Lattice-gauge theory and Duflo-Weyl quantization Alexander Stottmeister joint work with Arnaud Brothier University of Rome “Tor Vergata” Department of Mathemtics Göttingen February 2, 2018
Inhalt Motivation 1 Time-dependent Born-Oppenheimer approximation Operator-algebraic approaches to lattice-gauge theory Unitary representations of Thompson’s groups The basic construction 2 A projective phase space for lattice-gauge theories 3 Constructions in 1 + 1 dimensions and the infinite tensor product 4 Construction of states and type classification of algebras 5 A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 1 / 27
Motivation 1 Time-dependent Born-Oppenheimer approximation Operator-algebraic approaches to lattice-gauge theory Unitary representations of Thompson’s groups The basic construction 2 A projective phase space for lattice-gauge theories 3 Constructions in 1 + 1 dimensions and the infinite tensor product 4 Construction of states and type classification of algebras 5 A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 2 / 27
Time-dependent Born-Oppenheimer approximation How to extract QFTs on curved backgrounds from quantum gravity? Problems Mathematical framework? → beyond � − → 0+ → Hamiltonian approach Approximate dynamics? → systematics beyonds O ( t ) → no fibered Hamiltonians � ⊕ H ε = dξ H 0 ( ξ ) + f ( − iε ∇ ) ⊗ 1 Main idea Utilize/adapt space-adiabatic perturbation theory [Panati, Spohn,Teufel; 2003] . Basic ingredient Suitable pseudo-differential calculus ( → Equivariant Duflo-Weyl quantization). A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 3 / 27
Time-dependent Born-Oppenheimer approximation Space-adiabatic perturbation theory Wish list (1) Coupled quantum dynamical system ( H , ( ˆ H, D ( ˆ H ))) (2) Splitting of the dynamics (controlled by parameter ε ) H = H slow ⊗ H fast (3) ε -dependent deformation (de)quantization symbols ���� . ε : S ∞ ( ε, , B ( H fast )) ⊂ C ∞ (Γ , B ( H fast )) − � Γ → L ( H ) ���� slow phase space (4) Asymptotic expansion of Hamiltonian symbol (up to smoothing operators S −∞ ) � ∞ ε k H k , H k ∈ S ρ − k H ε ∼ k =0 ε H = � ˆ H ε (5) Conditions on the (point-wise) spectrum σ ∗ ( H 0 ) = { σ ( H 0 ( γ )) } γ ∈ Γ A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 4 / 27
Time-dependent Born-Oppenheimer approximation Space-adiabatic perturbation theory Upshot Construct effective dynamics in H π 0 ( ε -independent subspace). A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 5 / 27
Motivation 1 Time-dependent Born-Oppenheimer approximation Operator-algebraic approaches to lattice-gauge theory Unitary representations of Thompson’s groups The basic construction 2 A projective phase space for lattice-gauge theories 3 Constructions in 1 + 1 dimensions and the infinite tensor product 4 Construction of states and type classification of algebras 5 A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 6 / 27
Operator-algebraic approaches to lattice-gauge theory Hamiltonian formulation [Kogut, Susskind; 1975] Operator-algebraic formulations Mathematical framework → fixed finite lattices [Kijowski, Rudolph; 2002] → fixed infinite lattice [Grundling, Rudolph; 2013] → inductive limit over finite lattices [Arici, Stienstra, van Suijlekom; 2017] Common aspect → Replace the classical edge phase space T ∗ G by the C ∗ -algebra C ( G ) ⋊ G ( G -version of CCR). Problem C ( G ) ⋊ G is not unital. This complicates constructions. Observation Equivariant Duflo-Weyl quantization is related to C ( G ) ⋊ G as well. It requires a unital extension to be well-defined. A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 7 / 27
Motivation 1 Time-dependent Born-Oppenheimer approximation Operator-algebraic approaches to lattice-gauge theory Unitary representations of Thompson’s groups The basic construction 2 A projective phase space for lattice-gauge theories 3 Constructions in 1 + 1 dimensions and the infinite tensor product 4 Construction of states and type classification of algebras 5 A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 8 / 27
Unitary representations of Thompson’s groups Reconstruction of CFTs from subfactors [Jones; 2014] 1+1 dimensional chiral CFTs {A ( I ) } I ⊂ S 1 (conformal net of type III factors) A ( I ) ⊂ B ( I ) , extensions give subfactors → Characterized by algebraic data (planar algebras). Main idea [Jones; 2014] Use planar-algebra data to reconstruct CFTs from subfactors. → Define a functor from binary planar forest to Hilbert spaces. Y �− → ( H 1 → H 2 ) ���� � �� � basic forest “spin doubling” → Gives discrete CFT models (Thompson group symmetry). Observation These discrete CFT models fit into the same framework as those defined by equivariant Duflo-Weyl quantization. Functor ← → Inductive limit over lattices/graphs A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 9 / 27
Motivation 1 Time-dependent Born-Oppenheimer approximation Operator-algebraic approaches to lattice-gauge theory Unitary representations of Thompson’s groups The basic construction 2 A projective phase space for lattice-gauge theories 3 Constructions in 1 + 1 dimensions and the infinite tensor product 4 Construction of states and type classification of algebras 5 A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 10 / 27
The basic construction The elementary phase space Some ingredients Γ will be modeled on T ∗ G . Pseudo-differential calculus for T ∗ G ? → Start from a strict deformation quantization [Rieffel; 1990], [Landsman; 1993] . T ∗ G ∼ = G × g , g = Lie ( G ) , n = dim( G ) . exp : g − → G is onto and locally one-to-one ( U → V ). → Use exp to relate the Haar measure on G and the Lebesgue measure on g : � � dX j ( X ) 2 f (exp( X )) , f ∈ C ∞ dg f ( g ) = c ( V ) V ⊂ G U ⊂ g � sin( α ( H ) / 2) j ( H ) = , H ∈ t ( restriction to a maximal torus ) α ( H ) / 2 α ∈ R + A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 11 / 27
The basic construction Operators from convolution kernels Fibre-wise Fourier transform For X h = exp − 1 ( h ) , σ ∈ C ∞ P W ,U ε ( g )ˆ ⊗ C ∞ ( G ) , U ε = ε − 1 U , define � dθ i F ε σ 1 ε θ ( X h ) σ ( θ, g ) , ε ∈ (0 , 1] . σ ( h, g ) = ˇ ε ( X h , g ) = (2 πε ) n e g ∗ σ ∈ C ∞ ( G )ˆ ⊗ C ∞ ( G ) gives the kernel of a Kohn-Nirenberg-type Ψ DO . → F ε Deform the construction to obtain a Duflo-Weyl-type Ψ DO : √ → Locally: F W,ε ( h, g ) = F ε σ ( h, h − 1 g ) . σ → Globally: Use the wrapping map Φ DW [Dooley, Wildberger; 1993] < Φ DW (ˇ 2 ( . )) g ) , j · exp ∗ f > g , f ∈ C ∞ ( G ) σ 1 σ 1 ε (exp( − 1 ε )( g ) , f > G = < ˇ → L ( L 2 ( G )) Duflo-Weyl formula for C ∞ ( T ∗ G ) − Operators are obtained from the integrated left-regular representation: ( σ ) f ) = < Φ DW (ˇ σ 1 ε )( g ) , ι ∗ R ∗ g f > G , f ∈ C ∞ ( G ) ( Q DW ε A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 12 / 27
The basic construction Properties of the quantization Theorem (generalization of [Landsman; 1993]) ) → K ( L 2 ( G )) ∼ Q DW : C ∞ P W ,U ( g )ˆ ⊗ C ∞ ( G ) − = C ( G ) ⋊ G ε is a non-degenerate strict deformation quantization on (0 , 1] w.r.t. to the canonical Poisson structure on T ∗ G . Furthermore, the G -CCR are satisfied: Q DW ( { σ f , σ f ′ } T ∗ G ) = i ε [ Q DW ( σ f ) , Q DW ( σ f ′ )] = 0 , ε ε ε Q DW ε [ Q DW ( σ X ) , Q DW ( { σ X , σ f } T ∗ G ) = i ( σ f )] = R X f, ε ε ε Q DW ( { σ X , σ Y } T ∗ G ) = i ε [ Q DW ( σ X ) , Q DW ( σ Y )] = iεR [ X,Y ] , ε ε ε for σ f ( θ, g ) = f ( g ) , f ∈ C ∞ ( G ) , and σ X ( θ, g ) = θ ( X ) , X ∈ g (momentum map of the Hamiltonian G -action). Pseudo-differential calculus The quantization Q DW allows for a pseudo-differential calculus on T ∗ G . ε → Symbol spaces, asymptotic completeness, star product, etc. → Complications due to the compactness of G . A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 13 / 27
Motivation 1 Time-dependent Born-Oppenheimer approximation Operator-algebraic approaches to lattice-gauge theory Unitary representations of Thompson’s groups The basic construction 2 A projective phase space for lattice-gauge theories 3 Constructions in 1 + 1 dimensions and the infinite tensor product 4 Construction of states and type classification of algebras 5 A. Stottmeister Lattice-gauge theory and Duflo-Weyl quantization Göttingen February 2, 2018 14 / 27
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