emergence of space, general relativity and gauge theory from tensor models
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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 !