Quantum gravity from the point of view of locally covariant QFT - PowerPoint PPT Presentation

Introduction Classical theory Quantization Quantum gravity from the point of view of locally covariant QFT Katarzyna Rejzner 1 INdAM (Marie Curie) fellow University of Rome Tor Vergata Wuppertal, 01.06.2013 1 Based on the joint work with Klaus Fredenhagen and Romeo Brunetti Katarzyna Rejzner Quantum Gravity 1 / 23

Introduction Classical theory Quantization Outline of the talk Introduction 1 Effective quantum gravity Local covariance Classical theory 2 Kinematical structure Equations of motion and symmetries BV complex Quantization 3 Deformation quantization Background independence

Introduction Classical theory Quantization Local covariance Intuitive idea In experiment, geometric structure is probed by local observations. We have the following data: Katarzyna Rejzner Quantum Gravity 2 / 23

Introduction Classical theory Quantization Local covariance Intuitive idea In experiment, geometric structure is probed by local observations. We have the following data: M Compact causally convex region O of spacetime where the measurement is performed, O Katarzyna Rejzner Quantum Gravity 2 / 23

Introduction Classical theory Quantization Local covariance Intuitive idea In experiment, geometric structure is probed by local observations. We have the following data: M Compact causally convex region O of spacetime where the measurement is performed, O An observable Φ , which we measure, Φ Katarzyna Rejzner Quantum Gravity 2 / 23

Introduction Classical theory Quantization Local covariance Intuitive idea In experiment, geometric structure is probed by local observations. We have the following data: M Compact causally convex region O of spacetime f where the measurement is performed, An observable Φ , which we measure, O We don’t measure the scalar curvature at a point, but we have some smearing related to the Z Φ ( f ) experimantal setting: Φ ( f ) = f ( x ) R ( x ) , supp ( f ) ⇢ O . Katarzyna Rejzner Quantum Gravity 2 / 23

Introduction Classical theory Quantization Local covariance Intuitive idea In experiment, geometric structure is probed by local observations. We have the following data: ( M , g ) Compact causally convex region O of spacetime f where the measurement is performed, An observable Φ , which we measure, O We don’t measure the scalar curvature at a point, but we have some smearing related to the Z Φ ( O , g ) ( f )[ h ] experimantal setting: Φ ( f ) = f ( x ) R ( x ) , supp ( f ) ⇢ O . We can think of the measured observable as a perturbation of the fixed background metric: a tentative split into: ˜ g µ ⌫ = g µ ⌫ + h µ ⌫ . Katarzyna Rejzner Quantum Gravity 2 / 23

Introduction Classical theory Quantization Local covariance Intuitive idea In experiment, geometric structure is probed by local observations. We have the following data: ( M , g ) O 0 Compact causally convex region O of spacetime f where the measurement is performed, An observable Φ , which we measure, O We don’t measure the scalar curvature at a point, but we have some smearing related to the Z Φ ( O , g ) ( f )[ h ] experimantal setting: Φ ( f ) = f ( x ) R ( x ) , supp ( f ) ⇢ O . We can think of the measured observable as a perturbation of the fixed background metric: a tentative split into: ˜ g µ ⌫ = g µ ⌫ + h µ ⌫ . Diffeomorphism transformation: move our experimental setup to a different region O 0 . Katarzyna Rejzner Quantum Gravity 2 / 23

Introduction Classical theory Quantization Local covariance How to implement it? To compare Φ ( O , g ) ( f ) and Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) we ↵ ( M , g ) ↵ ⇤ f need to know what does it mean to have "the same observable in a different region". O 0 f O Φ ( O , g ) ( f ) Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) Katarzyna Rejzner Quantum Gravity 3 / 23

Introduction Classical theory Quantization Local covariance How to implement it? To compare Φ ( O , g ) ( f ) and Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) we ↵ ( M , g ) ↵ ⇤ f need to know what does it mean to have "the same observable in a different region". O 0 f A good language to formalize it is the category O theory. We need following categories: Φ ( O , g ) ( f ) Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) Katarzyna Rejzner Quantum Gravity 3 / 23

Introduction Classical theory Quantization Local covariance How to implement it? To compare Φ ( O , g ) ( f ) and Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) we ↵ ( M , g ) ↵ ⇤ f need to know what does it mean to have "the same observable in a different region". O 0 f A good language to formalize it is the category O theory. We need following categories: Loc where the objects are all four-dimensional, globally hyperbolic oriented and time-oriented spacetimes M = ( M , g ) . Morphisms: isometric Φ ( O , g ) ( f ) Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) embeddings preserving orientation, time-orientation and the causal structure. Katarzyna Rejzner Quantum Gravity 3 / 23

Introduction Classical theory Quantization Local covariance How to implement it? To compare Φ ( O , g ) ( f ) and Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) we ↵ ( M , g ) ↵ ⇤ f need to know what does it mean to have "the same observable in a different region". O 0 f A good language to formalize it is the category O theory. We need following categories: Loc where the objects are all four-dimensional, globally hyperbolic oriented and time-oriented spacetimes M = ( M , g ) . Morphisms: isometric Φ ( O , g ) ( f ) Φ ( O 0 , ↵ ⇤ g ) ( ↵ ⇤ f ) embeddings preserving orientation, time-orientation and the causal structure. Vec with (small) topological vector spaces as objects and injective continuous homomorphisms of topological vector spaces as morphisms. Katarzyna Rejzner Quantum Gravity 3 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Kinematical structure Having the quantization in mind we formulate already the classical theory in the perturbative setting. Katarzyna Rejzner Quantum Gravity 4 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Kinematical structure Having the quantization in mind we formulate already the classical theory in the perturbative setting. We work off-shell, so for the effective theory of gravity the configuration space is E ( M ) = Γ (( T ⇤ M ) 2 ⌦ ) . The space of compactly supported configurations is denoted by E c ( M ) . Katarzyna Rejzner Quantum Gravity 4 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Kinematical structure Having the quantization in mind we formulate already the classical theory in the perturbative setting. We work off-shell, so for the effective theory of gravity the configuration space is E ( M ) = Γ (( T ⇤ M ) 2 ⌦ ) . The space of compactly supported configurations is denoted by E c ( M ) . We define a contravariant functor E : Loc ! Vec , which assigns to a spacetime the corresponding configuration space and acts on morphisms � : M ! N as E � = � ⇤ : E ( N ) ! E ( M ) . Katarzyna Rejzner Quantum Gravity 4 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Kinematical structure Having the quantization in mind we formulate already the classical theory in the perturbative setting. We work off-shell, so for the effective theory of gravity the configuration space is E ( M ) = Γ (( T ⇤ M ) 2 ⌦ ) . The space of compactly supported configurations is denoted by E c ( M ) . We define a contravariant functor E : Loc ! Vec , which assigns to a spacetime the corresponding configuration space and acts on morphisms � : M ! N as E � = � ⇤ : E ( N ) ! E ( M ) . In a similar way we define a covariant functor E c : Loc ! Vec by setting E � = � ⇤ , where: ⇢ ( � � 1 ) ⇤ h ( x ) x 2 � ( M ) , , � ⇤ h . = 0 else , Katarzyna Rejzner Quantum Gravity 4 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Functionals and dynamics We consider the space of smooth functionals on E ( M ) , i.e. C 1 ( E ( M ) , R ) . Katarzyna Rejzner Quantum Gravity 5 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Functionals and dynamics We consider the space of smooth functionals on E ( M ) , i.e. C 1 ( E ( M ) , R ) . The support of F 2 C 1 ( E ( M ) , R ) is defined as: supp F = { x 2 M | 8 neighbourhoods U of x 9 h 1 , h 2 2 E ( M ) , supp h 2 ⇢ U such that F ( h 1 + h 2 ) 6 = F ( h 1 ) } . Katarzyna Rejzner Quantum Gravity 5 / 23

Introduction Kinematical structure Classical theory Equations of motion and symmetries Quantization BV complex Functionals and dynamics We consider the space of smooth functionals on E ( M ) , i.e. C 1 ( E ( M ) , R ) . The support of F 2 C 1 ( E ( M ) , R ) is defined as: supp F = { x 2 M | 8 neighbourhoods U of x 9 h 1 , h 2 2 E ( M ) , supp h 2 ⇢ U such that F ( h 1 + h 2 ) 6 = F ( h 1 ) } . Z F is local if it is of the form: F ( h ) = f ( j x ( h ))( x ) , where f is a M density-valued function on the jet bundle over M and j x ( h ) is the jet of ' at the point x . Katarzyna Rejzner Quantum Gravity 5 / 23