Transcript Expanding (3+1)-dimensional universe from a Lorentzian

Expanding (3+1)-dimensional
universe from a Lorentzian matrix
model for superstring theory in
(9+1)-dimensions
Seminar at University of Tokyo,
2011.12.5 (Mon.)
Jun Nishimura (KEK, Sokendai)
Ref.) Kim-J.N.-Tsuchiya
arXiv:1108.1540, to appear in PRL
fundamental questions concerning our universe
1. Why (3+1)-dimensions ?
2. Why expanding ?
Answers from a nonperturbative formulation
of superstring theory in (9+1)-dimensions
Previous works
Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)
Euclideanized model with SO(10) sym. has been studied.
SSB of SO(10) studied by Gaussian Expansion Method
J.N.-Okubo-Sugino, JHEP1110(2011)135
 free energy of SO(d) symmetric vacua (d=2,3,4,5,6,7)
minimum at d=3
 extent of space-time
finite in all directions
motivated us to reconsider the formulation
Our new proposal :
regularize SO(9,1) Lorentzian symmetric model
without Wick rotation
Monte Carlo simulation
3 out of 9 spatial directions start to expand
at some “critical time”
Comments on related works
 Quantum Cosmology with minisuperspace approx.
Vilenkin (’82,’84), Hartle-Hawking (’83)
 Causal Dynamical Traingulation
Ambjorn-Jurkiewicz-Loll (’05)
 Matrix Cosmology
 Matrix Big Bang
 Emergent gravity
Freedman-Gibbons-Schnabl (’05)
Craps-Sethi-Verlinde (’05)
Steinacker (’11), Yang (’10)
We expect that what we are doing is essentially
a first-principle calculation of the unified theory
including quantum gravity.
Plan of the talk
1.
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Introduction
Matrix model for superstrings
Matrix model with SO(9,1) symmetry
Monte Carlo results
Mechanism of the SSB
Removing the cutoffs
Summary and discussions
IKKT model
Ishibashi-Kawai-Kitazawa-Tsuchiya (’96)
proposed as a nonperturbative definition of
type IIB superstring theory in 10 dim.
c.f.) Matrix Theory
Banks-Fischler-Shenker-Susskind (’96)
 matrix regularization of the Green-Schwarz
worldsheet action in the Schild gauge
 interactions between D-branes
 string field theory from SD eqs. for Wilson loops
Fukuma-Kawai-Kitazawa-Tsuchiya (’98)
Does our 4-dimensional space-time
appear from 10-dimensions ?
10
Hermitian matrices
4Ｄ
10Ｄ
 Derivation of low-energy effective theory
branched-polymer-like system Aoki-Iso-Kawai-Kitazawa-Tada (’99)
 Explicit calculations by the Gaussian expansion method
J.N.-Sugino (’02), J.N.-Okubo-Sugino,
Kawai, Kawamoto, Kuroki, Matsuo, Shinohara, Aoyama, Shibusa,…
Results of the Gaussian expansion method
J.N.-Okubo-Sugino, JHEP1110(2011)135
extended directions
shrunken directions
Minimum of the free energy
occurs at d=3
Extent of space-time
finite in all directions
SSB of SO(10) : interesting dynamical property of IKKT model,
but we still seem to miss some important ingredient…
Matrix model with SO(9,1) symmetry
Hermitian matrices
raised and lowered by the metric
Wick rotation
Euclidean model with SO(10) symmetry
Partition function of the Lorentzian model
connection to the worldsheet theory
opposite sign !
Unlike the Euclidean model,
the path integral is ill-defined !
Krauth-Nicolai-Staudacher (’98),
Austing-Wheater (’01)
c.f.) complex in the Euclidean model
the phase factor induces SSB of SO(10)
J.N.-Vernizzi (’00), Anagnostopoulos-J.N.(’02)
Regularizing the Lorentzian model
In order to separate space and time,
we “gauge fix” the boost invariance.
SO(9) symmetry is still manifest.
(1) IR cutoff in the temporal direction
Regularizing oscillating functions
convergence factor
inserting unity
Cure this divergence by imposing :
(2) IR cutoff in the spatial direction
Thus we arrive at
Yoneya ('97)
Monte Carlo simulation : Rational Hybrid Monte Carlo algorithm
no sign problem unlike in the Euclidean model
Extracting time evolution
SSB
“critical time”
Consider a simpler problem :
solution :
representation matrices of
a compact semi-simple Lie algebra
with d generators
Maximum is achieved for SU(2) algebra
(continuum limit)
(infinite volume limit)
The theory thus obtained has
no parameters other than one scale paramter !
Clear large-N scaling behavior observed with
(continuum limit)
The extent of time increases and
the size of the universe becomes very large at later time.
(infinite volume limit)
Summary
 A new proposal for the nonperturbative formulation
of type IIB superstring theory in ten dimensions.
instead of making Wick rotation,
we introduce the IR cutoffs
for both temporal and spatial directions
after “gauge fixing” the boost symmetry
 The two cutoffs can be removed in the large-N limit.
 The theory thus obtained has no parameters
other than one scale parameter.
 Integrating over the scale factor first,
we obtain a model without sign problem.
c.f.) Monte Carlo studies of Euclidean model
difficult due to sign problem
 Monte Carlo simulation revealed
SSB of SO(9) down to SO(3) at some critical time.
mechanism is totally different from
that for the SSB in the Euclidean model
the size of the 3d space increases with time
 Cosmological singularity is naturally avoided
due to noncommutativity.
Crucial role played by SUSY
c.f.) bosonic model
no expansion and no SSB !
Complementary studies based on
classical equations of motion
Kim-J.N.-Tsuchiya
arXiv:1110.4803
A class of SO(3) symmetric solutions
 the time-dependence compatible with the expanding universe
 noncommutativity of space time : OK
Speculations
classical solution
space-space noncommutativity
Monte Carlo
simulation
accelerating
expansion
size of the space
space-time
noncommutativity
time
tcr
SO(9)
present time
symmetry of space
SO(3)
Space-space NC disappears for some dynamical reason.
Future directions
 Does a local field theory on a commutative space-time
appear at later time ?
 How do 4 fundamental interactions and the matter fields
appear at later time ?
Monte Carlo simulation AND
Studies of classical solutions (+ quantum corrections)
We hope the Lorentzian matrix model
provides a new perspective on
particle physics beyond the standard model
cosmological models for inflation, modified gravity, etc..