Quantum cosmology with scalar fields: self-adjointness and cosmological scenarios

0.17 MB·English

by  C. R. Almeida

Quantum cosmology with scalar fields:
self-adjointness and cosmological scenarios
5
1
Carla R. Almeida∗, Antonio B. Batista†, Ju´lio C. Fabris‡
0
2 DF - UFES, Vit´oria, ES, Brazil

n
a
and
J

7
1 Paulo R.L.V. Moniz§

DF, Universidade da Beira Interior, Covilh˜a, Portugal
]
c
q
-
r January 20, 2015
g
[

1
v Abstract
0
7 We discuss the issue of unitarity in particular quantum cosmological
1 models with scalar field. The time variable is recovered, in this context,
4 by using the Schutz’s formalism for a radiative fluid. Two cases are con-
0 sidered: a phantom scalar field and an ordinary scalar field. For the first
.
1 case,itisshownthattheevolutionisunitaryprovidedaconvenientfactor
0 orderingandinnerproductmeasurearechosen;thesamehappensforthe
5 ordinary scalar field, except for some special cases for which the Hamil-
1 tonian isnot self-adjoint butadmitsaself-adjoint extension. Inall cases,
: even for those cases not exhibiting unitary evolution, the formal compu-
v
i tationoftheexpectationvalueofthescalefactorindicatesanon-singular
X
bounce. The importance of the unitary evolution in quantum cosmology
r is brieflydiscussed.
a
∗[email protected]
†[email protected]
‡[email protected]
§[email protected]
1
1 Introduction
Quantum cosmology faces difficulties that range from the technical to the con-
ceptual point of view. It is usually based on the ADM formulation of the
General Relativity theory [1, 2, 3, 4], leading to the Wheeler-de Witt equa-
tion, a functional equation defined in the superspace, the space of all possi-
ble three-dimensional metrics on the space-like hypersurface which foliates the
four-dimensionalspace-time. Technically, the foremostproblemis to determine
solutions of the Wheeler-de Witt equation in its complete form, since it is a
functional equation with an infinite number of degrees of freedom. This sug-
gests to freeze out an infinite number of such degrees of freedom, reducing a
system to a few numbers of variables, leading to the mini-superspace setting.
This drastic reduction implies that we ignore, for example, the violation of the
uncertaintyprincipletothosedegreesoffreedomthathavebeenfrozenout. On
theotherhand,theADMformulationofGeneralRelativityformsaconstrained
system, and the Hamiltonian constraints implies the absence of an explicit and
cleartimevariable. Ontheconceptualside,moreover,wemustfacetheproblem
of a unique system, the Universe at a whole, what leads to the inapplicability
of the usual Copenhagen interpretation of quantum mechanics.
Theissueofthetimevariablehasbeenaddressedinmanydifferentmanners.
One can, for example, identify an internal time, determining the evolution of
the system, through the deparametrisation procedure [5]. Or an external time
(from the point of view of the gravitational system) may be introduced, for
example, throughthe WKB approach(see, for example, reference [6] and refer-
encestherein)orbyconsideringmatterfields. Thislastprocedurehasbeenused
extensivelyintheliterature,andonepossibilityistoconsiderafluidwithinter-
naldegreesoffreedomusingSchutz’sdescription[7,8]. Thisapproachleadstoa
Schr¨odinger-likeequation,sincethemomentumassociatedtothefluidvariables
appears linearly in the resulting Hamiltonian [9].
The interpretation problem is more delicate. This very important issue will
not be treated in a direct way in the present text, but the reader can address
himself to many text about this subject in the literature. See, as example,
references [4, 10, 11, 12, 13] and references therein.
Our goal in this paper is to investigate another difficulty that appears even
when the mini-superspace approach is employed and the time variable is iden-
tified through, for example, the Schutz’s variable: the unitary evolution of re-
sultingquantumsystem. IthasbeenshownthatwhenjusttheEinstein-Hilbert
Lagrangian and a perfect fluid matter component is considered, the resulting
Hamiltoniancan be made self-adjoint (unitarity is assured)under the hypothe-
sis of a maximally symmetric spatial section [14]. However, when anisotropy is
taken into account, in the same context, the Hamiltonian is generally no more
self-adjoint [15]. However, recently, it has been shown that a convenient choice
of the ordering factor associated with the gravitational operators may restore
the self-adjoint character of the Hamiltonian [16].
The breakdownof the unitary evolutionwhenthe time variableis recovered
through the Schutz variable occurs also when scalar fields are considered to-
1
getherwiththeEinstein-HilbertLagrangian,evenintheisotropicehomogenous
case[17,18]. Curiously,inthiscase,andusingtheSchutzformalismtorecovera
Schr¨odinger-likeequation,aphantomscalarfield(with negativekinetic energy)
assures the positivity of the total energy, while for a normal scalar field (with
positive kinetic energy) this positivity of the energy is not assured, since the
Hamiltonianhasahyperbolicsignatureforthiscase. Itwillbeshown,however,
that a convenient factor ordering (similarly to what has been discussed in the
anisotropic case, reference [16]) together with a convenient measure associated
with the inner product guarantees a self-adjoint Hamiltonian (hence a unitary
evolution) for a phantom scalar field. The same occurs for the normal scalar
field,butforspecialcases(connectedwiththeorderingfactor)theHamiltonian
isnotself-adjointbutadmitsaself-adjointextension. We willnotconsiderhere
the issue of unitary evolution for other approaches to recover the time variable
as, for example, by using the scalar field itself as the time coordinate [19].
This paper is organised as follows. In next section with consider a scalar-
tensor theory - specifically, the Brans-Dicke theory - coupled with a radiative
fluid. By performing conformal transformation, the theory is re-written in the
so-calledEinstein’s frame. In section 3, solutions for the Schr¨odinger-likeequa-
tion are obtained, and it is shownthat they do not display a unitary evolution.
Even though, formal predictions for the evolution of the Universe are obtained
revealing a non-singular behaviour. In section 5 it is shown how to recover
unitarity at least for the anomalous, phantom configuration of the scalar field.
The general self-adjoint issue is considered in section 6. In section 7 we dis-
cuss the self-adjoint extension for the particular cases where the Hamiltonian
is not self-adjoint but admits such an extension. In section 8 we present our
conclusions.
2 Scalar-tensor model with radiative fluid
Theprototypeofthescalar-tensorgravityformulationistheBrans-Dicketheory
[20], represented by the Lagrangian,
φ φ;ρ
= g˜φ R˜ ω˜ ;ρ + , (1)
L − − φ2 Lm
(cid:26) (cid:27)
p
whereL isthematterLagrangian. WewillsupposethatthematterLagrangian
m
referstoaradiativefluid,havingconformalsymmetry. ThisLagrangianincludes
as a particular case the string dilatonic Lagrangianfor which ω˜ = 1 [21].
−
The Lagrangian (1) defines the theory in the Jordan’s frame. Performing a
conformal transformation such that g =φ 1g˜ , we transpose the action (1)
µν − µν
to the corresponding expression written in the Einstein’s frame [22]:
φ φ;ρ
=√ g R ω ;ρ + , (2)
L − − φ2 Lm
(cid:26) (cid:27)
where ω =ω˜ +3/2. Remark that the matter Lagrangianis not affected by the
conformal transformation since we are considering a radiative fluid.
2
We will consider from now on that the line element in the Einstein’s frame
can be written as,
ds2 =N(t)2dt2 a(t)2γ dxidxj (3)
ij
−
where N(t) is the lapse function, a(t) is the scale factor and γ is the induced
ij
metric of the homogeneous and isotropic spatial hypersurfaces with curvature
k =0, 1. For simplicity, we will fix k =0. With this metric, the gravitational
±
Lagrangianbecomes,
V a3 a¨ a˙ 2 a˙ N˙ φ˙2
0
= 6 + ω , (4)
LG N − a a −aN − φ2
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
where V is a constant and can be interpreted as the physical volume of the
0
compact universe (in this case a three-torus) divided by a3. Since there is an
identical multiplicative constant in front of the matter Lagrangianwe can drop
it from our analysis. Discarding a surface term, the gravitational Lagrangian
can be written as,
1 φ˙2
= 6aa˙2 ωa3 . (5)
LG N − φ2
(cid:26) (cid:27)
Defining,
σ = ω lnφ, (6)
| |
we obtain, p
1
= 6aa˙2 ǫa3σ˙2 , (7)
G
L N −
(cid:26) (cid:27)
where ǫ = 1 according ω is positive (upper sign) or negative (lower sign).
±
The canonicalmomenta associatedwith the scale factorandthe scalarfieldare
respectively:
aa˙ a3σ˙
p =12 , p = 2ǫ , (8)
a σ
N − N
leading to the following expression in terms of the conjugate momentum:
1 p2 p2
=p a˙ +p σ˙ N a ǫ σ . (9)
LG a σ − 24 a − 4a3
(cid:26) (cid:27)
Considering a radiative matter component (for the computation of the con-
jugate momentum associated with the fluid, see references [9, 14]), the total
Hamiltonian is:
1 p2 p2 p
H =N a ǫ σ T . (10)
24 a − 4a3 − a
(cid:26) (cid:27)
The resulting Schr¨odinger-like equation is,
∂2Ψ ǫ ∂2Ψ ∂Ψ
+ =i , (11)
− ∂a2 a2 ∂σ2 ∂T
where we made the redefinition σ σ and T T.
√6 → 24 →
3
3 Cosmological scenarios
Inthereference[18],thequantumcosmologicalmodeldefinedbytheSchr¨odinger-
typeequation(11)hasbeenstudied. Wewillreviewinthissectiontheprocedure
and results obtained in reference [18].
Inordertoobtaintreatableexpressions,theorderingambiguityofoperators
has been exploited, and equation (11) has been rewritten as,
∂2Ψ 1 ǫ ∂2Ψ ∂Ψ
∂ Ψ+ =i , (12)
− ∂a2 − a a a2 ∂σ2 ∂T
For ǫ = 1, a condition that assures the positivity of energy, equation (12)
−
admits a solution in terms of stationary states of energy E:
Ψ=AJ (√Ea)eikσe iET, ν =k , (13)
ν −
with A being a normalization constant and k is a separation constant.
A particular wavepacket can be obtained by a convenient superposition of
the constants E and k, as it is described in [18]. The final result is
a2
e−4B(T)
Ψ(a,σ,T)=C , (14)
B(T)g (a,B,σ)
α
where
a
B(T)=(γ+iT), g (a,σ,T)= α+ ln iσ, (15)
α
− 2B(T) ±
(cid:20) (cid:21)
γ and α being constants connected with the gaussian-type superposition, and
C is a normalisation constant.
The norm of the wavefunction (14) can be calculated explicitly:
∞ +∞ C2 ∞ e−γu2
N = Ψ∗Ψdadσ = π du
(BB )1/2
Z0 Z−∞ ∗ Z0 α+ln u2
(cid:18) (cid:19)
C2
= πI , (16)
(BB )1/2 1
∗
where I is the definite integral,
1
∞ e−γu2
I = du. (17)
1
Z0 α+ln u
2
(cid:18) (cid:19)
The norm is clearly time-dependent: the corresponding quantum model is not
unitary. Eventhough, the expectationvalue for the scalarfield canbe formally
computed, leading to the expression,
<a> (γ2+T2)1/2. (18)
∝
4
Thesameresultis,essentially,obtainedthroughthecomputationofthebohmian
trajectories. The expectation value for the scale factor indicates a non-singular
bounce. Asitisshowninreference[18],theexpectationvalueofthe scalarfield
σ is time-dependent, reading
T
<σ > arctan . (19)
∝ γ
(cid:18) (cid:19)
4 Recovering unitarity
Let us suppress the ordering factor introduced previously. The Schr¨odinger
equation is
ǫ
∂2Ψ+ ∂2Ψ=i∂ Ψ, (20)
− a a2 σ T
with ǫ= 1. For a stationary state,
±
Ψ=φe−iET, (21)
leading to
ǫ
∂2φ+ ∂2φ=Eφ. (22)
− a a2 σ
Using the separation variables method, such that
φ(a,σ)=X(a)Y(σ), (23)
the equation becomes,
X ǫ Y¨
′′
+ =E, (24)
− X a2Y
wheretheprimesmeanderivativewithrespecttoaandthedotsmeanderivative
with respect to σ. This equation can be rewritten as,
Y¨ X
ǫ =a2 E+ ′′ = ǫk2, (25)
Y X −
(cid:18) (cid:19)
where k is a constant of separation. In this case, the function Y satisfies the
equation,
Y¨ +k2Y =0, (26)
with the solution,
Y =Y eikσ. (27)
0
The equation for X reads,
k2
X + E+ǫ X =0. (28)
′′
a2
(cid:18) (cid:19)
5
The soluton is,
1
X(a)=√aJ (√Ea), ν = ǫk2. (29)
ν
4 −
r
The total wavefunction is,
Ψ=Ψ √aJ (√Ea)e i(kσ+ET). (30)
0 ν −
Let us construct the wavepacket. First we write x = √E. Then, we can
made the following superposition:
+ +
Ψ(a,σ)=√a ∞ ∞A(k)xν+1e Bx2J (xa)e ikσdkdx, (31)
− ν −
Z−∞ Z0
where, as before,
B =γ+iT, (32)
γ being a positive constant (in order to assure the convergence of the integral),
playing the same rˆole as the previous section. The function A(k) will remain
forthemomentunspecified,butitmustdecayexponentiallyforlargek inorder
also to assure the convergence of the wavepacket. The integral in x can be
performed leading to [23],
Ψ= +∞A(k)e ikσ aν+1/2 exp a2 dk. (33)
−
(2B)ν+1 − 4B
Z−∞ (cid:18) (cid:19)
From now on, let us consider the case ǫ = 1, such that ν is real. The
−
modulus of the wavefunction reads,
+ +
Ψ∗Ψ= ∞ ∞A(k)A∗(k′)e−i(k−k′)σB−ν−1B∗−ν′−1aν+ν′+1
Z−∞ Z−∞
γa2
exp dkdk , (34)
× − 2B¯ ′
(cid:20) (cid:21)
where B¯ =BB =γ2+T2.
∗
Theintegrationinσ intheinterval <σ <+ impliesadeltafunction,
−∞ ∞
δ(k k ). After integrating in k , we find
′ ′
−
+ +
∞Ψ∗Ψdσ = ∞ A(k)2B¯−2ν−1a2ν+1e−2aB2 dk. (35)
| |
Z−∞ Z−∞
Performing the change of variable,
a
y = ,
√B¯
we can write the norm as,
+ +
N = ∞ ∞ A(k)2e−γ2y2y2ν+1dydk. (36)
| |
Z0 Z−∞
6
The normis time-independent. After integratinginy,it canalsobe writtenas,
N = 1 +∞ A(k)2Γ 3+2ν dk. (37)
2 | | 4
Z−∞ (cid:18) (cid:19)
Due to the asymptotic properties of the Γ’s function, the integral converges if
A(k) behaves, for example, as e k2 for large values of k.
−
The expectation value for the scalar field is given by the expression,
1 +∞ +∞
<a>= Ψ∗aΨdσda. (38)
N
Z−∞ Z0
Using the same procedure as before, we end up with,
<a>= 1 B¯ +∞ +∞ A(k)2e−γ2y2y2(ν+1)dydk (γ2+T2)1/2, (39)
N | | ∝
p Z−∞ Z0
which is essentially the same prediction as in the previous non-unitary case.
We can also evaluate the expectation value for the scalar field. In fact, to
obtain the expectation value of σ we must compute
+
∞ ∞
<σ > Ψ(a,σ) σΨ(a,σ)dadσ. (40)
∗
∝
Z0 Z−∞
Using (33), we can write
σΨ=i +∞A(k)[∂ e ikσ] aν+1/2 exp a2 dk, (41)
k − (2B)ν+1 − 4B
Z−∞ (cid:18) (cid:19)
which can be integrated by parts. Using this expression, the integration in σ
leads again to delta function. Redefining, as before, the integration in a, we
obtain the following expression,
<σ >T = 2πi +∞ +∞ A(k)2e−γ2y2 Ak(k)
− | | A(k)
Z−∞ Z0 (cid:26)
2ν+1
y B y
+ ν ln ν ln ∗ dydk, (42)
k k
2 − B 2
(cid:18) (cid:19) (cid:18) r (cid:19)(cid:27)(cid:18) (cid:19)
wherethe subscriptk indicates derivativewithrespecttothis parameter. After
expressing B in polar form, we obtain,
T
<σ > =σ +σ arctan , (43)
T 0 1
γ
(cid:18) (cid:19)
where σ and σ are constants. This is essentially the same result obtained
0 1
in the previous section for the expectation value of σ. Remark, however, that
<σ >=0 if A(k) is an even function, such that A(k)=A( k).
−
7
5 Self-adjointness
The results shown before reveal that the self-adjointness property of the effec-
tive Hamiltonian may depend on the ordering factor. The overall situation is,
however,more involved. The Hamiltonian used in equation (12) is,
1 ǫ
H = ∂2 ∂ + ∂2. (44)
− a− a a a2 σ
When ǫ= 1,this operator lookslike that one of the two dimensional problem
−
writteninpolarcoordinates. Infact,if wemakethe identificationa r, σ θ,
≡ ≡
considering the variable θ as periodic, such that we can define the cartesian
coordinates x=rcosθ and y =rsinθ, the Hamiltonian takes the form,
H = ∂2 ∂2, (45)
− x− y
with <x,y <+ , which is clearly self-adjoint.
−∞ ∞
What makes the Hamiltonian (44) not self-adjoint? We could think that is
the fact that the coordinate σ there is not periodic. But, we remark that in
passing from cartesian to polar coordinates, there is a Jacobian factor that it
has not been used in computing the inner product in the Hilbert space for the
Hamiltonian (44). The clue to this problem seems to be in the ordering factor
and in the measure of the inner product, as point out in reference [16] in the
context of anisotropic models.
Let us be more specific. Let us consider the Hamiltonian with a ordering
factor labeled by q:
q ǫ
Hˆ = ∂2 ∂ + ∂2, (46)
− a− a a a2 σ
wherewehaverestoredthefactorǫ= 1. ThisHamiltonianissymmetricunder
±
the inner product defined by,
+
(φ,Ψ)= ∞ ∞φ∗Ψaqdadσ, (47)
Z0 Z−∞
ifthefunctionsφandΨ,aswellastheirfirstderivatives,arenullintheextreme
of the interval.
To be symmetric is a necessary but not sufficient condition for the operator
Hˆ to be self-adjoint: the domain of the operator and of is hermitian conjugate
must be also the same. The self-adjoint character of a symmetric Hamiltonian
can be obtained through the deficiencies indices of von Neumann [24, 25].
Thedeficienciesindicesaredefinedasfollows. Considertheeigenvalueprob-
lem,
HˆΨ= iΨ. (48)
±
Letuscalln andn thenumberofsquareintegrablesolutionsoftheeigenvalue
+
−
problem given by equation (48) when the upper and lower sign in the right
8
hand side is chosen, respectively. If n = n = 0, the operator Hˆ is already
+
−
self-adjoint; if n = n = 0, the operator is not self-adjoint but it admits self-
+
− 6
adjoint extensions given by some restrictions in the wavefunctions; if n = n
+
6 −
the operator is not self-adjoint and it does not admit any self-ajoint extension.
Hence, we must solve the eigenvalue equation,
q ǫ
∂2Ψ ∂ Ψ+ Ψ= iΨ=ηΨ, (49)
− a − a a a2 ±
where we havedefined η = i. The solutionsof this equationcanbe written as
±
Ψ=ap Hν(1)(√ηa)+Hν(2)(√ηa) e±ikσ, (50)
(cid:20) (cid:21)
where,
1 q
p= − , ν = p2 ǫk2, (51)
2 −
p
k is a separation constant which must be real (otherwise, in the corresponding
Schr¨odinger equation, we would have just non-integrable solutions, with diver-
gent norm), and H(1,2)(x) are Hankel’s functions of first and second kind.
ν
Let us write the functions,
Ψ(1) = apH(1)(ηa)eikσ, (52)
ν
±
Ψ(2) = apH(2)(ηa)eikσ. (53)
ν
±
These solutions contain the separation parameter k, which parametrises the
plane wave behaviour in terms of the coordinate σ. In order to give a physical
meaning to such structure is necessary to construct a wave packet related to
theseplanewavesolutions. Thisprocedurehasthe purposetoavoidunphysical
behaviour,but it has no major impact on the considerationson the self-adjoint
character of the operator. Hence, the solutions take the form,
+
Ψ(1) = ∞A(k)apH(1)(ηa)eikσdk, (54)
ν
±
Z−∞
+
Ψ(2) = ∞A(k)apH(2)(ηa)eikσdk, (55)
ν
±
Z−∞
where A(k), as before, is a superposition factor which satisfies the condition to
go to zero sufficiently fast at both infinities.
We must investigate the norm of the wave functions (54,55), given by
+ +
N(1,2) = ∞ ∞Ψ(1,2)(a,σ)Ψ(1,2)∗(a,σ)aqdadσ. (56)
± Z−∞ Z0 ± ±
First, let us consider the integration in σ. We have:
+
I = ∞Ψ(1,2)∗Ψ(1,2)dσ
σ
± ±
Z−∞
9