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 1 / 17

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 R d , 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”. 2 / 17

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) 3 / 17

What is NCG? “The correspondence between geometric spaces and commutative al- gebras is a familiar and basic idea of algebraic geometry. The purpose of 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 Comm. C ∗ -algebras C ( X ) Loc. comp. Haus. top. spaces 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 . ✵ ( M ) ✱ L ✷ ( M ✱ S ) ✱ D Motivating example: ( C ∞ / ) . Notice that ✶ ∈ C ∞ ✵ ( M ) iff M is compact. Additional structures: � Regular spectral triples: abstract pseudodifferential calculus, local index formulas, . . . � Real spectral triples: reconstruction theorem, . . . 4 / 17

Spectral triples II. A noncommutative example ( M N ( C ) ✱ M N ( 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 < ∞ . 5 / 17

The metric aspect of NCG. For a ∈ A let ◮ δa := a ⊗ ✶ − ✶ ⊗ a the universal differential of a ; ◮ Lip D ( 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. µ n → µ iff µ n ( 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 ) := sup a ∈ A s.a. µ ⊗ ν ( δa ) : Lip D ( 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 �→ µ t ∈ S ( A ) s.t. d ( µ t ✱ µ s ) = ( t − s ) d ( µ ✵ ✱ µ ✶ ) ∀ ✵ � s � t � ✶ ✳ For example, straight lines µ t = ( ✶ − t ) µ ✵ + tµ ✶ are geodesics. 6 / 17

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); ◮ Lip D coincides with the Lipschitz semi-norm Lip ρ associated to the Riemannian metric ρ of M , that is Lip ρ ( f ) := sup x � = 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 ) ✱ Lip ρ ) , X as the spectrum of the algebra and ρ from the formula � � ρ ( x ✱ y ) = sup f ( x ) − f ( y ) : Lip ρ ( f ) � ✶ ✳ This motivates the following definition. . . 7 / 17

Compact quantum metric spaces. If ✶ ∈ A ⊂ B ( H ) , the set A s.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 A s.a. equipped with a semi-norm ||| ✳ ||| : A s.a. → R such that i) ||| ✶ ||| = ✵ ; ii) the topology on S ( A ) induced by the distance � � ρ ( µ ✱ ν ) := sup a ∈ A s.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. ( M N ( C ) ✱ M N ( C ) ✱ ✵ ) has d ( µ ✱ ν ) = + ∞ ∀ µ � = ν , but S ( A ) is bounded in the weak* topology) 8 / 17

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 A θ := ( S ( R ✷ ) ✱ ∗ θ ) with � ✶ ✷ i θ ω ( y ✱ z ) ❞ ✷ y ❞ ✷ z ✱ ( f ∗ θ g )( x ) := f ( x + y ) g ( x + z ) e ( πθ ) ✷ with ω = standard symplectic form. Given a tempered distribution T ∈ S ′ ( R ✷ ) define: � T ✱ ✠ � � T ✱ g ∗ θ ✠ � � f ∗ θ T ✱ g � := f ∗ θ g � T ∗ θ f ✱ g � := ✱ f ✳ � � T : T ∗ θ f ✱ f ∗ θ T ∈ S ( R ✷ ) ∀ f ∈ S ( R ✷ ) The Moyal multiplier algebra is: M ( A θ ) := . It turns out that x ✶ ✱ x ✷ ∈ M ( A θ ) 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 R n (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 ✷ : � ✵ √ √ ∂ + � ∂ ± i ∂ D = − i ✷ ∂ ± := ✷ ✱ ✱ ∂x ✶ ∂x ✷ ∂ − ✵ and π θ : A θ → B ( H ) given by π θ ( f ) ψ = ( f ∗ θ ψ ✶ ✱ f ∗ θ ψ ✷ ) ∀ ψ = ( ψ ✶ ✱ ψ ✷ ) ∈ H ✳ Since L ✷ ( R ✷ ) ⊂ S ′ ( R ✷ ) , the map π θ is well defined. Proposition [Gayral et al., CMP 246, 2004] The datum ( A θ ✱ 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 ✱ π θ ( f )] is clearly bounded. The check of the compact resolvent condition is easier in the oscillator basis. . . 10 / 17