Metric Aspects of the Moyal Algebra ( with: E. Cagnache, P . - - PowerPoint PPT Presentation

Metric Aspects of the Moyal Algebra

( with: E. Cagnache, P . Martinetti, J.-C. Wallet — J. Geom. Phys. 2011 )

Francesco D’Andrea

Department of Mathematics and Applications, University of Naples Federico II P .le Tecchio 80, Naples, Italy

15/07/2011

Noncommutative Geometry Days in Istanbul – 12-15 July 2011

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Introduction to nc-geometry.

◮ Geometry. Ancient Greek: γεωµετρ´

ıα (geometr´ ıa), from γ˜ η (geo-, “earth, land”) + µετρ´ ıα (-metria, “measurement”).

◮ Quantum physics: manifolds are replaced by operator

• algebras. In typical examples, e.g. the phase space of

quantum mechanics, one has “spaces with no points”.

(from www.gps.oma.be) ◮ Noncommutative geometry provides the mathematical tools to study these “spaces”.

The aim of nc-geometry is to translate (differential) geometric properties into algebraic

• nes, that can be studied with algebraic tools and generalized to noncomm. algebras.

◮ How do we perform a measurement on a quantum space? We measure spectra.

A basic idea of nc-geometry is that the metric properties of spaces can be encoded into the spectrum of a special operator, called (generalized) Dirac operator.

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Spectral triples.

In nc-geometry (` a la Connes), spaces are replaced by spectral triples. . .

Definition

A spectral triple is given by:

◮ a separable Hilbert space H; ◮ an algebra A of (bounded) operators on H; ◮ a (unbounded) selfadjoint operator D on H

such that a(D + i)−1 is compact and [D, a] is bounded, for all a ∈ A. It is called unital if 1 ∈ A. Example: the unit 2-sphere S2

◮ H = L2(S2) ⊗ C2 ◮ A = C∞(S2) ◮ D = σ1J1 + σ2J2 + σ3J3 , where

σj’s are Pauli matrices and in cartesian coordinates: Jj = iǫjklxk ∂ ∂xl . Remarks: D is usually called “Dirac operator”; under some additional conditions any commutative spectral triple is of the form (C∞

0 (M), L2(M, S), D

/ ) [Connes, 2008]. Notice that 1 ∈ C∞

0 (M) iff M is compact.

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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 = 0 ⇒ 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. One way to select “interesting” D is to require that the index map is non-trivial. ! 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 ⊗ 1 − 1 ⊗ a the universal differential of a; ◮ LipD(a) := [D, a]op the norm of the 1-form [D, a] ∈ Ω1

D.

Let S(A) be the set of positive linear functionals on A with norm 1. 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: dA,D(µ, ν) := supa∈As.a.

• µ ⊗ ν(δa) : LipD(a) 1
• ,

µ, ν ∈ S(A) . (S(A), dA,D) is an extended metric space, i.e. dA,D(µ, ν) may be +∞ (e.g. A = C∞

0 (M)

with M disconnected). Connected components of S(A) are ordinary metric spaces.

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Spectral distance and representation theory.

There is a correspondence between states and (cyclic) representations

Gelfand-Naimark-Segal construction.

If ϕ : A → C is a state, the norm on Hϕ = L2(A, ϕ) is a2

ϕ = ϕ(a∗a) .

Can we compare Hϕ and Hψ? We have a2

ϕ − a2 ψ dA,D(ϕ, ψ) LipD(a∗a)

with dA,D(ϕ, ψ) ⇒ independent of a ∈ A; LipD(a) ⇒ independent of ϕ, ψ.

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A commutative example.

If A = C∞

0 (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, δx, δy, . . .); ◮ 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) ;

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

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) 1
• .

This motivates the following definition. . .

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Compact quantum metric spaces.

If 1 ∈ 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 L : As.a. → R such that i) L(1) = 0 ; ii) the topology on S(A) induced by the distance ρ(µ, ν) := supa∈As.a.

• µ ⊗ ν(δa) : L(a) 1
• is the weak* topology.

If L(a) = [D, a]op , then ρ(µ, ν) ≡ dA,D(µ, ν) is Connes’ distance. ⇒ i) is automatically satisfied by any unital spectral triple, but ii) may be not.

(e.g. (MN(C), MN(C), 0) has dA,D(µ, ν) = +∞ ∀ µ = ν, but S(A) is connected in the weak* topology)

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Spectral metric spaces.

Rieffel’s notion of compact quantum metric space has been adapted to the non-compact case, i.e. for non-unital algebras, by Latr´ emoli` ere [Taiwanese J. Math. 2007]. This leads to the recent definition of spectral metric space in: J.V. Bellissard, M. Marcolli and K. Reihani, Dynamical systems on spectral metric spaces, arXiv:1008.4617 [math.OA]. Quoting B-M-R: “A spectral metric space is a spectral triple (A, H, D) with additional properties which guaranty that the Connes metric induces the weak*-topology on the state space of A.”

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Noncommutativity and quantization

Balmer series (hydrogen emission spectrum in the visible region) ◮ Quantum physics: C0(M) → K(H) ◮ Quantum vs. noncommutative:

◮ Noncommutativity there are physical quantities that cannot be simultaneously

measured with arbitrary precision (e.g. ∆x ∆p |[x, p]| /2 = h/2 ).

◮ Compact operators have a discrete spectrum, and the corresponding

physical observables 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.

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The Moyal plane.

The most famous quantization of R2 is obtained by replacing x = (x1, x2) ∈ R2 with ˆ x1, ˆ x2 generators of the Heisenberg algebra of 1D quantum mechanics [ˆ x1, ˆ x2] = iθ . Bounded operator approach [Groenewold 1946, Moyal 1949]: let Aθ := (S(R2), ∗θ) with (f ∗θ g)(x) := 1 (πθ)2

• f(x + y)g(x + z)e

2i θ ω(y,z)d2y d2z ,

with ω = standard symplectic form. Given a tempered distribution T ∈ S′(R2) 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(R2) ∀ f ∈ S(R2)
• .

It turns out that x1, x2 ∈ M(Aθ) and x1 ∗θ x2 − x2 ∗θ x1 = iθ. Many names associated to ∗θ:

◮ Gracia-Bond´

ıa, V´ arilly “Algebras of distributions suitable for phase-space quantum mechanics”.

◮ Rieffel strict deformation quantization for action of Rn. ◮ θ-deformations Connes, Landi, Dubois-Violette, . . . ◮ “Moyal planes are NC-manifolds” Gayral, Gracia-Bond´

ıa, Iochum, Sch¨ ucker, Varilly.

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A spectral triple for Moyal plane.

Let H := L2(R2) ⊗ C2 , D = D / the classical Dirac operator of R2: D = −i √ 2 ∂+ ∂−

• ,

√ 2 ∂± := ∂ ∂x1 ± i ∂ ∂x2 , and πθ : Aθ → B(H) given by πθ(f)ψ = (f ∗θ ψ1, f ∗θ ψ2) ∀ ψ = (ψ1, ψ2) ∈ H . Since L2(R2) ⊂ S′(R2), 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.

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Moyal spectral triple in the oscillator basis.

A orthogonal basis {fmn} of L2(R2) (normalized as fmn2 = √ 2πθ ) is determined by H ∗θ fmn = θ(2m + 1)fmn , fmn ∗θ H = θ(2n + 1)fmn , m, n ∈ N , with H = 1

2(x2 1 + x2 2) the Hamiltonian of the harmonic oscillator. Moreover:

◮ fmn ∈ S(R2) and there is an isomorphism of Fr´

echet pre-C∗-algebras Aθ → S(ℓ2(N)) (with standard seminorms) given by a → amn = 1 2πθ

• a(x)fmn(x)d2x ,

(amn) → a =

m,namnfmn ;

◮ L2(R2) → L2(ℓ2(N)) with Hilbert-Schmidt inner product A, BHS := 2πθ Tr(A†B). ◮ ∂± becomes the operator A → ±[X±, A], where

X− := 1 √ θ     

. . . 1 . . . √ 2 . . . √ 3 . . . . . . . . . . . . . . .

     , X+ := tX− ;

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States on the Moyal plane.

Recall that Aθ = S(ℓ2(N)). Notice that K ⊃ ¯ Aθ ⊃ Aθ ⊃ M∞(C) :=

• k1 Mk(C)

where Mk(C) are identified with a = ((amn)) ∈ A such that amn = 0 if m k or n k. Thus ¯ Aθ = K is the C∗-algebra of compact operators on ℓ2(N). A density matrix R (on ℓ2(N)) is a positive trace-class operators on ℓ2(N) with trace 1. A normal state ωR ∈ S(Aθ) is a state that can be written as ωR(a) = Tr(Ra) . For ¯ Aθ = K, all states are normal and the weak* topology on S(Aθ) is equivalent to the uniform topology induced by the trace norm, T1 := Tr |T| for all traceclass T.

◮ S(Aθ) is path-connected for the weak* topology. ◮ Let (X, d) be an extended metric space. If d(µ, ν) = ∞, the points µ, ν ∈ X are in

different connected components.

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Some results on the Connes’ distance.

Pure states are rays in ℓ2(N). If ψ ∈ ℓ2(N) is a unit vector, ωψ(a) :=

• m,n

¯ ψmamnψn is a pure state (here R = ψψ†). 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 dA,D(ωm, ωn) =

• θ

2

n

• k=m+1

1 √ k .

• Proof. Three steps:

1

dA,D(ωm, ωn) = supa=a∗

• amm − ann : [X−, a]op 2− 1

2

;

2

amm − ann = n

k=m+1(ak−1,k−1 − akk) = n k=m+1

• θ

k [X−, a]k,k−1 ;

3

Aop = supp,q|Ap,q| if A has only one non-vanishing diagonal.

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Proposition

For any two unit vectors ψ, ψ′ ∈ ℓ2(N), dA,D(ωψ, ωψ′)

• θ

2

• p<kq

1 √ k (|ψpψ′

q|2 − |ψqψ′ p|2)

• .

Corollary

Consider the following two unit vectors: ψq = δq,0 , ψ′

q = (ζ(s)qs)− 1

2 ∀ q = 0 , ψ′

0 = 0 ,

where s > 1 and ζ(s) is Riemann zeta-function. If s 3/2, then dA,D(ωψ, ωψ′) = +∞.

• Proof. From the above lower bound we get

dA,D(ωψ, ωψ′) ζ(s)−1

• θ

2

• 1kq

1 qs√ k ζ(s)−1

• θ

2

• q1

q

1 2 −s.

This series in the r.h.s is divergent if s 3/2.

• ⇒ the topology induced by Connes’ distance is not the weak* topology (i.e. Moyal plane is

not a spectral metric space).

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Truncation of the Moyal spectral triple.

A spectral triple (MN(C), MN(C) ⊗ C2, DN) is given by DN = −i √ 2 D+

N

D−

N

• ,

D±

N(A) = ±[X± N, A] ,

with X−

N :=

1 √ θ          . . . 1 . . . √ 2 . . . . . . ... ... ... . . . . . . √ N − 1          , X+

N := tX− N .

Comparison with other examples:

◮ D = 0 (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 dA,D(µ, ν) < ∞ and (MN(C), LipDN) is a CQMS.

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Truncation of spectral triples.

In quantum field theory (on a closed Riemannian spin manifold), the partition function for fermions coupled with a gauge field is formally given by “ Z(D, A) = det(DA) ′′ where A is a Connes’ 1-form and DA = D + A + . . . the “gauged” Dirac operator. Regularization [Andrianov-Lizzi, JHEP 2010]: replace H with HN := PNH, where {PN}N1 is a sequence of finite-rank projections such that PNPN′ = PN′PN = PN ∀ N < N′ , PN → 1 in the weak operator topology . This means: HN ⊂ HN′ ∀ N < N′ , and PNv → v ∀ v ∈ H . Consider the subset XN ⊂ N(A) of states given by density matrices R such that RPN = PNR = R . Roughly speaking, the associated state ω ‘sees’ only the subspace HN = PNH. Do the XN converge to N(A) ?

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Some preliminary results.

Assume PN ∈ ¯ A (e.g. in Moyal case, ¯ A = K and any finite-rank operator belongs to K), so that AN = PN ¯ A PN is an algebra. Called DN = PNDPN, assume also that

• a ∈ AN : [DN, a] = 0
• = C · PN
• Then. . .

Theorem

i) (XN, dA,D) is a metric space; ii) (AN, HN, DN) gives a compact quantum metric space; iii) the distances dA,D and dAN,DN on XN are strongly equivalent; iv) XN → N(A) in the weak* topology.

Proof

i)-ii)-iii) All norms on a finite-dimensional vector space are equivalent. . .

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Gromov-Hausdorff distance.

There is a notion of convergence for metric spaces [Gromov, 1981]. Given two metric spaces (X, dX) and (Y, dY), the Gromov-Hausdorff distance is defined by dGH(X, Y) = inf          ε > 0

• (1)

∃ d : (X ⊔ Y) × (X ⊔ Y) → R, metric (2) d

• X×X = dX and d
• Y×Y = dY

(3) ∀ x ∈ X ∃ y ∈ Y : d(x, y) < ε (4) ∀ y ∈ Y ∃ x ∈ X : d(x, y) < ε          Thus, dGH(X, Y) is the infimum of possible ε for which there exists a metric on the disjoint union X ⊔ Y that extends the metrics of X and Y in such a way that any point of X is ε-close to some point of Y and vice versa. Remarks

1 The GH-distance is zero iff (X, dX) and (Y, dY) are isometric. 2 The collection of isometry classes of complete metric spaces, together with the

Gromov-Hausdorff distance, is an extended metric space.

3 The collection of isometry classes of compact metric spaces, together with the

Gromov-Hausdorff distance, is a metric space.

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Quantum Gromov-Hausdorff distance.

The Gromov-Hausdorff distance has been generalized to compact quantum metric spaces by Rieffel. Quoting Rieffel: There is the popular BFSS conjecture in string theory, which conjectures that the putative “M-theory”, which is supposed to unify the various versions of string theory, is a “suitable” limit of theories on matrix algebras [Banks-Fischler- Shenker-Susskind, P .R.D 1997]. One can wonder whether quantum Gromov-Hausdorff distance might have a bit to say in clarifying “suitable”.

Remark (XN, dA,D) are not convergent to (N(A), dA,D) for the quantum Gromov-Hausdorff

• distance. Counterexample: the canonical spectral triple of the manifold N.

In Moyal case, let W ⊂ S(A) be the connected component containing

N XN. Then

(XN, dA,D) are convergent to (W, dA,D) for the quantum Gromov-Hausdorff distance.

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Thank you for your attention.

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