The spectral distance on the Moyal plane ( with: E. Cagnache, P . - - PowerPoint PPT Presentation

The spectral distance on the Moyal plane

( with: E. Cagnache, P . Martinetti, J.-C. Wallet — arXiv:0912.0906 )

Francesco D’Andrea

International School for Advanced Studies (SISSA) Via Beirut 2-4, Trieste, Italy

24/02/2010

Laboratoire de Physique Théorique d’Orsay – 24 February 2010

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Introduction to NCG.

◮ Motivation: in many interesting cases, the description of a space as a set of

points (with some additional structure) is inadequate. Examples:

◮ Foliations: e.g. the Kronecker foliation (i.e. the “noncommutative torus”). ◮ “Bad” quotients Y = X/ ∼ , tilings of Rd, Pontrjagin dual of non-abelian G. ◮ more. . . (see e.g. Connes-Marcolli, A walk in the noncommutative garden). ◮ Quantum physics: C✵(M) → K(H) . ◮ Noncommutative there are physical quantities that cannot be

simultaneously measured with arbitrary precision (e.g. ∆x ∆p h/✷ ).

◮ Compact operators have a discrete spectrum, and physical observable

are quantized (e.g. absorption and emission spectra of atoms).

Moyal plane is both a noncommutative and a quantum space. It provides an interesting example to be studied from a “geometric” point of view.

◮ Noncommutative geometry (NCG) provides the tools to study these “spaces”.

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What is NCG?

“The correspondence between geometric spaces and commutative algebras is a familiar and basic idea of algebraic geometry. The pur- pose of this book is to extend the correspondence to the noncommu- tative case in the framework of real analysis.” (A. Connes)

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What is NCG?

“The correspondence between geometric spaces and commutative al- gebras is a familiar and basic idea of algebraic geometry. The purpose

• f this book is to extend the correspondence to the noncommutative

case in the framework of real analysis.” (A. Connes)

The aim of NCG is to translate (differential) geometric properties into algebraic ones, that can be studied with algebraic tools and generalized to noncomm. algebras. Dictionary:

Geometry “is dual to” Algebra

• Loc. comp. Haus. top. spaces X
• Comm. C∗-algebras C(X)

( Gel’fand, 1939 )

Vector bundles E over X Finite projective C(X)-modules

( Serre-Swan, 1962 )

. . . . . . Riemannian spin manifolds

• Comm. real spectral triples

( Connes, 1996 & 2008 ) 3 / 17

Spectral triples.

Definition [Connes, 1985; Connes, 1989; Connes,1995]

The datum (A✱ H✱ D) is called spectral triple iff:

◮ A ⊂ B(H) is an involutive algebra, H a (separable) Hilbert space; ◮ D is a selfadjoint operator on H, a(D + i)−✶ ∈ K(H) and [D✱ a] ∈ B(H) ∀ a ∈ A.

Remarks: D is usually called “Dirac operator”; a(D − λ)−✶ = a(D + i)−✶ ✶ + (λ + i)(D − λ)−✶ is compact for any λ / ∈ ❙♣(D); a spectral triple describes a compact NC-space if ✶ ∈ A. Motivating example: (C∞

✵ (M)✱ L✷(M✱ S)✱ D

/ ). Notice that ✶ ∈ C∞

✵ (M) iff M is compact.

Additional structures:

• Regular spectral triples: abstract pseudodifferential calculus, local index formulas, . . .
• Real spectral triples: reconstruction theorem, . . .

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Spectral triples II.

A noncommutative example (MN(C)✱ MN(C)✱ D) is a unital spectral triple for any choice of D. D = ✵ ⇒ SU(N) Einstein-Yang-Mills field theory [Chamseddine-Connes, 1997].

For a fixed A, there are many spectral triples (A✱ H✱ D). When is (A✱ H✱ D) “non-trivial” ?

◮ Topological condition: the “conformal class” of a spectral triple is a Fredholm

module, this can be paired with the K•(A) using the so-called index map. ! If dim H < ∞ any linear operator is compact, and the index map is identically zero.

◮ Metric condition: a spectral triple induces a metric on S(A). The study of metric

properties allows to select interesting D even when dim H < ∞.

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The metric aspect of NCG.

For a ∈ A let

◮ δa := a ⊗ ✶ − ✶ ⊗ a the universal differential of a; ◮ LipD(a) := ||[D✱ a]||op the norm of the ✶-form [D✱ a] ∈ Ω✶

D.

Let S(A) be the set of positive linear functionals on A with norm ✶. It is a convex set, expreme points are called “pure states”. S(A) with weak* topology (i.e. µn → µ iff µn(a) → µ(a) ∀ a) is a bounded subset of A′.

Definition [Connes, 1994]

A spectral triple (A✱ H✱ D) induces a distance on S(A) given by: d(µ✱ ν) := supa∈As.a.

• µ ⊗ ν(δa) : LipD(a) ✶
• ✱

µ✱ ν ∈ S(A) ✳ (S(A)✱ d) is a metric space, except that d(µ✱ ν) may be +∞ (e.g. A = C∞

✵ (M) with M

disconnected). It is always geodesically complete: ∀ µ✵✱ µ✶ ∃ [✵✱ ✶] ∋ t → µt ∈ S(A) s.t. d(µt✱ µs) = (t − s)d(µ✵✱ µ✶) ∀ ✵ s t ✶ ✳ For example, straight lines µt = (✶ − t)µ✵ + tµ✶ are geodesics.

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A commutative example.

If A = C∞

✵ (M) , with M a Riemannian spin manifold without boundary, and D = D

/ is the Dirac operator:

◮ states are probability distributions (normalized measures) on M; ◮ pure states are points x✱ y✱ ✳ ✳ ✳ ∈ M (delta distributions); ◮ LipD coincides with the Lipschitz semi-norm Lipρ associated to the Riemannian

metric ρ of M, that is Lipρ(f) := supx=y|f(x) − f(y)|/ρ(x✱ y) ❀

◮ d(x✱ y) ≡ ρ(x✱ y) coincides with the geodesic distance of M; ◮ if M is complete, d(µ✱ ν) is the minimum cost for a transport from µ to ν (Kantorovich).

More generally, any compact metric space (X✱ ρ) can be reconstructed from the pair (C(X✱ R)✱ Lipρ) , X as the spectrum of the algebra and ρ from the formula ρ(x✱ y) = sup

• f(x) − f(y) : Lipρ(f) ✶
• ✳

This motivates the following definition. . .

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Compact quantum metric spaces.

If ✶ ∈ A ⊂ B(H) , the set As.a. := {a = a∗ ∈ A} is an order-unit space. Any order-unit space arises in this way.

Definition [Rieffel, 1999]

A compact quantum metric space (CQMS) is an order-unit space As.a. equipped with a semi-norm ||| ✳ ||| : As.a. → R such that i) |||✶||| = ✵ ; ii) the topology on S(A) induced by the distance ρ(µ✱ ν) := supa∈As.a.

• µ ⊗ ν(δa) : |||a||| ✶
• is the weak* topology.

If |||a||| = ||[D✱ a]||op , then ρ(µ✱ ν) ≡ d(µ✱ ν) is the spectral distance. ⇒ i) is automatically satisfied, but ii) may be not.

(e.g. (MN(C)✱ MN(C)✱ ✵) has d(µ✱ ν) = +∞ ∀ µ = ν, but S(A) is bounded in the weak* topology)

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The Moyal plane (dim = ✷).

The most famous quantization of R✷ is obtained by replacing x = (x✶✱ x✷) ∈ R✷ with ✂ x✶✱ ✂ x✷ generators of the Heisenberg algebra of 1D quantum mechanics [✂ x✶✱ ✂ x✷] = iθ ✳ Bounded operator approach [Groenewold 1946, Moyal 1949]: let Aθ := (S(R✷)✱ ∗θ) with (f ∗θ g)(x) := ✶ (πθ)✷

• f(x + y)g(x + z)e

✷i θ ω(y✱z)❞✷y ❞✷z ✱

with ω = standard symplectic form. Given a tempered distribution T ∈ S′(R✷) define: f ∗θ T✱ g :=

• T✱ ✠

f ∗θ g

• ✱

T ∗θ f✱ g :=

• T✱ g ∗θ ✠

f

• ✳

The Moyal multiplier algebra is: M(Aθ) :=

• T : T ∗θ f✱ f ∗θ T ∈ S(R✷) ∀ f ∈ S(R✷)
• .

It turns out that x✶✱ x✷ ∈ M(Aθ) and x✶ ∗θ x✷ − x✷ ∗θ x✶ = iθ. Many names associated to ∗θ:

◮ Gracia-Bondía, Várilly “Algebras of distributions suitable for phase-space quantum mechanics”. ◮ Rieffel strict deformation quantization for action of Rn (generalized by Bieliavsky to some non-abelian G) ◮ θ-deformations Connes, Landi, Dubois-Violette, . . . ◮ Noncommutative quantum field theory e.g. Grosse, Wulkenhaar, Rivasseau, Wallet, . . . ◮ “Moyal planes are NC-manifolds” Gayral, Gracia-Bondía, Iochum, Schücker, Varilly. 9 / 17

A spectral triple for Moyal plane.

Let H := L✷(R✷) ⊗ C✷ , D = D / the classical Dirac operator of R✷: D = −i √ ✷ ✵ ∂+ ∂− ✵

• ✱

√ ✷ ∂± := ∂ ∂x✶ ± i ∂ ∂x✷ ✱ and πθ : Aθ → B(H) given by πθ(f)ψ = (f ∗θ ψ✶✱ f ∗θ ψ✷) ∀ ψ = (ψ✶✱ ψ✷) ∈ H ✳ Since L✷(R✷) ⊂ S′(R✷), the map πθ is well defined. Proposition [Gayral et al., CMP 246, 2004] The datum (Aθ✱ H✱ D) is a spectral triple. Notice that ∂±(f ∗θ g) = (∂±f) ∗θ g + f ∗θ (∂±g) ✱ i.e. [∂±✱ f∗θ] = (∂±f)∗θ and so [D✱ πθ(f)] is clearly bounded. The check of the compact resolvent condition is easier in the oscillator basis. . .

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Moyal triple in the oscillator basis.

A orthogonal basis {fmn} of L✷(R✷) (normalized as ||fmn||✷ = √ ✷πθ ) is determined by H ∗θ fmn = θ(✷m + ✶)fmn ✱ fmn ∗θ H = θ(✷n + ✶)fmn ✱ m✱ n ∈ N ✱ with H = ✶

✷(x✷ ✶ + x✷ ✷) the Hamiltonian of the harmonic oscillator. Moreover:

◮ fmn ∈ S(R✷) and there is an isomorphism of Fréchet pre-C∗-algebras Aθ → S(ℓ✷(N))

(with standard seminorms) given by a → amn = ✶ ✷πθ

• a(x)fmn(x)❞✷x ✱

(amn) → a =

m✱namnfmn ❀

◮ L✷(R✷) → L✷(ℓ✷(N)) with Hilbert-Schmidt inner product A✱ BHS := ✷πθ Tr(A†B). ◮ ∂± becomes the operator A → ±[X±✱ A], where

X− := ✶ √ θ     

✵ ✵ ✵ ✳ ✳ ✳ ✶ ✵ ✵ ✳ ✳ ✳ ✵ √ ✷ ✵ ✳ ✳ ✳ ✵ ✵ √ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

     ✱ X+ := tX− ❀

◮ S(ℓ✷(N)) = K and pure states are rays in ℓ✷(N) (also, a(D + i)−✶ ∈ K·B(H) = K).

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Some results on the spectral distance.

If ψ ∈ ℓ✷(N) is a unit vector, ωψ(a) :=

• m✱n

✠ ψmamnψn is a pure state. If ψ = en is the n-th basis vector, the associated state corresponds to the n-th energy level of the quantum harmonic oscillator ωn(a) = ann ✳ Proposition For all m < n d(ωm✱ ωn) =

• θ

✷

n

• k=m+✶

✶ √ k ✳

• Proof. Three steps:

1

d(ωm✱ ωn) = supa=a∗

• amm − ann : ||[X−✱ a]||op ✷− ✶

✷

❀

2

amm − ann = n

k=m+✶ ak−✶✱k−✶ − akk = n k=m+✶

• θ

k [X−✱ a]k✱k−✶ ❀

3

||A||op = maxp✱q|Ap✱q| if A has only one non-vanishing diagonal.

• 12 / 17

Proposition

For any two unit vectors ψ✱ ψ′ ∈ ℓ✷(N), d(ωψ✱ ωψ′)

• θ

✷

• p<kq

✶ √ k (|ψpψ′

q|✷ − |ψqψ′ p|✷)

• ✳

Corollary

Consider the following two unit vectors: ψq = δq✱✵ ✱ ψ′

q = (ζ(s)qs)− ✶

✷ ∀ q = ✵ ✱ ψ′

✵ = ✵ ✱

where s > ✶ and ζ(s) is Riemann zeta-function. If s ✸/✷, then d(ωψ✱ ωψ′) = +∞.

• Proof. From the above lower bound we get

d(ωψ✱ ωψ′) ζ(s)−✶

• θ

✷

• ✶kq

✶ qs√ k ζ(s)−✶

• θ

✷

• q✶

q

✶ ✷ −s✳

This series in the r.h.s is divergent if s ✸/✷.

• ⇒ the topology induced by the spectral distance is not the weak* topology.

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Truncation of the Moyal spectral triple.

A spectral triple (MN(C)✱ MN(C) ⊗ C✷✱ DN) is given by DN = −i √ ✷ ✵ D+

N

D−

N

✵

• ✱

D±

N(A) = ±[X± N✱ A] ✱

with X−

N :=

✶ √ θ          ✵ ✵ ✵ ✳ ✳ ✳ ✵ ✶ ✵ ✵ ✳ ✳ ✳ ✵ ✵ √ ✷ ✵ ✳ ✳ ✳ ✵ . . . ... ... ... . . . ✵ ✳ ✳ ✳ ✵ √ N − ✶ ✵          ✱ X+

N := tX− N ✳

Comparison with other examples:

◮ D = ✵ (Einstein-Yang-Mills system) any two states are at infinite distance; ◮ D as in [Iochum-Krajewski-Martinetti, 2001] some states are at infinite distance; ◮ D = DN d(µ✱ ν) < ∞ and (MN(C)✱ LipDN) is a CQMS.

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Proposition (MN(C)✱ LipDN) is a CQMS for any N ∈ N.

• Proof. Recall that

d(µ✱ ν) = supa∈VN

• µ(a) − ν(a) : ||D−

N(a)||op ✷− ✶

✷

where we can take a in the set VN ⊂ MN(C) of traceless selfadjoint matrices. Notice that a → ||D−

N(a)||op is a norm on VN (not just a seminorm).

By the Cauchy-Schwarts inequality, the weak* topology is induced by the distance ∆(µ✱ ν) = supa∈VN

• µ(a) − ν(a) : ||a||HS ✶
• ✳

Any two norms on a finite dimensional vector space VN are equivalent. ⇒ ||D−

N( ✳ )||op ∼ || ✳ ||HS ⇒ d ∼ ∆ .

• With some effort d(µ✱ ν) can be explicitly computed, for the N = ✷ case see the preprint:
• E. Cagnache, F. D’A., P

. Martinetti and J.-C. Wallet [arXiv:0912.0906].

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Conclusion.

• For particular states d on Moyal plane can be computed.
• It does not induce the weak* topology.
• For finite dimensional “approximations” we get CQMS.
• The distance between arbitrary states may be computed on M✷(C), M✸(C), etc.

What is the physical interpretation of d? For commutative A it is the minimum work of transport theory. Is there a similar interpretation if A is non-commutative? Do the MN(C) converge to something for N → ∞, e.g. to Moyal plane (w.r.t. the Gromov-Hausdorff distance, as explained in [Rieffel, Mem. AMS 168, 2004]) ? What about the large scale structure? Coherent states ϕx are parametrized by points x ∈ R✷. Is d(ϕx✱ ϕy) ∼ |x − y| when |x − y| ≫ θ✶/✷ ? (quantum effects should be relevant only at short distances)

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Thank you for your attention.

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