Deformations of Operator Algebras and the Construction of Quantum - PowerPoint PPT Presentation

Deformations of Operator Algebras and the Construction of Quantum Field Theories Gandalf Lechner Department of Physics, University of Vienna AQFT – The first 50 years, Uni G¨ ottingen Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 1 / 20

Algebraic Quantum Field Theory Algebraic QFT has in the past mainly focussed on the analysis of general, model-independent properties of quantum field theories / nets of algebras Many tools to extract physical data from a given net are available today (particle content, cross sections, charges, short distance behaviour, and many more ...) But, as in any approach to QFT, the rigorous construction of models is still a challenging problem (in particular in d = 4) Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 2 / 20

Some Constructive Approaches to QFT In perturbative setting, use classical Lagrangean as input, then perturbative renormalization [ → Fredenhagen’s talk] Good quantum description of possible interactions still missing Exception: Integrable models in d = 2 with S-matrix simple enough to be taken as an input [Schroer 97-01, GL 03, Buchholz/GL 04, GL 08] In algebraic QFT, individual models can be desribed by algebraic data (i.e. half-sided inclusions for conformal QFTs on the circle) [ → Longo’s talk] R d without In this talk, focus on the construction of models on I conformal symmetry. Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 3 / 20

Wedges In the following, wedge regions play a significant role. The right wedge R d : x 1 > | x 0 |} W R := { x ∈ I General wedge: Poincar´ e transform W = Λ W R + x . “Wedges are big enough to allow for simple observables being localized in them, but also small enough so that two of them can be spacelike separated” Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 4 / 20

Local nets and wedge algebras Local nets can be constructed from a single algebra (“wedge algebra”) and an action of the Poincar´ e group. Let B a C ∗ -algebra with automorphic Poincar´ e action α and C ∗ -subalgebra A ⊂ B such that α x , Λ ( A ) ⊂ A for ( x , Λ) with Λ W R + x ⊂ W R “isotony condition” α x , Λ ( A ) ⊂ A ′ for ( x , Λ) with Λ W R + x ⊂ W ′ “locality condition” R The system A ⊂ B , α will be called a wedge algebra. Then Λ W R + x �− → α x , Λ ( A ) is a well-defined, isotonous, local, covariant net of C ∗ -algebras. Extension to smaller regions: A ( � W n ) := � A ( W n ). Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 5 / 20

Task Given a wedge algebra A ⊂ B , α , (e.g. given by an interaction-free theory) satisfying the isotony and locality condition, construct a new wedge algebra ˆ A ⊂ ˆ B , ˆ α still satisfying these conditions, such that the associated net has non-trivial S-matrix. Deform A ⊂ B , α continuously from the free to the interacting case (“Perturbation theory for wedge algebras”) Keep α fixed (scattering theory) Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 6 / 20

Wedge-local Deformations in QFT Development of the subject: Deformation of free field theories on Minkowski space by transferring them to “noncommutative Minkowski space” (CCR techniques) [Grosse/GL 07] Generalization of this procedure to arbitrary QFTs by “warped convolutions” in an operator-algebraic setting [Buchholz/Summers 08] Deformation of Wightman QFTs by introducing a new product on the Borchers-Uhlmann testfunction algebra [Grosse/GL 08] [ → Yngvason’s talk] Connection between these two points of view: New product in the operator-algebraic setting → Rieffel deformations [Buchholz/Summers/GL, work in progress] Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 7 / 20

Rieffel Deformations Deformation procedure for C ∗ -algebras [Rieffel 93] Inspired by quantization, “strict deformation quantization” Setting: C ∗ -algebra B with strongly continuous automorphic action β R d . of I Deformation parameter: antisymmetric real ( d × d )-matrix θ On dense subalgebra B ∞ ⊂ B of smooth elements, define new product � � dx e − ipx β θ p ( A ) β x ( B ) A × θ B := (2 π ) − d dp Integral defined in an oscillatory sense This product was designed to deform a commutative C ∗ -algebra B into a noncommutative one, but it can also be applied to noncommutative B . Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 8 / 20

Rieffel Deformations � � dx e − ipx β θ p ( A ) β x ( B ) A × θ B := (2 π ) − d dp Main results about the product × θ [Rieffel 93] : A × 0 B = AB × θ is an associative product on B ∞ ( A × θ B ) ∗ = B ∗ × θ A ∗ A × θ 1 = A = 1 × θ A β is still automorphic w.r.t. × θ . smooth algebra B ∞ can be completed to a deformed C ∗ -algebra B θ θ Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 9 / 20

States and representations Deformation B → B θ introduces new positive cone, ( B ∗ × θ B ) ∈ B + B ∗ B ∈ B + , θ . A state on B is usually only a linear functional on B θ Each state on B can be deformed to a state on B θ [Kaschek/Neumaier/Waldmann 08] Here: Consider only translationally invariant states ω , i.e. R d . ω ◦ β x = ω , x ∈ I QFT examples: Vacuum states, KMS states Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 10 / 20

Let ω be a β -invariant state on B , and ( H , Ω , π ) the GNS data of ( B , ω ), with unitaries U ( x ) implementing β x on H . Then ω is also a state on B ∞ θ , and A , B ∈ B ∞ . ω ( A × θ B ) = ω ( AB ) , The GNS triple ( H θ , Ω θ , π θ ) of ( B ∞ θ , ω ) is H θ = H , Ω θ = Ω , π θ ( A ) π ( B )Ω = π ( A × θ B )Ω � � dx e − ipx U ( θ p ) π ( A ) U ( − θ p + x ) π ( B )Ω = (2 π ) − d dp In particular, π θ ( A )Ω = π ( A × θ 1)Ω = π ( A )Ω. Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 11 / 20

Warped Convolutions The formula � � dx e − ipx U ( θ p ) FU ( x − θ p )Ψ F θ Ψ := (2 π ) − d dp makes sense for any smooth F ∈ B ( H ) ∞ on smooth vectors Ψ. � dE ( k ) e ikx , With spectral resolution U ( x ) = � � � dx e − ipx U ( θ p ) FU ( − θ p ) F θ = (2 π ) − d dE ( k ) e ikx dp � = U ( θ k ) FU ( − θ k ) dE ( k ) warped convolution deformation [Buchholz/Summers 08] Important effect of state/representation: p -integration in Rieffel integral runs only over the spectrum Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 12 / 20

Warped Convolutions The formula � � dx e − ipx U ( θ p ) FU ( x − θ p )Ψ F θ Ψ := (2 π ) − d dp S makes sense for any smooth F ∈ B ( H ) ∞ on smooth vectors Ψ. S dE ( k ) e ikx , � With spectral resolution U ( x ) = � � � dx e − ipx U ( θ p ) FU ( − θ p ) F θ = (2 π ) − d dE ( k ) e ikx dp S S � = U ( θ k ) FU ( − θ k ) dE ( k ) S warped convolution deformation [Buchholz/Summers 08] Important effect of state/representation: p -integration in Rieffel integral runs only over the spectrum S Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 12 / 20

Application of Rieffel Deformations to QFT Consider a wedge algebra A ⊂ B , α , i.e. α x , Λ ( A ) ⊂ A for ( x , Λ) with Λ W R + x ⊂ W R α x , Λ ( A ) ⊂ A ′ for ( x , Λ) with Λ W R + x ⊂ W ′ R Rieffel’s deformation can be applied to B with action β := α | I R d . Consider deformed wedge algebra A θ generated by A 1 , ..., A n ∈ A ∞ A 1 × θ ... × θ A n , Lorentz transformations act according to α x , Λ ( A × θ B ) = α x , Λ ( A ) × Λ θ Λ T α x , Λ ( B ) To satisfy the isotony condition, need Λ θ Λ T = θ for Λ W R ⊂ W R . Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 13 / 20

Lemma [Grosse/GL 07] Let for d = 4 and d � = 4, respectively,   0 0 · · · 0 κ   0 κ 0 0 − κ 0 0 · · · 0   − κ 0 0 0   0 0 0 · · · 0   θ := θ :=  , .    κ ′  0 0 0  . . . .  ...  . . . .   . . . . − κ ′ 0 0 0   0 0 0 · · · 0 ( κ, κ ′ ∈ I R free parameters.) Then ⇒ Λ θ Λ T = θ , Λ W R ⊂ W R ⇐ ⇒ Λ θ Λ T = − θ . Λ W R ⊂ W ′ R ⇐ With θ chosen as above, the isotony condition is satisfied for the deformed system A θ ⊂ B , α . For locality condition, need to consider expressions like A × θ ( B × − θ C ) − B × − θ ( A × θ C ) Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 14 / 20

Compute A × θ ( B × − θ C ) − B × − θ ( A × θ C ) � � dx e − ipx α x / 2 ([ α θ p ( A ) , α − θ p ( B )]) α x ( C ) = (2 π ) − d dp In GNS-representation w.r.t. translationally invariant state ω : [ π ( A ) θ , π ( B ) − θ ] π ( C )Ω � � dx e − ipx U ( x = (2 π ) − d 2 ) π ([ α θ p ( A ) , α − θ p ( B )]) U ( x dp 2 ) π ( C )Ω S = ⇒ [ π ( A ) θ , π ( B ) − θ ] = 0 if [ α θ p ( A ) , α − θ p ( B )] = 0 for all p ∈ S . If κ ≥ 0, this condition is satisfied for a vacuum state since θ S ⊂ θ V + ⊂ W R [Buchholz/Summers 08] For this choice of θ , get deformed wedge algebra (in vac. rep.) π ( A θ ) ⊂ B ( H ) , ad U Gandalf Lechner (Uni Vienna) Deformations AQFT 50th birthday . 15 / 20