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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 ¡ a realization: Wick-Voros product

[x, y] = iθ, θ

✓ z = x + iy √ 2θ , ¯ z = x − iy √ 2θ ◆ (f ? g)(z, ¯ z) = exp ✓ @ @¯ z0 @ @z00 ◆ f(z0, ¯ z0)g(z00, ¯ z00)

• z0=z00=z

[z, ¯ z] = z ? ¯ z − ¯ z ? z = 1

Field Theory on noncommutative space

S = Z dtd2x ⇣ ∂zφ∂¯

zφ + m

2 φ2 + · · · ⌘ S = Z dtd2x ⇣ @z ? @¯

z + m

2 ? + · · · ⌘

commutative: noncommutative:

Fuzzy Objects

Nontrivial Solutions in Noncommutative Geometry

Noncommutative Solitons

• ¡ scalar field theory on NC plane: GMS solitons

[Gopakumar- Minwalla- Strominger, Kraus-Larsen, …] ¡ circular symmetric, connection to D-branes E = Z

D

d2zV?(Φ)

V?(Φ) = b2 2 Φ ? Φ + b3 3 Φ ? Φ ? Φ + · · ·

• 2

2

• 2

2 2 4 6 8 10

• 2

2

Weyl-Wigner Correspondence

Weyl projection function

• perator

f(¯ z, z) =

∞

X

n=0

f

Tay

mn¯

zmzn

ˆ f(ˆ a†, ˆ a) =

∞

X

n=0

f

Tay

mnˆ

a†mˆ an

Weyl-Wigner Correspondence

function: star product:

• perators which act
• n the Fock space
• f a harmonic oscillator

f(x, y) [x, y] = iθ ˆ f(ˆ x, ˆ y)

algebra

f ? g

• perator:

product with an ordering: algebra

ˆ f · ˆ g

[ˆ x, ˆ y] = iθ

[ˆ a, ˆ a†] = 1

Weyl-Wigner Correspondence

Weyl projection function

• perator

f(¯ z, z) =

∞

X

n=0

f

Tay

mn¯

zmzn

ˆ f(ˆ a†, ˆ a) =

∞

X

n=0

f

Tay

mnˆ

a†mˆ an

Weyl-Wigner Correspondence

inverse Weyl projection function

• perator

ˆ f(ˆ a†, ˆ a) =

∞

X

n=0

f

Tay

mnˆ

a†mˆ an

f(¯ z, z) = h z | ˆ f | z i

ˆ a | z i = z | z i

: coherent state

Weyl-Wigner Correspondence

In operator formalism:

ˆ N | n i = n | n i

Projection operators as NC solitons

¡ EOM: ¡ non-trivial sln: with

0 = @V? @Φ = b2Φ + b3Φ ? Φ + b4Φ ? Φ ? Φ + · · · projection operator

ˆ pn = | n i h n | Φ = λ∗pn(z, ¯ z)

p2

n(z, ¯

z) = pn(z, ¯ z) pn(r) = h z | n i h n | z i = e− r2

2θ

r2n n!(2θ)n

:circular symmetric [GMS (2000)]

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)

p0 + p1 + · · · + p9

=

9

X

n=0

e− r2

2θ

r2n n!(2θ)n

ˆ A10 = ˆ P10 ˆ A ˆ P10 ˆ P10 = ˆ p0 + ˆ p1 + · · · + ˆ p9

N=4 case

GMS solitons and fuzzy disc

ˆ p0 = | 0 i h 0 | ˆ p1

ˆ p2

ˆ p3

Another orthonormal basis : angle states?

¡ number basis: concentric cutting of space

¡ ~ radius

¡ another basis: radial cutting of space

¡ ~ angle

ˆ N

ˆ ϕ

( ˆ N ∼ p ˆ x2 + ˆ y2)

Angles in the fuzzy disc

SK-Asakawa, arXiv:1206.6602

Angle Operator and States

¡ The angle operator: ¡ Eigen states of the angle operator: ¡ Relation to the number state ¡ Orthonormality:

¡ Angular projection operators: → angular “delta function” peaked at

h ϕm | ϕn i = δmn

ˆ ϕ | ϕm i = ϕm | ϕm i ˆ ϕ =

N−1

X

m=0

ϕm | ϕm i h ϕm |

ϕm = 2π N m

ˆ πm = | ϕm i h ϕm |

| ϕm i = 1 p N

N−1

X

n=0

einϕm | n i with help of Pegg-Barnett phase operator

¡ baum-kuchen vs shortcake

• Two descriptions of fuzzy disc

ˆ p0 ˆ p1 ˆ p2 ˆ p3

ˆ π3 ˆ π2 ˆ π1 ˆ π0

N−1

X

n=0

ˆ pn =

N−1

X

m=0

ˆ πm

angular projection operators

π(N)

k

(r, ϕ) = 1 N

N−1

X

m,n=0

e− r2

2θ

rm+n p m!n!(2θ)m+n e−i(m−n)(ϕ−ϕk)

ˆ π3 ˆ π2 ˆ π1 ˆ π0

not concentric, but fan-shaped, like pieces of cake N=4 case

Other fuzzy objects: e.g.) fuzzy Annulus

ˆ P M

N := ˆ

pM + ˆ pM+1 + · · · + ˆ pM+N−1

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 ¡ The solution = a D0-brane (rank → same tension) ¡ Same thing can be said to our case: the solution also can be seen as a D0-brane, with very different shape (fan-shaped) ¡ Commutative limit (with Nθ fixed), angular NC soliton becomes thinner and thinner (A D0-brane is twisted into a string!)

Φ = λ∗ˆ pn

ˆ pn = 1

Φ = λ∗ˆ πm

• 2

• 2

• 2

Angular NC Solitons in Gravity

S = − Λ κ2 Z dtd2z E? E? = det?E = 1 3!✏µ⌫⇢✏abcEa

µ ? Eb ⌫ ? Ec ⇢

Ea

µ =

  E0 E1

1

E2

2

  =    α0π(N) α1π(N)

1

α2π(N)

2

   ds2 = −α2

0π(3) 0 dt2 + α2 1π(3) 1 dx2 + α2 2π(3) 2 dy2

π(3)

k (r, ϕ) = 1

3e−r2/θ " 1 + 2r θ1/2 cos(ϕ − ϕ(3)

k ) + r2

θ n 1 + √ 2 cos[2(ϕ − ϕ(3)

k )]

• +

√ 2r3 θ3/2 cos(ϕ − ϕ(3)

k ) + r4

θ2 #

✏µ⌫⇢✏abc{Eb

⌫, Ec ⇢}? = 0

Experiment with laser

¡ Gaussian beam

Experiment with laser

¡ Laguerre-Gaussian beam GMS solitons angular NC solitons

Noncommutative Solitons with Time-Dependence

Weyl-Wigner Correspondence

function: star product:

• perators which act
• n the Fock space
• f a harmonic oscillator

f(x, y)

[x, y] = iθ

ˆ f(ˆ x, ˆ y)

algebra

f ? g

• perator:

product with an ordering: algebra

ˆ f · ˆ g

[ˆ x, ˆ y] = iθ [ˆ

a, ˆ a†] = 1

no restriction as long as [ˆ

x, ˆ y] = iθ

Noncommutative Solitons with Time-Dependence

¡ Time-dependent Harmonic Oscillator e.g., → we can solve this system analytically by the LR method

ˆ H(x, p, t) = ˆ p2 2m(t) + 1 2m(t)ω2(t)ˆ x2

Lewis-Reisenfeld Method

¡ Time-dependent Schroedinger equation ¡ Invariant operator ¡ Eigenvalue problem i~ ∂ ∂tψ = ˆ Hψ

ψ(x, p, t) = ei✏(t)φ(x, p, t)

ˆ I

dˆ I dt = ∂ ˆ I ∂t + 1 i~[ˆ I, ˆ H] = 0

ˆ Iφn(x, p, t) = λnφn(x, p, t)

~˙ ✏ = h n(t) | ✓ i~ @ @t ˆ H ◆ | n(t) i

Time-Dependent NC Solitons

¡ general, quadratic, time-dependent Hamiltonian → apply to NC solitons ¡ creation and annihilation operator ˆ H(ˆ x, ˆ y, t) = A(t)ˆ x2 + B(t)(ˆ xˆ y + ˆ yˆ x) + C(t)ˆ y2 + D(t)ˆ x + E(t)ˆ y + F(t)

[Choi-Gweon (2004)]

ˆ a = r 1 θk1/2 ("√ k 2ρ + i2Bρ − ˙ ρ 2A # (ˆ x − xp(t)) + iρ(ˆ y − yp(t)) ) ˆ a† = r 1 θk1/2 ("√ k 2ρ − i2Bρ − ˙ ρ 2A # (ˆ x − xp(t)) − iρ(ˆ y − yp(t)) )

[ˆ a, ˆ a†] = 1

they satisfy

Time-Dependent NC Solitons

¡ time-dependent circular symmetric solitons

[SK, in progress]

| φn, t i h φn, t |

e−|ζ(t)|2 |ζ(t)|2 n!

|ζ(t)|2 = 1 θk1/2 ( k 4ρ2 (x − xp(t))2 + ✓2Bρ − ˙ ρ 2A ◆ (x − xp(t)) + ρ (y − yp(t)) 2)

Time-Dependent NC Solitons

e.g., Caldirola-Kanai oscillator

[SK, in progress]

ˆ H = eγtˆ x2 + e−γtˆ y2

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