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 !