Metric Aspects of the Moyal Algebra

Transcription

1 Metric Aspects of the Moyal Algebra ( with: E. Cagnache, P. Martinetti, J.-C. Wallet J. Geom. Phys ) 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 July / 22

2 Introduction to nc-geometry. Geometry. Ancient Greek: γεωµετρíα (geometría), from γ η (geo-, earth, land ) + µετρíα (-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 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

3 Spectral triples. In nc-geometry (à 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 S 2 H = L 2 (S 2 ) C 2 A = C (S 2 ) D = σ 1 J 1 + σ 2 J 2 + σ 3 J 3, where σ j s are Pauli matrices and in cartesian coordinates: J j = iɛ jkl x k. x l Remarks: D is usually called Dirac operator ; under some additional conditions any commutative spectral triple is of the form (C 0 (M), L 2 (M, S), D/ ) [Connes, 2008]. Notice that 1 C 0 (M) iff M is compact. 3 / 22

4 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

5 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) : LipD (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

6 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 Can we compare H ϕ and H ψ? a 2 ϕ = ϕ(a a). We have a 2 ϕ a 2 ψ d A,D (ϕ, ψ) Lip D (a a) with d A,D (ϕ, ψ) independent of a A; Lip D (a) independent of ϕ, ψ. 6 / 22

7 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

8 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

9 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 Latrémolière [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: [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

10 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

11 The Moyal plane. The most famous quantization of R 2 is obtained by replacing x = (x 1, x 2 ) R 2 with ˆx 1, ˆx 2 generators of the Heisenberg algebra of 1D quantum mechanics [ˆx 1, ˆx 2 ] = iθ. Bounded operator approach [Groenewold 1946, Moyal 1949]: let A θ := (S(R 2 ), θ ) with (f θ g)(x) := 1 f(x + y)g(x + z)e 2i (πθ) 2 θ ω(y,z) d 2 y d 2 z, with ω = standard symplectic form. Given a tempered distribution T S (R 2 ) define: f θ T, g := T, f θ g, T θ f, g := T, g θ f. The Moyal multiplier algebra is: M(A θ ) := { T : T θ f, f θ T S(R 2 ) f S(R 2 ) }. 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-Bondía, Várilly Algebras of distributions suitable for phase-space quantum mechanics. Rieffel strict deformation quantization for action of R n. θ-deformations Connes, Landi, Dubois-Violette,... Moyal planes are NC-manifolds Gayral, Gracia-Bondía, Iochum, Schücker, Varilly. 11 / 22

12 A spectral triple for Moyal plane. Let H := L 2 (R 2 ) C 2, D = D/ the classical Dirac operator of R 2 : and π θ : A θ B(H) given by D = i ( ) 0 + 2, 0 2 ± := ± i, x 1 x 2 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

13 Moyal spectral triple in the oscillator basis. A orthogonal basis {f mn } of L 2 (R 2 ) (normalized as f mn 2 = 2πθ ) is determined by H θ f mn = θ(2m + 1)f mn, f mn θ H = θ(2n + 1)f mn, m, n N, with H = 1 2 (x2 1 + x 2 2) the Hamiltonian of the harmonic oscillator. Moreover: f mn S(R 2 ) and there is an isomorphism of Fréchet pre-c -algebras A θ S(l 2 (N)) (with standard seminorms) given by a a mn = 1 2πθ a(x)f mn (x)d 2 x, (a mn ) a = m,n a mnf mn ; L 2 (R 2 ) L 2 (l 2 (N)) with Hilbert-Schmidt inner product A, B HS := 2πθ Tr(A B). ± becomes the operator A ±[X ±, A], where X := θ , X + := t X ; / 22

14 States on the Moyal plane. Recall that A θ = S(l 2 (N)). Notice that K Āθ A θ M (C) := M k(c) k 1 where M k (C) are identified with a = ((a mn )) A such that a mn = 0 if m k or n k. Thus Āθ = K is the C -algebra of compact operators on l 2 (N). A density matrix R (on l 2 (N)) is a positive trace-class operators on l 2 (N) with trace 1. A normal state ω R S(A θ ) is a state that can be written as ω R (a) = Tr(Ra). For Āθ = K, all states are normal and the weak* topology on S(A θ ) is equivalent to the uniform topology induced by the trace norm, T 1 := Tr T for all traceclass T. S(A θ ) 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. 14 / 22

15 Some results on the Connes distance. Pure states are rays in l 2 (N). If ψ l 2 (N) is a unit vector, ω ψ (a) := ψ m a mn ψ n m,n is a pure state (here R = ψψ ). If ψ = e n is the n-th basis vector, the associated state corresponds to the n-th energy level of the quantum harmonic oscillator ω n (a) = a nn. Proposition For all m < n d A,D (ω m, ω n ) = θ 2 n k=m+1 1 k. Proof. Three steps: } 1 d A,D (ω m, ω n ) = sup a=a { amm a nn : [X, a] op ; 2 a mm a nn = n k=m+1 (a k 1,k 1 a kk ) = n θ k=m+1 [X k, a] k,k 1 ; 3 A op = sup p,q A p,q if A has only one non-vanishing diagonal. 15 / 22

16 Proposition For any two unit vectors ψ, ψ l 2 (N), θ d A,D (ω ψ, ω ψ ) 2 Corollary Consider the following two unit vectors: p<k q 1 ( ψ p ψ q 2 ψ q ψ p 2 ). k ψ q = δ q,0, ψ q = (ζ(s)q s ) 1 2 q 0, ψ 0 = 0, where s > 1 and ζ(s) is Riemann zeta-function. If s 3/2, then d A,D (ω ψ, ω ψ ) = +. Proof. From the above lower bound we get θ d A,D (ω ψ, ω ψ ) ζ(s) 1 1 θ 2 q s ζ(s) 1 q 1 2 s. k 2 This series in the r.h.s is divergent if s 3/2. 1 k q q 1 the topology induced by Connes distance is not the weak* topology (i.e. Moyal plane is not a spectral metric space). 16 / 22

17 Truncation of the Moyal spectral triple. A spectral triple (M N (C), M N (C) C 2, D N ) is given by D N = i ( 0 D + ) N 2, D ± D N N 0 (A) = ±[X± N, A], with X N := 1 θ , X + N := t X N N 1 0 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 = D N d A,D (µ, ν) < and (M N (C), Lip DN ) is a CQMS. 17 / 22

18 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(d A ) where A is a Connes 1-form and D A = D + A +... the gauged Dirac operator. Regularization [Andrianov-Lizzi, JHEP 2010]: replace H with H N := P N H, where {P N } N 1 is a sequence of finite-rank projections such that P N P N = P N P N = P N N < N, P N 1 in the weak operator topology. This means: H N H N N < N, and P N v v v H. Consider the subset X N N(A) of states given by density matrices R such that RP N = P N R = R. Roughly speaking, the associated state ω sees only the subspace H N = P N H. Do the X N converge to N(A)? 18 / 22

19 Some preliminary results. Assume P N Ā (e.g. in Moyal case, Ā = K and any finite-rank operator belongs to K), so that A N = P N Ā P N is an algebra. Called D N = P N DP N, assume also that { a AN : [D N, a] = 0 } = C P N Then... Theorem i) (X N, d A,D ) is a metric space; ii) (A N, H N, D N ) gives a compact quantum metric space; iii) the distances d A,D and d AN,D N on X N are strongly equivalent; iv) X N N(A) in the weak* topology. Proof i)-ii)-iii) All norms on a finite-dimensional vector space are equivalent / 22

20 Gromov-Hausdorff distance. There is a notion of convergence for metric spaces [Gromov, 1981]. Given two metric spaces (X, d X ) and (Y, d Y ), the Gromov-Hausdorff distance is defined by (1) d : (X Y) (X Y) R, metric (2) d d GH (X, Y) = inf ε > 0 X X = d X and d Y Y = d Y (3) x X y Y : d(x, y) < ε (4) y Y x X : d(x, y) < ε Thus, d GH (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, d X ) and (Y, d Y ) 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. 20 / 22

21 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 (X N, d A,D ) are not convergent to (N(A), d A,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 X N. Then (X N, d A,D ) are convergent to (W, d A,D ) for the quantum Gromov-Hausdorff distance. 21 / 22

22 Thank you for your attention. 22 / 22