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Reduced phase space quantization of FRWuniverseTo cite this article Fumitoshi Amemiya and Tatsuhiko Koike 2011 J Phys Conf Ser 314 012053

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Reduced phase space quantization of FRW universe

Fumitoshi Amemiya and Tatsuhiko KoikeDepartment of Physics Keio University 3-14-1 Hiyoshi Kohoku-ku 223-8522 YokohamaJapan

E-mail famemiyarkphyskeioacjp koikephyskeioacjp

Abstract A gauge-invariant quantum theory of the flat Friedmann-Robertson-Walker (FRW)universe with dust is studied in terms of the Ashtekar variables We use the reduced phase spacequantization which has following advantages (i) fundamental variables are all gauge invariant(ii) there exists a physical time evolution of gauge-invariant quantities so that the problemof time is absent and (iii) the reduced phase space can be quantized in the same manner asin ordinary quantum mechanics Analyzing the dynamics of a wave packet we show that theclassical initial singularity is replaced by a big bounce in quantum theory

1 IntroductionOne of the motivations of quantum cosmology is to shed light on quantum nature of the initialsingularity However there exists potential problems that have not been completely resolvedyet A problem is about what should be interpreted as observables in classical and quantumgravity [1 2] A canonical formulation of general relativity (GR) is a constrained systemwith first-class constraints in which the spacetime diffeomorphisms are interpreted as gaugetransformations In gauge theories only gauge-invariant quantities are observables Howeverthere are technical and conceptual difficulties in the realization of the idea especially in GRIn many works gauge-variant quantities are used as observables This issue must be seriouslyconsidered especially in quantum gravity because it is substantially related to the problem oftime [3]

In this paper we shall construct and analyze a gauge-invariant quantum theory of the flatFRW universe with the Brown-Kuchar dust [4] in terms of the Ashtekar variables [5 6] Weuse the reduced phase space quantization method where the so-called relational formalism [7 8]is used to construct the classical reduced phase space spanned by gauge-invariant quantitiesand then the system is quantized in the same manner as in ordinary quantum mechanics Thequantization gives a possible resolution to the problem of time As for the dynamics of theuniverse we consider the motion of a wave packet and evaluate the expectation value of thescale factor It is shown that the expectation value has a non-zero minimum that is the initialsingularity is replaced by a big bounce in quantum theory The remarkable point is that thebig bounce mixes the states representing right-handed and left-handed systems See [9 10] fordetails of the work In this paper we adopt the unit in which c = 1

2 Reduced phase space of Friedmann-Robertson-Walker universe with dustIn the Ashtekar formulation [5 6] the variables (Ai

a Eai ) form a canonically conjugate pair where

Aia is a SU(2) connection and Ea

i is an orthonormal triad with density weight 1 In the flat FRW

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

Published under licence by IOP Publishing Ltd 1

model the Ashtekar variables can be written in terms of only one independent components cand p [11]

Aia = c(t)ωi

a Eai = p(t)Xa

i (1)

where ωi are bases of left invariant one-forms and Xi are invariant vector fields dual to theone-forms These variables have relations to the scale factor a such that

|p| = a2 c = sgn(p)γ

Na (2)

where γ is the so-called Barbero-Immirzi parameter N is the lapse function and the dot denotesthe derivative with respect to t Note that while the scale factor is restricted to be nonnegativep ranges over the entire real line carrying an orientation of triads determined by the sign ofp We here consider a compact universe to avoide the divergence of the three-space integraland in particular we only consider the case of three-dimensional torus where we take a cube ofcoordinate range 0 le x y z le V

13 and identify the opposite faces

If we define new variables as p = V23 p and c = V

13 c the total action for gravity plus the

Brown-Kuchar dust [4] is written as

Stot =int

dt

[3κγ

pc + PT T minus NHtot

] (3)

where κ = 8πG T is the proper time measured along the particle flow lines when the equationsof motion hold PT is its conjugate momentum and the Hamiltonian constraint takes the form

Htot = Hgrav + Hdust = minus 3κγ2

c2radic

|p| + PT = 0 (4)

The key observation of the relational formalism [7 8] to define gauge-invariant quantities isas follows Take two gauge-variant functions F and T on the phase space and choose one ofthe functions T as a clock Then the value of F at T = τ is gauge-invariant even if F and Tthemselves are gauge variant Suppose a phase space has a 2n-dimension (n ge 2) and there arecanonical coordinates (qa pa a = 1 middot middot middot n) such that qa pb = δa

b We will denote a first-classconstraint by H and a phase space point by y = (qa pa) Under the gauge transformationgenerated by H a point y is mapped to y 7rarr αt

H(y) where t is a gauge parameter That isαt

H(y) is a gauge flow generated by H starting from y Then we can define the gauge-invariantquantity Oτ

F (y) as

OτF (y) = F (αt

H(y))|T (αtH(y))=τ (5)

A constraint equation H = 0 is said to be of deparametrized form if it is written asH(qa T pa PT ) = PT + h(qa pa) = 0 with some phase space coordinates qa T pa PT In the deparametrized theories the reduced phase space is spanned by the gauge-invariantquantities

(Oτ

qa(y) Oτpa

(y))

associated with qa and pa with the simple symplectic structureOτ

qa(y) Oτpb

(y)

= δab The physical Hamiltonian Hphys is obtained by replacing qa and

pa in h(qa pa) with Oτqa(y) and Oτ

pa(y) Hphys

(Oτ

qa(y) Oτpa

(y))

= h(Oτ

qa(y) Oτpa

(y)) The

Hamiltonian generates the time evolution of the gauge-invariant quantity associated with afunction F which depends only on qa and pa

partOτF (y)partτ = Hphys O

τF (y)

In the present case it is natural to choose the function T as the clock variable Thenthe reduced phase space is coordinatized by the gauge-invariant quantities C(τ) = Oτ

c (y) andP (τ) = Oτ

p(y) associated with c and p with very simple symplectic structure

C(τ) P (τ) =κγ

3 (6)

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

2

Moreover we can obtain the physical Hamiltonian Hphys by replacing c and p in Hgrav(c p) withC and P

Hphys = minus 3κγ2

C(τ)2radic

|P (τ)| (7)

3 QuantizationIn this section we shall quantize the system on the reduced phase space obtained in the previoussection Now the physical variables are operators and the Poisson bracket bull bull is replaced withthe commutation relation (1i~)[bull bull] Thus (6) becomes the canonical commutation relation

[C P ] =iκγ~

3 (8)

Let us choose the ordinary Schrodinger representation in which the operators P and Crespectively act on a wave function Ψ(P ) in the following way PΨ(P ) = PΨ(P ) CΨ(P ) =i~κγ

3partΨ(P )

partP As a concrete example we choose the following operator ordering for theHamiltonian

Hphys = minus 3κγ2

radic|P |C2 (9)

Then the Schrodinger equation takes the simplest form

i~partΨpartτ

=κ~2

3

radic|P |part

2ΨpartP 2

(10)

Since the present Hamiltonian is different from the ordinary kinematical term we choose theHilbert space as H = L2(R |P |minus

12 dP ) in order to make the Hamiltonian (9) Hermitian up to

surface term Hphys is somewhat singular at the origin P = 0 it is indeed self-adjoint in H

4 Dynamics of the universeLet us now analyze the dynamics of a wave packet The procedure is as follows First we preparean initial wave packet Ψ(P 0) at some nonzero P Then we numerically evolve it backward intime by the Schrodinger equation (10) and evaluate the expectation value of |P | as a function ofthe internal time τ Here we consider |P | because both the positive and negative P correspondto the universe of the same size with different orientation of triads For simplicity we herechoose the initial wave function as a Gaussian wave packet

Ψ(P 0) = C0 exp(minus(P minus P0)2

2σ2minus ik0P

) (11)

where C0 is the normalization constant Figs 1 show the absolute value of the wave function asa function of P and τ and the expectation value of |P | is plotted as a function of the time τ

We can see from Fig 1 (a) that a part of the wave packet is reflected and the rest istransmitted at the origin We here remind that the sign of P determines an orientation oftriads which correspond to a right-handed and left-handed systems respectively Thus theresult indicates that if the present state of the universe is in a right-handed system the paststate is in superposition of the states of a right-handed and left-handed systems As for theexpectation value of |P | Fig 1 (b) indicates that the expectation value never goes to zero andbounces at a nonzero minimum That is the initial singularity is replaced by a big bounce inthe present model

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

3

0

05

1

15

-014

-012

-01

-008

-006

-004

-002

0

-10

-5

0

5

10

Ψ

τ [t ]Pl

P [(38π) l ]Pl223

(a)

0

05

1

15

2

25

3

35

4

45

5

55

-014 -012 -01 -008 -006 -004 -002 0

|P| [(

38

π)

l

]P

l2

23

τ [t ]Pl

lt gt(b)

Figure 1 Fig(a) shows the absolute value of the wave function as a function of τ and P Fig(b) shows the expectation values of |P | as a functions of τ

5 ConclusionsA gauge-invariant quantum theory of the flat FRW universe with dust has been studied interms of the Ashtekar variables We have first constructed the classical reduced phase spaceof the system by using the relational formalism and then have quantized the reduced systemThe advantages of the quantization method are as follows (i) fundamental variables are gauge-invariant quantities (ii) a natural time evolution of the gauge-invariant quantities exists sothat the problem of time is absent and (iii) the reduced phase space can be quantized in thesame manner as in ordinary quantum mechanics because there are no constraints in the reducedphase space In the obtained quantum theory we have analyzed the dynamics of a wave packetand have shown that the expectation value of P has a non-zero minimum that is the initialsingularity is replaced by a big bounce in quantum theory The interpretation of the wave packetis that if the present state of the universe is in a right-handed system the past state has beenin a superposition of the states of a right-handed and left-handed systems

AchnowledgementThis work was supported in part by Global COE Program ldquoHigh-Level Grobal Cooperation forLeading-Edge Platform on Access Spaces (C12)rdquo

References[1] P G Bergmann Rev Mod Phys 33 510 (1961)[2] C Rovelli Class Quant Grav 8 1895 (1991) Phys Rev D 65 124013 (2002)[3] For a comprehensive review on the problem of time see eg C J Isham arXivgr-qc9210011[4] J D Brown and K V Kuchar Phys Rev D 51 5600 (1995)[5] A Ashtekar Phys Rev Lett 57 2244 (1986) Phys Rev D 36 1587 (1987)[6] J F Barbero G Phys Rev D 51 5507 (1995)[7] B Dittrich Gen Rel Grav 39 1891 (2007)[8] T Thiemann Class Quant Grav 23 1163 (2006)[9] F Amemiya and T Koike Phys Rev D 80 103507 (2009)

[10] F Amemiya and T Koike Phys Rev D 82 104007 (2010)[11] A Ashtekar M Bojowald and J Lewandowski Adv Theor Math Phys 7 233 (2003)

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

4

model the Ashtekar variables can be written in terms of only one independent components cand p [11]

Aia = c(t)ωi

a Eai = p(t)Xa

i (1)

where ωi are bases of left invariant one-forms and Xi are invariant vector fields dual to theone-forms These variables have relations to the scale factor a such that

|p| = a2 c = sgn(p)γ

Na (2)

where γ is the so-called Barbero-Immirzi parameter N is the lapse function and the dot denotesthe derivative with respect to t Note that while the scale factor is restricted to be nonnegativep ranges over the entire real line carrying an orientation of triads determined by the sign ofp We here consider a compact universe to avoide the divergence of the three-space integraland in particular we only consider the case of three-dimensional torus where we take a cube ofcoordinate range 0 le x y z le V

13 and identify the opposite faces

If we define new variables as p = V23 p and c = V

13 c the total action for gravity plus the

Brown-Kuchar dust [4] is written as

Stot =int

dt

[3κγ

pc + PT T minus NHtot

] (3)

where κ = 8πG T is the proper time measured along the particle flow lines when the equationsof motion hold PT is its conjugate momentum and the Hamiltonian constraint takes the form

Htot = Hgrav + Hdust = minus 3κγ2

c2radic

|p| + PT = 0 (4)

The key observation of the relational formalism [7 8] to define gauge-invariant quantities isas follows Take two gauge-variant functions F and T on the phase space and choose one ofthe functions T as a clock Then the value of F at T = τ is gauge-invariant even if F and Tthemselves are gauge variant Suppose a phase space has a 2n-dimension (n ge 2) and there arecanonical coordinates (qa pa a = 1 middot middot middot n) such that qa pb = δa

b We will denote a first-classconstraint by H and a phase space point by y = (qa pa) Under the gauge transformationgenerated by H a point y is mapped to y 7rarr αt

H(y) where t is a gauge parameter That isαt

H(y) is a gauge flow generated by H starting from y Then we can define the gauge-invariantquantity Oτ

F (y) as

OτF (y) = F (αt

H(y))|T (αtH(y))=τ (5)

A constraint equation H = 0 is said to be of deparametrized form if it is written asH(qa T pa PT ) = PT + h(qa pa) = 0 with some phase space coordinates qa T pa PT In the deparametrized theories the reduced phase space is spanned by the gauge-invariantquantities

(Oτ

qa(y) Oτpa

(y))

associated with qa and pa with the simple symplectic structureOτ

qa(y) Oτpb

(y)

= δab The physical Hamiltonian Hphys is obtained by replacing qa and

pa in h(qa pa) with Oτqa(y) and Oτ

pa(y) Hphys

(Oτ

qa(y) Oτpa

(y))

= h(Oτ

qa(y) Oτpa

(y)) The

Hamiltonian generates the time evolution of the gauge-invariant quantity associated with afunction F which depends only on qa and pa

partOτF (y)partτ = Hphys O

τF (y)

In the present case it is natural to choose the function T as the clock variable Thenthe reduced phase space is coordinatized by the gauge-invariant quantities C(τ) = Oτ

c (y) andP (τ) = Oτ

p(y) associated with c and p with very simple symplectic structure

C(τ) P (τ) =κγ

3 (6)

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

2

Moreover we can obtain the physical Hamiltonian Hphys by replacing c and p in Hgrav(c p) withC and P

Hphys = minus 3κγ2

C(τ)2radic

|P (τ)| (7)

3 QuantizationIn this section we shall quantize the system on the reduced phase space obtained in the previoussection Now the physical variables are operators and the Poisson bracket bull bull is replaced withthe commutation relation (1i~)[bull bull] Thus (6) becomes the canonical commutation relation

[C P ] =iκγ~

3 (8)

Let us choose the ordinary Schrodinger representation in which the operators P and Crespectively act on a wave function Ψ(P ) in the following way PΨ(P ) = PΨ(P ) CΨ(P ) =i~κγ

3partΨ(P )

partP As a concrete example we choose the following operator ordering for theHamiltonian

Hphys = minus 3κγ2

radic|P |C2 (9)

Then the Schrodinger equation takes the simplest form

i~partΨpartτ

=κ~2

3

radic|P |part

2ΨpartP 2

(10)

Since the present Hamiltonian is different from the ordinary kinematical term we choose theHilbert space as H = L2(R |P |minus

12 dP ) in order to make the Hamiltonian (9) Hermitian up to

surface term Hphys is somewhat singular at the origin P = 0 it is indeed self-adjoint in H

4 Dynamics of the universeLet us now analyze the dynamics of a wave packet The procedure is as follows First we preparean initial wave packet Ψ(P 0) at some nonzero P Then we numerically evolve it backward intime by the Schrodinger equation (10) and evaluate the expectation value of |P | as a function ofthe internal time τ Here we consider |P | because both the positive and negative P correspondto the universe of the same size with different orientation of triads For simplicity we herechoose the initial wave function as a Gaussian wave packet

Ψ(P 0) = C0 exp(minus(P minus P0)2

2σ2minus ik0P

) (11)

where C0 is the normalization constant Figs 1 show the absolute value of the wave function asa function of P and τ and the expectation value of |P | is plotted as a function of the time τ

We can see from Fig 1 (a) that a part of the wave packet is reflected and the rest istransmitted at the origin We here remind that the sign of P determines an orientation oftriads which correspond to a right-handed and left-handed systems respectively Thus theresult indicates that if the present state of the universe is in a right-handed system the paststate is in superposition of the states of a right-handed and left-handed systems As for theexpectation value of |P | Fig 1 (b) indicates that the expectation value never goes to zero andbounces at a nonzero minimum That is the initial singularity is replaced by a big bounce inthe present model

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

3

0

05

1

15

-014

-012

-01

-008

-006

-004

-002

0

-10

-5

0

5

10

Ψ

τ [t ]Pl

P [(38π) l ]Pl223

(a)

0

05

1

15

2

25

3

35

4

45

5

55

-014 -012 -01 -008 -006 -004 -002 0

|P| [(

38

π)

l

]P

l2

23

τ [t ]Pl

lt gt(b)

Figure 1 Fig(a) shows the absolute value of the wave function as a function of τ and P Fig(b) shows the expectation values of |P | as a functions of τ

5 ConclusionsA gauge-invariant quantum theory of the flat FRW universe with dust has been studied interms of the Ashtekar variables We have first constructed the classical reduced phase spaceof the system by using the relational formalism and then have quantized the reduced systemThe advantages of the quantization method are as follows (i) fundamental variables are gauge-invariant quantities (ii) a natural time evolution of the gauge-invariant quantities exists sothat the problem of time is absent and (iii) the reduced phase space can be quantized in thesame manner as in ordinary quantum mechanics because there are no constraints in the reducedphase space In the obtained quantum theory we have analyzed the dynamics of a wave packetand have shown that the expectation value of P has a non-zero minimum that is the initialsingularity is replaced by a big bounce in quantum theory The interpretation of the wave packetis that if the present state of the universe is in a right-handed system the past state has beenin a superposition of the states of a right-handed and left-handed systems

AchnowledgementThis work was supported in part by Global COE Program ldquoHigh-Level Grobal Cooperation forLeading-Edge Platform on Access Spaces (C12)rdquo

References[1] P G Bergmann Rev Mod Phys 33 510 (1961)[2] C Rovelli Class Quant Grav 8 1895 (1991) Phys Rev D 65 124013 (2002)[3] For a comprehensive review on the problem of time see eg C J Isham arXivgr-qc9210011[4] J D Brown and K V Kuchar Phys Rev D 51 5600 (1995)[5] A Ashtekar Phys Rev Lett 57 2244 (1986) Phys Rev D 36 1587 (1987)[6] J F Barbero G Phys Rev D 51 5507 (1995)[7] B Dittrich Gen Rel Grav 39 1891 (2007)[8] T Thiemann Class Quant Grav 23 1163 (2006)[9] F Amemiya and T Koike Phys Rev D 80 103507 (2009)

[10] F Amemiya and T Koike Phys Rev D 82 104007 (2010)[11] A Ashtekar M Bojowald and J Lewandowski Adv Theor Math Phys 7 233 (2003)

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

4

Moreover we can obtain the physical Hamiltonian Hphys by replacing c and p in Hgrav(c p) withC and P

Hphys = minus 3κγ2

C(τ)2radic

|P (τ)| (7)

3 QuantizationIn this section we shall quantize the system on the reduced phase space obtained in the previoussection Now the physical variables are operators and the Poisson bracket bull bull is replaced withthe commutation relation (1i~)[bull bull] Thus (6) becomes the canonical commutation relation

[C P ] =iκγ~

3 (8)

Let us choose the ordinary Schrodinger representation in which the operators P and Crespectively act on a wave function Ψ(P ) in the following way PΨ(P ) = PΨ(P ) CΨ(P ) =i~κγ

3partΨ(P )

partP As a concrete example we choose the following operator ordering for theHamiltonian

Hphys = minus 3κγ2

radic|P |C2 (9)

Then the Schrodinger equation takes the simplest form

i~partΨpartτ

=κ~2

3

radic|P |part

2ΨpartP 2

(10)

Since the present Hamiltonian is different from the ordinary kinematical term we choose theHilbert space as H = L2(R |P |minus

12 dP ) in order to make the Hamiltonian (9) Hermitian up to

surface term Hphys is somewhat singular at the origin P = 0 it is indeed self-adjoint in H

4 Dynamics of the universeLet us now analyze the dynamics of a wave packet The procedure is as follows First we preparean initial wave packet Ψ(P 0) at some nonzero P Then we numerically evolve it backward intime by the Schrodinger equation (10) and evaluate the expectation value of |P | as a function ofthe internal time τ Here we consider |P | because both the positive and negative P correspondto the universe of the same size with different orientation of triads For simplicity we herechoose the initial wave function as a Gaussian wave packet

Ψ(P 0) = C0 exp(minus(P minus P0)2

2σ2minus ik0P

) (11)

where C0 is the normalization constant Figs 1 show the absolute value of the wave function asa function of P and τ and the expectation value of |P | is plotted as a function of the time τ

We can see from Fig 1 (a) that a part of the wave packet is reflected and the rest istransmitted at the origin We here remind that the sign of P determines an orientation oftriads which correspond to a right-handed and left-handed systems respectively Thus theresult indicates that if the present state of the universe is in a right-handed system the paststate is in superposition of the states of a right-handed and left-handed systems As for theexpectation value of |P | Fig 1 (b) indicates that the expectation value never goes to zero andbounces at a nonzero minimum That is the initial singularity is replaced by a big bounce inthe present model

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

3

0

05

1

15

-014

-012

-01

-008

-006

-004

-002

0

-10

-5

0

5

10

Ψ

τ [t ]Pl

P [(38π) l ]Pl223

(a)

0

05

1

15

2

25

3

35

4

45

5

55

-014 -012 -01 -008 -006 -004 -002 0

|P| [(

38

π)

l

]P

l2

23

τ [t ]Pl

lt gt(b)

Figure 1 Fig(a) shows the absolute value of the wave function as a function of τ and P Fig(b) shows the expectation values of |P | as a functions of τ

5 ConclusionsA gauge-invariant quantum theory of the flat FRW universe with dust has been studied interms of the Ashtekar variables We have first constructed the classical reduced phase spaceof the system by using the relational formalism and then have quantized the reduced systemThe advantages of the quantization method are as follows (i) fundamental variables are gauge-invariant quantities (ii) a natural time evolution of the gauge-invariant quantities exists sothat the problem of time is absent and (iii) the reduced phase space can be quantized in thesame manner as in ordinary quantum mechanics because there are no constraints in the reducedphase space In the obtained quantum theory we have analyzed the dynamics of a wave packetand have shown that the expectation value of P has a non-zero minimum that is the initialsingularity is replaced by a big bounce in quantum theory The interpretation of the wave packetis that if the present state of the universe is in a right-handed system the past state has beenin a superposition of the states of a right-handed and left-handed systems

AchnowledgementThis work was supported in part by Global COE Program ldquoHigh-Level Grobal Cooperation forLeading-Edge Platform on Access Spaces (C12)rdquo

References[1] P G Bergmann Rev Mod Phys 33 510 (1961)[2] C Rovelli Class Quant Grav 8 1895 (1991) Phys Rev D 65 124013 (2002)[3] For a comprehensive review on the problem of time see eg C J Isham arXivgr-qc9210011[4] J D Brown and K V Kuchar Phys Rev D 51 5600 (1995)[5] A Ashtekar Phys Rev Lett 57 2244 (1986) Phys Rev D 36 1587 (1987)[6] J F Barbero G Phys Rev D 51 5507 (1995)[7] B Dittrich Gen Rel Grav 39 1891 (2007)[8] T Thiemann Class Quant Grav 23 1163 (2006)[9] F Amemiya and T Koike Phys Rev D 80 103507 (2009)

[10] F Amemiya and T Koike Phys Rev D 82 104007 (2010)[11] A Ashtekar M Bojowald and J Lewandowski Adv Theor Math Phys 7 233 (2003)

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

4

0

05

1

15

-014

-012

-01

-008

-006

-004

-002

0

-10

-5

0

5

10

Ψ

τ [t ]Pl

P [(38π) l ]Pl223

(a)

0

05

1

15

2

25

3

35

4

45

5

55

-014 -012 -01 -008 -006 -004 -002 0

|P| [(

38

π)

l

]P

l2

23

τ [t ]Pl

lt gt(b)

Figure 1 Fig(a) shows the absolute value of the wave function as a function of τ and P Fig(b) shows the expectation values of |P | as a functions of τ

5 ConclusionsA gauge-invariant quantum theory of the flat FRW universe with dust has been studied interms of the Ashtekar variables We have first constructed the classical reduced phase spaceof the system by using the relational formalism and then have quantized the reduced systemThe advantages of the quantization method are as follows (i) fundamental variables are gauge-invariant quantities (ii) a natural time evolution of the gauge-invariant quantities exists sothat the problem of time is absent and (iii) the reduced phase space can be quantized in thesame manner as in ordinary quantum mechanics because there are no constraints in the reducedphase space In the obtained quantum theory we have analyzed the dynamics of a wave packetand have shown that the expectation value of P has a non-zero minimum that is the initialsingularity is replaced by a big bounce in quantum theory The interpretation of the wave packetis that if the present state of the universe is in a right-handed system the past state has beenin a superposition of the states of a right-handed and left-handed systems

AchnowledgementThis work was supported in part by Global COE Program ldquoHigh-Level Grobal Cooperation forLeading-Edge Platform on Access Spaces (C12)rdquo

References[1] P G Bergmann Rev Mod Phys 33 510 (1961)[2] C Rovelli Class Quant Grav 8 1895 (1991) Phys Rev D 65 124013 (2002)[3] For a comprehensive review on the problem of time see eg C J Isham arXivgr-qc9210011[4] J D Brown and K V Kuchar Phys Rev D 51 5600 (1995)[5] A Ashtekar Phys Rev Lett 57 2244 (1986) Phys Rev D 36 1587 (1987)[6] J F Barbero G Phys Rev D 51 5507 (1995)[7] B Dittrich Gen Rel Grav 39 1891 (2007)[8] T Thiemann Class Quant Grav 23 1163 (2006)[9] F Amemiya and T Koike Phys Rev D 80 103507 (2009)

[10] F Amemiya and T Koike Phys Rev D 82 104007 (2010)[11] A Ashtekar M Bojowald and J Lewandowski Adv Theor Math Phys 7 233 (2003)

Spanish Relativity Meeting (ERE 2010) Gravity as a Crossroad in Physics IOP PublishingJournal of Physics Conference Series 314 (2011) 012053 doi1010881742-65963141012053

4