Invariant states on Weyl algebras for the action of the symplectic group

Transcription

1 Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Invariant states on Weyl algebras for the action of the symplectic group Simone Murro Department of Mathematics University of Freiburg AQFT: WHERE OPERATOR ALGEBRA MEETS MICROLOCAL ANALYSIS Cortona, 8th of June 2018 Joint project with Federico Bambozzi and Nicola Pinamonti

2 Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 DYNAMICAL SYSTEM: (A unital -algebra, G group, Φ ergodic group of -automorphisms) QUESTION: How many invariant states we can nd? A Von Neumann, G compact and Abelian Størmer unique G-invariant state A Von Neumann, G compact and Abelian Høegh-Krohn Landstad, Størmer unique G-invariant state ANY MODELS ARE LEFT OUT FROM THIS SCENARIO? Orientation preserving automorphisms of the noncommutative torus (A nc T, Sl(2, Z), Φ) Symplectomorphim of the quantum Hall eect (A QHE, Sp(2g, Z), Φ) POSSIBLE OBSTRACTION! innite Sl(2, Z)-invariant states ω on the (commutative) C -algebra A T following Waldmann's ideas we can "deform" ω so that ω W is state on A NC T 2 the noncommutative torus A NC T 2 is -isomorphic to Weyl C -algebra GOAL: CLASSIFY Sp(2g, Z)-INVARIANT STATES ON WEYL ALGEBRAS

3 Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Outline of the Talk Weyl algebras (I) Construction for Z 2g (II) Automorphism induced by Sp(2g, Z) Sp(2g, Z)-invariant states (I) Denition and main theorem (II) Sketch of the proof Based on: Invariant states on Weyl algebras for the action of the symplectic group Federico Bambozzi, SM, Nicola Pinamonti - (arxiv: [mathoa])

4 Weyl algebras Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Weyl algebras I: Construction for Z 2g Fix h R st := h/2π R \ Q Choose a skew-symmetric, bilinear map σ : Z 2g Z 2g Z ( ) 0 1g g σ := 1 g g 0 Z 2 m W m linear operator on C 0 (Z 2g, C) dened by (W mv)(n) := e ıhσ(m,n) v(n + m) Weyl -algebra A is obtained by endowing V= span C W m m Z 2g } with ( ) W mw n = e ıhσ(m,n) W (n + m) ( ) (W m) = W m Remark: Weyl C -algebra A = (A, ) where the C -norm is given by a := sup ω(a a) ω S A By setting g = 1 and W (1,0) = U, W (0,1) = V, we obtain NC torus UV = e 2ıh VU Since Z 2 Z 2g by m = (m 1, m 2 ) m = (m 1, 0, 0, m 2, 0,, 0), we set g = 1

5 Weyl algebras Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Weyl algebra II: automorphism induced by Sp(2, Z) Symplectic group Sp(2, Z) ( Sl(2, Z) ) acts on Z 2 m Θm, with det Θ=1 Sp(2, Z) Θ Φ θ Aut(A) by linearity: Φ Θ W m = W Θm Set of xed points = (0, 0) Z 2 } = the action of Φ Θ is ergodic on A ( ) Φ Θ λw(0,0) = λw(0,0) for any λ C, Θ Sp(2, Z) Proposition: characterization of Sp(2, Z)-orbits (1) set of orbits of the symplectic group Sp(2, Z) } E := (0, j) j N} (2) Every Sp(2, Z)-orbit of Z 2 contains an element of the form (j, j) with j N Sketch of the proof - part (1): ( a b - Θ Sp(2, Z) Sl(2, Z) takes the form Θ = c d - q = (q 1, q 2 ) Z 2 we have Θq = (aq 1 + bq 2, cq 1 + dq 2 ) ), with ad bc = 1 q1,q2 ===== a,b,c,d Θq = ( ) 0 q 3 > 0 - Let be n i = (0, m i ) st m i 0 and m 1 m 2 and assume Θ Sp(2, Z) st Θn 1 = n 2 - Θn 1 = n 2 = b = 0 det Θ=1 ===== a = d = 1 Θn 1=n2 ===== Θ = Id = m 1 = m 2 m i >0

6 Weyl algebras Weyl algebra II: automorphism induced by Sp(2, Z) Symplectic group Sp(2, Z) ( Sl(2, Z) ) acts on Z 2 m Θm, with det Θ=1 Sp(2, Z) Θ Φ θ Aut(A) by linearity: Φ Θ W m = W Θm Set of xed points = (0, 0) Z 2 } = the action of Φ Θ is ergodic on A ( ) Φ Θ λw(0,0) = λw(0,0) for any λ C, Θ Sp(2, Z) Proposition: characterization of Sp(2, Z)-orbits (1) set of orbits of the symplectic group Sp(2, Z) } E := (0, j) j N} (2) Every Sp(2, Z)-orbit of Z 2 contains an element of the form (j, j) with j N Sketch of the proof - part (2): - Let O be a Sp(2, Z)-orbit - (1) = j = (0, j) O - Choosing Θ = ( ) 1 1, we have Θ j= (j, j) O 0 1 Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10

7 Invariant states Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Sp(2, Z)-invariant states Denitions: A Sp(2, Z)-invariant state ω if for any Φ Θ Aut (A), Θ Sp(2, Z) it holds - ω(a a) 0 - ω(1 A ) = 1 - ω Φ Θ = ω NB: To construct ω, it is enough to prescribe its values on the generators of A 1 if m = (0, 0) ω(w m) = p (m) C else for a sequence of values p (m) and then extend it by linearity to any a A Theorem The only Sp(2, Z)-invariant state on A is the trace state τ dened by 1 if m = (0, 0) τ(w m) = 0 else τ is obviously invariant: τ(φ Θ W m) = τ(w θm ) = 1 if m = (0, 0) 0 else

8 Invariant states Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Sketch of the proof I Assume by contraddiction there exists Sp(2, Z)-invariant state ω (dierent from τ!) 1 if χ = (0, 0) ω(w χ) = p (χ) C else Goal: Sp(2, Z)-invariance p (χ) = 0 for any χ 0= ω τ (1) By positivity of ω = ω(w χ) = ω(w χ ) (2) By Sp(2, Z)-invariance = ω(w χ) = ω(w χ ) = ω(w χ ) = ω(w Idχ ) = ω(w χ) (3) Choosing a = W 0 + W χ ω(a a) 0 ====== 1 p 2 0 (4) Hence, any Sp(2, Z)-invariant states reads as 1 if χ = (0, 0) ω(w χ) = p (χ) [ 1, 1] else Next, we choose a more suitable χ without loosing of generality (5) Fix χ 0 (6) By previous Prop: For any χ there exists Θ Sp(2, Z) st Θχ = ξ = (ξ 1, ξ 1 ) (7) In particular Sp(2, Z)-invariance = ω(w χ) = ω(w ξ ) = p

9 Invariant states Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Sketch of the proof II m n (8) Let m, n N st N and n > 1 and consider V ξ; m,n A with elements of the form a = α 0 W (0,0) + ( 1 + m m (n 1)j α j W Θj ξ with Θ j := n j ) Sp(2, Z) n 1 1 j 1 (9) For any V ξ; m,n, the map a ω(a a) is a quadratic form ω(a a) = α t H α 1 p p p p p p 1 q me ıϕm,n q 2m e 2ıϕm,n q 3m e 3ıϕm,n q 4m e 4ıϕm,n p q me ıϕm,n 1 q me ıϕm,n q 2m e 2ıϕm,n q 3m e 3ıϕm,n p q H = 2m e 2ıϕm,n q me ıϕm,n 1 q me ıϕm,n q 2m e 2ıϕm,n p q 3m e 3ıϕm,n q 2m e 2ıϕm,n q me ıϕm,n 1 q me ıϕmn p q 4m e 4ıϕm,n q 3m e 3ıϕm,n q 2m e 2ıϕm,n q me ıϕm,n 1 where q (i j)m := ω(w Θi ξ Θ j ξ) and ϕ m,n := hmnξ 2 2 (10) Notation: H = [ p, 1, q me ıϕm,n, q 2m e ı2ϕm,n, q 3m e ı3ϕm,n, ]

10 Invariant states Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Sketch of the proof III (11) On (d + 1)-D subspace V d;n V ξ; m,n the restriction of H to V d;n reads H n = [ p, 1, qmeı 2π d n, q 2m e 2ı 2π d n, q 3m e 3ı 2π d n,, q (d 1)m e (d 1)ı 2π d n ] + ε ( idea: := h m is irrational m N big enough st 2π d! N and (h m ξ2 2 ) We can now argue that p has to be 0: (12) We can notice that the set of positive Hermitian matrices form a convex cone (13) The matrix P d := [p; 1; 0; 0; ; 0] can be obtained as the convex combination P d = d n=1 1 d H n mod (2π) 2π d < ε ) 4d 2 (14) For d big enough det(p d ) = 1 dp2 < 0 P n not positive H n not positive (15) Hence p = 0 is a necessary condition for ω being an Sp(2, Z)-invariant state (16) This holds for every m Z 2 = therefore the only Sp(2, Z)-invariant state is τ

11 Outlook Simone Murro (Universität Freiburg) Sp(2, Z)-invariant states AQFT, / 10 Resume & Outlook Weyl -algebra useful used in QM and noncommutative geometry A = span C W m m, n Z 2g, W mw n = e ıhσ(m,n) W (n + m), (W m) = W m } Unique Sp(2, Z)-invariant states on Weyl algebras is 1 if m = 0 τ(w m) = 0 else What comes next? - Weyl -algebra for presymplectic abelian group - Other noncommutative spaces: Moyal space, Connes-Landi Sphere, and most importantly: KLAUS BOOK! THANKS for your attention!