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 1 / 22
Introduction to nc-geometry. ◮ Geometry. Ancient Greek: γεωµετρ ´ ı α (geometr´ ıa), from γ ˜ η (geo-, “earth, land”) + µετρ ´ ı α (-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 ones, 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. 2 / 22
Spectral triples. In nc-geometry (` a la Connes), spaces are replaced by spectral triples. . . Definition Example: the unit 2 -sphere S 2 A spectral triple is given by: ◮ H = L 2 ( S 2 ) ⊗ C 2 ◮ a separable Hilbert space H ; ◮ A = C ∞ ( S 2 ) ◮ an algebra A of (bounded) operators on H ; ◮ D = σ 1 J 1 + σ 2 J 2 + σ 3 J 3 , where ◮ a (unbounded) selfadjoint operator D on H σ j ’s are Pauli matrices and in such that a ( D + i ) − 1 is compact and [ D , a ] is cartesian coordinates: bounded, for all a ∈ A . ∂ J j = i ǫ jkl x k . It is called unital if 1 ∈ A . ∂x l Remarks: D is usually called “Dirac operator”; under some additional conditions � any commutative spectral triple is of the form 0 ( M ) , L 2 ( M , S ) , D ( C ∞ / ) [Connes, 2008]. Notice that 1 ∈ C ∞ 0 ( M ) iff M is compact. 3 / 22
Spectral triples II. A noncommutative example ( M N ( C ) , M N ( 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 < ∞ . 4 / 22
The metric aspect of NCG. For a ∈ A let ◮ δa := a ⊗ 1 − 1 ⊗ a the universal differential of a ; ◮ Lip D ( a ) := � [ D , a ] � op the norm of the 1 -form [ D , a ] ∈ Ω 1 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. 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 A , D ( µ , ν ) := sup a ∈ A s.a. µ ⊗ ν ( δa ) : Lip D ( a ) � 1 , µ , ν ∈ S ( A ) . ( S ( A ) , d A , D ) is an extended metric space, i.e. d A , D ( µ , ν ) may be + ∞ (e.g. A = C ∞ 0 ( M ) with M disconnected). Connected components of S ( A ) are ordinary metric spaces. 5 / 22
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 H ϕ = L 2 ( A , ϕ ) is � a � 2 ϕ = ϕ ( a ∗ a ) . Can we compare H ϕ and H ψ ? We have ψ � d A , D ( ϕ , ψ ) Lip D ( a ∗ a ) � a � 2 ϕ − � a � 2 with d A , D ( ϕ , ψ ) ⇒ independent of a ∈ A ; Lip D ( a ) ⇒ independent of ϕ , ψ . 6 / 22
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, δ x , δ y , . . . ); ◮ 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 A , D ( x , y ) ≡ ρ ( x , y ) coincides with the geodesic distance of M ; ◮ if M is complete, d A , 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 ) , 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 ) � 1 . This motivates the following definition. . . 7 / 22
Compact quantum metric spaces. If 1 ∈ 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 L : A s.a. → R such that i) L ( 1 ) = 0 ; ii) the topology on S ( A ) induced by the distance � � ρ ( µ , ν ) := sup a ∈ A s.a. µ ⊗ ν ( δa ) : L ( a ) � 1 is the weak* topology. If L ( a ) = � [ D , a ] � op , then ρ ( µ , ν ) ≡ d A , D ( µ , ν ) is Connes’ distance. ⇒ i) is automatically satisfied by any unital spectral triple, but ii) may be not. (e.g. ( M N ( C ) , M N ( C ) , 0 ) has d A , D ( µ , ν ) = + ∞ ∀ µ � = ν , but S ( A ) is connected in the weak* topology) 8 / 22
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 .” 9 / 22
Noncommutativity and quantization Balmer series (hydrogen emission spectrum in the visible region) ◮ Quantum physics: C 0 ( 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. 10 / 22
The Moyal plane. The most famous quantization of R 2 is obtained by replacing x = ( x 1 , x 2 ) ∈ R 2 with ˆ x 1 , ˆ x 2 generators of the Heisenberg algebra of 1D quantum mechanics [ ˆ x 1 , ˆ x 2 ] = iθ . Bounded operator approach [Groenewold 1946, Moyal 1949]: let A θ := ( S ( R 2 ) , ∗ θ ) with � 1 2 i θ ω ( y , z ) d 2 y d 2 z , ( f ∗ θ g )( x ) := f ( x + y ) g ( x + z ) e ( πθ ) 2 with ω = standard symplectic form. Given a tempered distribution T ∈ S ′ ( R 2 ) 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 2 ) ∀ f ∈ S ( R 2 ) The Moyal multiplier algebra is: M ( A θ ) := . It turns out that x 1 , x 2 ∈ M ( A θ ) and x 1 ∗ θ x 2 − x 2 ∗ θ x 1 = iθ . Many names associated to ∗ θ : ◮ Gracia-Bond´ arilly � “Algebras of distributions suitable for phase-space quantum mechanics”. ıa, V´ ◮ Rieffel � strict deformation quantization for action of R n . ◮ θ -deformations � Connes, Landi, Dubois-Violette, . . . ◮ “Moyal planes are NC-manifolds” � Gayral, Gracia-Bond´ ıa, Iochum, Sch¨ ucker, Varilly. 11 / 22
A spectral triple for Moyal plane. Let H := L 2 ( R 2 ) ⊗ C 2 , D = D / the classical Dirac operator of R 2 : � 0 √ √ ∂ + � ∂ ± i ∂ D = − i 2 , 2 ∂ ± := , ∂x 1 ∂x 2 ∂ − 0 and π θ : A θ → B ( H ) given by π θ ( f ) ψ = ( f ∗ θ ψ 1 , f ∗ θ ψ 2 ) ∀ ψ = ( ψ 1 , ψ 2 ) ∈ H . Since L 2 ( R 2 ) ⊂ S ′ ( R 2 ) , 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. 12 / 22
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