Wheeler-DeWitt quantization and singularities

0.45 MB·English

by  Felipe Tovar Falciano

Wheeler-DeWitt quantization and singularities
F. T. Falciano,1,∗ N. Pinto-Neto,1,† and W. Struyve2,‡
1CBPF - Centro Brasileiro de Pesquisas F´ısicas,
Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, Brazil.
2Department of Physics, University of Liege, 4000 Liege, Belgium.
(Dated: January 20, 2015)
WeconsideraBohmianapproachtotheWheeler-DeWittquantizationoftheFriedmann-Lemaˆıtre-
Robertson-Walker model and investigate the question whether or not there are singularities, in the
sense that the universe reaches zero volume. We find that for generic wave functions (i.e., non-
classical wave functions), there is a non-zero probability for a trajectory to be non-singular. This
should be contrasted to the consistent histories approach for which it was recently shown by Craig
andSinghthatthereisalwaysasingularity. Thisresultillustratesthatthequestionofsingularities
depends much on which version of quantum theory one adopts. This was already pointed out
by Pinto-Neto et al., albeit with a different Bohmian approach. Our current Bohmian approach
5 agreeswiththeconsistenthistoriesapproachbyCraigandSinghforsingle-timehistories,unlikethe
1
one studied earlier by Pinto-Neto et al. Although the trajectories are usually different in the two
0
Bohmian approach, their qualitative behavior is the same for generic wave functions.
2

n PACSnumbers: 98.80.Cq,04.60.Ds,98.80.Qc
a
J
7 I. INTRODUCTION However, while the dynamics of the Bohmian model is
1 very natural, there is no natural probability distribution
onthesetofhistories. Assuch,thereisnoimmediateway
] Recently, some of us investigated the issue of singular- of making probabilistic statements, like about the prob-
c ities in the Wheeler-DeWitt approach to quantum cos-
q ability for a history to have a singularity for a particular
mologyandshowedthattheanswerdependsstronglyon
- wavefunction. Ontheotherhand,CraigandSingh[2,3]
r theversionofquantummechanicsthatoneconsiders[1].
g showed that probabilistic statements can be made in the
Two versions were compared: the consistent (or deco-
[ consistent histories approach. This might seem puzzling
herent) histories and the Bohmian approach (also called
since, at least in the context of non-relativistic systems,
1 the deBroglie-Bohm or pilot-wave approach). Both ap-
it is often claimed that consistent histories and Bohmian
v proaches have a way of introducing possible histories of
mechanics yield the same predictions for outcomes of
1
the universe. In the Bohmian approach, there is an ac-
8 measurements [4, 5]. The reason for this mismatch is
tualconfigurationwhosetimeevolutionisdeterminedby
1 that in the consistent histories approach the scalar field
the wave function. The possible paths of the configura-
4 is treated (at least formally) as a time variable, whereas
0 tion are the histories. On the other hand, in the consis- it is not in the Bohmian approach. The goal of this pa-
. tent histories approach the (coarse-grained) histories are
1 peristoexploreanotherBohmiandynamicswhichisalso
sequences of propositions at different moments in time.
0 based on treating the scalar field as a time variable. In
5 (For example, a possible proposition could be that the this case, there is a natural probability distribution for
1 system occupies a certain region in space). the scale factor which agrees with the one of the con-
:
v A simple quantum cosmological model was studied, sistent histories approach when single-time histories are
i namely the quantized Friedmann-Lemaˆıtre-Robertson- considered. Hence, this allows for a better comparison
X
Walker model, with a homogeneous scalar field. For betweentheBohmianapproachandtheconsistenthisto-
r
the consistent histories approach, Craig and Singh [2, 3] riesapproachconcerningthequestionofsingularities. In
a
showed that the possible histories are always singular ir- particular, given a wave function, we can now calculate
respective of the given wave function. Either they start the probability for a trajectory to run into a singularity.
from a singularity or they end in a singularity. On the We will find that although the trajectories are different
otherhand,accordingtotheBohmianapproachofPinto- from the ones in the model proposed in [1], they are of-
Neto et al. [1], there are wave functions for which there tenqualitativelythesame. Inparticular,wewillfindthat
are also histories that display a bounce (which corre- some wave functions allow for trajectories that are never
sponds to a universe that contracts until it reaches a singular and instead display a bounce. More precisely,
minimum volume and then starts expanding) and hence we will see that for a non-classical wave function there is
do not have a singularity. always a non-zero probability for a bounce (while trajec-
tories corresponding to a classical wave function always
have a singularity). This should be contrasted with the
fact that in the consistent histories approach the proba-
∗Electronicaddress: [email protected] bility for a singularity is always one, for any wave func-
†Electronicaddress: [email protected] tion.
‡Electronicaddress: [email protected]
2
The outline of the paper is as follows. We start integration of (3) yields a = eα = [3(cτ +c˜)]1/3, where
by recalling the Wheeler-DeWitt quantization of the c˜is an integration constant, so that a = 0 for τ = −c˜/c
Friedmann-Lemaˆıtre-Robertson-Walker model and the (andthereisabigbangifc>oandabigcrunchifc<0).
Bohmian approach of [1]. In section III, we will discuss This means that the universe reaches the singularity in
possible choices of Hilbert space that were considered in finite proper time.
the consistent histories approach. In section IV, we will Canonical quantization of this theory leads to the
develop and discuss the alternative Bohmian dynamics. Wheeler-DeWitt equation
InsectionV,wecalculatetheprobabilityforatrajectory
to be singular. Finally, in section VI, we consider the ∂2ψ−∂2ψ =0. (5)
φ α
probabilities for singularities in the consistent histories
approach, extending the work of Craig and Singh from In the corresponding Bohmian theory [1], there is an ac-
two-time histories to many-time histories, and compare tual scalar field φ and scale factor a=eα, which satisfy
these to those predicted by the Bohmian approach.
N N
φ˙ = ∂ S, α˙ =− ∂ S, (6)
e3α φ e3α α
II. WHEELER-DEWITT QUANTIZATION AND
BOHMIAN MECHANICS where ψ = |ψ|eiS. The function N is again the lapse
function, which is arbitrary, and, as in the classical case,
implies that the dynamics is time reparameterization in-
AclassicalflatFriedmann-Lemaˆıtre-Robertson-Walker
variant.2
space-time is described by a metric
The Wheeler-De Witt equation implies
ds2 =N(t)2dt2−e3α(t)γ dxidxj, (1)
ij (∂ S)2−(∂ S)2+Q=0, (7)
φ α
where N is the lapse function, a = eα is the scale fac-
tor, and γ is the flat spatial 3-metric. Considering the
matterconitjentdescribedbyafreemasslesshomogeneous ∂φ(cid:0)|ψ|2∂φS(cid:1)−∂α(cid:0)|ψ|2∂αS(cid:1)=0, (8)
scalar field φ(t), the Lagrangian1 of the system can be
written as [6] where
(cid:32) (cid:33) 1 1
φ˙2 α˙2 Q=− ∂2|ψ|+ ∂2|ψ| (9)
L=Ne3α − , (2) |ψ| φ |ψ| α
2N2 2N2
isthequantumpotential. IfQ=0, then(7)impliesthat
where a dot means derivative with respect to coordinate (∂ S)2 = (∂ S)2. In addition, ∂ S is then conserved
φ α α
time t. The corresponding equations of motion lead to along a Bohmian trajectory. Hence, in this case, the
Bohmian motion is reduced to the classical motion given
N N
φ˙ =± c, α˙ = c, (3) by (3).
e3α e3α Equation (8) is a continuity equation and implies that
the Bohmian dynamics preserves |ψ|2. However, since
where c is an integration constant. N remains an arbi-
|ψ|2 is not normalizable it can not be straightforwardly
trary function of time. This implies that the dynamics
used to make statistical predictions. In this paper, we
is time reparameterization invariant. Different choices of
will consider an alternative Bohmian dynamics, which
N merely correspond to different choices of the parame-
allows for immediate statistical predictions and which
terization of the paths in (φ,α) space. In the case c=0,
allows for a direct comparison with consistent histories
the universe is static. For c(cid:54)=0, we have
approaches discussed in [2, 3, 14].
α=±φ+c¯, (4)
with c¯another integration constant.
Theuniversereachesthesingularity(i.e.,zerovolume) 2 Notethatthetimereparameterizationinvarianceisaspecialfea-
when a=eα =0. For a non-static universe the singular- ture of Bohmian approaches to mini-superspace models [7, 8].
For the usual formulation of Bohmian dynamics for the full
ity is reached for either φ→−∞ or φ→∞. So the uni-
Wheeler-DeWitt theory of quantum gravity, a particular space-
verseeitherstartswithabigbangorendsinabigcrunch. like foliation of space-time or, equivalently, a particular choice
Intermsofpropertimeτ,whichisdefinedbydτ =Ndt, of “initial” space-like hypersurface and lapse function, needs to
beintroduced. Differentfoliations(orlapsefunctions)yielddif-
ferent Bohmian theories [9–12]. So, in this case, the dynamics
is not invariant under space-time diffeomorphisms. (Yet, while
theusualBohmianformulationisnotdiffeomorphisminvariant,
1 For simplicity, we have dropped the constant factor 4πG/3 in this may perhaps be achieved with alternative approaches. For
frontofα˙2. Thisfactorcouldberemovedbyasuitablerescaling a discussion of analogous issues concerning special relativity in
ofthescalarfield. Bohmianmechanics,see[13].)
3
φ which satisfy3
(cid:112)
i∂φψ± =H(cid:98)±ψ± =∓ −∂α2ψ±. (15)
Theleft-andright-movingcomponentsofψ willbede-
±
α noted by ψ and ψ . If one uses the scalarfield φ as
L,± R,±
a time variable, equation (15) represents a Schr¨odinger-
like equation and given the initial wave function ψ0 at
ψR ±
ψL time φ0, the (formal) solution is given by
√
ψ±(α,φ)=e±i −∂α2(φ−φ0) ψ±0(α). (16)
FIG. 1: The wave functions ψ and ψ respectively move to
R L The first choice of Hilbert space is H = L2(R,C2)
the right and to the left without change in shape. 1
[2, 3, 15]. We have that H = H ⊕ H , with
1 1,+ 1,−
H = L2(R,C) the positive and negative frequency
1,±
III. WHEELER-DEWITT EQUATION AND Hilbert spaces. We write Ψ(α) = (ψ (α),ψ (α)) with
+ −
HILBERT SPACES Ψ∈H and ψ ∈H .4 The inner product is given by
1 ± 1,±
(cid:90)
So far we have not introduced a Hilbert space for the (cid:104)Ψ|Φ(cid:105) = dα(cid:0)ψ∗φ +ψ∗φ (cid:1) . (17)
1 + + − −
Wheeler-DeWitt equation (5). We will discuss two pos-
sible choices of Hilbert space that have appeared in the
The dynamics for the positive and negative frequency
literature [2, 3, 14–16]. Both are motivated by consid-
ering φ (at least formally) as a time variable. However, states is given by (15). Since the Hamiltonians H(cid:98)± are
Hermitian operators on H , this dynamics preserves
as we shall see, for the purpose of assigning histories to 1,±
the inner product.
the scale factor, either in the consistent histories or in
The natural observable corresponding to the scale fac-
the Bohmian framework, both choices can be considered
tor is given by the projection-valued measure (PVM)
equivalent,assumingasuitablechoiceof“observable”for
the scale factor in each case.
First, consider a general solution to the Wheeler- P(cid:98)1(dα)=P(cid:98)1,+(dα)⊕P(cid:98)1,−(dα), (18)
DeWitt equation (5), which is of the form
where
ψ(α,φ)=ψ (α+φ)+ψ (α−φ), (10)
L R P(cid:98)1,±(dα)=|α(cid:105)(cid:104)α|dα, (19)
where the indices L and R denote respectively “left-
moving” and “right-moving”. This terminology stems with |α¯(cid:105) = δ(α−α¯). (Since (cid:82) P(cid:98)1,±(dα) = 1(cid:98)±, with 1(cid:98)±
from the fact that if φ is regarded as time, then ψ , as theidentityoperatoronH ,itimmediatelyfollowsthat
L 1,±
a function of α, moves to the “left” over time, without (cid:82) P(cid:98)1(dα)=1(cid:98), the identity operator on H1.)
change in shape, whereas ψR moves to the right, see fig- For a state Ψ = (ψ+,ψ−), the corresponding density
ure 1. In terms of the Fourier transform, we have for α is
1 (cid:90) ∞
ψ(α,φ)=√ dk ψ (k)eik(α+φ) ρ(α,φ)=|ψ (α,φ)|2+|ψ (α,φ)|2, (20)
L + −
2π
−∞
+ √1 (cid:90) ∞ dk ψ (k)eik(α−φ). (11) i.e., (cid:104)Ψ|P(cid:98)1(dα)|Ψ(cid:105)1 = ρ(α,φ)dα. (The Hilbert space is
2π R thesameasthatofthePaulitheoryforanon-relativistic
−∞
The general solution can also be decomposed into a pos- spin-1/2 particle, with P(cid:98)1(dα) being the analogue of the
position PVM in the Pauli theory.)
itive and negative frequency part
The second choice of Hilbert space H is again the
2
ψ =ψ +ψ , (12) direct sum of positive and negative frequency Hilbert
+ −
where
1 (cid:90) ∞
ψ =√ dk ψ (k)eik(α+φ)
+ 2π L 3 For a wave function ψ(α) = 1 (cid:82)∞ dkψ(k)eikα, the
0 2π −∞
+ √12π (cid:90)−0∞dk ψR(k)eik(α−φ), (13) 4 aA21cπctti(cid:82)uo−∞na∞llyo,dfkin|kt[|h2ψe,(3k,H)e1ai5km]αoi.lntolynitahnepiossitdievfienferedquaesncy(cid:112)s−ec∂toα2rψo(fαH) 1=is
1 (cid:90) 0 considered, byappealtosuperselection. However, theBohmian
ψ− =√ dk ψL(k)eik(α+φ) trajectorieswilldependonwhetherthewavefunctionisasuper-
2π −∞ position of both frequencies or not. (Following the terminology
1 (cid:90) ∞ of[17],onecouldsaythatfrequencyisweaklysuperselected,but
+ √ dk ψR(k)eik(α−φ), (14) notstrongly.) Therefore,ageneralanalysisshouldincludeboth
2π 0 frequencysectors.
4
spaces, i.e., H = H ⊕ H , with H = H In the consistent histories approach, (coarse-grained)
2 2,+ 2,− 2,+ 2,−
[14, 16]. States in H are maps from R to C2 and histories for α can be introduced, by choosing a Hilbert
2
states in H are maps from R to C. We write Ψ¯(α)= space, a PVM (or positive operator-valued measure
2,±
(ψ¯ (α),ψ¯ (α)) with Ψ¯ ∈H and ψ¯ ∈H (the bar is (POVM)) and a Hamiltonian. The results will be iden-
+ − 2 ± 2,±
usedinordertodistinguishthestatesfromthoseinH1). tical when choosing either (H1,P(cid:98)1,H(cid:98)) or (H2,P(cid:98)2,H(cid:98)),
The inner product on H is given by
2,± whereH(cid:98) istheHamiltonianoperatordeterminedby(15)
(cid:90) (cid:90) (the former is considered in [2, 3], albeit just for positive
(cid:104)ψ¯|φ¯(cid:105)2,± =2 dαψ¯∗(∓H(cid:98)±)φ¯=2 dαψ¯∗(cid:112)−∂α2φ¯, frequencies, the latter in [14]).6 The reason is that there
(21) is a unitary mapping H2 → H1, which maps P(cid:98)2 → P(cid:98)1
and leaves the Hamiltonian invariant. This mapping
so that the inner product for H reads5
2 is defined by ψ¯ (α) → ψ (α) = (cid:104)α,NW|ψ¯ (cid:105) (or
± ± ± 2,±
(cid:104)Ψ¯|Φ¯(cid:105)2 =2(cid:90) dα(cid:16)ψ¯+∗(cid:112)−∂α2φ¯++ψ¯−∗(cid:112)−∂α2φ¯−(cid:17) . (22) |ψα,(Nk)W=(cid:105) (cid:112)→2||kα|(cid:105)ψ¯in(kt)erimnsteorfmbsaosfisFsotuartieesr,coormψp¯±on(ken)ts→).
± ±
Thismapisunitarysince(cid:104)Ψ|Φ(cid:105) =(cid:104)Ψ¯|Φ¯(cid:105) (whichfollows
1 2
The dynamics for the positive and negative frequency from the fact that (cid:82) P(cid:98)2(dα) = 1(cid:98), the identity operator
states is given again by (15). Since the Hamiltonians
H(cid:98)± are Hermitian operators on H2,±, the dynamics also on H2). Since it takes P(cid:98)2 →P(cid:98)1, we have
preFsoerrvtehsetnhoisrminnoefrapwroadvuectfu.nction Ψ¯, the integrand on (cid:104)Ψ¯|P(cid:98)2(dα)|Ψ¯(cid:105)2 =(cid:104)Ψ|P(cid:98)1(dα)|Ψ(cid:105)1 =ρ(α,φ)dα. (27)
the right hand side of (22) reads As such, we clearly get the same result for probabilities
2(cid:16)ψ¯∗(cid:112)−∂2ψ¯ +ψ¯∗(cid:112)−∂2ψ¯ (cid:17) . (23) ofWhiestocrainesawlsitohd(Hev1e,loP(cid:98)p1)aorB(oHh2m,Pi(cid:98)a2n).approach given a
+ α + − α −
Hilbert space, a POVM and a Hamiltonian. For a
One might be tempted to regard this quantity as the given wave function, the distribution corresponding to
densityofαatagiventimeφforthestateΨ¯. However,it the PVM will be the Bohmian equilibrium distribution.
isnotalwayspositivedefinite,evenifwerestrictourselves Thedynamicscanthenbedefinedinsuchawaythatthis
to positive frequencies [18, 19]. distribution is preserved [22, 23]. In the next section,
One possibility to obtain a positive density is by using we will consider a dynamics that preserves the density
the Newton-Wigner states [14, 20] (20). So also in the Bohmian approach, the two choices
1 (cid:90) ∞ dk (H1,P(cid:98)1,H(cid:98)) and (H2,P(cid:98)2,H(cid:98)) yield an identical dynamics
|α¯,NW(cid:105)= eik(α−α¯). (24) for the scale factor.
(cid:112)
2π 2|k|
−∞
These states are normalized according to IV. ALTERNATIVE BOHMIAN DYNAMICS
(cid:104)α,NW|α¯,NW(cid:105) = δ(α − α¯) and form a basis of
2
H . The PVM then reads
2,± The density ρ(α,φ), given in (20), is not preserved by
theBohmiandynamics(6),i.e.,ifthedensityofαisgiven
P(cid:98)2(dα)=P(cid:98)2,+(dα)⊕P(cid:98)2,−(dα), (25) by ρ(α,φ ) at a certain time φ , then the Bohmian dy-
0 0
namicsingeneralleadstoadensitydifferentfromρ(α,φ)
where now
atothertimesφ[1]. However,onecanpostulateadiffer-
ent dynamics that preserves ρ [23–25]. According to this
P(cid:98)2,±(dα)=|α,NW(cid:105)(cid:104)α,NW|dα. (26)
dynamics, possible trajectories α(φ) satisfy
Note that, as before, (cid:82) P(cid:98)2(dα) = 1(cid:98), being understood (cid:90) α(φ) (cid:90) α(0)
that the action of these operators on the states in the dα¯ρ(α¯,φ)= dα¯ρ(α¯,0), (28)
Hilbert space H2 is through the inner product defined in −∞ −∞
Eq. (22).
or, equivalently,
dα 1 (cid:90) α
=v (α,φ)=− dα¯∂ ρ(α¯,φ). (29)
dφ 2 ρ(α,φ) φ
−∞
5 Perhaps a more familiar way of writing the inner products (21)
and(22)isintermsofsolutionstothewaveequations(15)[14]. Since ρ satisfies the continuity equation
The inner product (21) then corresponds to the Klein-Gordon
innerproduct(cid:104)ψ¯|φ¯(cid:105)KG=−i(cid:82) dα(cid:0)ψ¯∗∂φφ¯−φ¯∂φψ¯∗(cid:1)fortwopos- ∂φρ+∂α(v2ρ)=0, (30)
itive frequency solutions to (15). The inner product (22) then
correspondsto(cid:104)Ψ¯|Φ¯(cid:105)2 =(cid:104)ψ¯+|φ¯+(cid:105)KG−(cid:104)ψ¯−|φ¯−(cid:105)KG. Notethat
thisinnerproductispositivedefinite,unliketheusualdefinition
fortheinnerproductoftwosolutionstotheKlein-Gordonequa-
tion,whichdoesnotcontaintherelativeminussignbetweenthe 6 See [21] for similar considerations on equivalence of quantum
positiveandnegativefrequencypart. theories.
5
it is preserved by the Bohmian dynamics. functionandhencewedonothaveastraightforwarddef-
Let us compare this dynamics to the one formulated inition of proper time. So, it is not immediately clear
in section II. First, consider a real wave function ψ, i.e. whetherthesingularitiesarereachedinfinitepropertime
Imψ = 0.7 According to the dynamics of section II, ornot. Nonetheless,sincethesetrajectoriesareclassical,
the possible solutions are α(t) = α(0), φ(t) = φ(0), so it seems natural to introduce a proper time that agrees
that the universe is static, with a constant scale factor with that of the classical theory. As such, the conclusion
and constant scalar field. Since these solutions can not isagainthatthesingularitiesarereachedinfiniteproper
be expressed as α(φ) they must be different from the time.
trajectories given by (28). This dynamics seems less natural than the one given
NowassumethatthetrajectoriesforthefirstBohmian earlier. The reason is of course that φ plays the role
dynamics can be expressed as α(φ) (at least locally). If of time in this dynamics. However, there seems to be no
∂ S (cid:54)=0, then the vector field tangent to these trajecto- reasontogiveitthatdistinguishedrole. Forexample,we
φ
ries is given by could equally well have taken α to play the role of time.
This would have resulted in completely different paths.
∂ S
v (α,φ)=− α . (31) Of course, this is only a simplified model. In a more
1 ∂φS serious model, we expect that some of the matter fields
will represent time on an effective level. But this should
Thevelocityfieldsv andv aregenericallynotthesame.
1 2 follow from analyzing a more fundamental Bohmian dy-
Using (7) and (8), to evaluate the integral in (29), we
namics, similar to (6) (see e.g. [9, 26, 27]), rather than
have that
be postulated a priori.
1(cid:90) α ρ (cid:104)
v =v − dα¯ ∂ Q
2 1 ρ 2(∂ S)2 φ
−∞ φ V. SINGULARITIES AND BOHMIAN
2∂ S (cid:105)
+ φ {∂ (ρ∂ S)−∂ (ρ∂ S)} . MECHANICS
ρ φ φ α α
(32)
In this section, we consider the possibility of singu-
larities (zero scale factor) for trajectories given by the
So a wave function ψ will lead to the same trajectories
Bohmian dynamics of section IV . In particular, using
only if
the equilibrium distribution (20) as probability distribu-
2∂ S tion,wecalculatetheprobabilityforatrajectorytohave
∂ Q+ φ {∂ (ρ∂ S)−∂ (ρ∂ S)}=0. (33)
φ ρ φ φ α α a singularity, for a given wave function.
Westartbyconsideringtheasymptoticbehaviorofthe
Generically the above condition will not hold. For ex- Bohmiantrajectories. First,supposethatthewavefunc-
ample, consider a positive or negative frequency wave tion has only right-moving components, i.e., Ψ(α,φ) =
function. Then the term in curly brackets is zero (be- Ψ (α − φ) = (ψ (α − φ),ψ (α − φ)). Then the
R R,+ R,−
cause the density ρ then equals |ψ|2 and hence satisfies support of the wave function is localized on α < 0 for
(8)) and the condition is reduced to ∂φQ = 0, which is φ→−∞ and α>0 for φ→+∞. This follows from the
generically not satisfied. fact that
On the other hand, if the wave function ψ is ei-
(cid:90) 0
ther only left- or right-moving, i.e., ψ = ψL(α+φ) or lim dαρΨR(α,φ)
ψ =ψR(α−φ), then the condition (33) is automatically φ→−∞ −∞
satisfied. In this case, the trajectories are the same for (cid:90) 0
bothBohmianapproaches. Inaddition,thequantumpo- = lim dα(cid:0)|ψR,+(α−φ)|2+|ψR,−(α−φ)|2(cid:1)
tential (9) is zero, so that the trajectories are classical. φ→−∞ −∞
(Conversely, if Q=0, then the wave function ψ is either = lim (cid:90) −φdν(cid:0)|ψ (ν)|2+|ψ (ν)|2(cid:1)
left-orright-movingandthetrajectoriesarethesamefor R,+ R,−
φ→−∞ −∞
both Bohmian approaches.) As shown in section II, the
=||Ψ ||2 =1 (34)
classical trajectories are given by α = ±φ+c, with c a R 1
constant. Thepositiveandnegativesignrespectivelycor-
and, similarly, for the other limit we have
respond to a right- and left-moving wave function. The
trajectories reach the singularity a=eα =0 respectively (cid:90) ∞
lim dαρΨR(α,φ)
at φ→−∞ and φ→∞. Note, however, that in our sec-
φ→+∞ 0
ond Bohmian approach, we have not introduced a lapse (cid:90) ∞
= lim dα(cid:0)|ψ (α−φ)|2+|ψ (α−φ)|2(cid:1)
R,+ R,−
φ→+∞ 0
(cid:90) ∞
= lim dν(cid:0)|ψ (ν)|2+|ψ (ν)|2(cid:1)
R,+ R,−
7 NotethatinsectionII,wewrotethewavefunctionasψ=ψ++ φ→+∞ −φ
ψ−, whereasinsectionIIIwewroteitasΨ=(ψ+,ψ−). These =||Ψ ||2 =1. (35)
notationsareofcourseequivalent. R 1
6
In the same manner, for a wave function that has only φ
left-moving components, i.e., Ψ(α,φ) = Ψ (α + φ) =
L
(ψ (α+φ),ψ (α+φ)), we have that
L,+ L,−
(cid:90) ∞
lim dαρΨL(α,φ)
φ→−∞ 0
(cid:90) ∞ α
= lim dα(cid:0)|ψ (α+φ)|2+|ψ (α+φ)|2(cid:1)
L,+ L,−
φ→−∞ 0
(cid:90) ∞
= lim dν(cid:0)|ψ (ν)|2+|ψ (ν)|2(cid:1)
L,+ L,−
φ→−∞ φ
=||Ψ ||2 =1 (36)
L 1
FIG. 2: Asymptotic classical behavior of the Bohmian tra-
and
jectories for an arbitrary wave function.
(cid:90) 0
lim dαρΨL(α,φ)
φ→+∞ −∞ 3. starts with a big bang and ends in a big crunch
(cid:90) 0
= lim dα(cid:0)|ψ (α+φ)|2+|ψ (α+φ)|2(cid:1)
L,+ L,−
φ→+∞ −∞ 4. undergoes a bounce, i.e., contracts until it reaches
(cid:90) φ a minimal size and then expands again.
= lim dν(cid:0)|ψ (ν)|2+|ψ (ν)|2(cid:1)
L,+ L,−
φ→+∞ −∞ Only trajectories of type 4 are non-singular. As we shall
=||ΨL||21 =1, (37) see, allfourtypesoftrajectoriesmayoccur. Though, for
a given wave function, there can not be trajectories of
sothatthesupportisnowlocalizedonα>0forφ→−∞
both types 1 and 2. This is an immediate consequence
and α<0 for φ→+∞.
of the no-crossing property of the Bohmian dynamics
So in summary, we have that
(whichstatesthattrajectoriescannotcrossatequaltimes
 inconfigurationspace),sincesuchtrajectoriesmustcross
α<0 for φ→−∞
 each other.
ΨR has support on , Intheprevioussection,weshowedthatwavefunctions
α>0 for φ→∞ that are purely left- or right-moving give rise to classical
 trajectories, which are either of type 1 or type 2, and
α>0 for φ→−∞
 which are singular. There are also wave functions for
ΨL has support on . which there are trajectories of type 3 or 4.
α<0 for φ→∞ As a simple example, consider a wave function which
is even or odd in α. For such a wave function, the ve-
locity field v is odd in α. As such, for every possible
This analysis shows that for any wave function Ψ = 2
trajectory α(φ), also −α(φ) is a possible trajectory. Be-
Ψ +Ψ , its left- and right-moving components will be-
R L cause of the no-crossing property, this implies that no
come asymptotically non-overlapping in α-space. As a
trajectory will cross the φ-axis.9 In other words, trajec-
consequence, asymptotically, Bohmian trajectories will
tories that have positive value of α at any given time φ,
be determined either by the Ψ or Ψ . Hence, asymp-
R L must have α>0 at all times and hence avoid the singu-
totically,thetrajectoriesareclassical,givenbyα=φ+c
larity. These trajectories correspond to a bouncing uni-
or α = −φ+c, see figure 2.8 (The same holds for the
verse. Theyasymptoticallystartwithα=−φ+c ,reach
BohmiandynamicsofsectionII,inthecasethatthetra- i
aminimalvalueα andthenasymptoticallyevolveac-
jectories case be expressed as α(φ).) min
cording to α=φ+c for φ→∞. See figure 3 for an ex-
This asymptotic classical behavior implies that there f
ample. Conversely,trajectoriesthatasymptoticallystart
may be four different types of trajectories. Namely, tra-
with α=φ+c˜ originate from a singularity. Eventually
jectories that represent a universe that: i
they reach a maximal vale α and then for φ → ∞
max
they move again to the singularity according to the tra-
1. startswithabigbangandkeepsexpandingforever
jectory α = −φ+c˜ . So these trajectories are big bang
f
2. keeps contracting until a big crunch - big crunch solutions.
8 There might also be a trajectory that does not display this 9 Their might also be the trajectory α(φ) = 0 which acts as a
asymptotic behavior and acts as a bifurcation line between tra- bifurcation line between trajectories with different asymptotic
jectorieswithdifferentpossibleasymptoticbehavior. behavior.
7
φ the probability for trajectories coming from α > 0 to
keep contracting towards the singularity is P =
contracting
0. Similarly, for the case that P > P , one can
R,i R,f
show that P = P , P = P − P ,
bounce R,f contracting L,f L,i
P =P and P =0.
recollapsing L,i expanding
In summary, for an arbitrary state Ψ, we have
α
P =P =min(P ,P )=min(P ,P ),
bounce recollapsing L,i L,f R,i R,f
P =max(P −P ,0)=max(P −P ,0),
expanding L,i L,f R,f R,i
P =max(P −P ,0)=max(P −P ,0).
contracting L,f L,i R,i R,f
(40)
ThisimpliesthattheprobabilityP =1−P
FIG. 3: Some trajectories for a wave function that is sym- singularity bounce
to run into a singularity satisfies
metric under α→−α. The trajectories on the left represent
universeswhichstartwithabigbangandendinabigcrunch. 1
Thetrajectoriesontherightrepresentuniversesthatbounce. 2 (cid:54)Psingularity (cid:54)1, (41)
i.e., the probability to run into a singularity is at least
Let us now analyze the general case and establish the one half. Maximum probability is reached when there
probability to have singular trajectories for a general are only left- or right-moving components. In that case,
wavefunctionΨ=Ψ +Ψ . CallP andP theprob- every trajectory either runs into a singularity or starts
R L L,i R,i
ability for the universe to start respectively with α < 0 from a singularity. On the other hand, for a superpo-
(from the left) and α > 0 (from the right) in the limit sition of left- or right-moving components, there is al-
φ→−∞. Similarly, P and P denote the probabil- ways a non-zero probability for a bounce. For a state
L,f R,f
ities for the universe to end up respectively with α < 0 Ψ, the maximum probability that can ever be attained
and α>0 in the limit φ→∞. Since Ψ and Ψ do not foratrajectorytobenon-singularis1/2, whichhappens
R L
overlap asymptotically, we have when P =P . This happens for the examples of the
L,i L,f
symmetric and anti-symmetric states discussed before.10
(cid:90) 0
P = lim dαρΨ(α,φ)=||Ψ ||2, The trajectories for the first Bohmian approach, that
L,i φ→−∞ −∞ R 1 was presented in section II, are different from those con-
(cid:90) ∞ sidered here. However, the trajectories in the first ap-
PR,i = lim dαρΨ(α,φ)=||ΨL||21, proachthatareexpressibleasα(φ)havethesameasymp-
φ→−∞ 0 totic behavior as those considered here and hence are
(cid:90) 0
P = lim dαρΨ(α,φ)=||Ψ ||2, qualitatively the same. However, there is no natural
L,f L 1
φ→∞ −∞ probability distribution over trajectories which allows to
(cid:90) ∞ calculate the probability for trajectories to be singular.
P = lim dαρΨ(α,φ)=||Ψ ||2, (38)
R,f R 1
φ→∞ 0
and VI. SINGULARITIES AND CONSISTENT
HISTORIES
P =1−P =P =1−P . (39)
L,i L,f R,f R,i
According to the consistent histories approach the
We can now find the probability for the universe to
probability for a singularity is always one. Coarse-
start or to end in a singularity for a given wave func-
grainedhistoriesareeitheroftype1or2. Thiswasshown
tion Ψ. First, assume that P < P . In this case,
R,i R,f for two-time histories in [2, 3], with the times being the
trajectories starting from α > 0 as φ → −∞ can not
infinite past and future.
end up moving to α < 0 as φ → ∞. If a trajec-
In [1], some of us argued that introducing a third in-
tory did do that, then all the trajectories that started
termediate time would not lead to a family of consistent
on the α < 0 would must also end up with α < 0 as
φ → ∞ because of the no-crossing property. This im-
plies that P (cid:62) P and hence, because of (39), that
L,f L,i
P (cid:62)P , but this contradicts our assumption. Thus,
R,i R,f 10 Theexamplethatwasusedin[4,5]tocomparetheBohmianap-
inthecasethatP <P ,theprobabilityforabounce
R,i R,f proachtotheconsistenthistoriesapproachinthecontextofnon-
is Pbounce = PR,i. In addition, it implies that the prob- relativisticquantummechanicsisactuallycompletelyanalogous
ability for trajectories to start from a singularity and to the case of a symmetric state in our cosmological model. In
to keep expanding is P = P −P and the [4,5],thewavefunctioniscomposedoutoftwopacketsthatcross
expanding R,f R,i
eachotherintime. Thewavefunctionisconsideredtobecom-
probability for trajectories to start and end in a singu-
pletely symmetric so that Bohmian trajectories will never cross
larity (i.e., trajectories with a big bang and big crunch)
thesymmetryaxis,whilethehistoriesintheconsistenthistories
is Precollapsing = PL,i −(PR,f −PR,i) = PR,i. Finally, approachcorrespondtoeitherleft-orright-movingpaths.
8
histories, unless the state is classical (i.e. the state is a Given a wave function Ψ (and given that φ is suf-
1
left- or right-moving wave function). As such, for non- ficiently far in the past or φ sufficiently far in the fu-
n
classical states, the consistent histories approach would ture), we canalwaysfind acollections ofsets {∆k } such
ik
be unable to deal with the properties of the universe at that the decoherence condition is satisfied. The reason
intermediate times. This would of course undermine the is that the wave function is a superposition of a left-
consistenthistoriesapproachforthiscosmologicalmodel. moving and a right-moving packet. For each time φ ,
k
However,theargumentin[1]isincomplete,mainlybe- we can choose the regions {∆k } such that support of
ik
cause only one particular family of histories was consid- Ψ and Ψ are each approximately within one of the
L R
ered. That is, the class of propositions (which are repre- ∆k (but not necessarily with the same i ). We then
ik k
saesnttheadtbiynp[2r,o3je]cftoironalloptherreaetotrism)ews.asAtnadkefnortothbisectlhaesss,atmhee have that for each time φk that P(cid:98)∆kkikΨL(α,φk) ≈ 0 for
historiesarenotconsistent(unlesstheintermediatetime all but one ∆kik, which we denote ∆kL. For ∆kL, we have
is chosen sufficiently early or late). However, this does thatP(cid:98)∆kkΨL(α,φk)≈ΨL(α,φk). Wethenalsohavethat
not exclude other possible families of histories for which L
thehistoriesareconsistent. Wewillshowthatasuitable C(cid:98)hLΨL(α,φ0)≈ΨL(α,φn), with hL =(∆1L,...,∆nL). In
other words, there is a coarse-grained history that ap-
class of propositions can be chosen such that consistent
proximatelyfollowsthetrackofthepacketΨ . Similarly,
historiescanbeconsideredforanarbitrarynumberofin- L
there is a coarse-grained history h that approximately
termediate times.11 With this choice, the probability for R
follows the track of the packet Ψ .
a singularity is always one. This generalizes the results R
If one of the φ is such that Ψ and Ψ are approxi-
in [2, 3] to an arbitrary number of times. k L R
matelynon-overlappingatthattime(forexamplebytak-
To setup the consistent histories framework, we will
ing φ sufficiently early or φ sufficiently late, since we
1 n
use the triplet (H1,P(cid:98)1,H(cid:98)). As explained in section III, know that the overlap vanishes in the limit φ → ±∞.),
(H2,P(cid:98)2,H(cid:98)) would lead to the same results. then
A coarse-grained history is a sequence of regions
∆1,...,∆n in α-space at a sequence of times φ1,...,φn. C(cid:98)hLΨ(α,φ0)≈ΨL(α,φn), C(cid:98)hRΨ(α,φ0)≈ΨR(α,φn)
(46)
These are obtained by considering an exhaustive set of
regions {∆kik}, ik = 1,2,..., of α-space for each time and
φ , k = 1,...,n. For each of these regions, there is a
k
projection operator C(cid:98)hΨ(α,φ0)≈0 , for all h(cid:54)=hL,hR. (47)
(cid:90) Since12
P(cid:98)∆kkik = ∆kik P(cid:98)1(dα) (42) (cid:104)C(cid:98)hLΨ|C(cid:98)hRΨ(cid:105)≈0, (48)
we have that the decoherence condition (44) is satisfied.
defined in terms of the PVM P(cid:98)1(dα). In terms of these Theonlytwocoarse-grainedhistorieswithnon-zeroprob-
projectionoperators,theclassoperatorforahistoryh=
ability are h and h , with probabilities
(∆1 ,...,∆n ) is L R
i1 in P ≈||Ψ ||2 , P ≈||Ψ ||2. (49)
C(cid:98)h =P(cid:98)∆nninU(cid:98)(φn,φn−1)P(cid:98)∆n−nin−1−11U(cid:98)(φn−1,φn−2) ··· In orderhLto consLide1r the questiohnR whethRer1there is a
··· P(cid:98)∆22i2U(cid:98)(φ2,φ1)P(cid:98)∆11i1U(cid:98)(φ1,φ0), (43) sTinhgenula(4ri6t)y-,(4w9e) hshoolduladndta,kuesiφn1g a→si−m∞ilarannodtaφtnion→as∞in.
the previous section, we have
where φ0 is some initial time and U(cid:98)(φk,φk−1) =
e−iH(cid:98)(φk−φk−1) is the unitary time evolution from time Pbounce =Precollapsing ≈0,
φkI−n1otrodφerk.to associate probabilities to a family of histo- Pexpanding =PhL ≈||ΨL||21,
P =P ≈||Ψ ||2. (50)
ries, the decoherence condition contracting hR R 1
Theprobabilityforabounceisnegligiblysmall. Itcan
(cid:104)C(cid:98)h(cid:48)Ψ|C(cid:98)hΨ(cid:105)1 ≈0, for h(cid:48) (cid:54)=h, (44) actually be made arbitrarily small by suitably choosing
needs to be satisfied. The probability for a history h is the set {∆1 }. So according to the consistent histories
i1
then given by approach, one can always find a consistent family of his-
tories where the probability for a singularity is one.
Ph =||C(cid:98)hΨ||21. (45)
12 Eq. (48) follows from the fact that a left-moving packet and
11 Theanalysisiscompletelyanalogoustothatof[4,5]. Thereason right-movingpacketareorthogonal. Sincetheyhavenocommon
is that, as already noted in footnote 10, we are considering two support at the far past they are clearly orthogonal then. Since
packetsthatmoveacrosseachother,justlikeintheexampleof theinnerproductispreservedovertime,theymustbeorthogonal
[4,5]. atalltimes.
9
VII. CONCLUSION DeWitt quantization. It would be interesting to also
study the Bohmian approach to loop quantization. For
We have analyzed a Bohmian approach to the the consistent histories approach to loop quantization, it
Wheeler-DeWitt quantization of the Friedmann- was recently shown that histories do not have singulari-
Lemaˆıtre-Robertson-Walker model. This Bohmian tiesforgenericwavefunctions[28]. Itisunclearwhether
approach agrees with the consistent histories approach this is also true for a Bohmian approach.
concerning the probabilities for single-time histories.
However, it makes different predictions for the probabil-
ity of trajectories or histories to have a singularity. In
Acknowledgments
theconsistenthistoriesapproach,atleastforthefamilies
considered in the present paper, the probability for a
history to have a singularity is one. On the other hand, F.T.F.andN.P.-N.wouldliketothankCNPqofBrazil
in the Bohmian approach, for generic wave functions for financial support. W.S. acknowledges support from
(i.e., non-classical wave functions), there is a non-zero the Actions de Recherches Concert´ees (ARC) of the Bel-
probability for a trajectory to be non-singular and have gium Wallonia-Brussels Federation under contract No.
a bounce. 12-17/02. W.S. carried out part of this work at Rutgers
So, as was already emphasized in [1], where a different University, with a grant from the John Templeton Foun-
Bohmian model was considered to make the compari- dation. The opinions expressed in this publication are
son with the consistent histories approach, the question those of the authors and do not necessarily reflect the
whetherornottheWheeler-DeWittquantizationleadsto views of the John Templeton Foundation. W.S. is also
singularities depends very much on the version of quan- grateful to CBPF where part of this work was carried
tum theory one adopts. out and to Sheldon Goldstein and Roderich Tumulka for
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