Sigma models in algebraic QFT Local Quantum Physics and Beyond In - PowerPoint PPT Presentation

Gandalf Lechner partly joint work with Sabina Alazzawi Stefan Hollands Sigma models in algebraic QFT Local Quantum Physics and Beyond In Memoriam Rudolf Haag Hamburg 27 September 2016

Book “ Local Quantum Physics Fields, Particles, Algebras ” Describes QFT via families of local algebras 1/20 instead of quantum fields Algebraic QFT ■ Rudolf Haag was one of the founding fathers of an operator-algebraic approach to QFT .

instead of quantum fields 1/20 Algebraic QFT ■ Rudolf Haag was one of the founding fathers of an operator-algebraic approach to QFT . ■ Book “ Local Quantum Physics − Fields, Particles, Algebras ” ■ Describes QFT via families of local algebras O �− → A ( O )

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT)

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT)

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT)

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT)

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT)

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT)

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT) Study maps O �→ A ( O ) of spacetime regions to von Neumann algebras.

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT) Study maps O �→ A ( O ) of spacetime regions to von Neumann algebras.

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT) Study maps O �→ A ( O ) of spacetime regions to von Neumann algebras.

2/20 Sketch of AQFT setuing (on Minkowski space): Algebraic QFT (AQFT) Study maps O �→ A ( O ) of spacetime regions to von Neumann algebras.

2/20 Sketch of AQFT setuing (on Minkowski space): “Axioms”: automorphisms Algebraic QFT (AQFT) Study maps O �→ A ( O ) of spacetime regions to von Neumann algebras. ■ Isotony: Inclusions of regions give inclusions of algebras ■ Locality: Algebras of spacelike separated regions commute ■ Covariance: The isometry group of spacetime acts covariantly by ■ further axioms regarding states (vacuum …. )

Local algebras are less singular than local quantum fields: Local algebras are more invariant than local quantum fields: The algebraic approach brings new mathematical tools into QFT: AQFT has led to deep conceptual insights. Examples: 3/20 Advantages of the algebraic approach: bounded operators vs unbounded operator-valued distributions many difgerent quantum fields correspond to the same physics, and to the same net of local algebras operator-algebraic tools, e.g. Tomita-Takesaki modular theory Doplicher-Haag-Roberts theory of localized charges / global gauge theories Haag-Ruelle scatuering theory Formulation of thermal equilibrium (KMS) states [Haag, Hugenholtz, Winnink ’67] Algebraic QFT (AQFT)

Local algebras are more invariant than local quantum fields: The algebraic approach brings new mathematical tools into QFT: AQFT has led to deep conceptual insights. Examples: 3/20 Advantages of the algebraic approach: many difgerent quantum fields correspond to the same physics, and to the same net of local algebras operator-algebraic tools, e.g. Tomita-Takesaki modular theory Doplicher-Haag-Roberts theory of localized charges / global gauge theories Haag-Ruelle scatuering theory Formulation of thermal equilibrium (KMS) states [Haag, Hugenholtz, Winnink ’67] Algebraic QFT (AQFT) Local algebras are less singular than local quantum fields: ■ bounded operators vs unbounded operator-valued distributions

The algebraic approach brings new mathematical tools into QFT: AQFT has led to deep conceptual insights. Examples: 3/20 Advantages of the algebraic approach: to the same net of local algebras operator-algebraic tools, e.g. Tomita-Takesaki modular theory Doplicher-Haag-Roberts theory of localized charges / global gauge theories Haag-Ruelle scatuering theory Formulation of thermal equilibrium (KMS) states [Haag, Hugenholtz, Winnink ’67] Algebraic QFT (AQFT) Local algebras are less singular than local quantum fields: ■ bounded operators vs unbounded operator-valued distributions Local algebras are more invariant than local quantum fields: ■ many difgerent quantum fields correspond to the same physics, and

AQFT has led to deep conceptual insights. Examples: 3/20 Advantages of the algebraic approach: to the same net of local algebras Doplicher-Haag-Roberts theory of localized charges / global gauge theories Haag-Ruelle scatuering theory Formulation of thermal equilibrium (KMS) states [Haag, Hugenholtz, Winnink ’67] Algebraic QFT (AQFT) Local algebras are less singular than local quantum fields: ■ bounded operators vs unbounded operator-valued distributions Local algebras are more invariant than local quantum fields: ■ many difgerent quantum fields correspond to the same physics, and The algebraic approach brings new mathematical tools into QFT: ■ operator-algebraic tools, e.g. Tomita-Takesaki modular theory

3/20 Advantages of the algebraic approach: to the same net of local algebras theories Winnink ’67] Algebraic QFT (AQFT) Local algebras are less singular than local quantum fields: ■ bounded operators vs unbounded operator-valued distributions Local algebras are more invariant than local quantum fields: ■ many difgerent quantum fields correspond to the same physics, and The algebraic approach brings new mathematical tools into QFT: ■ operator-algebraic tools, e.g. Tomita-Takesaki modular theory AQFT has led to deep conceptual insights. Examples: ■ Doplicher-Haag-Roberts theory of localized charges / global gauge ■ Haag-Ruelle scatuering theory ■ Formulation of thermal equilibrium (KMS) states [Haag, Hugenholtz,

This situation has changed a lot since the beginnings of AQFT. talk by Rejzner talk by Longo talks by Doplicher, Gérard this talk low-dimensional AQFT models spacetimes AQFT on curved or quantum conformal AQFT 4/20 tools of AQFT: aiming at building models with the There now exist several programmes Disadvantages of the algebraic approach: More abstract formulation, perturbative AQFT Models in AQFT not clear at first sight how to build models (examples).

talk by Rejzner talk by Longo talks by Doplicher, Gérard this talk low-dimensional AQFT models spacetimes AQFT on curved or quantum conformal AQFT 4/20 tools of AQFT: aiming at building models with the There now exist several programmes Disadvantages of the algebraic approach: More abstract formulation, perturbative AQFT Models in AQFT not clear at first sight how to build models (examples). This situation has changed a lot since the beginnings of AQFT.

4/20 spacetimes Disadvantages of the algebraic approach: More abstract formulation, There now exist several programmes aiming at building models with the tools of AQFT: Models in AQFT not clear at first sight how to build models (examples). This situation has changed a lot since the beginnings of AQFT. ■ perturbative AQFT → talk by Rejzner ■ conformal AQFT → talk by Longo ■ AQFT on curved or quantum → talks by Doplicher, Gérard ■ low-dimensional AQFT models → this talk

(masses, spins) , and representation V of global gauge group (charges) localization: also encoded in U (modular localization) (wedges) h J is “localized in W ” if h t U it t : boosts into W 5/20 Here it is useful to first look at special . U This also defines a single particle TCP operator J . on a single particle Hilbert space particle spectrum: fixed by representation U of Poincaré group free QFT = particle spectrum + localization + second quantization Starting point: interaction-free models h Models in AQFT

(masses, spins) , and representation V of global gauge group (charges) localization: also encoded in U (modular localization) (wedges) h J is “localized in W ” if h t U it t : boosts into W 5/20 Here it is useful to first look at special . U This also defines a single particle TCP operator J . on a single particle Hilbert space particle spectrum: fixed by representation U of Poincaré group free QFT = particle spectrum + localization + second quantization Starting point: interaction-free models h Models in AQFT

localization: also encoded in U (modular localization) (wedges) t : boosts into W h J is “localized in W ” if h t U it 5/20 free QFT = particle spectrum + localization + second quantization Here it is useful to first look at special Starting point: interaction-free models h Models in AQFT ■ particle spectrum: fixed by representation U 1 of Poincaré group (masses, spins) , and representation V 1 of global gauge group (charges) on a single particle Hilbert space H 1 . This also defines a single particle TCP operator J 1 = U 1 ( − 1) ⊗ Γ 1 .

5/20 t : boosts into W h J is “localized in W ” if h t U it h free QFT = particle spectrum + localization + second quantization Starting point: interaction-free models Models in AQFT ■ particle spectrum: fixed by representation U 1 of Poincaré group (masses, spins) , and representation V 1 of global gauge group (charges) on a single particle Hilbert space H 1 . This also defines a single particle TCP operator J 1 = U 1 ( − 1) ⊗ Γ 1 . ■ localization: also encoded in U (modular localization) Here it is useful to first look at special O (wedges)