Emergence of space, general relativity and gauge theory from tensor models

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Emergence of space, general relativity and gauge
theoryfrom tensor models

• Naoki Sasakura
• Yukawa Institute for Theoretical Physics

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Kawamoto-sans 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

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Kawamoto-san loves discussions

• 1330 Class starts
• 1500 Continue (Official end)
• 1700 Continue (End for most classes)
• 1900 End of the class
• 1900 Go to drink at Izakaya
• Various discussions on physics and
  non-physics
• 2200 Go to Kawamoto-sans home
• Discussions continue
• 600 Back home

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Spacetime is lattice (literally)
--- Kawamoto-sans philosophy ---
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-sans 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

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Random surface
Numerical Simulation
Matrix model
2D quantum gravity
Kawamoto, Kazakov, Watabiki,
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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
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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.

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A different interpretation of tensor models
--- My proposal ---

• 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.A211017-1028,2006
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Fuzzy space

• Defines algebraically a space. No coordinates.
• Points replaced with operators
• Includes noncommutative spaces
• Connect distinct topologies and dimensions

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Fuzzy space
Lattice
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• Symmetry of continuous relabeling of points

Total number of points
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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
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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

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Gaussian fuzzy space

• Ordinary continuum space
• Gaussian fuzzy space
• ß parameter of fuzziness

Sasai,Sasakura, JHEP 0609046,2006.
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• Gaussian fuzzy space
• Simplest fuzzy space
• Poincare symmetry ? Flat D-dimensional fuzzy
  space
• Can naturally generalize to curved space

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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.A233863-3890,2008.
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Construction of an action

• Demand has Gaussian fuzzy spaces as classical
  solutions
• Infinitely many such actions
• Generally very complicated and unnatural

--- Future problems

• 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.

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(Symmetric, positive definite)
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• 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
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Analysis of the small fluctuations around
Gaussian solutions
Eigenvalue and eigenmode analysis
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List of numerical analysis performed
Classical sol. (Gaussian) fuzzy flat
D-dimensional torus

• Emergence of general relativity
• D2 Results shown
• D1,3,4 Similar good results
• Kaluza-Klein mechanism
• D21 Results shown
• D11 Similar good results

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Emergence of general relativity
D2 , L10

• 3 states at P0
• 1 state at each P?0
• Zero eigenmodes

Sasakura, Prog.Theor.Phys.1191029-1040,2008.
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The three modes at P0
Tensor model
General Relativity
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The mode at P?0
One mode remains.
General relativity
Tensor model
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Kaluza-Klein mechanism

• In continuum theory

MS1 S1 with small radius
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Fuzzy Kaluza-Klein mechanism in tensor models
Classical solution
21 dimensional flat torus


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Numerical analysis of fluctuation modes
L3
L6

• Scalar mass does not scale
• Slopes of lines scale

Scalar
Vector
L ? Large
Supermassive scalar field (Planck order)
Gravity
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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.
Background independent
Supermassive scalar field in Kaluza-Klein
mechanism. Possible resolution to moduli
stabilization.

• Natural action ?
• Fermion ?

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Thank you very much for many suggestions ! And
Happy Birthday !