Construction of Hadamard states by pseudo-di ff erential calculus - PowerPoint PPT Presentation

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Construction of Hadamard states by pseudo-di ff erential calculus Micha l Wrochna → arXiv:1209.2604 joint work with Christian G´ erard 31.05.13

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times 1 Introduction 2 Quasi-free Hadamard states 3 Model Klein-Gordon equation 4 General space-times

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Physical states for QFT on curved space-times • On general curved space-times, no notion of vacuum state. • Substitute for vacuum state: Hadamard states, characterized by the singularity structure of their two-point functions [Kay, Wald, etc. ’70-’80] ; • Most important property: quantum stress-energy tensor can be renormalized w.r.t. a Hadamard state; • Since [Radzikowski ’96] , Hadamard condition formulated in terms of wave front set. • Problem: few examples of Hadamard states have been constructed.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Physical states for QFT on curved space-times • On general curved space-times, no notion of vacuum state. • Substitute for vacuum state: Hadamard states, characterized by the singularity structure of their two-point functions [Kay, Wald, etc. ’70-’80] ; • Most important property: quantum stress-energy tensor can be renormalized w.r.t. a Hadamard state; • Since [Radzikowski ’96] , Hadamard condition formulated in terms of wave front set. • Problem: few examples of Hadamard states have been constructed.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Physical states for QFT on curved space-times • On general curved space-times, no notion of vacuum state. • Substitute for vacuum state: Hadamard states, characterized by the singularity structure of their two-point functions [Kay, Wald, etc. ’70-’80] ; • Most important property: quantum stress-energy tensor can be renormalized w.r.t. a Hadamard state; • Since [Radzikowski ’96] , Hadamard condition formulated in terms of wave front set. • Problem: few examples of Hadamard states have been constructed.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Physical states for QFT on curved space-times • On general curved space-times, no notion of vacuum state. • Substitute for vacuum state: Hadamard states, characterized by the singularity structure of their two-point functions [Kay, Wald, etc. ’70-’80] ; • Most important property: quantum stress-energy tensor can be renormalized w.r.t. a Hadamard state; • Since [Radzikowski ’96] , Hadamard condition formulated in terms of wave front set. • Problem: few examples of Hadamard states have been constructed.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Overview • We reconsider the construction of Hadamard states on space-times with metric well-behaved at spatial infinity; • Working on a fixed Cauchy surface, we can use rather standard pseudo-di ↵ erential analysis. • We construct a large class of Hadamard states with DO two-point functions, in particular all pure Hadamard states. • We give a new construction of Hadamard states on general globally hyperbolic space-times.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Overview • We reconsider the construction of Hadamard states on space-times with metric well-behaved at spatial infinity; • Working on a fixed Cauchy surface, we can use rather standard pseudo-di ↵ erential analysis. • We construct a large class of Hadamard states with DO two-point functions, in particular all pure Hadamard states. • We give a new construction of Hadamard states on general globally hyperbolic space-times.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Overview • We reconsider the construction of Hadamard states on space-times with metric well-behaved at spatial infinity; • Working on a fixed Cauchy surface, we can use rather standard pseudo-di ↵ erential analysis. • We construct a large class of Hadamard states with DO two-point functions, in particular all pure Hadamard states. • We give a new construction of Hadamard states on general globally hyperbolic space-times.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Overview • We reconsider the construction of Hadamard states on space-times with metric well-behaved at spatial infinity; • Working on a fixed Cauchy surface, we can use rather standard pseudo-di ↵ erential analysis. • We construct a large class of Hadamard states with DO two-point functions, in particular all pure Hadamard states. • We give a new construction of Hadamard states on general globally hyperbolic space-times.

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Quasi-free states: neutral case Let ( X , σ ) be a symplectic space and A its Weyl CCR C ⇤ -algebra, generated by elements W ( f ), f 2 X , with W ( f ) ⇤ = W ( � f ) , W ( f ) W ( g ) = e � i σ ( f , g ) / 2 W ( f + g ) , f , g 2 X . A state ω on A is quasi-free if there is a symmetric form η s.t. ω ( W ( f )) = e � 1 2 η ( f , f ) , f 2 X . • A symmetric form η on X defines a quasi-free state i ↵ the two-point function λ = η + i satisfies λ � 0 . 2 σ • A symmetric form λ on X is the two-point function of a quasi-free state i ↵ λ � 0 and λ � i σ . The field operators φ ( f ) in the GNS rep. of ω satisfy [ φ ( f ) , φ ( g )] = i σ ( f , g ) , ω ( φ ( f ) φ ( g )) = λ ( f , g ) .

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Quasi-free states: neutral case Let ( X , σ ) be a symplectic space and A its Weyl CCR C ⇤ -algebra, generated by elements W ( f ), f 2 X , with W ( f ) ⇤ = W ( � f ) , W ( f ) W ( g ) = e � i σ ( f , g ) / 2 W ( f + g ) , f , g 2 X . A state ω on A is quasi-free if there is a symmetric form η s.t. ω ( W ( f )) = e � 1 2 η ( f , f ) , f 2 X . • A symmetric form η on X defines a quasi-free state i ↵ the two-point function λ = η + i satisfies λ � 0 . 2 σ • A symmetric form λ on X is the two-point function of a quasi-free state i ↵ λ � 0 and λ � i σ . The field operators φ ( f ) in the GNS rep. of ω satisfy [ φ ( f ) , φ ( g )] = i σ ( f , g ) , ω ( φ ( f ) φ ( g )) = λ ( f , g ) .

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Quasi-free states: neutral case Let ( X , σ ) be a symplectic space and A its Weyl CCR C ⇤ -algebra, generated by elements W ( f ), f 2 X , with W ( f ) ⇤ = W ( � f ) , W ( f ) W ( g ) = e � i σ ( f , g ) / 2 W ( f + g ) , f , g 2 X . A state ω on A is quasi-free if there is a symmetric form η s.t. ω ( W ( f )) = e � 1 2 η ( f , f ) , f 2 X . • A symmetric form η on X defines a quasi-free state i ↵ the two-point function λ = η + i satisfies λ � 0 . 2 σ • A symmetric form λ on X is the two-point function of a quasi-free state i ↵ λ � 0 and λ � i σ . The field operators φ ( f ) in the GNS rep. of ω satisfy [ φ ( f ) , φ ( g )] = i σ ( f , g ) , ω ( φ ( f ) φ ( g )) = λ ( f , g ) .

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Quasi-free states: neutral case Let ( X , σ ) be a symplectic space and A its Weyl CCR C ⇤ -algebra, generated by elements W ( f ), f 2 X , with W ( f ) ⇤ = W ( � f ) , W ( f ) W ( g ) = e � i σ ( f , g ) / 2 W ( f + g ) , f , g 2 X . A state ω on A is quasi-free if there is a symmetric form η s.t. ω ( W ( f )) = e � 1 2 η ( f , f ) , f 2 X . • A symmetric form η on X defines a quasi-free state i ↵ the two-point function λ = η + i satisfies λ � 0 . 2 σ • A symmetric form λ on X is the two-point function of a quasi-free state i ↵ λ � 0 and λ � i σ . The field operators φ ( f ) in the GNS rep. of ω satisfy [ φ ( f ) , φ ( g )] = i σ ( f , g ) , ω ( φ ( f ) φ ( g )) = λ ( f , g ) .

Plan Introduction Quasi-free Hadamard states Model Klein-Gordon equation General space-times Quasi-free states: charged case Let ( Y , σ ) be a complex symplectic space (with some complex structure j ) and A the Weyl CCR C ⇤ -algebra of ( Y , Re σ ). A quasi-free state ω on A is gauge-invariant if ω ( W ( y )) = ω ( W ( e j θ y )) , 0  θ < 2 π , y 2 Y . Let φ ( y ) be the (‘neutral’) field operators in the GNS rep. of ω . The charged fields: ψ ( y ) · 1 ψ ⇤ ( y ) · 1 · = 2 ( φ ( y ) + i φ ( j y )) , · = 2 ( φ ( y ) � i φ ( j y )) p p ω ( ψ ( y 1 ) ψ ⇤ ( y 2 )) = · [ ψ ( y 1 ) , ψ ⇤ ( y 2 )] = i σ ( y 1 , y 2 ) , · λ ( y 1 , y 2 ) ) λ ( y 1 , y 2 ) � i σ ( y 1 , y 2 ) = ω ( ψ ⇤ ( y 2 ) ψ ( y 1 )) • A symmetric form λ on Y is the two-point function of a gauge-invariant quasi-free state i ↵ λ � 0 and λ � i σ .