monte carlo study on the birth of our universe by a ... · on the birth of our universe by a...
TRANSCRIPT
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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
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Motivation
� Cosmology is another frontier for high energy particle physics.
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Motivation
� Cosmology is another frontier for high energy particle physics.
� Many interesting questions in the early universe :
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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
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� 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), …
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� 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.
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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.
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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.
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Lorentzian Matrix Model
� Pfaffian is real.
� Let’s avoid Wick rotation to study real time evolution.
� Pfaffian is real.
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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.
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Lorentzian Matrix Model
� We can regularize oscillating phase by
a) introduce damping term,
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Lorentzian Matrix Model
� We can regularize oscillating phase by
a) introduce damping term,
b) insert identity,
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Lorentzian Matrix Model
� We can regularize oscillating phase by
a) introduce damping term,
c) and integrate out scale with
b) insert identity,
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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 :
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Lorentzian Matrix Model
� We study following model by Monte Carlo method :
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Lorentzian Matrix Model
� We study following model by Monte Carlo method :
� Note that Lorentzian� Note that Lorentzian
noncommutative : Lie algebraic, …
Euclidean
commutative
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Results : Time Eigenvalues
� Let be eigenvalues of .
Thanks to SUSY, they are smoothly extended as temporal cutoff increases.
0
0
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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
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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
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Results : SSB of SO(9) symmetry
order parameter : 9x9 real sym.
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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
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Mechanism of SSB
� Large kappa for fixed N is described by
� EOM is
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Mechanism of SSB
� Large kappa for fixed N is described by
� EOM is
� Let’s try an Ansatz
3d
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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)
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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.
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VDM model
� In ’s diagonal basis,
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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.
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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
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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
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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
æ æ æ ææ
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ææ æ
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à ààà
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ì ì
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.
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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
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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.
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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
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BackupBackup
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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
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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
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Classical solution 2
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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)
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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”.
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IIB Matrix Model
NxN hermitian matrices (N→∞)NxN hermitian matrices (N→∞)
[Ishibashi, Kawai, Kitazawa, Tsuchiya 96]
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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
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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.
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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.
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IR cutoff in spatial direction
identityidentityidentityidentity
rescale
diverges when
scale fixedboost sym fixed
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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.
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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
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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
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Mechanism of SSB
� Can we understand SSB in the large kappa ?
Maximize with