Classical and Quantum Aspects of the Inhomogeneous Mixmaster Chaoticity

0.13 MB·English

by  Riccardo Benini

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7 Classical and Quantum Aspects of the Inhomogeneous Mixmaster
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Chaoticity
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R.Benini12†andG.Montani23‡
n
a 1Dipartimento di Fisica - Universit`adi Bologna and INFN
J Sezione di Bologna, viaIrnerio 46, 40126 Bologna, Italy
2ICRA—International CenterforRelativistic Astrophysics c/o Dipartimento di Fisica (G9)
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Universit`adi Roma “LaSapienza”, Piazza A.Moro 5 00185 Roma, Italy
1 3ENEA C.R. Frascati (U.T.S. Fusione), Via Enrico Fermi 45, 00044 Frascati, Roma, Italy

†[email protected]
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‡[email protected]
v
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9 WerefineMisner’sanalysisoftheclassicalandquantumMixmasterinthefullyinhomo-
0 geneouspicture;webothconnectthequantumbehaviortotheensemblerepresentation,
1 bothdescribethepreciseeffectoftheboundaryconditionsonthestructureofthequan-
0 tumstates.
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Nearthecosmologicalsingularity,thedynamicsofagenericinhomogeneouscos-
0
mological model is reduced by an ADM procedure to the evolution of the physical
/
c degrees of freedom, i.e. the anisotropies of the Universe. In fact asymptotically to
q
- the Big-Bangthe space points dynamically decouple1 because the spatialgradients
gr of the dynamical variables become of higher order2 and we can model the inhomo-
: geneous Mixmaster via the reduced action:
v
i
X
I = d3xdτ(p ∂ u+p ∂ v ǫ), ǫ=v p2 +p2, (1)
ar u τ v τ − u v
Z
p
The dynamics of such a model is equivalent to (the one of) a billiard-ball on a
Lobachevsky plane; this can be shown by the use of the Jacobi metric.1 The man-
ifold described turns out to have a constant negative curvature, where the Ricci
scalaris givenby R= 2/E2:the complex dynamicsofthe genericinhomogeneous
−
model results in a collection of decoupled dynamical systems, one for each point of
the space,andallofthemequivalentto abilliardproblemonaLobachevskyplane.
We wantto investigatethe relationexisting betweenthe classicalandthe semiclas-
sical dynamics, and from this analysis we will derive the correct operator ordering
to be used when quantizing the system.
Let’s write down the Hamilton-Jacobi equation for the system
2 2
2 2 δ 0 δ 0
ǫ =v S + S (2)
δu δv
(cid:18) (cid:19) (cid:18) (cid:19) !
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This can be explicitly solved by separation of constants
ǫ+ ǫ2 k2(ya)v2
S0(u,v)=k(ya)u+ ǫ2−k2(ya)v2−ǫln 2 p ǫ−2v !+c(ya). (3)
p
The semiclassical analysis can be developed furthermore, and the stationary con-
tinuity equation for the distribution function can be worked in order to obtain
informations about the statistical properties of the model; as soon as we restrict
the dynamics to the configuration space, we get the following equation for the dis-
tribution function w˜
∂w˜(u,v;k) E 2 ∂w˜(u,v;k) E2 2k2v2 w˜(u,v;k)
+ 1 + − =0 (4)
∂u s kv − ∂v kv2 E2 (kv)2
(cid:18) (cid:19) −
This can be solved, and the exact distribution function capn be obtained as soon as
we eliminate by integration the constant k
Ev g u+v kE2v22 −1
w˜(u,v)= (cid:18) q (cid:19)dk (5)
−E v√E2 k2v2
Z v −
Itisworthnothinghowinthecaseg =const,themicrocanonicalLiouvillemeasure
w (u,v)= π is recovered.
mc v2
Weexpectthatthedistributionfunctionw˜(u,v)isre-obtainedassoonasthequan-
tum dynamics is investigated to the first order in ~. This can be easily done as a
WKBapproximationtothe quantumdynamicsisconstructed;assoonasweretain
only the lower order in ~, we obtain that: i) the phase S(u,v) coincides with the
Hamilton-Jacobifunction,andii)the probabilitydensityfunction r(u,v) obeysthe
following equation:
∂r(u,v) E 2 ∂r(u,v) a(E2 k2v2) E2
k + k2 + − − r(u,v)=0. (6)
∂u s v − ∂v v2√E2 k2v2
(cid:18) (cid:19) −
that coincides with (4) for a particular choice for the operator-orderingonly, i.e.
∂ ∂
vˆ2pˆ2 ~2 v2 . (7)
v → − ∂v ∂v
(cid:18) (cid:19)
With this result, the problem of the full quantization of the system can be taken
in consideration. The main problem is the presence of the root square function
in the definition of the Hamiltonian (1), but well grounded motivations exist3 to
assume that the real Hamiltonian and the squared one have same eigenfunctions
and squared eigenvalues. This way the solution of the eigenvalue equation can be
obtained
Ψ(u,v)= anKs−1/2(2nπv)sin(2nπu) (8)
n>0
X
The spectrumisobtainedasDiricheletboundaryconditionsonthe domainΓ (see
Q
Fig.1) are taken into account. The condition on the vertical lines can be imposed
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exactly, but the one on the semicircle cannot be solved exactly; so we approximate
it as in Fig.2 with a straight line v =1/π (it preserves the domain area µ=π).
All these imply that s = 1/2+t, with t , and that the spectrum assumes the
∈ ℜ
3
2.5
2
v 1.5
1
0.5
0-2 -1.5 -1 -0.5 0 0.5 1
u
Fig.1. TherealdomainΓQ Fig.2. Theapproximatedomain
following form
1
E2 =( +t2)~2 (9)
t 4
The values of the real parameter t have to be numerically evaluated by solving the
equation K (2n) = 0 for every natural n. This condition implies a discrete but
it
quite complicated shape for the spectrum.
Asymptotic expansions for high occupation numbers can be derived for different
regions of the parameters (t, n)4
References
1. R.Benini, G. Montani, Phys. Rev. D 70, 103527 (2004)
2. A.A. Kirillov, Zh. Eksp. Theor. Fiz. 103, 721-729 (1993)
3. R.Puzio, Class. Quantum Grav. 11, 609-620 (1994)
4. R.Benini, G. Montani, Class. Quantum Grav. in pubblication