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OPEN
Loop quantum cosmology and
singularities
Ward Struyve
Received: 20 April 2017
Accepted: 14 June 2017
Published: xx xx xxxx
Loop quantum gravity is believed to eliminate singularities such as the big bang and big crunch
singularity. This belief is based on studies of so-called loop quantum cosmology which concerns
symmetry-reduced models of quantum gravity. In this paper, the problem of singularities is analysed
in the context of the Bohmian formulation of loop quantum cosmology. In this formulation there
is an actual metric in addition to the wave function, which evolves stochastically (rather than
deterministically as the case of the particle evolution in non-relativistic Bohmian mechanics). Thus
a singularity occurs whenever this actual metric is singular. It is shown that in the loop quantum
cosmology for a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker space-time with
arbitrary constant spatial curvature and cosmological constant, coupled to a massless homogeneous
scalar field, a big bang or big crunch singularity is never obtained. This should be contrasted with the
fact that in the Bohmian formulation of the Wheeler-DeWitt theory singularities may exist.
According to general relativity space-time singularities such as a big bang or big crunch are generic. This is often
taken as signaling a limit on the validity of the classical theory of gravity. The hope is that a quantum theory for
gravity will eliminate the singularities. Several candidates for a quantum gravity theory have been proposed,
such as the Wheeler-DeWitt theory, loop quantum gravity, string theory, etc.1. These different proposals may
lead to different answers to the question of singularities. Much effort has gone into studying mini-superspace
models, which are symmetry-reduced versions of quantum gravity, and which are obtained by using the usual
quantization techniques on symmetry-reduced general relativity. In particular, in recent years, there has been a
comparison of the Wheeler-DeWitt theory and loop quantum gravity (called loop quantum cosmology (LQC)
in this context) in the case of a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW)
metric coupled to a scalar field. It was found (for a large class of wave functions) that the Wheeler-DeWitt theory
yields singularities, while LQC has no singularities2–5. However, there are some problems which have to do with
applying standard quantum theory in this case. First of all there is the measurement problem, which has to do
with the ambiguity of when exactly collapses happen. This problem carries over from non-relativistic quantum
mechanics and is especially severe in the context of quantum cosmology. Namely, the aim is to describe the whole
universe (albeit with simplified models) and hence there are no outside observers or measurement devices that
could collapse the wave function. In addition, the aim is also to describe for example the early universe and then
there are no observers or measurement devices present even within the universe. Second, there is the problem of
time1, 6, 7. In both the Wheeler-DeWitt theory and LQC, the wave function is static. So how can time evolution
can be explained in terms of such a wave function? How can we tell from the theory whether the universe is
expanding or contracting or running into a singularity? Finally, there is the problem of what it means to have a
space-time singularity. In both theories, the universe is described solely by a wave function, but there is no actual
metric. Various definitions of what a singularity could mean have been explored2–5, 8, 9: that the wave function has
support on singular metrics, that the wave function is peaked around singular metrics, that the expectation value
of the metric operator is singular, etc. Although these definitions may have something so say about the occurrence
or non-occurrence singularities, neither of these is completely satisfactory. In fact, since there is merely the wave
function, one might even consider the question about space-time singularities as off-target, since it is the dynamics of the wave function that needs to be well-defined.
Various possible solutions have been explored to solve (some) of these problems. In particular, a number of
solutions to the measurement problem exist, such as for example the Many Worlds theory, spontaneous collapse
models and Bohmian mechanics. There also exist a number of approaches to solving the problem of time, for a
recent overview see ref. 10. Solving one problem may also lead to the solution of another one. For example, in
spontaneous collapse models the collapses are objective processes. But the collapses entail change and hence may
Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333, München, Germany.
Correspondence and requests for materials should be addressed to W.S. (email: ward.struyve@gmail.com)
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solve the problem of time. The question of singularities in the context of both the Wheeler-DeWitt theory and
LQC has been discussed in great detail for the Consistent Histories approach to quantum mechanics11–14.
In this paper, we consider Bohmian mechanics. Bohmian mechanics is an alternative to standard quantum
mechanics that solves the aforementioned problems. In non-relativistic Bohmian mechanics there are particles
in addition to the wave function15–17. The wave function determines the motion of the particles in a way that
is similar to the way the Hamiltonian determines the motion of classical particles. There is no measurement
problem since there is no collapse of the wave function. Outcomes of measurements are determined by the positions of the actual particles. We explore the question of singularities in the mini-superspace model of a FLRW
space-time coupled to a homogeneous scalar field in the context of Bohmian mechanics. In the Bohmian versions
of the Wheeler-DeWitt theory there is an actual FLRW metric and scalar field whose dynamics is determined by
the wave function in a deterministic way18–21. In the Bohmian version of LQC, which is developed here, there
are also an actual FLRW metric and scalar field, but now the dynamics of the metric is stochastic rather than
deterministic. While the wave function is static, the actual metric and scalar field generically evolve in time. The
wave function does not collapse, although it may at an effective level, so that there is no measurement problem.
Finally, it is also clear in this case what is meant by a singularity: there are singularities whenever the actual metric
becomes singular.
In previous work11, 22, the question of singularities was studied for the Bohmian version of the Wheeler-DeWitt
theory for mini-superspace. It was found that there may or may not be singularities; it depends on the wave function and the initial conditions of the actual fields. In particular, there are wave functions for which there are no
singularities for any of the initial conditions and there wave functions for which there are always singularities. In
this paper, we develop a Bohmian theory for LQC and consider the question of singularities. We consider some
common models for LQC which correspond to different wave equations (arising from operator ordering ambiguities) and find that big bang or big crunch singularities do not occur for any value of the spatial curvature and
cosmological constant.
The outline of the paper is as follows. First, we consider the Wheeler-DeWitt quantization of the
mini-superspace model, the corresponding Bohmian theory and the results for singularities. Then in section 3, we
present some common models for LQC. In section 4, we present their Bohmian versions and show that there is no
big bang or big crunch singularity. In section 5, we discuss how the problem of time is usually addressed in LQC
and compare it to the Bohmian solution. Finally, in section 6, we consider a modified Wheeler-DeWitt equation
inspired by loop quantum cosmology which also has the potential to eliminate singularities.
Wheeler-DeWitt quantization
A classical FLRW space-time is described by a metric
ds 2 = N (t )2 dt 2 − a(t )2 dΩk2 ,
(1)
where N > 0 is the lapse function, a = eα is the scale factor, and dΩ2k is the spatial line-element on three-space with
constant curvature k. The coupling to a homogeneous scalar field φ is described by the Lagrangian
φ2
α 2
−
L = Ne 3ακ2 2 − κ2VM −
V
,
G
2
2N
2N
(2)
where κ = 4πG/3 , with G the gravitational constant, VM is the potential for the scalar field, VG = − 1 ke−2α
2
1
+ 6 Λ, and Λ is the cosmological constant23, 24. The classical equations of motion are
d e3αφ
+ N e3α∂φVM = 0,
dt N
(3)
φ2
α 2
= 2κ2 2 + VM + 2VG .
2
2N
N
(4)
The latter equation is the Friedmann equation. The Friedmann acceleration equation follows from (3) and (4). N
remains an arbitrary function of time. This implies that the dynamics is time reparameterization invariant.
Canonical quantization of the classical theory leads to the Wheeler-DeWitt equation
2
− 1 ∂ φ2 + κ ∂ 2α + e3αVM + 1 VGψ(φ, α) = 0 .
3α
2
2e3α
2e
κ
(5)
1, 6, 7
In the context of standard quantum theory, this equation is hard to interpret due to the problem of time
.
Namely, the wave function is static. So how can the apparent time evolution of the universe be accounted for?
In the Bohmian theory22, 25, there is an actual scalar field φ and an actual FLRW metric of the form (1), whose
time evolution is determined by
N
φ = 3α ∂φS ,
e
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α = −
N 2
κ ∂αS ,
e 3α
(6)
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where ψ = |ψ|e iS. The function N is again the lapse function, which is arbitrary, and, just as in the classical case,
implies that the dynamics is time reparameterization invariant.
As usual, the Bohmian dynamics can be motivated by the conservation equation
∂φJφ + ∂αJα = 0,
(7)
where
Jφ = ∂φS ψ 2 ,
Jα = −∂αS ψ 2 .
(8)
We then take (φ, α ) ∼ (Jφ , Jα). The natural proportionality constant is given by N/e3α|ψ|2, since it follows from the
equations of motion (6) that
d e3αφ
+ N e3α∂φ(VM + Q M ) = 0,
dt N
(9)
2
α 2
2 φ
2 + (VM + Q M ) + 2(VG + QG ),
2
=
κ
2
2N
N
(10)
where
QM = −
2
1 ∂φ ψ
,
2e6α ψ
2
QG =
κ4 ∂ α ψ
2e6α ψ
(11)
are respectively the matter and the gravitational quantum potential. As such, the classical equations are obtained,
with addition of the quantum potentials to the classical potentials. The guidance equations can also be obtained
from the classical Hamilton equations by replacing the conjugate momenta πα and πφ by respectively ∂αS and ∂φS
(a procedure that works for Hamiltonians that are at most quadratic in the momenta26).
Even though the wave function is static, the Bohmian scale factor generically depends on time. As such there
is no problem of time. We will discuss this further in section 5.
The so-called quantum equilibrium measure is e3α|ψ(φ, α)|2dφdα (or a2|ψ(φ, a)|2dadφ). This measure is
preserved by the Bohmian dynamics. However, it is non-normalizable (i.e., no probability measure), so it can
not straightforwardly be used to extract probabilities for possible histories (while in non-relativistic Bohmian
mechanics the equilibrium measure gives rise to Born’s law). Probabilities are only secondary, with the primary
role of the wave function to determine the evolution of the metric and the scalar field. For that reason, it is also not
important to introduce a Hilbert space. We just need to assume that the wave function is such that the Bohmian
dynamics is well-defined.
Since it is rather unclear what the Wheeler-DeWitt equation means in the context of standard quantum
mechanics, there is also no straightforward comparison between the Bohmian predictions and those of standard
quantum theory possible. This is unlike the situation in non-relativistic quantum mechanics, where it can be
shown that Bohmian mechanics reproduces the predictions of standard quantum theory (provided the latter are
unambiguous). For example, the Hartle-Hawking wave function which is studied in great detail is empirically
inadequate from the Bohmian point of view since it is a real wave function and implies a stationary universe18.
Let us now turn to the question of singularities. In the classical theory, there is a big bang or big crunch singularity when a = 0. This singularity is obtained for generic solutions. For example, in the case of VM = VG = 0, the
classical equations lead to
N
φ = 3α c ,
e
α = ±
N 2
κ c,
e3α
(12)
where c is an integration constant. In the case c = 0, the universe is static and described by the Minkowski metric.
In this case there is no singularity. For c ≠ 0, we have
α = ± κ2 φ + c ,
(13)
with c another integration constant. In terms of proper time τ for a co-moving observer (i.e. moving with the
expansion of the universe), which is defined by dτ = Ndt, integration of (12) yields a = e α = [3(cτ + c )]1/3,
where c is an integration constant, so that a = 0 for τ = − c /c (and there is a big bang if c > 0 and a big crunch if
c < 0). This means that the universe reaches the singularity in finite proper time.
In the standard quantum mechanical approach to the Wheeler-DeWitt theory, the complete description is
given by the wave function and as such, as mentioned in the introduction, the notion of a singularity becomes
ambiguous. Not so in the Bohmian theory. The Bohmian theory describes the evolution of an actual metric and
hence there are singularities whenever this metric is singular. The question of singularities in the special case
where VM = VG = 0 was considered in refs 11, 22. In this case, the Wheeler-DeWitt equation is
1 2
1
∂ ψ − κ2 2 ∂a(a∂aψ) = 0,
3 φ
a
a
(14)
or in terms of α:
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∂ φ2 ψ − κ2∂ 2αψ = 0 .
(15)
ψ = ψR(κφ − α) + ψL(κφ + α) .
(16)
The solutions are
The actual metric might be singular; it depends on the wave function and on the initial conditions. For example,
for a real wave function, S = 0, and hence the universe is static, so that there is no singularity. Assuming that
limx →±∞ ψR , L(x ) = 0, it can be shown (work in preparation with D. Dürr and H. Ochner) that the only wave
functions for which there are no singularities are of the form
ψ(φ, α) = ψR(κφ − α) + ψL(κφ + α) e iθ
(17)
(up to an irrelevant constant phase factor) with θ a constant. On the other hand, for wave functions ψ = ψR,L the
solutions are always classical, i.e., they are either static (if ∂αS(κφ(0) − α(0)) = 0, with (φ(0), α(0)) the initial configuration) or they reach a singularity in finite proper time τ for a co-moving observer (if ∂αS(κφ(0) − α(0)) ≠ 0).
Wave functions with ψR = ψL satisfy ψ(φ, α) = −ψ(φ, −α) and lead to trajectories that do not cross the plane
α = 0 in (φ, α)-space. As such trajectories starting with α(0) > 0 will not have singularities.
In comparison we note that in the context of the consistent histories approach to quantum mechanics, it was
shown that singularities are always obtained for this system11–14.
Finally, in order to compare to LQC, we introduce the variable ν = Ca3, with = ±1 and C > 0 a constant
given in (21). The Wheeler-DeWitt equation then reads
1 2
∂ φψ − 9κ2∂ν ( ν ∂ν ψ) = 0
ν
(18)
and the guidance equations read
NC
φ =
∂φS ,
ν
ν = − N 9Cκ2 ν ∂ν S .
(19)
2
The quantum equilibrium measure is |ψ(φ, ν)| dφdν. In analogy with LQC we can further assume ψ(φ,
−ν) = ψ(φ, ν). (While ψ(φ, −ν) = ψ(φ, ν) actually introduces the boundary condition ∂νψ(φ, 0) = 0, this is not
important since we will be making the comparison only for large |ν|.)
Loop quantum cosmology
Loop quantization is a different way to quantize general relativity27, 28. Application of this quantization method
to the mini-superspace model considered here results in the following. States are functions ψν(φ) of a continuous
variable φ and a discrete variable
ν = Ca3,
(20)
with
C=
V0
,
2πGγ
(21)
where = ± 1 is the orientation of the triad (which is used in passing from the metric representation of general
relativity to the connection representation), V0 is the fiducial volume (which is introduced to make volume integrations finite) and γ is the Barbero-Immirzi parameter. ν is discrete as it is given by ν = 4nλ with n ∈ and
λ2 = 2 3 πγG . The value ν = 0, which corresponds to the singularity, is included. One could also take
ν = + 4nλ, with ∈ (0, 4λ ). This does not include the value ν = 0 and as such the singularity would automatically be avoided in the corresponding Bohmian theory (because, as will be discussed in the next section, in the
Bohmian theory the possible values the scale factor can take are given by the discrete values of ν on which ψ has
its support).
As usual, the quantization is not unique. Because of operator ordering ambiguities different wave
equations may be obtained. We discuss 4 different ones that are commonly used in the literature: the
Ashtekar-Pawlowski-Singh (APS) model4, 5, the simplified APS model5, called sLQC, the Martín-Benito–
Mena-Marugán–Olmedo (MMO) model29 and the simplified MMO model29, called sMMO30, 31. A comparison
of these models can be found in ref. 30. (The APS model is an improved version of an earlier model of Ashtekar,
Pawlowski and Singh2, 3. Unlike the earlier version, the APS model tends to yield a bounce when the matter
density enters the Planck regime rather than higher energies. The Bohmian singularity analysis of this model
would not differ much from that of the APS model. In the limit for large ν this model does not give rise to the
Wheeler-DeWitt equation (18) but to one that differs from it by a factor ordering. The models for LQC that we
consider here will all give rise to the Wheeler-DeWitt equation (18).)
In all models, the wave equation is of the form
Bν ∂ 2φψν (φ) +
∑Kν ,ν ′ψν ′(φ) = 0,
ν′
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(22)
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with ψν = ψ−ν and Bν and Kν,ν′ = Kν′,ν are real. The gravitational part, determined by K, is not a differential equation but a difference equation. For now, we do not consider a non-zero curvature or a cosmological constant. This
will be done at the end of this section. Just as in the case of the Wheeler-DeWitt theory, we will not worry about a
suitable Hilbert space for the wave equation.
In the APS model, the wave equation is
Bν ∂ φ2 ψν (φ) − 9κ2D2λ ( ν D2λ ψν (φ)) = 0,
(23)
where
Dhψν =
ψν +h − ψν −h
,
2h
(24)
so that
Kν , ν ±4λ = −
9κ2
ν ± 2λ ,
16λ2
Kν , ν = − Kν , ν +4λ − Kν , ν −4λ
and the other Kν,ν′ are zero. Various choices for Bν exist, again due to operator ordering ambiguities
choice is4:
(25)
32, 33
. One
3
Bν = ν 3Dλ ν
1/3 3
ν + λ 1/3 − ν − λ 1/3
= ν 3
.
2λ
(26)
Another one is5:
Bν =
3
Dλ ν 2/3
2
3
3
=
2/3
− ν − λ 2/3
3 ν+λ
.
2
2λ
(27)
All choices of Bν share the important properties that B0 = 0 and that for |ν| ≫ λ (taking the limit λ → 0, or equivalently, taking the Barbero-Immirzi parameter or the area gap to zero), Bν → 1/|ν|.
In sLQC, the wave equation takes the form (23), but with
Bν =
1
.
ν
(28)
In this case, B0 ≠ 0. This is a simplification that was introduced to make the model exactly solvable.
In the MMO model, the wave equation is
Bν ∂ 2φψν (φ) − 9κ2 Bν (Gν D2λ Gν )2 Bν ψν (φ) = 0,
(29)
ν 1/3 Bν−1/6 if ν ≠ 0
Gν =
0
if ν = 0,
(30)
where
with Bν given by one of the choices for the APS model. Hence
Kν , ν ±4λ = −
Kν , ν =
9κ2
Bν Gν Gν2±2λGν ±4λ Bν ±4λ ,
16λ2
(31)
9κ2
Bν Gν Gν2+2λGν −2λ Bν −2λ + Gν2−2λGν +2λ Bν +2λ .
16λ2
(
)
(32)
This model has the special property that Kν,0 = K0,ν = 0. This implies that ψ0(φ) is dynamically decoupled from the
ψν(φ) with ν ≠ 0 and moreover that ψ0(φ) is arbitrary.
The sMMO model is obtained from the MMO model by replacing Gν by ν in (29) (which amounts to replacing Bν by 1/|ν| in (30)). This results in the wave equation
Bν ∂ φ2 ψν (φ) − 9κ2 Bν (
ν D2 λ
2
ν )
Bν ψν (φ) = 0,
(33)
ν ± 4λ Bν ±4λ ,
(34)
so that
Kν , ν ±4λ = −
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9κ2
16λ2
ν Bν ν ± 2λ
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Kν , ν =
9κ 2
16λ2
(
ν Bν ν + 2λ
ν − 2λ Bν −2λ + ν − 2λ
)
ν + 2λ Bν +2λ .
(35)
As in the MMO model, Kν,0 = K0,ν = 0.
For ν λ , all these models reduce to the Wheeler-DeWitt equation (18).
Bohmian loop quantum cosmology
In the Bohmian theory there is again an actual scalar field and an actual metric of the form (1). Since the gravitational part of the wave equation (22) is now a difference operator, rather than a differential operator, we need to
develop a Bohmian theory which results in a jump process rather than a deterministic process. Such processes
have been introduced in the context of quantum field theory to account for particle creation and annihilation34–36.
In the Bohmian theory, the scalar field will evolve continuously, while the scale factor a, which will be expressed
in terms of ν using (20), takes discrete values, determined by ν = 4nλ with n ∈ .
The wave equation (22) implies the continuity equation
∂φJν (φ) =
∑Jν ,ν ′(φ),
(36)
ν′
where
Jν (φ) = Bν ∂φSν (φ),
Jν , ν ′(φ) = − Kν , ν ′Im(ψν (φ)ψν⁎′(φ)) .
(37)
Jν,ν′ is anti-symmetric and non-zero only for ν′ = ν ± 4λ for the LQC models considered above. Writing
∑Jν ,ν ′ = ∑(Tν ,ν ′ |ψ ν ′|2
ν′
− Tν ′, ν |ψν |2 ),
ν′
(38)
where
Jν , ν ′(φ)
Tν , ν ′(φ) = |ψ ν ′(φ)|2
0
if Jν , ν ′(φ) > 0
,
otherwise
(39)
we can introduce the following Bohmian dynamics which preserves the quantum equilibrium distribution
|ψν(φ)|2dφ. The scalar field satisfies the guidance equation
φ = NCBν ∂φSν ,
(40)
iS ν
where ψν = ψν e . The variable ν, which determines the scale factor, may jump ν′ → ν with transition rates given
by Tν , ν ′(φ) = NCTν , ν ′(φ). That is, Tν,ν′(φ) is the probability to have a jump ν′ → ν in the time interval (t, t + dt).
Note that the jump rates at a certain time depend on both the wave function and on the value of φ at that time. The
properties of Jν,ν′ imply that for a fixed ν either Tν,ν +4λ or Tν,ν−4λ may be non-zero (not both). The jump rates are
“minimal”, i.e., they correspond to the least frequent jump rates that preserve the quantum equilibrium distribution36. Just as in the classical case and the Bohmian Wheeler-DeWitt theory, the lapse function is arbitrary, which
guarantees time-reparameterization invariance.
Since the evolution of the scale factor is no longer deterministic like in the Wheeler-DeWitt theory, but stochastic, the metric is no longer Lorentzian. Namely, once there is a jump, the metric becomes discontinuous. The
metric is only “piece-wise” Lorentzian, i.e., Lorentzian in between two jumps.
For ν λ (taking the limit λ → 0), this Bohmian theory reduces to the one of the Wheeler-DeWitt equation
(using similar arguments as in ref. 37). That is why we have chosen the particular form (40) for the guidance
equation for φ.
Let us now turn to the question of singularities. If T0,±4λ = 0, then the scale factor a (or ν) can never jump
to zero, so a big crunch is not possible. If T±4λ,0 = 0, then the scale factor can not jump from zero to a non-zero
value, so a big bang is not possible. Hence there are no singularities if J0,±4λ = 0. If the boundary condition ψ0 = 0
is imposed, then J0,±4λ = 0 for all the LQC models that are considered here. However, except for sLQC, J0,±4λ = 0
without imposing this boundary condition. To see this, consider the wave equation evaluated for ν = 0. Since
B0 = 0 in the APS, MMO and sMMO models, this results in
K0,4λψ4λ + K0, −4λψ−4λ + K0,0ψ0 = 0 .
(41)
Using the properties K0,ν = K0,−ν and ψν = ψ−ν, we obtain that
Im(ψ0⁎K0, ±4λψ±4λ ) = 0
(42)
and hence that J0,±4λ = 0 (even if ψ0 ≠ 0). This argument can not be used in sLQC since B0 ≠ 0 in that case. In any
case, the sLQC model is considered only as a simplification of the APS model; it does not follow from applying the
loop quantization techniques to mini-superspace. In summary, Bohmian loop quantum cosmology models for
which the wave equation (22) has the properties that B0 = 0, K0,ν = K0,−ν and ψν = ψ−ν, do not have singularities.
Importantly, no boundary conditions need to be assumed.
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In the case that ψ is real, both φ and a are static. For other possible solutions, the wave equation needs to be
solved first. This is rather hard, but can perhaps be done in sLQC since the eigenstates of the gravitational part
of the Hamiltonian are known in this case. Something can be said about the asymptotic behaviour however.
Since for large ν this Bohmian theory reduces to the one for the Wheeler-DeWitt theory, the trajectories will
tend to be classical in this regime. Namely consider solutions (16) to the Wheeler-DeWitt equation for which the
functions ψR and ψL go to zero at infinity. Then for α → ∞, the wave functions ψR and ψL become approximately
non-overlapping in (φ, α)-space. As such the Bohmian motion will approximately be determined by either ψR
or ψL and hence classical motion is obtained. This implies an expanding or contracting (or static) universe. We
expect that a bouncing universe will be the generic solution.
So far we assumed k = Λ = 0. In the case k = ±1 or Λ ≠ 0 there is an extra potential term Wν in (22), which
amounts to the replacement
Kν , ν ′ → Kν , ν ′ + Wν δν , ν ′
(43)
compared to the free case. Λ > 0 is discussed in refs 3, 38, Λ < 0 in ref. 3, k = 1 in refs 39, 40, k = −1 in refs 41, 42.
In each case, the extra term is merely a potential term so that it does not contribute to the Bohmian jump process.
So the same results hold concerning singularities as in the free case.
On the role of time
In the Bohmian Wheeler-DeWitt theory and LQC there is time-reparameterization invariance, so that time is
relational. An actual clock should be modeled in terms of the other variables. In the case of the Wheeler-DeWitt
theory, if either a or φ is monotonically increasing with time (at least for some period of time), it could play the
role of a clock (for that period of time). One can then express the evolution of the other variable in terms of the
clock variable. In the case of LQC, the same is true, except if the scale factor is taken as a clock variable, then it
will be a discrete one.
The situation is different in the standard quantum mechanical approach to the Wheeler-DeWitt theory and
loop quantum theory. There one has to deal with the notorious problem of time1, 6, 7. In both cases, the wave equation does not depend on time and hence the wave function is static. So how does one account for apparent time
evolution? One attempt for a solution is to take one of the variables, say φ, as time and to take the square root of
the wave equation (22), which results in a Schrödinger-like equation
i∂φψ ±(φ, ν ) = ±Θψ ±(φ, ν )
(44)
in the case of the Wheeler-DeWitt theory and in
i∂φψν±(φ) = ± ∑Θν , ν ψν±(φ)
′ ′
ν′
(45)
2–5
in the case of LQC . However, also the scale factor could be taken as clock variable. In particular, in the case of
the Wheeler-DeWitt theory without potentials, the wave equation is completely symmetric with respect to interchange of α and φ, and therefore there seems to be no reason to prefer either one as a time variable. The argument
to take φ as time variable in refs 2–5 is that classically it is monotonic (just as the scale factor). However, we believe
that the classical behaviour of some variables should have no implications concerning the suitability to act as time
variables in the quantum case. Different time variables also lead to different theories (characterized by different
Hilbert spaces) and in particular to different conclusions concerning the presence of singularities.
In the Bohmian treatments no such ambiguities arise. In particular, whether variables act as clock variables
depends on their quantum behaviour, not on their classical behaviour. The wave equations (44) and (45) (or similar ones) could in principle be derived as effective equations (using the conditional wave function17, 43, 44), when
the Bohmian variable φ behaves as a clock variable.
So on the fundamental level, the scalar field is not regarded as a time variable in the Bohmian theory. However,
for the sake of comparison to the analysis of singularities in the Wheeler-DeWitt theory in the context of standard
quantum mechanics2–5 and consistent histories12–14, 45, for which the starting point is (44), an alternative Bohmian
model was considered11 where the variable φ is also regarded as a time variable from the start. In this model, the
equilibrium measure |ψ(φ, α)|2dα is normalizable which hence directly allows for probabilistic statements. It was
found that the probability Ps for a trajectory α(φ) to develop a singularity satisfies 1/2 ≤ Ps ≤ 1. So, just as in the
fundamental Bohmian theory, singularities may or may not exist depending on the wave function and the initial
conditions.
One can do a similar analysis for LQC. Let us first consider the Hilbert space. The Hilbert space is given by
+
= − the positive and negative frequency Hilbert spaces. The inner product on ± is
= + ⊕ −, with
given by ψ χ = ∑ν ∈ 4λ ψν⁎Bν χν . (The operator Θ needs to be properly defined in order to be self-adjoint with
respect to this inner product46.) By taking this inner product, states can not have support on ν = 0. This is because
if B0 = ∞, like in sLQC then, states ψ0 have infinite norm and if B0 = 0, like in the other models, then the states ψ0
have zero norm. So for two states Ψ = (ψ+, ψ−) and X = (χ+, χ−) the inner product in is
ΨX =
∑
ν ∈ 4λ
[(ψν+)⁎ Bν χν+ + (ψν−)⁎ Bν χν−] .
(46)
Since states have no support on ν = 0, singularities are immediately eliminated in the context of standard quantum mechanics. (This point is not really emphasized in refs 2–5. There, the issue of singularities is analyzed by
considering whether the wave function is peaked around the singularity2–4 or whether the expectation value of the
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volume operator becomes zero5.) Also in the Bohmian theory the singularities are eliminated. Because of the
choice of inner product it is natural to take the quantum equilibrium distribution to be
2
Pν (φ) = Bν ( ψν+(φ) + ψν−(φ) 2 ), which is now normalized to one. This distribution satisfies the continuity
equation
∂φPν =
∑Jν ,ν ′ ,
ν′
(47)
where now
Jν , ν ′ = 2Bν Im[(ψν+)∗ Θν , ν ′ψ ν+′ − (ψν−)∗ Θν , ν ′ψ ν−′] .
(48)
The jump rates to have a jump ν′ → ν in the time interval (φ, φ + dφ) are given by
Jν , ν ′(φ)
if Jν , ν ′(φ) > 0
Tν , ν ′(φ) = Pν ′(φ)
otherwise.
0
(49)
For large ν (λ → 0), this Bohmian model reduces to the one introduced in ref. 11 for the square root of the
Wheeler-DeWitt equation, at least for the square-root expressions given by Ashtekar et al.4, 5. In refs 29, 31,
Martín-Benito et al. consider a different square-root expression, which leads to a different square root of the
Wheeler-DeWitt equation.
There will be no big bang or big crunch singularity if T0,ν = Tν,0 = 0 and this is trivially guaranteed since
ψ0+ = ψ0− = 0.
Modified Wheeler-DeWitt equation
In section 3, we found that Bohmian loop quantum cosmology models for which the wave equation (22)
has the properties that B0 = 0, K0,ν = K0,−ν and ψν = ψ−ν, do not have singularities. Consider now a modified
Wheeler-DeWitt equation
B(ν )∂ 2φψ − 9κ2∂ν ( ν ∂ν ψ) = 0,
(50)
where ψ(φ, −ν) = ψ(φ, ν) and where B(ν) is some function that approximates 1/|ν| in the limit of large |ν| and
satisfies B(0) = 0. The guidance equations in this case are
φ = NCB(ν )∂φS ,
ν = − N 9Cκ2 ν ∂ν S .
(51)
Is such a modification sufficient to eliminate the singularities? Or is the discreteness also essential? Or does it
depend on the choice of the function B?
As a simple example we can consider a function B which is zero for |ν| < ν0 for some ν0 > 0, which we can take
arbitrarily small. Then for ν ≤ ν0 , the wave equation reduces to ∂ ν (|ν|∂ ν ψ) = 0, which implies
ψ(φ, ν ) = χ1(φ)sgn(ν )ln( ν ) + χ2(φ), with the χi some functions of φ. Since ψ(φ, −ν) = ψ(φ, ν), it must be that
χ1 = 0 and therefore the guidance equation for ν implies that ν must be static. Hence in this case there are no
singularities.
Note that the condition ψ(φ, −ν) = ψ(φ, ν) actually entails the boundary condition ∂νψ(φ, 0) = 0. If we consider
the equations (50) and (51) just for ν ≥ 0, which would be natural, then without the condition ∂νψ(φ, 0) = 0 singularities are still possible. Namely, for ν < ν0, (51) implies that φ is constant, while ν = − N 9Cκ2Im(χ2⁎(φ)χ1(φ))
/ χ1(φ)ln(ν ) + χ2(φ) 2. Taking for example χ1(φ) = −i and χ2(φ) = 1 for the constant value of φ that is obtained,
integration of guidance equation for ν yields 9Cκ2τ = ν(3 + ln ν(ln ν − 2)) + 9Cκ2c, with τ the proper time for
a co-moving observer (see section 2) and c an integration constant. But this implies that the singularity is reached
(i.e. ν = 0) for τ = c.
In summary, singularities can be eliminated by assuming the above mentioned modifications to the
Wheeler-DeWitt equation. It is unclear whether this is so for any choice of B which satisfies the above mentioned
assumptions.
Finally, from the guidance equations (51) and the modified Wheeler-DeWitt equation (50) the modified
Friedmann equation follows
φ2
a 2
2
2
+ Q M + 2QG ,
κ
=
2
2 2
2N νB
Na
(52)
where
QM = −
C 2B 2
∂φ ψ ,
2ν ψ
QG =
9C 2κ 4
∂ν ( ν ∂ν ψ )
2ν ψ
(53)
are respectively the matter and graviational quantum potential. The presence of the quantum potentials allows a
to become zero and hence for the universe to undergo a bounce. Regarding the present model as a continuum
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approximation of LQG one can perhaps deduce an effective equation that illustrates a generic bouncing behaviour, in analogy with the one considered in ref. 8.
Conclusion
We have developed a Bohmian version for some common models of LQC for homogeneous and isotropic FLRW
space-time coupled to a homogeneous scalar field, with arbitrary values of the constant spatial curvature and
cosmological constant. The Bohmian theory describes an actual metric, whose sole degree of freedom in this case
is the scale factor, and a scalar field, whose dynamics depends on the wave function. While the scalar field evolves
continuously in time, the scale factor changes stochastically (unlike in the case of the Wheeler-DeWitt theory
where it changes deterministically). We showed that a non-zero scale factor never jumps to zero and conversely
that a zero scale-factor never becomes non-zero. This means that there is no big bang or big crunch singularity.
This result was obtained without assuming any boundary conditions; it follows solely from the structure of the
wave equation. (Sometimes boundary conditions are considered to avoid singularities, like for example the condition that the wave function vanishes on singular metrics1, 9.)
It is to be expected that similar results hold for other space-times, such as anisotropic space-times, like for
example the Kantowski-Sachs space-time, which is particularly interesting since it can be used to describe the
interior of a black hole and hence can be used to study the question of black-hole singularities47.
So far we have restricted our attention merely to the question of singularities. While we have established that
there are no singularities, further work is required to learn the generic behaviour of solutions.
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Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft. It is a pleasure to thank Alejandro Corichi,
Felipe Falciano, Claus Kiefer, Hannah Ochner, Nelson Pinto-Neto, Parampreet Singh, and especially Detlef Dürr,
for useful discussions and comments.
Additional Information
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