Emergence of space, general relativity and gauge theory
from tensor models
Naoki SasakuraYukawa Institute for Theoretical Physics
Kawamoto-san’s educationA 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
• 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 loves discussions
Spacetime is lattice (literally)
• 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 structuresKawamoto-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
--- Kawamoto-san’s philosophy ---
Not new but has potential to solve problems in the frontiers.
Random surface
2D quantum gravity
Kawamoto, Kazakov, Watabiki, …
Matrix model
Numerical Simulation
Tensor models
• Generalization of matrix models
Random surface Random volume
Master thesis under Kawamoto-san (1990)
Matrix model Tensor model
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.
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.
--- My proposal ---
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
LatticeFuzzy space
• Symmetry of continuous relabeling of “points”
: Total number of “points”
Relabeling symmetry → Origin of local gauge symmetries
A background fuzzy space causes symmetry breaking
Non-linearly realized local symmetry → Gauge symmetry (& Gen.Coord.Trans.Sym.)
The symmetry contains local transformations.
Ferrari, Picasso 1971Borisov, Ogievetsky 1974
• 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
Contents of the following talk
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• 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.
--- Future problems
(Symmetric, positive definite)
This action does not depend explicitly on D
All the dimensional Gaussian fuzzy spaces are the classical solutions of this single action.
--- An aspect of background independence
A cartoon for the action
Analysis of the small fluctuations around Gaussian solutions
Eigenvalue and eigenmode analysis
List of numerical analysis performed
• 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
Classical sol. : (Gaussian) fuzzy flat D-dimensional torus
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×S 1 : S 1 with small radius
Fuzzy Kaluza-Klein mechanism in tensor models
Classical solution
2+1 dimensional flat torus
==
Numerical analysis of fluctuation modes
Scalar
Vector
Gravity
L=6L=3
• Scalar mass does not scale
• Slopes of lines scale
Supermassive scalar field (“Planck” order)
L Large
Summary and future problems
Tensor models are physically interesting
Tensor models seem physically interesting.
・ Emergence of •Space•General relativity•Gauge theory•Gauge symmetry (Gen.Cood.Trans.Sym.)
from one single dynamical variable Cabc.
• Natural action ?• Fermion ?
・ Supermassive scalar field in Kaluza-Klein mechanism. Possible resolution to moduli stabilization.
・ Background independent
Thank you very much for many suggestions !
And
Happy Birthday !