a consistent semi-
classical approximation. We compare this semi-classical approximation with the usual
5
one (which is the equivalent of (1) and (2) for mini-superspace) and show that the latter
gives better results.
2 Bohmian mechanics
Non-relativistic Bohmian mechanics (also called pilot-wave theory or de Broglie-Bohm
theory) is a theory about point-particles in physical space moving under the influ-
ence of the wave function [7–10]. The equation of motion for the configuration X =
(X1, . . . ,Xn) of the particles, called the guidance equation, is given by1
X(t) = vψ(X(t), t) , (7)
where vψ = (vψ1 , . . . ,vψn ), with
vψk =1
mkIm
(∇kψ
ψ
)=
1
mk∇kS (8)
and ψ = |ψ|eiS . The wave function ψ(x, t) = ψ(x1, . . . ,xn) itself satisfies the non-
relativistic Schrodinger equation
i∂tψ(x, t) =
(−
n∑k=1
1
2mk∇2k + V (x)
)ψ(x, t) . (9)
For an ensemble of systems all with the same wave function ψ, there is a distinguished
distribution given by |ψ|2, which is called the quantum equilibrium distribution. This
distribution is equivariant. That is, it is preserved by the particles dynamics (7) in the
sense that if the particle distribution is given by |ψ(x, t0)|2 at some time t0, then it is
given by |ψ(x, t)|2 at all times t. This follows from the fact that any distribution ρ that
is transported by the particle motion satisfies the continuity equation
∂tρ+n∑k=1
∇k · (vψk ρ) = 0 (10)
and that |ψ|2 satisfies the same equation, i.e.,
∂t|ψ|2 +n∑k=1
∇k · (vψk |ψ|2) = 0 , (11)
as a consequence of the Schrodinger equation. It can be shown that for a typical initial
configuration of the universe, the (empirical) particle distribution for an actual ensemble
of subsystems within the universe will be given by the quantum equilibrium distribu-
tion [9, 10, 31]. Therefore for such a configuration Bohmian mechanics reproduces the
standard quantum predictions.
1Throughout the paper we assume units in which ~ = c = 1.
6
Note that the velocity field is of the form jψ/|ψ|2, where jψ = (jψ1 , . . . , jψn) with
jψk = Im(ψ∗∇kψ)/mk is the usual quantum current. In other quantum theories, such
as for example quantum field theories, the velocity can be defined in a similar way by
dividing the appropriate current by the density. In this way equivariance of the density
will be ensured. (See [32] for a treatment of arbitrary Hamiltonians.)
This theory solves the measurement problem. Notions such as measurement or
observer play no fundamental role. Instead measurement can be treated as any other
physical process.
There are two aspects of the theory that are important for deriving the semi-classical
approximation. Firstly, Bohmian mechanics allows for an unambiguous analysis of the
classical limit. Namely, the classical limit is obtained whenever the particles (or at least
the relevant macroscopic variables, such as the center of mass) move classically, i.e.,
satisfy Newton’s equation. By taking the time derivative of (7), we find that
mkXk(t) = −∇k(V (x) +Qψ(x, t))∣∣x=X(t)
, (12)
where
Qψ = −n∑k=1
1
2mk
∇2k|ψ||ψ|
(13)
is the quantum potential. Hence, if the quantum force −∇kQψ is negligible compared to
the classical force −∇kV , then the k-th particle approximately moves along a classical
trajectory.
Another aspect of the theory is that it allows for a simple and natural definition
of the wave function of a subsystem [9, 31]. Namely, consider a system with wave
function ψ(x, y) where x is the configuration variable of the subsystem and y is the
configuration variable of its environment. The actual configuration is (X,Y ), where X
is the configuration of the subsystem and Y is the configuration of the other particles.
The wave function of the subsystem χ(x, t), called the conditional wave function, is then
defined as
χ(x, t) = ψ(x, Y (t), t). (14)
This is a natural definition since the trajectory X(t) of the subsystem satisfies
X(t) = vψ(X(t), Y (t), t) = vχ(X(t), t) . (15)
That is, for the evolution of the subsystem’s configuration we can either consider the
conditional wave function or the total wave function (keeping the initial positions fixed).
(The conditional wave function is also the wave function that would be found by a
natural operationalist method for defining the wave function of a quantum mechanical
subsystem [33].) The time evolution of the conditional wave function is completely
determined by the time evolution of ψ and that of Y . The conditional wave function
does not necessarily satisfy a Schrodinger equation, although in many cases it does.
This wave function collapses according to the usual text book rules when an actual
measurement is performed.
7
We will also consider semi-classical approximations to quantum field theories. More
specifically, we will consider bosonic quantum field theories. In Bohmian theories it
is most easy to introduce actual field variables rather than particle positions [11, 34].
To illustrate how this works, let us consider the free massless real scalar field (for the
treatment of other bosonic field theories see [34]). Working in the functional Schrodinger
picture, the quantum state vector is a wave functional Ψ(ϕ) defined on a space of scalar
fieldsn ϕ(x) in 3-space and it satisfies the functional Schrodinger equation
i∂tΨ(ϕ, t) =1
2
∫d3x
(− δ2
δϕ(x)2+ ∇ϕ(x) ·∇ϕ(x)
)Ψ(ϕ, t) . (16)
The associated continuity equation is
∂t|Ψ(ϕ, t)|2 +
∫d3x
δ
δϕ(x)
(δS(ϕ, t)
δϕ(x)|Ψ(ϕ, t)|2
)= 0 , (17)
where Ψ = |Ψ|eiS . This suggests the following guidance equation for an actual scalar
field φ(x, t):
φ(x, t) =δS(ϕ, t)
δϕ(x)
∣∣∣∣ϕ(x)=φ(x,t)
. (18)
Taking the time derivative of this equation results in
φ(x, t) = −δQΨ(ϕ, t)
δϕ(x)
∣∣∣∣ϕ(x)=φ(x,t)
, (19)
where
QΨ = − 1
2|Ψ|
∫d3x
δ2|Ψ|δϕ(x)2
(20)
is the quantum potential. The classical limit is obtained whenever the quantum force,
i.e., the right-hand side of eq. (19), is negligible. Then the field approximately satisfies
the classical field equation φ = 0.
One can also consider the conditional wave functional of a subsystem. A subsystem
can in this case be regarded as a system confined to a certain region in space. The
conditional wave functional for the field confined to that region is then obtained from
the total wave functional by conditioning over the actual field value on the complement
of that region. However, in this paper we will not consider this kind of conditional wave
functional. Rather, there will be other degrees of freedom, like for example other fields,
which will be conditioned over.
This Bohmian theory is not Lorentz invariant. The guidance equation (18) is formu-
lated with respect to a preferred reference frame and as such violates Lorentz invariance.
This violation does not show up in the statistical predictions given quantum equilib-
rium, since the theory makes the same predictions as standard quantum theory which
are Lorentz invariant.2 The difficulty in finding a Lorentz invariant theory resides in
2Actually, this statement needs some qualifications since regulators need to be introduced to make
the theory and its statistical predictions well defined [34].
8
the fact that any adequate formulation of quantum theory must be non-local [35]. One
approach to make the Bohmian theory Lorentz invariant is by introducing a foliation
which is determined by the wave function in a covariant way [36]. In this paper, we
will not attempt to maintain Lorentz invariance. As such, the Bohmian semi-classical
approximations will not be Lorentz invariant, (very likely) not even concerning the sta-
tistical predictions. This is in contrast with the usual semi-classical approach like the
one for gravity given by (1) and (2) which is Lorentz invariant. However, this does
not take away the expectation that the Bohmian semi-classical approximation will give
better or at least equivalent results compared to the usual approach.
3 Non-relativistic quantum mechanics
3.1 Usual versus Bohmian semi-classical approximation
Consider a composite system of just two particles. The usual semi-classical approach
goes as follows. Particle 1 is described quantum mechanically, by a wave function
χ(x1, t), which satisfies the Schrodinger equation
i∂tχ(x1, t) =
[− 1
2m1∇2
1 + V (x1,X2(t))
]χ(x1, t) , (21)
where the potential is evaluated at the position of the second particle X2, which satisfies
Newton’s equation
m2X2(t) = −⟨χ∣∣∣∇2V (x1,x2)
∣∣x2=X2(t)
∣∣∣χ⟩=
∫d3x1|χ(x1, t)|2[−∇2V (x1,x2)]
∣∣∣x2=X2(t)
. (22)
So the force on the right-hand-side is averaged over the quantum particle.
An alternative semi-classical approach based on Bohmian mechanics was proposed
independently by Gindensperger et al. [14] and Prezhdo and Brooksby [15]. In this
approach there is also an actual position for particle 1, denoted by X1, which satisfies
the equation
X1(t) = vχ(X1(t), t) , (23)
where
vχ =1
m1Im
∇χ
χ, (24)
and where χ satisfies the Schrodinger equation (21). But instead of eq. (22), the second
particle now satisfies
m2X2(t) = −∇2V (X1(t),x2)∣∣x2=X2(t)
, (25)
where the force depends on the position of the first particle. So in this approximation
the second particle is not acted upon by some average force, but rather by the actual
9
particle of the quantum system. This approximation is therefore expected to yield a
better approach than the usual approach, in the sense that it yields predictions closer to
those predicted by full quantum theory, especially in the case where the wave function
evolves into a superposition of non-overlapping packets. This is indeed confirmed by a
number of studies, as we will discuss below.
Let us first mention some properties of this approximation and compare them to the
usual approach. In the usual field approach, the specification of an initial wave function
χ(x, t0), an initial position X2(t0) and velocity X2(t0) determines a unique solution for
the wave function and the trajectory of the classical particle. In the Bohmian approach
also the initial position X1(t0) of the particle of the quantum system needs to be specified
in order to uniquely determine a solution. Different initial positions X1(t0) yield different
evolutions for the wave function and the classical particle. This is because the evolution
of each of the variables X1,X2, χ depends on the others. Namely, the evolution of χ
depends on X2 via (21), whose evolution in turn depends on X1 via (25), whose evolution
in turn depends on χ via (23). (This should be contrasted with the full Bohmian theory,
where the wave function acts on the particles, but there is no back-reaction from the
particles onto the wave function.)
The initial configuration X1(t0) should be considered random with distribution
|χ(x, t0)|2. However, this does not imply that X1(t) is random with distribution |χ(x, t)|2
for later times t. It is not even clear what the latter statement should mean, since dif-
ferent initial positions X1(t0) lead to different wave function evolution; so which wave
function should χ(x, t) be?
This semi-classical approximation has been applied to a number of systems. Prezhdo
and Brooksby studied the case of a light particle scattering off a heavy particle [15].
They considered the scattering probability over time and found that the Bohmian semi-
classical approximation was in better agreement with the exact quantum mechanical
prediction than the usual approximation. The Bohmian semi-classical approximation
gives probability one for the scattering to have happened after some time, in agreement
with the exact result, whereas the probability predicted by the usual approach does
not reach one. The reported reason for the better results is that the wave function
of the quantum particle evolves into a superposition of non-overlapping packets, which
yields bad results for the usual approach (since the force on the classical particle contains
contributions from both packets), but not for the Bohmian approach. These results were
confirmed and further expanded by Gindensperger et al. [16]. Other examples have been
considered in [14, 17, 18]. In those cases, the Bohmian semi-classical approximation gave
very good agreement with the exact quantum or experimental results. It was always
either better or comparable to the usual approach. These results give good hope that
the Bohmian semi-classical approximation will also give better results than the usual
approximation in other domains such as quantum gravity.
10
3.2 Derivation of the Bohmian semi-classical approximation
The Bohmian semi-classical approach can easily be derived from the full Bohmian the-
ory.3 Consider a system of two particles. In the Bohmian description of this system, we
have a wave function ψ(x1,x2, t) and positions X1(t),X2(t), which respectively satisfy
the Schrodinger equation
i∂tψ =
[− 1
2m1∇2
1 −1
2m2∇2
2 + V (x1,x2)
]ψ (26)
and the guidance equations
X1(t) = vψ1 (X1(t),X2(t), t) , X2(t) = vψ2 (X1(t),X2(t), t) . (27)
The conditional wave function χ(x1, t) = ψ(x1,X2(t), t) for particle 1 satisfies the equa-
tion
i∂tχ(x1, t) =
(− ∇
21
2m1+ V (x1,X2(t))
)χ(x1, t) + I(x1, t) , (28)
where
I(x1, t) =
(− ∇
22
2m2ψ(x1,x2, t)
) ∣∣∣∣∣x2=X2(t)
+ i∇2ψ(x1,x2, t)∣∣∣x2=X2(t)
·vψ2 (X1(t),X2(t), t) .
(29)
So if I is negligible in (28), up to a time-dependent factor times χ,4 we are led to the
Schrodinger equation (21). This will for example be the case if m2 is much larger than
m1 (I is inversely proportional to m2) and if the wave function slowly varies as a function
of x2. We also have that
m2X2(t) = −∇2
[V (X1(t),x2) +Qψ(X1(t),x2, t)
] ∣∣∣∣∣x2=X2(t)
, (30)
with Qψ the quantum potential. We obtain the classical equation (25) if the quantum
force is negligible compared to the classical force.
In this way we obtain the equations for a semi-classical formulation. In addition, we
also have the conditions under which they will be valid. For other quantum theories,
such as quantum gravity, we can follow a similar path to find a Bohmian semi-classical
approximation.
3The derivation is very close to the one followed by Gindensperger et al. [14]. A difference is that
they also let the wave function of the quantum system depend parametrically on the position of the
classical particle. This leads to a quantum force term in the eq. (25) for particle 2. However, this does
not seem to lead to a useful set of equations. In particular, they can not be numerically integrated by
simply specifying the initial wave function and particle positions. In any case, Gindensperger et al. drop
this quantum force when considering examples [14, 16, 17], so that the resulting equations correspond to
the ones presented above.4If I contains a term of the form f(t)χ, then it can be eliminated by changing the phase of χ by a
time-dependent term.
11
4 The Abraham model
The next theory we consider is the Abraham model, which on the classical level de-
scribes extended, rigid, charged particles, which interact with an electromagnetic field
[37]. We will consider the Bohmian theory in two gauges: the Coulomb gauge and the
temporal gauge. These Bohmian theories are equivalent. An important difference be-
tween the gauges is that the former gauge completly fixes the gauge invariance, while the
second does not. From the full Bohmian theories, we will derive semi-classical approxi-
mations, where the charges are treated quantum mechanically and the electromagnetic
field classically. In both cases no problems with consistency arise (like those mentioned
in the introduction). The apparent reason is that the gauge transformations only act
on the electromagnetic field and not on the charges. Nevertheless, we will see that in
the case of the temporal gauge a further separation of gauge degrees of freedom from
gauge-invariant degrees of freedom will be desired in order to have a better semi-classical
approximation. In section 5, we will consider scalar electrodynamics in those gauges. In
that case, because the gauge transformations act on both matter and electromagnetic
field, a complete gauge fixing will be necessary to obtain a consistent semi-classical ap-
proximation. We end this section with a comparison of the semi-classical theory with
similar ones, including one by Kiessling [38], which are not derived from the full Bohmian
theories.
4.1 Classical theory
In the classical Abraham model, the charge distribution of the i-th particle is centered
around a position Qi and is given by eb(x−Qi), where the function b is assumed to be
smooth, radial, with compact support and normalized to 1. The particles move under
the Lorentz force law
mQi(t) = e[Eb(Qi(t), t) + Qi(t)×Bb(Qi(t), t)
], (31)
where E(x, t) and B(x, t) are respectively the electric and magnetic field and the sub-
script b denotes the convolutions
Eb(x, t) = (E ∗ b)(x, t) =
∫d3yE(y, t)b(x− y), (32)
Bb(x, t) = (B ∗ b)(x, t) =
∫d3yB(y, t)b(x− y). (33)
In terms of the electromagnetic potential Aµ = (A0,A), the electric and magnetic field
are given by E = −∂tA −∇A0 and B = ∇ × A. The electromagnetic field satisfies
Maxwell’s equations
∂µFµν(x, t) = jν(x, t) , (34)
with Fµν = ∂µAν − ∂νAµ and charge current jν given by
j0(x, t) =
n∑i=1
eb(x−Qi(t)) , j(x, t) =
n∑i=1
eQi(t)b(x−Qi(t)) . (35)
12
Note that although we have employed a covariant notation this model is not covariant,
due to the rigidity of the particles. There is a gauge symmetry
Aµ → Aµ − ∂µα . (36)
We will consider two gauges, namely the Coulomb gauge ∇ ·A = 0 and the temporal
gauge A0 = 0. For later comparison, we present here the Maxwell’s equation in those
gauges. In the Coulomb gauge, the equations read
AT = jT , AL = 0 , ∇2A0 + j0 = 0 , ∇A0 = jL . (37)
The superscripts T and L respectively refer to the tranverse and longitudinal part of
the vectors.5 In the temporal gauge, they read,
A + ∇∇ ·A = j , −∇ · A = j0 . (38)
4.2 Coulomb gauge
4.2.1 Bohmian theory
The quantization of the classical model in the Coulomb gauge ∇ · A = 0 is straight-
forward [37]. The wave functional Ψ(q,AT ), where q = (q1, . . . ,qn) and AT is the
transverse part of the vector potential,6 satisfies the Schrodinger equation7
i∂tΨ =
[− 1
2m
n∑i=1
(∂
∂qi− ieAT
b (qi)
)2
+ Vc(q) +
∫d3x
(−1
2
δ2
δAT (x)2+
1
2(∇×AT (x))2
)]Ψ,
(39)
where Vc is the Coulomb potential, which, using
ρ(x, q) =n∑i=1
eb(x− qi) (40)
is given by:
Vc(q) = −1
2
∫d3xρ(x, q)
1
∇2ρ(x, q) =
e2
8π
n∑i,j=1
∫d3xd3y
b(x)b(y)
|x− y + qi − qj |. (41)
So for the electromagnetic field we have used the functional Schrodinger representation
(see e.g. [34] for more details).
5A vector V can be written as V = VT +VL, with VT = V−∇ 1∇2∇ ·V and VL = ∇ 1
∇2∇ ·V re-
spectively the transverse and longitudinal part of the vector potential, where 1∇2 f(x) = − 1
4π
∫d3y f(y)
|x−y| .
6As before, we will distinguish the arguments q and AT of the wave functional from the actual
configuration Q and AT .7We used the notation δ
δAT = δδA−∇ 1
∇2∇ · δδA
and δδAL = ∇ 1
∇2∇ · δδA
.
13
In the Bohmian theory the configuration Q and the field AT satisfy
Qi(t) =1
m
[∂S(q,AT , t)
∂qi− eAT
b (qi)
] ∣∣∣∣∣Q(t),AT (x,t)
, (42)
AT (x, t) =δS(q,AT , t)
δAT (x)
∣∣∣∣∣Q(t),AT (x,t)
, (43)
where Ψ = |Ψ|eiS . Taking the time derivative, we find
mqi(t) = e [Eb(qi(t), t) + qi(t)×Bb(qi(t), t)]−∂Q(q,AT , t)
∂qi
∣∣∣∣∣Q(t),AT (x,t)
, (44)
∂µFµν(x, t) = jν(x, t) + jνQ(x, t) , (45)
where jµ is the classical expression for the current given in (35) and
QΨ = − 1
2m|Ψ|
n∑i=1
∂2|Ψ|∂q2
i
− 1
2|Ψ|
∫d3x
δ2|Ψ|δAT (x)2
, jµQ(x, t) =
(0,− δQΨ
δAT (x)
∣∣∣∣q(t),AT (x,t)
)(46)
are respectively the quantum potential and what can be called a quantum charge current,
which enters Maxwell’s equation in addition to the usual current. Both currents are
conserved, i.e., ∂µjµ = ∂µj
µQ = 0. These equations are written in terms of Aµ and it is
assumed that the Coulomb gauge ∇ ·A = 0 holds. So the equations (45) are equivalent
to
AT = jT + jQ , AL = 0 , ∇2A0 + j0 = 0 , ∇A0 = jL . (47)
The first equation follows from differentiating (43) with respect to time, the second one
is implied by the Coulomb gauge, and the third (and the fourth) defines A0 in terms
of the charge density. Compared to the classical equations (37), only the first equation
has changed through the addition of the quantum current.
4.2.2 Semi-classical approximation
We can consider a semi-classical approximation where either the charges or the elec-
tromagnetic field behave approximately classically. Let us consider the latter case. We
assume the quantum current jνQ to be negligible in (45). The conditional wave function
for the charges χ(q, t) = ψ(q,AT (x, t)), where (Q(t),AT (x, t)) is a particular solution
to the guidance equations, then satisfies the equation
i∂tχ =
[− 1
2m
n∑i=1
(∂
∂qi− ieAT
b (qi, t)
)2
+ Vc(q)
]χ+ I , (48)
where
I =
∫d3x
(−1
2
δ2Ψ
δAT (x)2+
1
2(∇×AT (x))2Ψ + i
δΨ
δAT (x)· AT (x, t)
) ∣∣∣∣∣AT (x)=AT (x,t)
.
(49)
14
Whenever I is negligible, up to a time-dependent factor times χ (cf. footnote 4), the
wave equation reduces to
i∂tχ =
[− 1
2m
n∑i=1
(∂
∂qi− ieAT
b (qi, t)
)2
+ Vc(q)
]χ . (50)
Together with
Qi(t) =1
m
∂S(q, t)
∂qi
∣∣∣∣∣Q(t)
− eATb (Qi(t), t)
, ∂µFµν(x, t) = jν(x, t) (51)
and the Coulomb gauge, this defines a consistent semi-classical approximation. That is,
due to the anti-symmetry of Fµν , we have ∂µ∂νFµν = 0 so that the current jν must be
conserved, which is actually the case by definition (irrespective of the dynamics of the
charges).
Actually, any current of the form (35) is conserved, irrespective of the dynamics of
the charges. The second-order equation for the charges is still of the form (44), but now
with quantum potential
Qχ = − 1
2m|χ|
n∑i=1
∂2|χ|∂q2
i
. (52)
4.3 Temporal gauge
4.3.1 Bohmian theory
The classical theory can also be quantized using the temporal gauge, which is given
by A0 = 0. This gauge does not completely fix the gauge. It still allows for gauge
transformations
A→ A + ∇θ , (53)
with θ time-independent. Quantization yields the Schrodinger equation8
i∂tΨ =
[− 1
2m
n∑i=1
(∂
∂qi− ieAb(qi)
)2
+
∫d3x
(−1
2
δ2
δA(x)2+
1
2(∇×A(x))2
)]Ψ
(54)
for the wave function Ψ(q,A), together with the constraint (corresponding to Gauss’s
law)
i∇ · δΨ
δA(x)− ρ(x, q)Ψ = 0 , (55)
with ρ as defined in (40). The constraint is preserved by the Schrodinger dynamics.
That is, if it is satisfied at some time, then it is satisfied at all times. The constraint
implies the gauge invariance
Ψ(q,A) = e−ie∑ni=1 θb(qi)Ψ(q,A + ∇θ) (56)
8This follows from the usual canonical quantization techniques explained for example in [39].
15
for time-independent θ.
The guidance equations are
Qi(t) =1
m
[∂S(q,A, t)
∂qi− eAb(qi)
] ∣∣∣∣∣Q(t),A(x,t)
, (57)
A(x, t) =δS(q,A, t)δA(x)
∣∣∣∣∣Q(t),A(x,t)
. (58)
Taking the time derivative, we find
mQi(t) = e[Eb(Qi(t), t) + Qi(t)×Bb(Qi(t), t)
]− ∂QΨ(q,AT , t)
∂qi
∣∣∣∣∣Q(t),AT (x,t)
, (59)
∂µFµν = jν + jνQ , (60)
where Aµ = (0,A), jµ is the classical expression for the current given in (35), and
QΨ = − 1
2m|Ψ|
n∑i=1
∂2|Ψ|∂q2
i
− 1
2|Ψ|
∫d3x
δ2|Ψ|δA(x)2
, jµQ(x, t) =
(0,− δQΨ
δA(x)
∣∣∣∣Q(t),A(x,t)
).
(61)
More in detail, (60) can be written as
A + ∇∇ ·A = j + jQ , −∇ · A = j0 , (62)
where the first equation follows from taking the time derivative of (58) and the second
equation follows from (58) and the constraint (55). The difference with the classical
equations (38) is the presence of the quantum current jQ.
Because of the identity (56), the Bohmian theory is invariant under the time-
independent gauge transformations (53).
This Bohmian theory is equivalent to the one formulated in the Coulomb gauge. To
see this, write A = AT + AL and define
Ψ′(q,AT ) = e−i∫d3x∇·A(x) 1
∇2 ρ(x,q)Ψ(q,A + ∇θ) = e−ie∑ni=1
1∇2∇·Ab(qi)Ψ(q,A + ∇θ) .
(63)
The constraint (55) then implies that δΨ′/δAL = 0 so that Ψ′ is indeed independent
of AL. It is easy to verify that Ψ′ satisfies the Schrodinger equation in the Coulomb
gauge. The appearance of the Coulomb potential Vc(q) follows from the constraint (55)
which implies that
−1
2
∫d3x
δ2Ψ
δAL(x)2= −1
2
∫d3xρ(x, q)
1
∇2ρ(x, q)Ψ = Vc(q)Ψ . (64)
The guidance equations (57) and (58) respectively reduce to (42) and (43). The guidance
equation (58) also yields −∇ · AL = j0. This means that AL is determined by j0 up to
time-independent gauge transformation. As such, AL is not an independent dynamical
degree of freedom and could be ignored. Also after quantization in the Coulomb gauge,
there is no degree of freedom correpsonding to AL and hence we can conclude that both
gauges lead to equivalent Bohmian theories.
16
4.3.2 Semi-classical approximation
Again, the derivation of the semi-classical approximation can be carried out by consid-
ering the conditional wave function χ(q, t) = ψ(q,A(x, t)). There is one term in i∂tχ
that requires special attention, namely
I =
∫d3x
(−1
2
δ2Ψ
δA(x)2+ i
δΨ
δA(x)· A(x, t)
) ∣∣∣∣∣A(x)=A(x,t)
. (65)
One might be tempted to ignore this term. However, because of the constraint (55),
further analysis is needed. Writing the vector potential in terms of its tranverse and
longitudinal component and using the identities (64) and∫d3xi
δΨ
δAL(x)
∣∣∣∣∣A(x)=A(x,t)
· AL(x, t) =
∫d3xρ(x, q)
1
∇2j0(x, t)χ , (66)
which both follow from the constraint, we get that
I =
∫d3x
(−1
2
δ2Ψ
δAT (x)2+ i
δΨ
δAT (x)· AT (x, t)
) ∣∣∣∣∣AT (x)=AT (x,t)
+
∫d3xρ(x, q)
1
∇2
(j0(x, t)− 1
2ρ(x, q)
)χ . (67)
While the first term should be ignored in the semi-classical approximation, the second
term is best kept. As a result the wave equation reads
i∂tχ = − 1
2m
n∑i=1
(∂
∂qi− ieAb(qi, t)
)2
χ+
∫d3xρ(x, q)
1
∇2
(j0(x, t)− 1
2ρ(x, q)
)χ .
(68)
Since j0(x, t) = ρ(x, Q(t)), the second term may presumably be approximated by −Vc(q)for practical purposes.
The semi-classical theory is completed by the following equations for the configura-
tion Q and the potential Aµ = (0,A):
Qi(t) =1
m
∂S(q, t)
∂qi
∣∣∣∣∣Q(t)
− eAb(Qi(t), t)
, ∂µFµν(x, t) = jν(x, t) . (69)
This theory is consistent since jν is conserved by definition. It is invariant under the
gauge symmetry A → A + ∇θ, Ψ(q) → eie∑ni=1 θb(qi)Ψ(q). It is also equivalent to
the semi-classical theory in the Coulomb gauge. This is shown similarly as for the full
Bohmian theory, by considering again the decomposotion A = AT +AL and by defining
Ψ′(q, t) = e−i∫d3x∇·A(x,t) 1
∇2 ρ(x,q)Ψ(q, t) = e−ie∑ni=1
1∇2∇·Ab(qi,t)Ψ(q, t) . (70)
which satisfies (50).
17
There are two important observations to make. First, we did not run into any
problems of the type mentioned in the introduction concerning the consistency of the
equations of motion, even though the full Bohmian theory is a gauge theory, with gauge
symmetry given by (53). The reason seems to be that the gauge symmetry acts solely on
the electromagnetic field and leaves the matter degrees of freedom invariant, rather than
mixing the degrees of freedom. Therefore we can consider a consistent approximation
where the electromagnetic field is treated classically (even if we had ignored the potential
term in (68)). In the next section, we will consider a matter field instead of particles.
In that case we run into problems with consistency if the gauge is not completely fixed
or if the gauge-invariant degrees are not separated from the gauge degrees of freedom.
Second, to get the appropriate Schrodinger equation we have used the separation
into transverse and longitudinal parts of the vector potential. The transverse part is
gauge invariant, while the longitudal part is a gauge degree of freedom. So in effect,
we have separated the gauge degrees of freedom form the gauge-independent degrees of
freedom. While this was not necessary for consistency, it seems necessary to obtain a
better semi-classical approximation.
4.4 Alternative models
We can actually consider different semi-classical models that are consistent. Namely,
consider
i∂tχ =
[− 1
2m
n∑i=1
(∂
∂qi− ieAb(qi, t)
)2
+ Vb(q, t)
]χ , (71)
Qi(t) =1
m
∂S(q, t)
∂qi
∣∣∣∣∣Q(t)
− eAb(Qi(t), t)
, ∂µFµν(x, t) = jν(x, t) , (72)
where Vb(q, t) =∑n
i=1 eA0b(qi, t). Kiessling considered such a model assuming the
Lorenz gauge ∂µAµ [38].9 We could also consider the Coulomb gauge ∇ · A = 0.
However, in the latter case, the semi-classical approximation is different from the one
presented in section 4.2.2. The difference lies in the scalar potential in the Schrodinger
equation. In the latter case, the potential is given Vc(q), which is a function independent
of the actual positions Qi(t), whereas in the former case, the potential Vb(q, t) depends
on the actual positions because of Gauss’s law ∇2A0(x, t) + j0(x, t) = 0. The difference
may however be negligible for practical applications. In the case of the temporal gauge,
we do not get the potential term that is in (68). So here we see the virtue of starting
from the full Bohmian theory in order to derive a semi-classical approximation.
5 Scalar electrodynamics
Scalar electrodynamics describes a scalar field interacting with an electromagnetic field.
There is a gauge symmetry which transforms both the scalar field and electromagnetic
9Kiessling also considers charges with spin and possible alternative guidance equations.
18
field. There exist various different but equivalent formulations of Bohmian scalar elec-
trodynamics [34]. These formulations can be found either by considering different gauges
or by working with different choices of gauge-independent variables (which more or less
amounts to the same thing [34, 40]). The gauges we will consider here are the Coulomb
gauge, the unitary gauge and the temporal gauge. However, while the Coulomb and the
unitary gauge completely fix the gauge freedom, the temporal gauge does not. We will
see that semi-classical approximations can straightforwardly be found in the case of the
Coulomb and the unitary gauge, while problems with consistency arise in the case of the
temporal gauge. The problem seems to originate from the fact that there is a remaining
gauge symmetry for the scalar and electromagnetic fields, such that the gauge invariant
content lies in a combination of both. So it seems necessary to completely eliminate the
gauge freedom (or at least separate gauge degrees of freedom from gauge-independent
degrees of freedom).
5.1 Coulomb gauge
5.1.1 Classical theory
The Lagrangian fpr scalar electrodynamics is
L =
∫d3x
((Dµφ)∗Dµφ−m2φ∗φ− 1
4FµνFµν
), (73)
where Dµ = ∂µ + ieAµ is the covariant derivative, with Aµ = (A0,A) and Fµν =
∂µAν − ∂νAµ. The corresponding field equations are
DµDµφ+m2φ = 0 , ∂µF
µν = jν , (74)
where
jν = ie (φ∗Dνφ− φDν∗φ∗) (75)
is the charge current. The theory has a local gauge symmetry
φ→ eieαφ , Aµ → Aµ − ∂µα . (76)
In this section we will focus on the Coulomb gauge ∇ ·A = 0. Writing A = AT +AL
(cf. footnote 5), this gauge corresponds to AL = 0 and the equations of motion reduce
to (+ 2ieA0∂t + ieA0 − e2A2
0 + 2ieAT ·∇ + e2AT2 +m2)φ = 0 , (77)
AT + ∇A0 = j , (78)
−∇2A0 = j0 , (79)
with
j = ie(φ∇φ∗ − φ∗∇φ)− 2e2AT |φ|2 , (80)
j0 = ie(φ∗φ− φφ∗)− 2e2A0|φ|2 . (81)
19
Eq. (79) can be written as(∇2 − 2e2|φ|2
)A0 = −ie(φ∗φ− φφ∗) . (82)
This is not a dynamical equation but rather a constraint on A0; it could be used to solve
for A0 in terms of φ.
5.1.2 Bohmian theory
The classical theory can easily be quantized using the Coulomb gauge. The same quan-
tum field theory can be obtained by eliminating the gauge degrees of freedom on the
classical level and then quantizing the remaining gauge-independent degrees of freedom
[34]. In the resulting Bohmian theory, the actual fields φ and AT are guided by a wave
functional Ψ(ϕ,AT , t) which satisfies the functional Schrodinger equation10
i∂tΨ =
∫d3x
(− δ2
δϕ∗δϕ+|(∇−ieAT )ϕ|2+m2|ϕ|2−1
2C 1
∇2C−1
2
δ2
δAT2+
1
2(∇×AT )2
)Ψ ,
(83)
where
C(x) = e
(ϕ∗(x)
δ
δϕ∗(x)− ϕ(x)
δ
δϕ(x)
)(84)
is the charge density operator in the functional Schrodinger picture. The first three
terms in the Hamiltonian correspond to the Hamiltonian of a scalar field minimally
coupled to a transverse vector potential. The fourth term corresponds to the Coulomb
potential and the remaining terms to the Hamiltonian of a free electromagnetic field.
The guidance equations are
φ =
(δS
δϕ∗− eϕ 1
∇2CS) ∣∣∣∣∣
φ,AT
, AT =δS
δAT
∣∣∣∣∣φ,AT
, (85)
with Ψ = |Ψ|eiS . Defining
A0 = −i1
∇2CS
∣∣∣∣∣φ,AT
, (86)
we can rewrite the guidance equation for the scalar field as
D0φ =δS
δϕ∗
∣∣∣∣∣φ,AT
. (87)
With this definition of A0, the classical equation (82) is satisfied.
10The wave functional should be understood as a functional of the real and imaginary part of ϕ. In
addition, writing ϕ = (ϕr + iϕi)/√
2, we have that the functional derivatives are given by Wirtinger
derivatives δ/δϕ = (δ/δϕr − iδ/δϕi)/√
2 and δ/δϕ = (δ/δϕr + iδ/δϕi)/√
2.
20
Taking the time derivative of the guidance equations we obtain, after some calcula-
tion, that
DµDµφ−m2φ = −δQ
Ψ
δϕ∗
∣∣∣∣∣φ,AT
, (88)
∂µFµν = jν + jνQ , (89)
where Aµ = (A0,AT ). The quantum potential is given by
QΨ = − 1
|Ψ|
∫d3x
(δ2
δϕ∗δϕ+
1
2C 1
∇2C +
1
2
δ2
δAT2
)|Ψ| . (90)
The current jµ is given by the classical expression (80) and (81). jνQ = (0, jQ), with
jQ =
(i∇ 1
∇2CQΨ − δQΨ
δAT
) ∣∣∣∣∣φ,AT
, (91)
can be considered an additional quantum current. The total current jν+jνQ is conserved,
i.e., ∂ν(jν + jνQ) = 0, as a consequence of (88). This is required for consistency of (89),
since ∂ν∂µFµν ≡ 0 (due to the anti-symmetry of Fµν). In contrast, the current jν
satisfies ∂νjν = −iCQΨ|φ,AT and is hence not necessarily conserved.
5.1.3 Semi-classical approximation: classical electromagnetic field
Let us first consider a semi-classical approximation by treating the electromagnetic field
classically and the scalar field quantum mechanically. Similarly as before, it can be
found by considering the conditional wave functional χ(ϕ, t) = Ψ(ϕ,AT (t), t) for the
scalar field and conditions under which the electromagnetic field approximately behaves
classically. We will not present this procedure here, but just give the resulting equations.
The wave functional χ(ϕ, t) satisfies the functional Schrodinger equation for a quan-
tized scalar field moving in an external classical transverse vector potential and Coulomb
potential:
i∂tχ =
∫d3x
(− δ2
δϕ∗δϕ+ |(∇− ieAT )ϕ|2 +m2|ϕ|2 − 1
2C 1
∇2C)χ , (92)
where C is defined as before. The actual scalar field satisfies
D0φ =δS
δϕ∗
∣∣∣∣∣φ
, (93)
where A0 is defined as before, with S now the phase of χ. The vector potential Aµ =
(A0,AT ) satisfies Maxwell’s equations
∂µFµν = jν + jνQ , (94)
21
where jνQ = (0, jQ) is an additional quantum current, with
jQ = i∇ 1
∇2CQχ
∣∣∣∣∣φ
(95)
and
Qχ = − 1
|χ|
∫d3x
(δ2
δϕ∗δϕ+
1
2C 1
∇2C)|χ| . (96)
So the equations (92)-(94) define the semi-classical approximation.
We still have that ∂µ(jµ + jµQ) = 0, so that the Maxwell equations (94) are consis-
tent. The correct source term in Maxwell’s equations was found by considering the full
Bohmian theory. Just using the current jµ would yield an inconsistent set of equations
since ∂νjν = −iCQχ|φ. Adding an extra current to jµ so that the total one would be
conserved would still leave an ambiguity since one could always add a vector that is
conserved.
Another essential ingredient in our derivation was that the gauge freedom was elim-
inated. We will see in section 5.3 that without such an elimination it does not seem
possible to derive a semi-classical approximation.
5.1.4 Semi-classical approximation: classical scalar field
We can also consider a semi-classical approximation where the scalar field is treated clas-
sically and the electromagnetic field quantum mechanically. In this case, the functional
Schrodinger equation for the wave function χ(AT , t) is
i∂tχ =
∫d3x
(− 1
2
δ2
δAT2+
1
2(∇×AT )2 +ieAT ·(φ∗∇φ− φ∇φ∗)+e2AT2|φ|2
)χ . (97)
The actual potential AT satisfies the guidance equation
AT =δS
δAT
∣∣∣∣∣AT
(98)
and the scalar field satisfies the classical equation
DµDµφ−m2φ = 0 , (99)
where again Aµ = (A0,AT ) with A0 defined by (82).
5.2 Unitary gauge
There are other possible semi-classical approximations that can be considered. We have
4 independent field degrees of freedom, namely the two transverse degrees of freedom of
the vector potential and the two (real) degrees of freedom of the scalar field, and any
of these or combinations thereof could in principle be assumed classical. Here, we will
22
consider two different semi-classical approximations. One where the amplitude of the
scalar field is assumed classical and another one where all degrees of freedom but the
amplitude are assumed classical. While we could consider these approximations using
the Bohmian formulation in terms of the Coulomb gauge, it is more elegant to consider it
in a different yet equivalent formulation which can be obtained by imposing the unitary
gauge. As in the case of the Coulomb gauge no consistency problems arise.
5.2.1 Classical theory
The unitary gauge is given by φ = φ∗. Writing
φ = ηeiθ/√
2 , (100)
with η =√
2|φ| (which can be done where φ 6= 0), this gauge amounts to θ = 0. The
classical equations become
η +m2η− e2AµAµη = 0 , ∂µFµν = jν , (101)
where now jν = −e2η2Aν . Maxwell’s equations become
A + ∇∇ ·A + e2η2A + ∇A0 = 0 , (102)(∇2 − e2η2
)A0 + ∇ · A = 0 . (103)
The last equation is again a constraint rather than a dynamical equation; it can be used
to express A0 in terms of A.
5.2.2 Bohmian theory
Quantization of the classical theory leads to the following functional Schrodinger equa-
tion for Ψ(η,A):
i∂tΨ =1
2
∫d3x
(− 1
η
δ
δη
(ηδ
δη
)+ (∇η)2 +m2η2 + e2A2η2
− δ2
δA2 −1
e2η2
(∇ · δ
δA
)2
+ (∇×A)2
)Ψ . (104)
The guidance equations are
η =δS
δη
∣∣∣∣∣η,A
, A =
[δS
δA −∇(
1
e2η2∇ · δS
δA
)] ∣∣∣∣∣η,A
. (105)
Defining
A0 =1
e2η2∇ · δS
δA
∣∣∣∣∣η,A
, (106)
the latter equation can be written as A = (δS/δA)|η,A−∇A0. With this definition the
classical equation (103) also holds for the Bohmian dynamics.
23
The second-order equations are
η +m2η− e2AµAµη = −δQΨ
δη
∣∣∣∣∣η,A
, ∂µFµν = jν + jνQ , (107)
where jνQ = (0, jQ), with
jQ = −δQΨ
δA
∣∣∣∣∣η,A
, (108)
and
QΨ = − 1
2|Ψ|
∫d3x
(1
η
δ
δη
(ηδ
δη
)+
δ2
δA2 +1
e2η2
(∇ · δ
δA
)2)|Ψ| . (109)
This Bohmian theory is equivalent to the one considered in the previous section.
This can easily be checked by using the field transformation (φ,AT )→ (η,A) which is
obtained by using the polar decomposition (100) for φ and A = AT + 1e∇θ (with inverse
transformation θ = e 1∇2∇ ·A).11
5.2.3 Semi-classical approximation: classical electromagnetic field
When the vector potential A evolves approximately classically, we have the following
semi-classical approximation. The wave functional χ(η, t) satisfies
i∂tχ =1
2
∫d3x
(− 1
η
δ
δη
(ηδ
δη
)+ (∇η)2 +m2η2 − e2AµA
µη2
)χ (110)
and the guidance equation
η =δS
δη
∣∣∣∣∣η
. (111)
The electromagnetic field satisfies
∂µFµν = −e2η2Aν . (112)
The appearance of the term containing A0 in the Schrodinger equation requires
some explanation. As before, the equation can be obtained by assuming that certain
terms negligible when considering the time derivative of the conditional wave functional
χ(η, t) = Ψ(η,A(t), t). In this case, i∂tχ contains the term
−∫d3x
1
2e2η2(x)
(∇ · δS
δA(x)
)2∣∣∣∣∣A(x,t)
χ =
∫d3x
e2η2(x)
2
−A20(x, t) +
A20(x, t)− 1
e4η4(x)
(∇ · δS
δA(x)
)2∣∣∣∣∣A(x,t)
χ . (113)
11The equivalence of the Schrodinger picture for the Coulomb gauge and the unitary gauge was
considered before in [41] (but seems to require a small correction for the kinetic term for η).
24
The term within the big square brackets on the right hand side would be zero when it
is evaluated for the actual field η(x, t) because of (106). Otherwise, it is not necessarily
zero. In our semi-classical approximation we assume that this term is negligible. As
such, the resulting Schrodinger equation (110) corresponds to the one of a quantized
scalar field minimally coupled to a classical electromagnetic field.
Note that there is no consistency issue in this case. Just as in the classical or the full
Bohmian case, the equation ∂µjµ = ∂µ(−e2η2Aµ) = 0 does not follow from the equation
of η, but from the Maxwell equations themselves.
5.2.4 Semi-classical approximation: classical scalar field
In the case the scalar field behaves approximately classically, we have the following
semi-classical approximation. The wave functional χ(A, t) satisfies
i∂tχ =1
2
∫d3x
(− δ2
δA2 −1
e2η2
(∇ · δ
δA
)2
+ (∇×A)2 + e2η2A2
)χ (114)
and guides the field A through the guidance equation
A =δS
δA
∣∣∣∣∣A
−∇A0 , (115)
where A0 is defined as in (106). The field η satisfies the classical equation
η +m2η− e2AµAµη = 0 . (116)
The Schrodinger equation corresponds to a spin-1 field with mass squared e2η2 [34].
5.3 Temporal gauge
In this section, we consider a Bohmian theory for scalar electrodynamics where not all
gauge freedom is eliminated. It will appear that we need to deal with the remaining
gauge freedom in order to get an adequate Bohmian semi-classical approximation.
5.3.1 Classical theory
The temporal gauge is given by A0 = 0. It does not completely fix the gauge. There is
still a residual gauge symmetry given by the time-independent transformations :
φ→ eieθφ , A→ A + ∇θ , (117)
with θ = 0.
In this gauge, the classical equations of motion read
φ−D2φ+m2φ = 0 , A + ∇∇ ·A = j , −∇ · A = j0 , (118)
where D = ∇− ieA, j = ie(φD∗φ∗ − φ∗Dφ) and j0 = ie(φ∗φ− φφ∗).
25
5.3.2 Bohmian theory
The Schrodinger equation for Ψ(ϕ,A, t) is12
i∂tΨ =
∫d3x
(− δ2
δϕ∗δϕ+ |(∇− ieA)ϕ|2 +m2|ϕ|2 − 1
2
δ2
δA2 +1
2(∇×A)2
)Ψ , (119)
together with the constraint
∇ · δΨδA + ie
(ϕ∗
δΨ
δϕ∗− ϕδΨ
δϕ
)= 0 , (120)
for the wave functional Ψ(ϕ,A, t). The constraint expresses the fact that the wave
functional is invariant under time-independent gauge transformations, i.e., Ψ(ϕ,A) =
Ψ(eieθϕ,A + ∇θ), with θ time-independent. The constraint is compatible with the
Schrodinger equation: if it is satisfied at one time, it is satisfied at all times.
The actual configurations φ and A satisfy [43]:
φ =δS
δϕ∗
∣∣∣∣∣φ,A
, A =δS
δA
∣∣∣∣∣φ,A
. (121)
These equations are invariant under the time-independent gauge transformations (117)
because of the constraint (120).
The corresponding second-order equations are
φ−D2φ+m2φ = −δQΨ
δϕ∗
∣∣∣∣∣φ,A
, A + ∇∇ ·A = j + jQ , (122)
where
QΨ = − 1
|Ψ|
∫d3x
(δ2
δϕ∗δϕ+
1
2
δ2
δA2
)|Ψ| (123)
and
jQ = −δQΨ
δA
∣∣∣∣∣φ,A
. (124)
The constraint (120) further implies that
−∇ · A = j0 , (125)
so that, assuming the gauge A0 = 0, (122) and (125) can be written as
DµDµφ−m2φ = −δQ
Ψ
δϕ∗
∣∣∣∣∣φ,A
, ∂µFµν = jν + jνQ , (126)
where jνQ = (0, jQ).
12In [42] this equation is considered in order to study the semi-classical approximation in the context
of standard quantum theory.
26
This Bohmian theory is equivalent to the one formulated using the Coulomb gauge
(and hence also to the one formulated using the unitary gauge). To see this, consider
the field transformation φ,A→ φ′,AT ,AL defined by φ′ = φ exp(−ie 1∇2∇ ·A) and the
usual decomposition of A into transverse and longitudinal part. In terms of the new
variables the wave function is Ψ′(ϕ′,AT ,AL) = Ψ(ϕ,A). Because of the constraint
(120) we have δΨ′/δAL = 0 and hence the wave functional does not depend on AL. For
a solution Ψ to the Schrodinger equation (119), Ψ′ will be a solution to the Schrodinger
equation (83) in the Coulomb gauge. (The latter equivalence is discussed in detail in
[41].) The guidance equations (121) also reduce to the ones in the Coulomb gauge.
They yield the extra equation −∇ ·AL = j0. This means that j0 determines AL up to a
time-independent gauge transformation. As such, AL is not an independent dynamical
degree of freedom and can be ignored. In the Coulomb gauge there is no field AL and
hence the Bohmian formulations are equivalent. (See section 4.3.1 and [34] for a similar
comparison respectively in the case of the Abraham model and the free electromagnetic
field.)
5.3.3 Usual semi-classical approximation
In the framework of standard quantum theory, there is a natural semi-classical ap-
proximation that treats the vector potential classically and the scalar field quantum
mechanically. The scalar field is described by a wave functional χ(ϕ, t) which satisfies
i∂tχ =
∫d3x
(− δ2
δϕ∗δϕ+ |Dϕ|2 +m2|ϕ|2
)χ (127)
and the electromagnetic field Aµ = (0,A) in the temporal gauge satisfies the classical
Maxwell equations
∂µFµν = 〈χ|jν |χ〉 , (128)
where
〈χ|j0|χ〉 =
∫Dϕχ∗Cχ = e
∫Dϕχ∗
(ϕ∗
δχ
δϕ∗− ϕδχ
δϕ
),
〈χ|j|χ〉 = ie
∫Dϕ|χ|2 (ϕD∗ϕ∗ − ϕ∗Dϕ) , (129)
where the operator C was defined in (84).
This theory is consistent since ∂µ〈χ|jµ|χ〉 = 0, as a consequence of the Schrodinger
equation (127). (It is also invariant under the time-independent gauge transformations
A→ A′ = A + ∇θ, χ(ϕ)→ χ′(ϕ) = χ(e−ieθϕ).)
5.3.4 Bohmian semi-classical approximation
A natural guess for a Bohmian semi-classical approximation similar to the usual one is
the following. An actual field φ is introduced that satisfies φ = (δS/δϕ∗)|φ, where the
wave functional satisfies (127), and the Maxwell equations are ∂µFµν = jν , where jµ is
27
the classical expression for the charge current. However, the second-order equation for
the Bohmian field is
φ−D2φ+m2φ = −δQχ
δϕ∗
∣∣∣∣∣φ
, (130)
where Qχ = − 1|χ|∫d3x
(δ2|χ|δϕ∗δϕ
). As a consequence, we have that ∂µj
µ = −iCQχ|φ and
hence Maxwell’s equations imply that CQχ = 0 or Qχ = Qχ(|ϕ|2). This is a constraint
on the wave functional that was absent in the usual semi-classical theory. It also seems
to be a rather strong condition. It will for example be satisfied if the scalar field evolves
classically (i.e., when the right-hand side of (130) is zero) but it is unclear whether there
are other solutions.
We arise to a similar conclusion from a more careful approach trying to derive a
semi-classical approximation from the full Bohmian theory. If we want a semi-classical
approximation with A classical, then we should require that jQ = 0 (since we want
the wave functional to depend solely on the scalar field and no longer on the vector
potential). However, due to the constraint (120), we have that ∇ · jQ|φ,A = iCQΨ|φ,Aand hence CQΨ|φ,A = 0.
So the conclusion seems to be that if we assume A classical, then φ should also
behave classically. This is not surprising since the gauge symmetry implies that the
physical (i.e., gauge-invariant) degrees of freedom are some combination of the fields A
and φ. So one can not just assume A classical and keep φ fully quantum.
A possible way to develop a semi-classical approximation is thus to separate gauge
degrees of freedom from gauge-independent ones and assume only some of the latter
to be classical. One way to do this is as follows. Writing A = AT + AL, we can
assume AT to behave classically and not AL, since AT does not change under a gauge
transformation, whereas AL does. We could fully eliminate AL by introducing the field
variable φ′ = φ exp(−ie 1∇2∇ ·A), which would lead to the full Bohmian theory and the
semi-classical approximation of section 5.1.3. However, we can also formulate a semi-
classical approximation by keeping the variable AL, but which is still equivalent to the
one of section 5.1.3. The functional Schrodinger equation for Ψ(ϕ,AL) is
i∂tΨ =
∫d3x
(− δ2
δϕ∗δϕ+ |[∇− ie(AT + AL)]ϕ|2 +m2|ϕ|2 − 1
2
δ2
δAL2
)Ψ (131)
and the constraint
∇ · δΨδAL
+ ie
(ϕ∗
δΨ
δϕ∗− ϕδΨ
δϕ
)= 0 . (132)
The guidance equations are
φ =δS
δϕ∗
∣∣∣∣∣φ,AL
, AL =δS
δAL
∣∣∣∣∣φ,AL
. (133)
The equation of motion for AT is
AT = jT . (134)
28
Together these equations imply
DµDµφ−m2φ = − δQ
δϕ∗
∣∣∣∣∣φ,AL
, ∂µFµν = jν + jνQ (135)
in the temporal gauge, where jνQ = (0, jQ), with
jQ = − δQΨ
δAL
∣∣∣∣∣φ,AL
, (136)
and quantum potential
QΨ = − 1
|Ψ|
∫d3x
(δ2
δϕ∗δϕ+
1
2
δ2
δAL2
)|Ψ| . (137)
We still have gauge invariance under time independent gauge transformations. This
semi-classical approximation is equivalent to the one of section 5.1.3 with the Coulomb
gauge. Similarly as in the case of the full Bohmian theories (cf. section 5.3.2), this
can be shown by applying the field transformation (φ,AL) → (φ′,AL) with φ′ =
φ exp(−ie 1∇2∇ ·AL).
6 Gravity
6.1 Canonical quantum gravity
In canonical quantum gravity, the state vector is a functional of a spatial metric hij(x)
on a 3-dimensional manifold and the matter degrees of freedom, say a scalar field ϕ(x).
The wave functional is static and merely satisfies the constraints [2]:
HΨ(h, ϕ) = 0 , (138)
HiΨ(h, ϕ) = 0 , i = 1, 2, 3 . (139)
Their explicit forms are not important here. The latter constraints express the fact
that the wave functional is invariant under infinitesimal diffeomorphisms of 3-space
(i.e., spatial coordinate transformations). The former equation is the Wheeler-DeWitt
equation. It is believed that this equation contains the dynamical content of the theory.
However, it is as yet not clear how this dynamical content should be extracted. This is
the problem of time [2, 44].
In the Bohmian theory, there is an actual 3-metric and a scalar field, whose dynamics
depends on the wave function [12, 45–47]. The dynamics expresses how the Bohmian
configuration changes along a succession of 3-dimensional space-like surfaces, which
forms a foliation of space-time.13 Although the wave function is stationary, the Bohmian
13The succession of the surfaces is determined by the lapse function and different choices of lapse
function lead to a different Bohmian dynamics. This is analogous to the fact that in Minkowski space-
time, the Bohmian dynamics (at least in the usual formulation) depends on the choice of a preferred
reference frame or foliation.
29
configuration will change along these surfaces for generic wave functions. This is how
the Bohmian theory solves the problem of time.
The structure of the theory is similar to that of scalar electrodynamics in the tempo-
ral gauge, which was discussed in section 5.3. Namely, in both cases there is a constraint
on the wave functional which expresses invariance under infinitesimal gauge transfor-
mations: spatial diffeomorphisms in the case of gravity and U(1) transformations in the
case of scalar electrodynamics. Therefore we may encounter similar complications in
developing a consistent Bohmian semi-classical approximation. Indeed, as we will see
below, the Bohmian energy-momentum tensor (for the matter) will not be covariantly
conserved and hence can not be used as the source term in the classical Einstein equa-
tions. This feature is analogous to what we saw in the case of scalar electrodynamics
in the temporal gauge. In that case, the Bohmian charge current was not conserved so
that it could not enter as the source in Maxwell’s equations. The possible solution to
the problem is presumably similar to that in the case of electrodynamics, namely the
gauge invariance should be eliminated by working with gauge-invariant degrees of free-
dom or by choosing a gauge. However, this is a notoriously hard problem in the case of
gravity. This problem does not appear in simplified symmetry-reduced models of quan-
tum gravity called mini-superspace models since the spatial diffeomorphism invariance
is eliminated. We will discuss such a model in section 6.5.
First, we will consider the problem in formulating a Bohmian semi-classical approx-
imation in more detail. In sections 6.2 and 6.3 we will respectively consider a matter
field and matter particles. Instead of starting from the Bohmian formulation for canon-
ical quantum gravity, it will be simpler to start from quantized matter on an classical
curved space-time. This already allows us to find a natural candidate for the energy-
momentum tensor that could be used in the classical Einstein field equations. However,
in both cases the natural energy-momentum tensor is not conserved. This suggest that
we should start from the full quantum theory with some appropriate elimination of the
gauge freedom. In section 6.4 we will consider a Bohmian semi-classical theory in terms
of non-relativistic point-particles coupled to Newtonian gravity, which is consistent. For
the mini-superspace model, we will derive the semi-classical approximation from the full
quantum theory.
6.2 Semi-classical gravity: matter field
The formulation of quantum field theory on a classical curved space-time in the func-
tional Schrodinger picture was detailed in [48, 49]. We assume that the space-time
manifold M is globally hyperbolic so that it can be foliated into space-like hypersur-
faces. M is then diffeomorphic to R × Σ, with Σ a 3-surface. We choose coordinates
xµ = (t,x) such that the time coordinate t labels the leaves of the foliation and x are
coordinates on Σ. In terms of these coordinates the space-time metric and its inverse
30
can be written as
gµν =
(N2 −NiN
i −Ni
−Ni −hij
), gµν =
(1N2
−N i
N2
−N i
N2N iNj
N2 − hij
), (140)
where N is the lapse function and Ni = hijNj are the shift functions. hij is the induced
Riemannian metric on the leaves of the foliation. The unit vector field normal to the
leaves is nµ = (1/N,−N i/N)T .
For a real mass-less scalar field, the Schrodinger equation for this space-time back-
ground reads
i∂Ψ
∂t=
∫Σd3x
(NH+N iHi
)Ψ , (141)
where Ψ is a functional on the space of fields ϕ on Σ and
H =1
2
√h
(−1
h
δ2
δϕ2+ hij∂iϕ∂jϕ
), (142)
Hi = − i
2
(∂iϕ
δ
δϕ+
δ
δϕ∂iϕ
). (143)
This equation describes the action of the classical metric onto the quantum field.
The usual way to introduce a back-reaction is by using the expectation value of the
energy-momentum tensor operator as the source term in Einstein’s field equations, i.e.,
Gµν(g) = 〈Ψ|Tµν(ϕ, g)|Ψ〉 . (144)
This equation is consistent since ∇µ〈Ψ|Tµν(ϕ, g)|Ψ〉 = 0 because of (141).
In the Bohmian theory, the actual scalar field φ satisfies the guidance equation
φ =N√h
δS
δϕ
∣∣∣∣∣φ
+N i∂iφ (145)
or, equivalently,
nµ∂µφ =1√h
δS
δϕ
∣∣∣∣∣φ
. (146)
This dynamics depends on the foliation (unlike the wave function dynamics). That is,
different foliations will lead to different evolutions of the scalar field.
Taking the time derivative of this equation, we obtain
∇µ∇µφ = − 1√−g
δQΨ
δϕ
∣∣∣∣∣φ
, (147)
with
QΨ(ϕ, t) = − 1
2|Ψ(ϕ, t)|
∫d3x
N(x, t)√h(x, t)
δ2|Ψ(ϕ, t)|δϕ2(x)
(148)
the quantum potential.
31
To find the energy-momentum tensor for the Bohmian field, we consider the action
SB =1
2
∫d4x√−g (gµν∂µφ∂νφ)−
∫dtQΨ(φ) (149)
from which the equation of motion (147) can be derived. We can then use the usual
definition of the energy-momentum tensor
Tµν = − 2√−g
δSBδgµν
(150)
to obtain
Tµν = ∂µφ∂νφ− 1
2gµνgαβ∂αφ∂βφ+ TµνQ , (151)
which has the form of the classical expression with a quantum contribution
TµνQ =2√−g
δ∫dtQΨ(φ)
δgµν= − 1
2h|Ψ|δ2|Ψ|δϕ2
∣∣∣∣∣φ
(nµnν + δµi δ
νj h
ij). (152)
The expression (151) seems to be the natural one for the Bohmian energy-momentum
tensor, which could be used in the classical Einstein field equations. However, by taking
the covariant derivative, we find
∇µTµν = − 1√−g
δQΨ
δϕ∂νϕ+∇µTµνQ , (153)
which is generically different from zero, so that Einstein’s equations would not be con-
sistent with this energy-momentum tensor as matter source. Note that even without
the contribution TµνQ , the energy-momentum tensor is not conserved.
The energy-momentum tensor was derived here in the semi-classical context. In a
more precise derivation, we should start from the Bohmian theory of quantum gravity
and consider the modified Einstein field equations [13] to find the energy-momentum
tensor for the matter field. However, unless the spatial diffeomorphism invariance is
eliminated, this would not help in finding a consistent semi-classical approximation.
Once this invariance is eliminated one can expect a further contribution to the energy-
momentum tensor that makes it conserved.
6.3 Semi-classical gravity: matter particles
A similar consistency problem arises when point-particles are considered. To illustrate
this, we will consider a single Klein-Gordon particle. (The extension to many particles
is straightforward.) The wave equation in an external gravitational field is
gµν∇µ∇νψ(x) +m2ψ(x) = 0 . (154)
The guidance equation for a point-particle with space-time coordinates Xµ is
dXµ
ds= − 1
m∇µS|X . (155)
32
Note that this guidance equation determines a particular parameterization of the world-
line. The proper-time τ is given by dτ =√
1 + 2Qψ(X(s))/mds, where
Qψ =1
2mR∇µ∇µR , R = |ψ| , (156)
is the quantum potential. The parameterization by s will be most convenient to express
the energy-momentum tensor. (As is well-known, even in the case of flat space-time
the Bohmian trajectories are not always time-like, not-even when restricting to positive
energies [8, 50]. However, this problem is unrelated to finding a consistent semi-classical
theory. Instead of the Klein-Gordon theory one could also consider the Dirac theory,
where the trajectories are time-like. However, this would entail unnecessary complica-
tions.)
The guidance equation implies the following second-order equation:
d2Xµ
ds2+ Γµαβ
dXα
ds
dXβ
ds=
1
m∂µQψ(X) . (157)
This equation can be derived from the following action
SB = −∫ds
(m
2gµν(X)
dXµ
ds
dXν
ds+Qψ(X)
). (158)
Using again the definition (150) for the energy-momentum tensor, we obtain
Tµν(x) =1√−g(x)
∫dsδ(x−X)m
dXµ
ds
dXν
ds+ TµνQ (x)
=1√−g(x)
1
m∇µS(x)∇νS(x)
∫dsδ(x−X) + TµνQ (x) , (159)
which is the classical expression with a quantum contribution
TµνQ (x) =2√−g(x)
δ∫dsQψ
δgµν(x). (160)
This expression could be further worked out. However, just as in the case of a matter
field, the energy-momentum tensor is not conserved but gives
∇µTµν(x) =1√−g(x)
∂νQψ(x)
∫dsδ(x−X) +∇µTµνQ (x) . (161)
In the case of the Abraham model, which also conserns point-particles, no inconsis-
tencies were encountered. But the important different is that in the case of the Abraham
model, as noted at the end of section 4.3.2, a gauge transformation does not act on the
particles, but just on the electromagnetic field, while in the case of gravity a gauge trans-
formation (i.e. diffeomorphism transformation) acts on both the particle and the metric
field. Presumably, that is the reason why a consistent semi-classical approximation does
not follow directly in the latter case.
33
6.4 Bohmian analogue of the Schrodinger-Newton equation
In the previous section, we explained the problem in finding a consistent Bohmian semi-
classical approximation by coupling classical gravity to relativistic point-particles. If we
consider the Newtonian limit of gravity and the non-relativistic limit for the quantum
matter, we obtain a consistent Bohmian semi-classical theory.
Without explicitly performing these limits to the semi-classical theory outlined in
the previous section (one could follow the analysis of [51]), the resulting theory is given
by the Schrodinger equation
i∂tψ(x, t) =
[− 1
2m∇2 +mΦ(x, t)
]ψ(x, t) , (162)
where Φ is the gravitational potential, which satisfies the Poisson equation
∇2Φ(x, t) = 4πGmδ(x−X(t)) , (163)
and where X is the position of the Bohmian particle, which satisfies the guidance equa-
tion
X(t) = vψ(X(t), t) . (164)
So the Bohmian particle acts as the gravitational source. Solving for the potential, one
can write (162) as
i∂tψ(x, t) =
[− 1
2m∇2 − Gm2
|x−X(t)|
]ψ(x, t) . (165)
The generalization to many particles is straightforward. In this case, the gravita-
tional potential generated by the Bohmian particles is
Φ(x, t) = −G∑k
mk1
|x−Xk(t)|, (166)
so that the potential in the many-particle Schrodinger equation is given by
V (x) =∑k
mkΦ(xk, t) . (167)
This semi-classical approximation is similar to the Schrodinger-Newton equation [52],
which can be derived from (141) and (144) [53], and which is given by (162) together
with
∇2Φ = 4πGm|ψ|2 . (168)
So in this case the mass density m|ψ|2 acts as the gravitational source. Solving for the
potential, the following non-linear Schrodinger equation
i∂tψ(x, t) =
[− 1
2m∇2 −Gm2
∫d3y|ψ(y, t)|2
|x− y|
]ψ(x, t) (169)
is obtained.
34
While the Bohmian theory is now consistent, it should be noted that it might not
directly follow from a Bohmian quantum gravity with point-particles, because of the
lessons that were drawn in the case of the Abraham model in section 4. For example, in
the case of the Coulomb gauge, we found (slightly) different theories by starting from
a natural guess for a semi-classical theory (cf. section 4.4) or by deriving it from the
full Bohmian theory for the Abraham model (cf. section 4.2.2). In the former case,
the Coulomb potential that appears in the Schrodinger equation was generated by the
Bohmian charges, while in the latter case it is not. In the latter case the Coulomb
potential just takes the usual form as a function on configuration space. In the case of
gravity, we have similarly merely guessed the semi-classical approximation rather than
having derived it from a Bohmian quantum gravity. This being said, the difference
between the two version in the case of the Abraham model seem minor. Similarly, the
Bohmian analogue of the Schrodinger-Newton equation might form a good semi-classical
approximation. (It could perhaps also be seen as an alternative to collapse models. A
model similar in spirit was studied in [54].)
6.5 Mini-superspace model
In this section, we consider a symmetry-reduced model of quantum gravity where homo-
geneity and isotropy are assumed. In this model, the spatial diffeomorphism invariance
is eliminated and we can straightforwardly develop a Bohmian semi-classical approxi-
mation.
In the classical mini-superspace model, the universe is described by the Friedmann-
Lemaıtre-Robertson-Walker (FLRW) metric
ds2 = N(t)2dt2 − a(t)2dΩ23 , (170)
where N is the lapse function, a = eα is the scale factor14 and dΩ23 is the metric on 3-
space with constant curvature k. Assuming matter that is described by a homogeneous
scalar field φ, the Lagrangian is [55, 56]
L = Ne3α
[−κ(α2
2N2+ VG(α)
)+
φ2
2N2− VM (φ)
], (171)
where κ = 3/4πG.
VG(α) = −1
2ke−2α +
1
6Λ (172)
is the gravitational potential, with Λ the cosmological constant, and VM is the potential
for the matter field. The corresponding equations of motion are, after imposing the
14The reason for introducing the variable α is that it is unbounded, unlike the scale factor, which
satisfies a > 0.
35
gauge N = 1:15
1
2α2 =
1
κ
(1
2φ2 + VM (φ)
)+ VG(α) , (173)
φ+ 3αφ+ ∂φVM (φ) = 0 . (174)
(The second-order equation for α which arises from variation with respect to α is re-
dundant since it can be derived from the other two equations.)
Using the canonical momenta
πN = 0 , πα = −κe3α α
N, πφ = e3α φ
N, (175)
we can pass to the Hamiltonian formulation. This leads to the Hamiltonian constraint
(which is just eq. (173))
− 1
2κe3απ2α +
1
2e3απ2φ + e3α [κVG(α) + VM (φ)] = 0 . (176)
Quantization yields the Wheeler-DeWitt equation:
(HG + HM )ψ(ϕ,α) = 0 , (177)
where
HG =1
2κe3α∂2α + κe3αVG(α) , HM = − 1
2e3α∂2ϕ + e3αVM (ϕ) . (178)
In the corresponding Bohmian theory [56], the FLRW metric (of the form (170)) and
the scalar field satisfy the guidance equations
α = − N
κe3α∂αS
∣∣ϕ=φ,α=α
, φ =N
e3α∂ϕS
∣∣ϕ=φ,α=α
, (179)
where N(t) is an arbitrary lapse function.16 In the gauge N = 1, these equations imply
1
2α2 =
1
κ
(1
2φ2 + VM (φ) +QψM (φ, α)
)+ VG(α) +QψG(φ, α) , (180)
φ+ 3αφ+ ∂ϕ(VM +QψM + κQψG)∣∣ϕ=φ,α=α
= 0 , (181)
where
QψG =1
2κ2e6α
∂2α|ψ||ψ|
, QψM = − 1
2e6α
∂2ϕ|ψ||ψ|
. (182)
15The theory is time-reparamaterization invariant. Solutions that differ only by a time-
reparameterization are considered physically equivalent. Choosing the gauge N = 1 corresponds to
a particular time-parameterization.16Just as the classical theory, the Bohmian theory is time-reparameterization invariant. This is a
special feature of mini-superspace models [57, 58]. As mentioned before, for the usual formulation of the
Bohmian dynamics for the Wheeler-DeWitt theory of quantum gravity, a particular space-like foliation
of space-time or, equivalently, a particular choice of “initial” space-like hypersurface and lapse function,
needs to be introduced. Different foliations (or lapse functions) yield different Bohmian theories.
36
We will now look for a semi-classical approximation where the scale factor behaves
approximately classical. In order to do so, we assume again the gauge N = 1 and we
consider the conditional wave function χ(ϕ, t) = ψ(ϕ, α(t)), given a set of trajectories
(α(t), φ(t)). Using
∂tχ(ϕ, t) = ∂αψ(ϕ,α)∣∣α=α(t)
α(t) , (183)
we can write
i∂tχ = HMχ+ I , (184)
where17
I =1
αi∂tχ
(α+
1
κe3α∂αS
∣∣α=α
)+
1
2κe3α
[(∂αS)2 + i∂2
αS] ∣∣∣
α=αχ+κe3α(VG+QψG)
∣∣∣α=α
χ .
(185)
When I is negligible (up to a real time-dependent function times χ), (184) becomes the
Schrodinger equation for a homogeneous matter field in an external FLRW metric. We
can further assume the quantum potential QψG to be negligible compared to other terms
in eq. (180). As such, we are led to the semi-classical theory:
i∂tχ = HMχ , (186)
φ =1
e3α∂ϕS
∣∣ϕ=φ
, (187)
1
2α2 =
1
κ
(1
2φ2 + VM (φ) +QχM (φ, α)
)+ VG(α) ≡ − 1
κe3α∂tS∣∣ϕ=φ
+ VG(α) . (188)
Let us now consider when the term I will be negligible. The quantity in brackets
in the first term would be zero when evaluated for the actual trajectory φ(t) (because
of the guidance equation for α). As such, the first term will be negligible if the actual
scale factor evolves approximately independently of the scalar field. The second term
will be negligible if S varies slowly with respect to α or if the term in square brackets
is approximately independent of ϕ. In the latter case, the second term becomes a
time-dependent function times χ, which can be eliminated by changing the phase of
χ. Similarly, if QψG VG then the third term also becomes a time-dependent function
times χ.
In the usual semi-classical approximation, one has (186) and
1
2α2 =
1
κe3α〈χ|HM (α)|χ〉+ VG(α) , (189)
with χ normalized to one (which is the analogue of (1) and (2) for mini-superspace).
In the next section, we will compare this approximation with the Bohmian one for a
particular example. It will appear that the latter gives better results than the usual
approximation. (Note that Vink himself, in his seminal paper [56] which introduces the
17To obtain this equation, note that ∂2αψ = [(∂αS)2 + i∂2
αS + ∂2α|ψ|/|ψ|]ψ + 2i∂αS∂αψ, so that
∂2αψ|α=α(t) = [(∂αS)2 + i∂2
α + ∂2α|ψ|/|ψ|]|α=αχ + 2i∂αS|α=α∂tχ/α. Using this equation together with
(177) we obtain (184).
37
Bohmian theory for quantum gravity, considers a derivation of the usual semi-classical
approximation, rather than the Bohmian one. But he hinted on the Bohmian semi-
classical approximation in [59].)
6.6 Example in mini-superspace
In this section, we will work out a simple example to compare the Bohmian and usual
semi-classical approximations to the full Bohmian result.
We put κ = 1 and assume VG = VM = 0. It will also be useful to introduce the time
parameter τ , defined by dτe3α = dt (which corresponds to choosing N = e3α instead
of N = 1). Derivatives with respect to τ will be denoted by primes. In this way, the
classical equations (173) and (174) reduce to
α′2 = φ′2 , φ′′ = 0 . (190)
The possible solutions are
α = c1τ + c2 , φ = ±c1τ + c3 , (191)
where ci, i = 1, 2, 3, are constants. In the case c1 = 0, the scale factor is constant and
we have Minkowski space-time. If c1 > 0 the universe starts from a big bang and keeps
expanding forever. If c1 < 0 the universe contracts until a big crunch. If c1 6= 0, the
corresponding paths in (φ, α)-space are given by
α = ±φ+ c , (192)
with c constant.
6.6.1 Full Bohmian analysis
In the full quantum case, we have the Wheeler-DeWitt equation
(∂2α − ∂2
ϕ)ψ(ϕ,α) = 0 (193)
and guidance equations
α′ = −∂αS∣∣ϕ=φ,α=α
, φ′ = ∂ϕS∣∣ϕ=φ,α=α
. (194)
For the wave function
ψR(ϕ,α) = exp
[iu(ϕ− α)− (ϕ− α)2
4σ2
], (195)
the guidance equations read α′ = φ′ = u, so that we have only classical solutions
α = uτ + c1 , φ = uτ + c2 (196)
or
α = φ+ c . (197)
38
Figure 1: Some trajectories for ΨR, σ =
1.
Figure 2: Some trajectories for ΨL, σ =
1.
See fig. 1 for some trajectories.
Similarly, for the wave function
ψL(ϕ,α) = exp
[−iv(ϕ+ α)− (ϕ+ α)2
4σ2
](198)
the solutions are also classical:
α = vτ + c1 , φ = −vτ + c2 , (199)
or
α = −φ+ c , (200)
see fig. 2.
Consider now the superposition ψ = ψR + ψL and assume v > u 0. For α→ ±∞the wave functions ψR and ψL are non-overlapping functions of ϕ so that, asymptotically,
the Bohmian dynamics is either determined by ψR or ψL. This means that asymptoti-
cally, i.e., for α→ ±∞, we have classical motion, given either by (197) or (200). Some
trajectories are plotted in figs. 3 and 4. (Trajectories for the case u = −v were plotted
in [60].) We see that trajectories starting from the “left”, i.e. φ < 0, for α → −∞will end up on the left, while trajectories that start from the “right”, i.e. φ > 0, might
either end up moving to the left or to the right. (If u = v, then trajectories starting on
the left stay on the left and trajectories starting on the right stay on the right.) Some
trajectories are closed. They correspond to cyclic universes (which oscillate between a
minimum and maximum scale factor), see fig. 4.
For trajectories with classical asymptotic behavior, there is possible non-classical
behavior in the region of overlap, where α ≈ 0. For trajectories starting on the left
there is a transition from α = uφ to α = −vφ (which is impossible classically). For some
trajectories starting on the right, there is the opposite transition.
39
Figure 3: Some trajectories (or pieces
thereof) for ΨR + ΨL, u = 1, v = 5,
σ = 1. Three trajectories are high-
lighted which illustrate the possible be-
haviors for trajectories which asymptot-
ically go like α = ±φ.
Figure 4: Some trajectories (or pieces
thereof) for ΨR+ΨL, u = 1, v = 5, σ =
1. Three trajectories are highlighted.
Two of these correspond to closed loops,
which correspond to cyclic universes.
Note that there is no natural measure on the set of trajectories [61], so we can not
make any probabilistic statements like about the probability for a trajectory to move
from left to right.
6.6.2 Usual semi-classical approximation
Let us first consider the usual semi-classical approximation and compare it to the full
Bohmian theory. Eqs. (186) and (189) reduce to
i∂τχ = hMχ = −1
2∂2ϕχ , (201)
α′2 = 2〈χ|hM |χ〉 = −〈χ|∂2ϕ|χ〉 . (202)
The wave equation corresponds to that of a single particle of unit mass in one dimension.
Hence, 〈χ|∂2ϕ|χ〉 is time-independent and we have that α′ is constant.
The initial wave function which we will use in the semi-classical approximation is
given by the conditional wave function χ(ϕ, τ0) = Nψ(ϕ, α(τ0)) at some time τ0, with
N a normalization constant. The conditional wave functions corresponding to ψR and
ψL are
χR(ϕ, τ0) = NRψR(ϕ, α0) = (2πσ2)−1/4 exp
[iu(ϕ− α0)− (ϕ− α0)2
4σ2
],
χL(ϕ, τ0) = NLψL(ϕ, α0) = (2πσ2)−1/4 exp
[−iv(ϕ+ α0)− (ϕ+ α0)2
4σ2
], (203)
40
where α0 = α(τ0). χR is a Gaussian packet centered around α0 and average momentum
u, while χL is a Gaussian packet centered around −α0 and average momentum −v. The
solutions to the Schrodinger equation (201), with these conditional wave functions as
initial conditions, are given by [8]:
χR(ϕ, τ) = [2πs2(τ)]−1/4 exp[iu(ϕR − uτ/2)− (ϕR − uτ)2/4s(τ)σ
],
χL(ϕ, τ) = [2πs2(τ)]−1/4 exp[−iv(ϕL + vτ/2)− (ϕL + vτ)2/4s(τ)σ
], (204)
where
ϕR = ϕ− α0 , ϕL = ϕ+ α0 , τ = τ − τ0 , s(τ) = σ(1 + iτ /2σ2) . (205)
In the following, we assume that
• τ0 is small enough, so that α0 < 0 and |α0| σ (so that χR and χL have little
overlap initially),
• α′(τ0) > 0 ((202) determines α′ only up to a sign),
• v > u 0,
• 1 σu.
The expectation value 〈χ|hM |χ〉 is time-independent. So we can calculate it at time
τ0. We have that
2〈χR|hM |χR〉 = u2 +1
4σ2. (206)
Hence α′ =√u2 + 1
4σ2 . Since 1 σu, we have that α′ ≈ u, so that we approximately
obtain the classical solution for the scale factor.
Similarly, for χL, we have that
2〈χL|hM |χL〉 = v2 +1
4σ2(207)
and since 1 σu < σv, we have α′ ≈ v.
Now consider the superposition χ = (χR + χL)/√
2. Since χR and χL have ap-
proximately negligible overlap initially, because |α0| σ, this state is approximately
normalized to one and
2〈χ|hM |χ〉 ≈ 〈χR|hM |χR〉+ 〈χL|hM |χL〉 =1
2(u2 + v2) +
1
4σ2. (208)
Hence, for 1 σu, we have that α′ ≈√
(u2 + v2)/2. As such, the semi-classical
approximation is very close to the exact result, given that u ≈ v. But if u is very
different from v, the semi-classical approximation is not so good. In particular, we do
not get the asymptotic behavior that α′ = u or α′ = v for early or late times (i.e., τ → τ0
or τ →∞).
41
6.6.3 Bohmian semi-classical approximation
The equations of motion in the Bohmian semi-classical approximation are
i∂τχ = −1
2∂2ϕχ , (209)
φ′ = ∂ϕS∣∣ϕ=φ
, (210)
α′2 = φ′2 −∂2ϕ|χ||χ|
∣∣∣∣∣ϕ=φ
≡ −2∂τS∣∣ϕ=φ
. (211)
For the state χR, the solutions of the guidance equation are [8]:
φ− α0 = u(τ − τ0) + (φ0 − α0)|s(τ − τ0))|
σ(212)
or
φR = uτ + φR,0|s(τ)|σ
, (213)
where φ0 = φ(τ0) and φR,0 = φR(τ0) = φ0−α0 and the notation (205) is used. With the
same assumptions about the constants as in the previous section and taking |φR,0| . σ,
i.e., that the initial value φR,0 does not lie too far outside the bulk of the Gaussian
packet, we have that
φR ≈ uτ + φR,0 . (214)
This is because the difference between the trajectories is
|φR,0|∣∣∣∣ |s(τ)|
σ− 1
∣∣∣∣ = |φR,0|∣∣∣√1 + τ2/4σ4 − 1
∣∣∣ 6 |φR,0| τ2σ2 uτ . (215)
So we approximately get classical motion and hence the full Bohmian result.
Using
S = −1
2tan−1
( τ
2σ2
)+ u
(ϕR −
1
2uτ
)− (ϕR − uτ)2
8σ2|s(τ)|2, (216)
the classical equation for the scale factor becomes
α′2 = u2 +1
2|s(τ)|2+
φR,0uτ
2σ3|s(τ)|+φ2R,0
4σ4
(τ2
4σ2|s(τ)|2− 1
). (217)
We have that∣∣∣∣∣ 1
2|s(τ)|2+
φR,0uτ
2σ3|s(τ)|+φ2R,0
4σ4
(τ2
4σ2|s(τ)|2− 1
)∣∣∣∣∣≤ 1
2|s(τ)|2+u |φR,0τ |2σ3|s(τ)|
+φ2R,0
4σ4
∣∣∣∣( τ2
4σ2|s(τ)|2− 1
)∣∣∣∣ ≤ 1
2σ2+u|φR,0|σ2
+φ2R,0
4σ4, (218)
where we used |τ |/2σ|s(τ)| 6 1 for the last two terms in order to obtain the last in-
equality. Using the assumptions that 1 σu and |φR,0| . σ, we find that
α′2 ≈ u2 . (219)
42
So similarly as in the case of the usual semi-classical approximation (just with the extra
condition on the initial value φR,0), we obtain the full quantum results. For the wave
function ψL similar results hold.
Consider now the superposition χ = (χR +χL)/√
2. This superposition corresponds
to two Gaussian packets that move across each other. Initially and finally they are
approximately non-overlapping.18 This means that before and after the wave packets
cross, the motions of φ and α are approximately classically, in agreement with the full
Bohmian results, since they will be determined by either χR or χL. This is unlike the
usual semi-classical approximation.
A trajectory for the scalar field starting from the left (i.e., φ < 0) will end up on the
left, while a trajectory starting on the right will end up on the right. The reason is that
because of equivariance of the distribution |χ|2, the |χ|2-probability to start from the
left equals the probability to end up on the left (which equals 1/2). Since trajectories do
not cross (since the dynamics is given by a first-order differential equation in time), the
probability for trajectories to start on the left (right) and end up on the right (left) must
be zero. This means that for all trajectories starting on the left, we have a transition
from α = uφ to α = −vφ and the opposite transition for trajectories starting on the
right. This is unlike the full Bohmian analysis where trajectories exist where such a
transition does not occur.
In conclusion, we see that the Bohmian semi-classical approximation is in better
agreement with the exact Bohmian results than the usual semi-classical approximation.
We came to this conclusion by making the comparison on the level of the actual tra-
jectories for α and φ. We did not attempt to make a comparison in the context of
standard quantum theory, since the standard quantum interpretation of the Wheeler-
DeWitt equation is problematic due to the problem of time. But it is clear that for
approaches to quantum theory that would associate (approximately) the classical evolu-
tions (197) and (200) to the superposition ΨR+ΨL (like perhaps the Consistent Histories
or Many Worlds theory) the usual semi-classical approximation would fare worse than
the Bohmian semi-classical approximation.
7 Conclusion
We have shown how semi-classical approximations can be developed using Bohmian
mechanics. We have obtained these approximations from the full Bohmian theory by
assuming certain degrees of freedom to evolve approximately classically. This was illus-
trated for non-relativistic systems. If there is a gauge symmetry, like in electrodynamics
or gravity, then extra care is required in order to obtain a consistent semi-classical
theory. By eliminating the gauge symmetry (either by imposing a gauge or by work-
ing with gauge-independent degrees of freedom), we were able to find a semi-classical
18The latter statement follows from the fact that the spread of the wave function, which equals
|s(τ)| = σ√
1 + τ2/4σ4, goes like τ /2σ for large time, which is much smaller than the distance between
the centers of the Gaussians, which equals (u+ v)τ .
43
approximation in the case of scalar quantum electrodynamics. For quantum gravity,
eliminating the gauge symmetry (more precisely the spatial diffeomorphism invariance)
is notoriously hard. We have only considered the simplified mini-superspace approach
to quantum gravity, which describes an isotropic and homogeneous universe, and where
the diffeomorphism invariance is explicitly eliminated. More general cases in quantum
gravity still need to be studied. For example, for the case of inflation theory, where one
usually considers quantum fluctuations on a classical isotropic and homogeneous uni-
verse, it should not be too difficult to develop a Bohmian semi-classical approximation.
Apart from possible applications in quantum cosmology, such as inflation theory, it
might also be interesting to consider potential applications in quantum electrodynamics
or quantum optics. In particular, since the results may be compared to the predictions
of full quantum theory, this could give us a handle on where to expect better results
for the Bohmian semi-classical approximation compared to the usual one in the case of
quantum gravity where the full quantum theory is not known. That is, it might give us
better insight in which effects are truly quantum and which effect are merely artifacts
of the approximation.
Further developments may include higher order corrections to the semi-classical ap-
proximation. One way of doing this might be by following the ideas presented in [62, 63].
As explained there, one might introduce extra wave functions for a subsystem in addi-
tion to the conditional wave function. These wave functions interact with each other and
the Bohmian configuration. By including more of those wave functions one presumably
obtains better approximations to the full quantum result.
Finally, although we regard the Bohmian semi-classical approximation for quantum
gravity as an approximation to some deeper quantum theory for gravity, one could also
entertain the possibility that it is a fundamental theory on its own. At least, there is
presumably as yet no experimental evidence against it (unlike the usual semi-classical
approximation which is ruled out as fundamental theory by the experiment of Page and
Geilker).
8 Acknowledgments
It is a pleasure to thank Detlef Durr, Sheldon Goldstein, Christian Maes, Travis Norsen,
Nelson Pinto-Neto, Daniel Sudarsky and Hans Westman for useful discussions and com-
ments. This work was done partly at the University of Liege, with support from the
Actions de Recherches Concertees (ARC) of the Belgium Wallonia-Brussels Federa-
tion under contract No. 12-17/02, at the LMU, Munich, with support of the Deutsche
Forschungsgemeinschaft. Currently the support is by the Research Foundation Flanders
(Fonds Wetenschappelijk Onderzoek, FWO), Grant No. G066918N.
44
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