Combinatorics of the Star Product in AQFT Eli Hawkins Kasia Rejzner - PowerPoint PPT Presentation

Combinatorics of the Star Product in AQFT Eli Hawkins Kasia Rejzner The University of York July 2, 2019

Classical Field Theory

Classical Field Theory • M spacetime

Classical Field Theory • M spacetime • E vector bundle

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections • S action

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell).

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function S ( ϕ ; x , y ) . • ∆ A = ∆ R S ( ϕ ; y , x ) advanced

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function S ( ϕ ; x , y ) . • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S

Classical Field Theory • M spacetime • E vector bundle • E . = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function S ( ϕ ; x , y ) . • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S • Peierls bracket of F , G : E → C : { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) �

Classical Field Theory Quantization of Free Theory • M spacetime S 0 quadratic ⇒ ∆ R S 0 ( ϕ ; x , y ) = ∆ R = S 0 ( x , y ) . • E vector bundle • E . = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function S ( ϕ ; x , y ) . • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S • Peierls bracket of F , G : E → C : { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) �

Classical Field Theory Quantization of Free Theory • M spacetime S 0 quadratic ⇒ ∆ R S 0 ( ϕ ; x , y ) = ∆ R = S 0 ( x , y ) . • E vector bundle Denote m ( F , G )( ϕ ) . • E . = F ( ϕ ) G ( ϕ ) = Γ( M , E ) smooth sections • S action S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function S ( ϕ ; x , y ) . • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S • Peierls bracket of F , G : E → C : { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) �

Classical Field Theory Quantization of Free Theory • M spacetime S 0 quadratic ⇒ ∆ R S 0 ( ϕ ; x , y ) = ∆ R = S 0 ( x , y ) . • E vector bundle Denote m ( F , G )( ϕ ) . • E . = F ( ϕ ) G ( ϕ ) = Γ( M , E ) smooth sections A distribution K on M × M defines • S action D K ( F ⊗ G )( ϕ 1 , ϕ 2 ) . = � K , F (1) ( ϕ 1 ) , G (1) ( ϕ 2 ) � S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). • ∆ R S retarded Green’s function S ( ϕ ; x , y ) . • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S • Peierls bracket of F , G : E → C : { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) �

Classical Field Theory Quantization of Free Theory • M spacetime S 0 quadratic ⇒ ∆ R S 0 ( ϕ ; x , y ) = ∆ R = S 0 ( x , y ) . • E vector bundle Denote m ( F , G )( ϕ ) . • E . = F ( ϕ ) G ( ϕ ) = Γ( M , E ) smooth sections A distribution K on M × M defines • S action D K ( F ⊗ G )( ϕ 1 , ϕ 2 ) . = � K , F (1) ( ϕ 1 ) , G (1) ( ϕ 2 ) � S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). and an exponential star product • ∆ R S retarded Green’s function F ⋆ K G . S ( ϕ ; x , y ) . hD K ( F ⊗ G ) . = m ◦ e ¯ • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S • Peierls bracket of F , G : E → C : { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) �

Classical Field Theory Quantization of Free Theory • M spacetime S 0 quadratic ⇒ ∆ R S 0 ( ϕ ; x , y ) = ∆ R = S 0 ( x , y ) . • E vector bundle Denote m ( F , G )( ϕ ) . • E . = F ( ϕ ) G ( ϕ ) = Γ( M , E ) smooth sections A distribution K on M × M defines • S action D K ( F ⊗ G )( ϕ 1 , ϕ 2 ) . = � K , F (1) ( ϕ 1 ) , G (1) ( ϕ 2 ) � S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). and an exponential star product • ∆ R S retarded Green’s function F ⋆ K G . S ( ϕ ; x , y ) . hD K ( F ⊗ G ) . = m ◦ e ¯ • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S If K ( x , y ) − K ( y , x ) = i ∆ S 0 ( x , y ), then • Peierls bracket of F , G : E → C : this is a quantization for S 0 . { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) �

Classical Field Theory Quantization of Free Theory • M spacetime S 0 quadratic ⇒ ∆ R S 0 ( ϕ ; x , y ) = ∆ R = S 0 ( x , y ) . • E vector bundle Denote m ( F , G )( ϕ ) . • E . = F ( ϕ ) G ( ϕ ) = Γ( M , E ) smooth sections A distribution K on M × M defines • S action D K ( F ⊗ G )( ϕ 1 , ϕ 2 ) . = � K , F (1) ( ϕ 1 ) , G (1) ( ϕ 2 ) � S ′′ ( φ ) is linearized equation of motion operator about ϕ ∈ E (off shell). and an exponential star product • ∆ R S retarded Green’s function F ⋆ K G . S ( ϕ ; x , y ) . hD K ( F ⊗ G ) . = m ◦ e ¯ • ∆ A = ∆ R S ( ϕ ; y , x ) advanced • ∆ S . = ∆ R S − ∆ A S If K ( x , y ) − K ( y , x ) = i ∆ S 0 ( x , y ), then • Peierls bracket of F , G : E → C : this is a quantization for S 0 . { F , G } S ( ϕ ) . = � ∆ S [ ϕ ] , F (1) ( ϕ ) ⊗ G (1) ( ϕ ) � Changing K by a smooth, symmetric function gives an equivalent star product.

Quantization Maps

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”.

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”. It induces a star product by Q ( F ⋆ G ) = Q ( F ) Q ( G ) .

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”. It induces a star product by Q ( F ⋆ G ) = Q ( F ) Q ( G ) . Equivalent star products come from different choices of Q .

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”. It induces a star product by Q ( F ⋆ G ) = Q ( F ) Q ( G ) . Equivalent star products come from different choices of Q . Ordering Product K i 2 ∆ S 0 = i � ∆ R S 0 − ∆ A � Symmetric Moyal-Weyl S 0 2

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”. It induces a star product by Q ( F ⋆ G ) = Q ( F ) Q ( G ) . Equivalent star products come from different choices of Q . Ordering Product K 2 ∆ S 0 = i i � ∆ R S 0 − ∆ A � Symmetric Moyal-Weyl S 0 2 ∆ + S 0 = i Normal 2 ∆ S 0 + H Wick

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”. It induces a star product by Q ( F ⋆ G ) = Q ( F ) Q ( G ) . Equivalent star products come from different choices of Q . Ordering Product K 2 ∆ S 0 = i i � ∆ R S 0 − ∆ A � Symmetric Moyal-Weyl S 0 2 ∆ + S 0 = i Normal 2 ∆ S 0 + H Wick − i ∆ A Time-ordered ⋆ T S 0

Quantization Maps A quantization map Q : { Classical observables } → { Quantum observables } is a “choice of operator ordering”. It induces a star product by Q ( F ⋆ G ) = Q ( F ) Q ( G ) . Equivalent star products come from different choices of Q . Ordering Product K i 2 ∆ S 0 = i � ∆ R S 0 − ∆ A � Symmetric Moyal-Weyl S 0 2 ∆ + S 0 = i Normal 2 ∆ S 0 + H Wick − i ∆ A Time-ordered ⋆ T S 0 The product ⋆ T will be the most useful here.

Star Product by Graphs

Star Product by Graphs A graph describes a multidifferential operator.

Star Product by Graphs A graph describes a multidifferential operator. • The vertex j represents (a derivative of) the j ’th argument.

Star Product by Graphs A graph describes a multidifferential operator. • The vertex j represents (a derivative of) the j ’th argument. • = ∆ A S 0 .

Star Product by Graphs A graph describes a multidifferential operator. • The vertex j represents (a derivative of) the j ’th argument. • = ∆ A S 0 . 2 = m , i.e., m ( F , G ) = F · G . E.g., 1