MONTE CARLO STUDY

ON THE BIRTH OF OUR UNIVERSE

BY A LORENTZIAN MATRIX MODEL

SANG-WOO KIM (OSAKA UNIVERSITY)

LATTICE 2012 @ CAIRNS

BY A LORENTZIAN MATRIX MODEL FOR SUPERSTRING THEORY

LATTICE 2012 @ CAIRNS

BASED ON 1108. 1540 (PRL 108 (2012) 011601)

BY SWK, JUN NISHIMURA, ASATO TSUCHIYA

Motivation

� Cosmology is another frontier for high energy particle physics.

Motivation

� Cosmology is another frontier for high energy particle physics.

� Many interesting questions in the early universe :

Motivation

� Cosmology is another frontier for high energy particle physics.

Initial singularity problem

� Many interesting questions in the early universe :

Spacetime dimensionality

Inflation, effective model

CMB spectrum

Dark energy, etc

� Many works so far :

String gas cosmology,

Quantum cosmology based on Wheeler-DeWitt eq,

Brandenberger, Vafa (’89), …

Vilenkin (’82), Hartle, Hawking (’83), …

D-brane + Perturbative analysis, etc.

Brandenberger, Vafa (’89), …

Herdeiro, Hirano, Kallosh (’01), …

� Many works so far :

String gas cosmology,

Quantum cosmology based on Wheeler-DeWitt eq,

Brandenberger, Vafa (’89), …

Vilenkin (’82), Hartle, Hawking (’83), …

D-brane + Perturbative analysis, etc.

Brandenberger, Vafa (’89), …

Herdeiro, Hirano, Kallosh (’01), …

� Today’s talk features :

Based on a matrix model proposal in string theory.Based on a matrix model proposal in string theory.

Unique time history is revealed by Monte Carlo method.

Expanding 3d spaces emerge in real time.

Local property for late time is suggested.

Matrix Model

� Our starting point is 0d Matrix Model.Ishibashi, Kawai, Kitazawa, Tsuchiya (’96)

A nonperturbative formulation proposed for superstring theory, as lattice QCD is for QCD.

Ishibashi, Kawai, Kitazawa, Tsuchiya (’96)

Obtained from Green-Schwarz action in string theory.

N=2 SUSY on matrix eigenvalues.

Matrix Model

� Our starting point is 0d Matrix Model.Ishibashi, Kawai, Kitazawa, Tsuchiya (’96)

A nonperturbative formulation proposed for superstring theory, as lattice QCD is for QCD.

Ishibashi, Kawai, Kitazawa, Tsuchiya (’96)

Obtained from Green-Schwarz action in string theory.

N=2 SUSY on matrix eigenvalues.

1d Matrix Quantum Mechanics,Banks, Fischler, Shenker, Susskind (’96)

2d Matrix String TheoryDijkraaf, Verlinde, Verlinde (’97)

� cf.

� Euclidean IKKT by K. Anagnostopoulos at yesterday’s parallel.

� More general review by M. Hanada at tomorrow’s plenary.

Lorentzian Matrix Model

� Pfaffian is real.

� Let’s avoid Wick rotation to study real time evolution.

� Pfaffian is real.

Lorentzian Matrix Model

� Pfaffian is real.

� Let’s avoid Wick rotation to study real time evolution.

� Pfaffian is real.

� Noncompact temporal direction requires a IR cutoff.

� This breaks SUSY and SO(9,1) symmetry.

Lorentzian Matrix Model

� We can regularize oscillating phase by

a) introduce damping term,

Lorentzian Matrix Model

� We can regularize oscillating phase by

a) introduce damping term,

b) insert identity,

Lorentzian Matrix Model

� We can regularize oscillating phase by

a) introduce damping term,

c) and integrate out scale with

b) insert identity,

Lorentzian Matrix Model

� We can regularize oscillating phase by

a) introduce damping term,

c) and integrate out scale with

b) insert identity,

d) Need to introduce L :

Lorentzian Matrix Model

� We study following model by Monte Carlo method :

Lorentzian Matrix Model

� We study following model by Monte Carlo method :

� Note that Lorentzian� Note that Lorentzian

noncommutative : Lie algebraic, …

Euclidean

commutative

Results : Time Eigenvalues

� Let be eigenvalues of .

Thanks to SUSY, they are smoothly extended as temporal cutoff increases.

0

0

Results : Band Diagonal Structure

� Band diagonal structure appears dynamically for

in ’s diagonal basis.in ’s diagonal basis.

small

0.125

0.25

0.5

1

|QIJ

/QN

/2,N

/2|1/

2

(I+J)/2=2(I+J)/2=4(I+J)/2=6(I+J)/2=8

large

small 0.125

-16 -12 -8 -4 0 4 8 12 16

I-J

Results : Band Diagonal Structure

� Band diagonal structure appears dynamically for

in ’s diagonal basis.in ’s diagonal basis.

small

0.125

0.25

0.5

1

|QIJ

/QN

/2,N

/2|1/

2

(I+J)/2=2(I+J)/2=4(I+J)/2=6(I+J)/2=8

subblock matrix represents space structure at given time.

small 0.125

-16 -12 -8 -4 0 4 8 12 16

I-J

Results : SSB of SO(9) symmetry

order parameter : 9x9 real sym.

Results : SSB of SO(9) symmetry

order parameter : 9x9 real sym.

0.3

0.4

0.5

eige

nval

ues

of T

ij(t)

Critical time

0

0.1

0.2

-2.5 -2 -1.5 -1 -0.5 0

eige

nval

ues

of T

t

Mechanism of SSB

� Large kappa for fixed N is described by

� EOM is

Mechanism of SSB

� Large kappa for fixed N is described by

� EOM is

� Let’s try an Ansatz

3d

Continuum / Infinite volume limit

2

3

4

5

R(t

)2 /R(t

c)2

N=8N=12N=16

4

6

8

10

R(t

)2 /R(t

c)2

κ=2.0κ=4.0κ=8.0

0

1

-2 -1 0 1 2(t-tc)/R(tc)

0

2

-4 -3 -2 -1 0 1 2 3 4(t-tc)/R(tc)

Continuum / Infinite volume limit

2

3

4

5

R(t

)2 /R(t

c)2

N=8N=12N=16

4

6

8

10

R(t

)2 /R(t

c)2

κ=2.0κ=4.0κ=8.0

0

1

-2 -1 0 1 2(t-tc)/R(tc)

0

2

-4 -3 -2 -1 0 1 2 3 4(t-tc)/R(tc)

They seem to converge to a single curve.

VDM model

� In ’s diagonal basis,

VDM model

� In ’s diagonal basis,

� Bosonic model with fermionic interactions on temporal eigenvalues. eigenvalues.

� It is like quenched QCD for QCD. Much faster than full SUSY model.

� Interesting properties such as SSB to 3d, expansion are kept.

Preliminary results on VDM model

continuumExponential ÈAiiÈ�RHt_cL

infinite vol

Exponentialexpansion

-40 -20 20 40 60 80Ht_i-t_cL�RHt_cL

2.0

3.0

1.5

H L

Preliminary results on VDM model

continuumExponential ÈAiiÈ�RHt_cL

infinite vol

Exponentialexpansion

1.5

ÈAijÈ

-40 -20 20 40 60 80Ht_i-t_cL�RHt_cL

2.0

3.0

1.5

ÈAiiÈ�RHt_cL

3.0

ÈAijÈ with i+ j=N

Line : N=32 kappa=1.68Dot : N=64 kappa=2

Effective band size decreases for late time.

-40 -20 0 20 40t_i-t_j

1.0

1.5

È È

-100 -50 0 50 100t_i-t_j

1.0

2.0

1.5

Dot : N=64 kappa=2Dash : N=32 kappa=3.36DotDash : N=64 kappa=4

Preliminary results on VDM model

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-4 -2 2 4 6Ht-t_cL�RHt_cL

1.2

1.4

1.6RHtL�RHt_cL

N=32, Κ=1.68

Blue Cirlce : n=3

Red Square : n=5

Yellow Diamond : n=7

Green Triangle : n=9

-30 -20 -10 10 20 30i

-20

20

40

60

t_i

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à ààà

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H L

0.8

H L

-60

-40

20

� Band size dependence is small in expanding region.

� Time eigenvalues are uniform in contrast to SUSY case.

More effective methods

� Let’s test whether we can ignore off-diagonal elements with

� At very late time, the interactionfor different time block is likelyto be ignored. 0

0

More effective methods

� Let’s test whether we can ignore off-diagonal elements with

� At very late time, the interactionfor different time block is likelyto be ignored. 0

0� Physics for is just quantum

mechanics, which emerges fromLorentzian matrix model.

This QM will be very effective for studying late time.

Summary

0-d matrix model Two IR cutoffs

� Unique time history

� SSB from 9d to 3d spaces

with SO(9,1)Two IR cutoffs

8

10

κ=2.0κ=4.0� SSB from 9d to 3d spaces

� Exponential expansion

� Noncommutative mechanism

� Local property for late time 0

2

4

6

8

-4 -3 -2 -1 0 1 2 3 4

R(t

)2 /R(t

c)2

(t-tc)/R(tc)

κ=4.0κ=8.0

SO(9)

SO(3)

Early Universe

BackupBackup

Alternative Approach

� The SSB and expansion relies on space-space noncommutativity.

� Does our model allow commutative spacetime in late time ?� Does our model allow commutative spacetime in late time ?

� Direct numerical study become more difficult for future.

� We look for classical solutions consistent with previous result.

Equation of MotionTwo IR cutoffs

Classical solution 1

0.5

1

R(t

)/R

(0)

N=16N=32N=64

N=128

0.4

0.6

χ

N=16N=32N=64

N=128

0-8 -4 0 4 8

t

0

0.2

-8 -4 0 4 8

t

Classical solution 2

Phase quenched Euclidean model

� Pfaffian is complex in Euclidean signature, which give rise to the sign problem.

[Ambjorn, Anagnostopoulos, Bietenholz, Hotta, Nishimura 2000]

� Without complex phase, there is no SSB.

(Origin of Euclidean SSB is fermionic)

Lorentz symmetry

� The cutoff restricts boost symmetry

and we found that the thermalized 4.0025

4.003

and we found that the thermalized

configurations have minimum

under Lorentz transformation.

� Therefore we may equivalently use Lorentz invariant cutoff,

4

4.0005

4.001

4.0015

4.002

4.0025

0 20 40 60 80 100

κ

number of boosts

� Therefore we may equivalently use Lorentz invariant cutoff, which act on configurations in “minimum frame”.

IIB Matrix Model

NxN hermitian matrices (N→∞)NxN hermitian matrices (N→∞)

[Ishibashi, Kawai, Kitazawa, Tsuchiya 96]

Interpretation of Matrix

If eigenvalues of bosonic matrix = spacetime coordinate,

� Note that bosonic action is positive definite in Euclidean � Note that bosonic action is positive definite in Euclidean signature, and prefers commuting configurations.

0

0Dynamically generated

N discrete spacetime points

Problem of Euclidean model

� A study by GEM for Euclidean IIB matrix model.[Nishimura, Okubo, Sugino 2011][Nishimura, Okubo, Sugino 2011]

d=3 has minimum d=3 has minimum free energy

3d spacetimeis too small

� Free energy prefers ,

and spacetime is too small compared to extra dimensions.

IR cutoff in temporal direction

� Bosonic part of the action is problematic due to the indefinite signature.

� Note that the Euclidean model is well defined and temporal direction is the source of the problem. We need to mod out boost transformation from integration measure.

� Let’s introduce a cutoff in temporal direction, which “gauge fix” SO(9,1) to SO(9) in general.Let’s introduce a cutoff in temporal direction, which “gauge fix” SO(9,1) to SO(9) in general.

� Important question is whether we can remove this constraint in the large N limit.

IR cutoff in spatial direction

identityidentityidentityidentity

rescale

diverges when

scale fixedboost sym fixed

Comparison with lattice regularization

Lattice

Lorentzian matrix model

� In contrast to lattice, SUSY is broken only by IR cutoffs.

� After continuum limit (~N) and infinite volume limit (~L),

only one parameter (~kappa) remains.

Mechanism of SSB

� Without any Ansatz

1

-0.5

0

0.5

1ei

genv

alue

s

L2

L3i[L1,L2]+L3

� 2x2 representation of SU(2) algebra gives the maximum, which explains 3 expanding spaces.

-1 1 2 3 4 5 6 7 8

n-th

Lorentzian vs Euclidean

� Let’s consider a solution to simple equatiion.

Euclidean :

� In our case,

Euclidean :

Lorentzian : light-like solutions

Lorentzian

noncommutative : Lie algebraic, …

� Wick rotation can not reproduce these solutions !

noncommutative : Lie algebraic, …

Euclidean

commutative

Mechanism of SSB

� Can we understand SSB in the large kappa ?

Maximize with