INHOMOGENEOUS QUANTUM COSMOLOGY: THE GOWDY T MODELrelativity.phys.lsu.edu/ilqgs/mena103106.pdf · Guillermo A. Mena Marugán. CSIC. (P3) CMB analysis calls for the development of

Guillermo A. Mena Marugán. IEM, CSIC (Madrid)

ILQGS. 31 October 2006

INHOMOGENEOUS QUANTUM COSMOLOGY:

THE GOWDY T3 MODEL

In collaboration with Alejandro Corichi, Jerónimo Cortez,and Zé Velhinho.

Outline

Guillermo A. Mena Marugán. CSIC. (P2)

Motivation

The model

Standard quantization and unitarity problems

New field parametrization and unitary quantization

Uniqueness of the quantization

Remarks and discussion

Motivation


CMB analysis calls for the development of inhomogeneous cosmology. Reliable perturbative quantum formalisms ought to be derived from inhomogeneous quantum cosmology. The extension of LQC to inhomogeneous spacetimes would validate present results and provide new predictions. Inhomogeneous quantum cosmological models would serve for comparison of techniques and interpretations. The linearly polarized Gowdy T3 model is the simplest of all inhomogeneous cosmological systems. It can be treated as 2+1 gravity coupled to a free massless scalar fieldwith axial symmetry, an interesting system from the viewpoint of QFT in curved backgrounds and parametrized field theory. The model is non-stationary and infinite dimensional. Issues of unitarity and uniqueness of the quantization are relevant.

It descibes vacuum spacetimes with the spatial topology of a three-torus and two commuting, axial, and hypersurface orthogonal Killing vector fields.

Classical solutions are inhomogenuous spacetimes with a cosmological singularity.

After gauge fixing:

t >0 is a positive time and are Killing vector fields.

and .

Q and P=ln p are canonically conjugate.

Deparametrization has been achieved by setting

There is some arbitrariness in the choice of time and of the field

The Gowdy T3 model


ds2=ee−/ p −dt 2d 2e−/ p t 2 p2 d 2e/ p d 2 .

, ,∈S 1 . ∂a={∂ ,∂}=Q , p , , Pp=−∮ P /20

det [∂ai hij ∂b

j ]=t2 p2 .

.

The reduced model


There is a homogeneous constraint left:

This constraint affects only the field sector .

The reduced action is (with 4G= )

Q and P are constants of motion.

The phase space can be decomposed as , corresponding to a “point particle” and the fieldlike degrees of freedom. We focus on .

The field is subject to the wave equation

S r=∫t i

t f dt P Q[∮P ]−H r , H r=12t∮ [P

2t2 ' 2] .

r=0

, P

C0=12∮P ' .

t−' '=0 .

The reduced model (standard description)


This is the Klein-Gordon equation of an axially symmetric free massless scalar field propagating in the fictitious 2+1 background :

Smooth solutions have the form

One obtains the covariant phase space , with symplectic structure

The basis of solutions is “orthonormal” in the product

M≃ℝ×T 2 , g Bg abB=−dt a dtbd a d bt2 d a d b .

f 0t =1−i ln t4

, f nt =H 0∣n∣t 8

n≠0.

f n t ,= f nt ei n ∀ n ,t ,=∑n=−∞

∞

[An f nt ,An∗ f n

∗t ,] .

S

S 1 ,2=∮ [2 t ∂t1−1 t ∂t2 ] .

f m , f n=−i S f m∗ , f n .

{ f n , f n∗}

Given any Cauchy surface (a constant time section t=t0), there is a natural isomorphism [such that ].

The evolution between Cauchy surfaces is given by

Let us consider a given and define

with Then In this sense are constants of motion. Note the explicit dependence of on

A complex structure is a real linear map on phase space such that It is -compatible if it preserves and is positive definite.

We choose

The reduced model (standard description)


t0

J 0 [Ant 0]=i An

t0 , J 0 [An∗t 0]=−i An

∗t0 .

I t0: S =t 0, , P =t0∂tt0,

V t1, t0:= I t1

−1 I t0: .

Ant 1[ , P]=∮ ie−i n

8 [∣n∣t1 H 1∗∣n∣t1H 0

∗∣n∣t1P] ,t0

Ant 0[ , P]=An=V t1, to An

t1[ , P] .Ant 1An

t 0 t1 .An∗t1=[An

t 1]∗.

J J 2=−1. J . , .

Standard quantization


provides the projector on the positive frequency sector.

One can then construct the “one-particle” Hilbert space , the symmetric Fock space , and the physical Hilbert space (by imposing the homogeneous constraint ).

Let be a symplectic transformation [e.g. ] and .

is unitarily implementable on is Hilbert-Schmidt is Hilbert-Schmidt and provide unitarilyequivalent quantizations is Hilbert-Schmidt.

The antilinear part of the Bogoliubov transformation that implements theevolution

has a single contribution for each mode (in terms

of ), that can be deduced employing the “orthonormalization” of the set in the product .{ f n , f n

∗}

n≠0

C0

1−i J 0

2

F H 0H 0

V V t1, t0 J V=V J 0 V −1

V F H 0⇔ J V− J 0

J V− J 0 J V J 0−1 ⇔ J V

J 0

⇔

⇔ V J 0 V J 0

V t1, t0Ant 0

f m , f n=−i S f m∗ , f n .

Standard quantization (unitarity problems)


The Bogoliubov coefficients for the evolution are

From Hankel's asymptotic expansions, it follows that for large n,

and hence the Bogoliubov coefficients are not square summable.

In fact, the unitarity problems persist on the physical Hilbert space (Torre).

The problems can be traced back to the appearance of a factor t in the symplectic structure

n t1 , t 0=i 2[ f n∗t 0 t1∂t f n

∗t1− f n∗t1 t0∂t f n

∗t 0]=

i∣n∣4 [H 0

∗∣n∣t1t0 H 1∗∣n∣t0−H 0

∗∣n∣t0 t1 H 1∗∣n∣t1] .

∣nt 1 , t0∣2≈t1−t 0

2

4t1 t0

V t1, t0

S 1 ,2=∮ [2 t ∂t1−1 t ∂t2 ] .

New field parametrization (preliminaries)

Guillermo A. Mena Marugán. CISC. (P10)

It it is worth noticing that, for large n, our basis of solutions behaves as

The corresponding solutions for behave like the standard modesof a free massless scalar field propagating in a “Minkowski” background in 1+1 (or 2+1) dimensions.Besides, the scaling of the field will absorb the problematic factor of t in the symplectic structure.

Therefore we introduce the TIME DEPENDENT canonical transformation

A generating functional for this transformation is, e.g.,

f n t ,=H 0 ∣n∣t 8

ei n≈ ei/4

4∣n∣te−i ∣n∣t−n.

=t

=t , P=1t

P .

F=−∮ P/ t .

New field parametrization (preliminaries)


If then After the time dependent

transformation, the new Hamiltonian is

We can expand and in Fourier series:

The Fourier coefficients are canonical pairs.

We introduce “annihilation” (and “creation”) variables for [corresponding to a massless field]:

However, the corresponding vacuum (characterized by ) does not belong to the domain of the Hamiltonian .

H r=H r∂tF=1

2∮ [P2' 2 P

t ]. P

=∑n=−∞

∞n

ei n

2, P=∑n=−∞

∞Pn ei n

2.

n , P−n

an=∣n∣niP

n

2∣n∣.

n≠0

H r

={ , H r}, ={ , H r}− t ddt 1t .

an∣0 ⟩=0

New field parametrization


The problem arises because of the term in the Hamiltonian.We eliminate it by modifying the canonical transformation:

The reduced action and Hamiltonian become

This Hamiltonian is in fact that of a scalar field propagating in a flat (two-dimensional) background, though subject to a time dependent potential that

tends to zero at large times. The field equations are

There is one constraint, which generates S1-translations:

∮ P/2t

=t , P=1t P

2 .

S r=∫t i

t f dt P Q∮ [P ]−H r , H r

=12∮ [P

2' 2 2

4t 2 ].

C0=12∮P

' ,

T : ∀∈S1 .

−' '

4t2=0.

New field parametrization: description


We expand in Fourier series, and introduce “annihilation” and “creation” variables for [corresponding to the “massless case”] :

We use these coordinates for the non-zero mode sector of phase space:

In terms of them, the S1-translations have a diagonal action:

Besides, the symplectic structure gets decomposed in 4x4 blocks:

n≠0

{Bm}; Bm=bm , b−m∗ , b−m , bm

∗ T ∀m∈ℕ .

bn=∣n∣niP

n

2∣n∣, b−n

∗ =∣n∣n−iP

n

2∣n∣.

, P

{Bm1}, {Bm

2}=∑mBm

1Tm Bm2 , m= 0 m

m 0 , m=0 −ii 0 .

T b±m=e±i mbm , T b±m∗ =e−±i mb±m

∗ .

New field parametrization: dynamics


For ANY SYSTEM whose classical evolution commutes with S1-transla-tions and with the -reversal transformation the

evolution between constant-t sections is block diagonal and has the form:

is a Bogoliubov transformation, with

For the evolution generated by in our case, and defining :

Bmt1=U mt1, t0Bmt0 ,

xmi =m t i

U m t1, t 0=um t1, t 0 00 um t1, t 0 , umt1, t0=mt1, t 0 m t1, t 0

m∗ t1, t 0 m

∗ t1, t 0.

c xmi = xm

i

8 [1 i2xm

i H 0 xmi−iH 1 xm

i] , d xmi = xm

i

2H 0∗ xm

i −c∗ xmi.

H r

U m t1, t0 ∣mt1, t0∣2−∣mt1, t0∣

2=1.

m t1, t0=c xm1c∗ xm

0−d xm1d∗ xm

0 , mt1, t0=d xm1c xm

0−c xm1d xm

0.

bm↔b−m , bm∗↔b−m

∗ ,

New field parametrization: quantization


We introduce the complex structure with

We call the one-particle Hilbert space determined by , thesymmetric Fock space, and the vacuum (which satisfies ).

is invariant under the S1-translations . One obtains an invariant unitary implementation of this gauge group (with ).

Furthermore, one can check that

Hence, the dynamical evolution is unitarily implemented on (and on the physical Hilbert space).

T

T ∣0 ⟩=∣0 ⟩ , T b±m

T −1=e±i m b±m , T

b±m∗ T

−1=e−±i m b±m∗

∑m∣mt1, t0∣

2∞ ∀ t1, t0≠0.

H 0

J 0 Bm= J 0mBm ,

J 0m=diag i ,−i , i ,−i .

J 0 F H 0∣0 ⟩

J 0

F H 0

b±m∣0 ⟩=0



Is this the only possible Fock quantization?We consider compatible complex structures that are invariant under the

group of S1-translations.

➢ Any such complex structure can be obtained as where is a symplectic transformation with the block diagonal form in the basis

➢ The evolution is unitarily implementable w.r.t. iff so is w.r.t. We get the blocks

J=K J J 0 K J−1 K J

{Bm}

K J m=k J m 00 k J m , k J m=m m

m∗ m

∗ , m0, ∣m∣2−∣m∣

2=1.

U t1, t0 JU J t1, t0:=K J

−1 U t1, t0K J J 0 .

U mJ t1, t0=um

J t1, t0 00 um

J t1, t0 , umJ t1, t0= m

J t1, t0 mJ t 1, t0

mJ∗t1, t0 m

J∗t1, t0 ,m

J t1, t0=∣m∣2mt1, t 0−∣m∣

2m∗ t1, t0m

∗m∗ mt1, t 0−mmm

∗ t1, t0 ,m

J t1, t0=2 iℑ[mt1, t0] m∗ mm

∗ 2m t1, t0−m2 m

∗ t1, t0.



From the expression of it is easy to see that,

The infimum of the coefficients of might be non-positive. For a free massless field, The idea is to average over on a subset of

✔ We assume that the evolution is unitary not only w.r.t. but also w.r.t. . ✔ We admit that and are measurable functions of on This is the case for the Gowdy model.

Then (using Egorov's theorem), one can show that, with Lebesgue measure and such that the integralover of the r.h.s. of the above inequality is bounded by

mJ t1, t0

∑m=1

N 2 a∣m∣2

1a {1−ℜ[m t1, t 0]2−a∣mt1, t0∣

2}≤∑m=1

N 2∣m t1, t 0∣2∣m

J t 1, t0∣2 .

∀ a≥0, N∈ℕ ,

∣m∣2

mt1, t0=ei mt1−t 0 , mt1, t 0=0.=t1−t0 [0,] .

JJ 0

t0 , t0 t0 , t 0 [0,] .

∀0, ∃E⊂[0,]E− ∃ I 0

E I ∀ N∈ℕ .



✔ We assume that, for (a given value of) such that, with that satisfies

This condition is fulfilled in the Gowdy model (and in the free massless case).

Integrating our inequality over the set (provided by Egorov's theorem),and taking , we then get

Hence, the symplectic transformation is implemented unitarily on

In conclusion, all compatible complex structures that are invariant under S1-translations and allow a unitary implementation of the dynamics

determine unitarily equivalent Fock representations.

t 0 , ∃∈0, ∀ E⊂[0,]E − , ∃E 0

∫Ed {1−ℜ[mt0 , t0]

2}E ∀m∈ℕ .

E0a2E/ I

∑m=1

N∣m∣

2≤1a I

a [2E−a I ]∞ ∀ N∈ℕ .

K J F H 0.

Remarks


The proof of uniqueness is valid for more general field models. This is the case, e.g., when the dynamics coincides with the free massless one in the entire future of a Cauchy surface. Uniqueness holds if one replaces the condition of unitary implementation of the dynamics by the requirement that the Fock vacuum belong to the domain of the Hamiltonian.

Apparently, compactness of the spatial sections plays an important role in the proof. In contrast, the spatial dimension does not seem to be so relevant.

Let be the natural complex structure corresponding to a free field with mass equal to the instantaneous value With compact topology, and lead to unitarily equivalent Fock representations.

However, the complex structures that induces by time evolution differ from

J 0

J M t0 M t0=1/4t02 .

J M t0

J M t0

J M t .

Remarks


Torre has recently shown that, in the Schrödinger representation of the standard field description of the Gowdy T3 model, Gaussian measures at different times are mutually singular. This problem disappears with the new field parametrization.

The new field parametrization can be extended to other models. In the Gowdy T3 model with general polarization, the reduced system consists of two coupled scalar fields; one of them propagates again in a flat (two-dimensional) background with a time dependent mass

In the case of the Einstein-Rosen waves (linearly polarized cylindrical waves) a similar scaling seems to cure the unitarity problems detected in parametrized field theory by Cho and Varadarajan. This scaling was already considered by Niedermaier, who defined the fundamental field on AdS2.

M t0=1/4t02 .

= r

Conclusion


We have introduced a new field parametrization for the Gowdy metric and completed it into a time dependent canonical transformation. In this way, we have redistributed the time dependence into an explicit and an implicit part, the latter being unitarily implementable in the quantum theory. There exists certain freedom in the choice of deparametrization (time gauge) and field parametrization for the model. Once these choices are made, the Fock representation is essentially unique if one demands unitary implementation of the dynamics and invariance under S1-translations.

The Fock vacuum is not left invariant by the time evolution. As a consequence, there is some “particle” production, that attenuates for asymptotically large times. However, these “particles” have no genuinely physical interpretation. It would be interesting to construct geometric operators, define a proper time and study the singularity quantum mechanically, comparing the results with those of other possible approaches to the quantization.

Documents

INHOMOGENEOUS QUANTUM COSMOLOGY: THE GOWDY T MODELrelativity.phys.lsu.edu/ilqgs/mena103106.pdf · Guillermo A. Mena Marugán. CSIC. (P3) CMB analysis calls for the development of