Transcript Spacetime physics from tensor models

Emergence of space,
general relativity and gauge theory
from tensor models
Naoki Sasakura
Yukawa Institute for Theoretical Physics
Kawamoto-san’s education
A class guided by Kawamoto-san
Text : the original BPZ paper on CFT
・Not allow superficial understanding
・Everything must be understood certainly
・Full of discussions
・No care about time
・Unusual members
Students and staff members from other universities
Russian style
Kawamoto-san loves discussions
• 13:30 Class starts
• 15:00 Continue (Official end)
• 17:00 Continue (End for most classes)
• 19:00 End of the class
• 19:00 Go to drink at Izakaya
Various discussions on physics and non-physics
• 22:00 Go to Kawamoto-san’s home
Discussions continue
• 6:00 Back home
--- Kawamoto-san’s philosophy ---
Spacetime is lattice (literally)
Not new but has potential to solve problems in the frontiers.
• Reduce degrees of freedom
Free from infinities
Incorporate minimal length
May prevent physically unwanted fields
(e.g. scalar massless moduli fields in string theory)
• Unified theory on lattice
Matter contents are related to lattice structures
Kawamoto-san’s talk at 13th Nishinomiya Yukawa Memorial
Symposium (1998)
“Non-String Pursuit towards Unified Model on the Lattice”
• Reconnection Dynamical spacetime
Possible route to quantum gravity
Intrinsically background independent
Random surface
Numerical Simulation
Matrix model
2D quantum gravity
Kawamoto, Kazakov, Watabiki, …
Tensor models
• Generalization of matrix models
Random surface
Random volume
Matrix model
Tensor model
Master thesis under Kawamoto-san (1990)
Sasakura, Mod.Phys.Lett.A6,2613,1991
Tensor models were not successful
• Continuum limit  Large volume 
Large Feynman diagram
But no analytical methods known for non-perturbative
computations in tensor models.
• Topological expansions not known.
Difficulty in physical interpretation of the partition function.
--- My proposal ---
A different interpretation of tensor models
Tensor models may be regarded as dynamical
theory of fuzzy spaces.
The structure constant
defining a fuzzy
space may be identified with the dynamical
variable of tensor models.
Sasakura, Mod.Phys.Lett.A21:1017-1028,2006
Fuzzy space
• Defines algebraically a space. No coordinates.
• “Points” replaced with operators
• Includes noncommutative spaces
• Connect distinct topologies and dimensions
Fuzzy
space
Lattice
• Symmetry of continuous relabeling of “points”
: Total number of “points”
The symmetry contains local transformations.
A background fuzzy space causes symmetry breaking
Non-linearly realized local symmetry →
Gauge symmetry (& Gen.Coord.Trans.Sym.)
Ferrari, Picasso 1971
Borisov, Ogievetsky 1974
Relabeling symmetry → Origin of local gauge symmetries
Contents of the following talk
• Gaussian fuzzy space （Flat D-dimensional fuzzy space)
• Construction of an action having Gaussian sol.
• Fluctuation mode analysis around the sol.
--- Emergence of general relativity
• Kaluza-Klein set up
--- Emergence of gauge theory
--- Emergent scalar field is supermassive (“Planck” order)
• Summary and future problems
Gaussian fuzzy space
• Ordinary continuum space
• Gaussian fuzzy space
β： parameter of fuzziness
Sasai,Sasakura, JHEP 0609:046,2006.
Gaussian fuzzy space
•Simplest fuzzy space
•Poincare symmetry  Flat D-dimensional fuzzy space
•Can naturally generalize to curved space
This metric-tensor correspondence derives DeWitt
supermetric from the configuration measure of tensor models.
Tensor models
DeWitt supermetric in general relativity
Used in the comparison of modes
Sasakura, Int.J.Mod.Phys.A23:3863-3890,2008.
Construction of an action
Demand : has Gaussian fuzzy spaces as classical solutions
• Infinitely many such actions
--- Future problems
• Generally very complicated and unnatural
The action in this talk ---- Convenient but singular
(There exists also non-singular but inconvenient one.)
• Least number of terms.
• The singular property will not harm the fluctuation analysis.
• The low-frequency property independent of the actions.
(Symmetric, positive definite)
This action does not depend explicitly on D
A cartoon for the action
All the dimensional Gaussian fuzzy spaces are the
classical solutions of this single action.
--- An aspect of background independence
Analysis of the small fluctuations
around Gaussian solutions
Eigenvalue and eigenmode analysis
List of numerical analysis performed
Classical sol. : (Gaussian) fuzzy flat D-dimensional torus
• Emergence of general relativity
D=2 : Results shown
D=1,3,4: Similar good results
• Kaluza-Klein mechanism
D=2+1 : Results shown
D=1+1 : Similar good results
Emergence of general relativity
D=2 , L=10
• 3 states at P=0
• 1 state at each P≠0
• Zero eigenmodes
Sasakura, Prog.Theor.Phys.119:1029-1040,2008.
The three modes at P=0
Tensor model
General Relativity
The mode at P≠0
One mode remains.
General relativity
Tensor model
Kaluza-Klein mechanism
In continuum theory
M×S１ : S１ with small radius
Fuzzy Kaluza-Klein mechanism in tensor models
Classical solution
2+1 dimensional flat torus
=
=
Numerical analysis of fluctuation modes
L=3
L=6
Scalar
• Scalar mass does not scale
• Slopes of lines scale
Vector
L  Large
Supermassive scalar field (“Planck” order)
Gravity
Summary and future problems
Tensor models seem physically interesting.
・Emergence of
•Space
•General relativity
•Gauge theory
•Gauge symmetry (Gen.Cood.Trans.Sym.)
from one single dynamical variable Cabc.
・ Background independent
・ Supermassive scalar field in Kaluza-Klein mechanism.
Possible resolution to moduli stabilization.
• Natural action ?
• Fermion ?
Thank you very much for many suggestions !
And
Happy Birthday !