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: ward.struyve@gmail.com

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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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