Fuzzy Objects in Noncommutative Geometry and Their Applications - PowerPoint PPT Presentation

Fuzzy Objects in Noncommutative Geometry and Their Applications � Shinpei Kobayashi (Gunma National College of Technology) 2012/11/12, JGRG22 @ RESCEU, University of Tokyo �

Motivation � ¡ Resolution of singularity ¡ String inspired ¡ non-BPS D2/D0-system ¡ 2-dimensional space with constant magnetic flux ¡ (an effective theory of ) quantum gravity

Realization of Noncommutativity � ¡ Noncommutativity between space coordinates : constant parameter [ x, y ] = i θ , θ ¡ a realization: Wick-Voros product ✓ @ ◆ @ � f ( z 0 , ¯ z 0 ) g ( z 00 , ¯ z 00 ) ( f ? g )( z, ¯ z ) = exp � @ ¯ @ z 00 z 0 � z 0 = z 00 = z [ z, ¯ z ] = z ? ¯ z − ¯ z ? z = 1 ✓ ◆ z = x + iy z = x − iy , ¯ √ √ 2 θ 2 θ

Field Theory on noncommutative space � commutative: � Z z φ + m ⇣ ⌘ 2 φ 2 + · · · dtd 2 x S = ∂ z φ∂ ¯ noncommutative: � Z z � + m ⇣ ⌘ dtd 2 x S = @ z � ? @ ¯ 2 � ? � + · · ·

Fuzzy Objects � Nontrivial Solutions in Noncommutative Geometry �

Noncommutative Solitons � ¡ scalar field theory on NC plane: GMS solitons [ Gopakumar- Minwalla- Strominger, Kraus-Larsen, …] Z d 2 zV ? ( Φ ) E = D V ? ( Φ ) = b 2 2 Φ ? Φ + b 3 3 Φ ? Φ ? Φ + · · · ¡ circular symmetric, connection to D-branes � 10 8 6 2 4 2 0 0 -2 -2 0 0 -2 2 2

Weyl-Wigner Correspondence � ∞ X function � f (¯ z, z ) = mn ¯ Tay f z m z n n =0 Weyl projection � ∞ ˆ X a † , ˆ a † m ˆ operator � f (ˆ a ) = mn ˆ Tay a n f n =0

Weyl-Wigner Correspondence � algebra � algebra � ˆ function: operator: f ( x, y ) f (ˆ x, ˆ y ) star product: product f ? g ˆ with an ordering: f · ˆ g a † ] = 1 [ x, y ] = i θ [ˆ a, ˆ [ˆ x, ˆ y ] = i θ operators which act on the Fock space of a harmonic oscillator �

Weyl-Wigner Correspondence � ∞ X function � f (¯ z, z ) = mn ¯ Tay f z m z n n =0 Weyl projection � ∞ ˆ X a † , ˆ a † m ˆ operator � f (ˆ a ) = mn ˆ Tay a n f n =0

Weyl-Wigner Correspondence � z, z ) = h z | ˆ function � f (¯ f | z i inverse Weyl projection � a | z i = z | z i : coherent state � ˆ ∞ ˆ X a † , ˆ a † m ˆ operator � f (ˆ a ) = mn ˆ Tay a n f n =0

Weyl-Wigner Correspondence � In operator formalism: ˆ N | n i = n | n i

Projection operators as NC solitons � [GMS (2000)] � 0 = @ V ? ¡ EOM: @ Φ = b 2 Φ + b 3 Φ ? Φ + b 4 Φ ? Φ ? Φ + · · · ¡ non-trivial sln: with p 2 Φ = λ ∗ p n ( z, ¯ z ) n ( z, ¯ z ) = p n ( z, ¯ z ) projection operator � p n = | n i h n | ˆ r 2 n p n ( r ) = h z | n i h n | z i = e − r 2 :circular symmetric � 2 θ n !(2 θ ) n

Fuzzy Disc [ Lizzi, ¡Vitale, ¡Zampini (2003)] � ¡ Def.: finite dim. truncation of a noncommutative plane ¡ Two parameters: ¡ noncommutativity: θ ¡ fuzzyness : N ¡ applications: ¡ matrix model ¡ quantum Hall effect

A Fuzzy Disc ( N=10, θ =1 ) � A 10 = ˆ ˆ P 10 ˆ A ˆ P 10 ˆ P 10 = ˆ p 0 + ˆ p 1 + · · · + ˆ p 9 p 0 + p 1 + · · · + p 9 9 r 2 n e − r 2 X = 2 θ n !(2 θ ) n n =0

GMS solitons and fuzzy disc � N=4 case � ˆ ˆ ˆ p 1 p 2 p 3 p 0 = | 0 i h 0 | ˆ

Another orthonormal basis : angle states? � ¡ number basis: concentric cutting of space ¡ ~ radius ˆ x 2 + ˆ ( ˆ p N y 2 ) ˆ N ∼ ¡ another basis: radial cutting of space ¡ ~ angle ˆ ϕ

Angles in the fuzzy disc � SK-Asakawa, arXiv:1206.6602 �

Angle Operator and States � N − 1 with help of X ¡ The angle operator: ϕ m | ϕ m i h ϕ m | ϕ = ˆ Pegg-Barnett m =0 phase operator � ¡ Eigen states of the angle operator: ϕ | ϕ m i = ϕ m | ϕ m i ˆ N − 1 1 ¡ Relation to the number state e in ϕ m | n i X | ϕ m i = p N n =0 ¡ Orthonormality: h ϕ m | ϕ n i = δ mn ¡ Angular projection operators: π m = | ϕ m i h ϕ m | ˆ ϕ m = 2 π N m → angular “delta function” peaked at �

� Two descriptions of fuzzy disc � ¡ baum-kuchen vs shortcake ˆ π 1 ˆ p 0 ˆ ˆ π 2 π 0 ˆ ˆ p 1 p 3 ˆ π 3 ˆ p 2 N − 1 N − 1 X X p n = ˆ ˆ π m n =0 m =0

angular projection operators � N − 1 r m + n ( r, ϕ ) = 1 e − r 2 π ( N ) X m ! n !(2 θ ) m + n e − i ( m − n )( ϕ − ϕ k ) 2 θ k p N m,n =0 ˆ π 1 ˆ ˆ π 0 π 2 ˆ π 3 N=4 case � not concentric, but fan-shaped, like pieces of cake �

Other fuzzy objects: e.g.) fuzzy Annulus � P M ˆ p M + ˆ p M +1 + · · · + ˆ p M + N − 1 N := ˆ any set of N orthonormal operators is allowed for truncation �

Noncommutative Solitons as D0-branes � ¡ scalar field on the NC plane = tachyon filed on a non-BPS D2-brane 10 ¡ The solution 8 Φ = λ ∗ ˆ 6 p n 2 4 2 = a D0-brane (rank → same tension) 0 p n = 1 ˆ 0 -2 -2 0 0 -2 2 2 ¡ Same thing can be said to our case: the solution also can be seen as a D0-brane, Φ = λ ∗ ˆ π m with very different shape (fan-shaped) ¡ Commutative limit (with N θ fixed), angular NC soliton becomes thinner and thinner (A D0-brane is twisted into a string!)

Angular NC Solitons in Gravity � S = − Λ E ? = det ? E = 1 Z 3! ✏ µ ⌫⇢ ✏ abc E a µ ? E b ⌫ ? E c dtd 2 z E ? ⇢ κ 2 ✏ µ ⌫⇢ ✏ abc { E b ⌫ , E c ⇢ } ? = 0  α 0 π ( N )   E 0  0 0 0 0 0 0 α 1 π ( N ) E a E 1  = µ = 0 0  0 0  1  1   E 2 α 2 π ( N ) 0 0 0 0 2 2 0 π (3) 1 π (3) 2 π (3) ds 2 = − α 2 0 dt 2 + α 2 1 dx 2 + α 2 2 dy 2 " √ # k ) + r 2 2 r 3 k ) + r 4 k ( r, ϕ ) = 1 1 + 2 r n √ o π (3) 3 e − r 2 / θ θ 1 / 2 cos( ϕ − ϕ (3) 2 cos[2( ϕ − ϕ (3) θ 3 / 2 cos( ϕ − ϕ (3) 1 + k )] + θ 2 θ

Experiment with laser � ¡ Gaussian beam �

Experiment with laser � ¡ Laguerre-Gaussian beam � GMS solitons � angular NC solitons �

Noncommutative Solitons with Time-Dependence �

Weyl-Wigner Correspondence � algebra � algebra � ˆ function: operator: f ( x, y ) f (ˆ x, ˆ y ) star product: product f ? g ˆ with an ordering: f · ˆ g a † ] = 1 [ x, y ] = i θ y ] = i θ [ˆ a, ˆ [ˆ x, ˆ operators which act on the Fock space no restriction as long as � [ˆ x, ˆ y ] = i θ of a harmonic oscillator �

Noncommutative Solitons with Time-Dependence � ¡ Time-dependent Harmonic Oscillator e.g., p 2 2 m ( t ) + 1 ˆ ˆ 2 m ( t ) ω 2 ( t )ˆ x 2 H ( x, p, t ) = → we can solve this system analytically by the LR method �

Lewis-Reisenfeld Method � ¡ Time-dependent Schroedinger equation i ~ ∂ ∂ t ψ = ˆ H ψ ¡ Invariant operator ˆ I d ˆ dt = ∂ ˆ ∂ t + 1 I I i ~ [ˆ I, ˆ H ] = 0 ¡ Eigenvalue problem ˆ I φ n ( x, p, t ) = λ n φ n ( x, p, t ) ψ ( x, p, t ) = e i ✏ ( t ) φ ( x, p, t ) ✓ ◆ i ~ @ @ t � ˆ ~ ˙ ✏ = h � n ( t ) | | � n ( t ) i H

Time-Dependent NC Solitons � ¡ general, quadratic, time-dependent Hamiltonian [Choi-Gweon (2004)] � → apply to NC solitons ˆ H (ˆ x, ˆ y, t ) x 2 + B ( t )(ˆ y 2 + D ( t )ˆ = A ( t )ˆ x ˆ y + ˆ y ˆ x ) + C ( t )ˆ x + E ( t )ˆ y + F ( t ) ¡ creation and annihilation operator � (" √ # ) r 1 2 ρ + i 2 B ρ − ˙ k ρ a = ˆ (ˆ x − x p ( t )) + i ρ (ˆ y − y p ( t )) θ k 1 / 2 2 A (" √ # ) r 1 2 ρ − i 2 B ρ − ˙ k ρ a † = ˆ (ˆ x − x p ( t )) − i ρ (ˆ y − y p ( t )) θ k 1 / 2 2 A they satisfy � a † ] = 1 [ˆ a, ˆ

Time-Dependent NC Solitons � [SK, in progress] � ¡ time-dependent circular symmetric solitons � | φ n , t i h φ n , t | e − | ζ ( t ) | 2 | ζ ( t ) | 2 n ! ( � 2 ) 1 ✓ 2 B ρ − ˙ ◆ k ρ | ζ ( t ) | 2 = 4 ρ 2 ( x − x p ( t )) 2 + ( x − x p ( t )) + ρ ( y − y p ( t )) θ k 1 / 2 2 A

Time-Dependent NC Solitons � [SK, in progress] � e.g., Caldirola-Kanai oscillator x 2 + e − γ t ˆ ˆ H = e γ t ˆ y 2 t = 1 / γ

Summary � ¡ The fuzzy disc: ¡ a disc-shaped, finite region in the NC plane ¡ a fuzzy approximation by θ ¡ Introduction of angles to the fuzzy disc ¡ angle projection operator and angle states ¡ directly relates the boundary to the bulk ¡ Application ¡ angular NC scalar solitons & fan-shaped D-branes ¡ angular NC gravitational solitons ¡ laser physics ¡ Time-dependent noncommutative solitons exist