The Early Time Expansion of the Heat Kernel Stefan Lippoldt - PowerPoint PPT Presentation

Motivation & Introduction Heat Kernel in FRG Early Time Expansion The Early Time Expansion of the Heat Kernel Stefan Lippoldt November 8, 2016 1 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Motivation & Introduction Heat Kernel origin: dissipation of heat arises naturally in many branches of mathematical physics specific tool for the analysis of the spectrum of a Laplacean (so far) the only tool for the evaluation of the flow equation on a generic(!) curved background within a curvature expansion 2 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Dissipation of Heat time evolution of heat distribution u ( x , s ) is described by initial condition: u ( x , s ց 0 ) = u 0 ( x ) PDE: ∂ s u ( x , s ) = − ∆ u ( x , s ) the unique solution reads � u ( x , s ) = e − s ∆ u 0 ( x ) = e − s ∆ δ ( x − y ) u 0 ( y ) y � �� � Heat Kernel � = K ( x , y ; s ) u 0 ( y ) y 3 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Heat Kernel: K ( x , y ; s ) = e − s ∆ δ ( x − y ) Definition as a solution to the heat equation initial condition: K ( x , y ; s ց 0 ) = δ ( x − y ) PDE: ∂ s K ( x , y ; s ) = − ∆ K ( x , y ; s ) using momentum representation of δ ( x − y ) we get � � ( 2 π ) d e i p ( x − y ) = d d p d d p ( 2 π ) d e − sp 2 + i p ( x − y ) K ( x , y ; s ) = e − s ∆ = e − ( x − y ) 2 4 s ( 4 π s ) d / 2 4 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Heat Kernel in FRG FRG in Position Space in FRG we have to calculate a functional trace: � � (Γ ( 2 ) ∂ k Γ k = 1 + R k ) − 1 ∂ k R k 2 Tr k What does “Tr” mean? ⇒ It is the trace over the considered space of functions � appearing in the path integral ( D ϕ ): � � � ¯ Φ n ( x ) M ( x , y )Φ n ( y ) Tr M = x y n for example in flat space: � � � � � ( 2 π ) d e − i px f (∆) δ ( x − y ) e i py = d d p d d p ( 2 π ) d f ( p 2 ) Tr f (∆) = x y x ∞ � � z ( d − 2 ) / 2 = d z ( 4 π ) d / 2 Γ( d / 2 ) f ( z ) x 0 5 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Laplace Transform consider a function of a positive argument f : R + → R ∞ � Laplace Transform ˜ d s ˜ f ( s ) e − sz f : R + → R : f ( z ) = 0 ∞ � d s s x − 1 e − as 1 e − sz , for example: ( z + a ) x = a , x > 0 Γ( x ) 0 generally one can show: ∞ ∞ � d s s − x ˜ � 1 d z z x − 1 f ( z ) f ( s ) = Γ( x ) 0 0 ∞ � d s s n ˜ f ( s ) = ( − 1 ) n lim z ց 0 f ( n ) ( z ) 0 6 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Heat Kernel in FRG simpler form of the trace: � � � � � � � � ¯ Φ n ( y )¯ Φ n ( x ) M ( x , y )Φ n ( y )= Φ n ( x ) Tr M = tr M ( x , y ) x y x y n n � = tr [ M ( x , y )] y = x x for example in flat space: e − s ∆ δ ( x − y ) = e − ( x − y ) 2 / ( 4 s ) ( 4 π s ) d / 2 ∞ � � � d s ˜ f ( s )[ e − s ∆ δ ( x − y ) Tr f (∆) = [ f (∆) δ ( x − y )] y = x = ] y = x � �� � x x 0 Heat Kernel ∞ ∞ � � � � ˜ f ( s ) z ( d − 2 ) / 2 = ( 4 π s ) d / 2 = ( 4 π ) d / 2 Γ( d / 2 ) f ( z ) d s d z x x 0 0 7 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Early Time Expansion The Coordinate independent Delta Distribution want to generalize the heat kernel to curved spaces partial to covariant derivatives: ∆ = − D µ D µ What to do with δ ( x − y ) ? ⇒ δ ( x , y ) is unit oprator in the considered space of functions: � � Φ n ( x )¯ Φ n ( y ) δ ( x , y )Φ( y ) = Φ( x ) ⇔ δ ( x , y ) = y n for example in flat space: � d d p ( 2 π ) d e i px e − i py δ ( x − y ) = 8 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion The World Function What to do with ( x − y ) 2 ? ⇒ geodesic distance d 2 g ( x , y ) introduce the “world function” σ ( x , y ) for convenience: d 2 g ( x , y ) 2 g µν ( D µ σ )( D ν σ ) = σ, 1 σ ( x , y ) := , 2 D µ D ν σ = g µν D µ σ = 0 , for example in flat space: 2 η µν � η µρ ( x − y ) ρ �� η νκ ( x − y ) κ � σ ( x , y ) = ( x − y ) 2 1 = σ ( x , y ) , , 2 ∂ µ η νρ ( x − y ) ρ = η µν η µρ ( x − y ) ρ = 0 , 9 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Heat Kernel in Curved Spaces generalized heat kernel: K ( x , y ; s ) = e − s ∆ δ ( x , y ) for example: a scalar on the one-sphere S 1 the metric g ϕϕ = r 2 , Laplacean ∆ = − 1 r 2 ∂ 2 ϕ eigenfunctions of the Laplacean: � { 1 } , n = 0 f n , l ( ϕ ) = e l · i n ϕ 2 π r , n ∈ N 0 , l ∈ D n = , √ {− 1 , 1 } , n > 0 λ n = n 2 ∆ f n , l = λ n · f n , l , r 2 Heat Kernel: ∞ e − sn 2 / r 2 e l · i n ϕ 1 e − l · i n ϕ 2 � � K ( ϕ 1 , ϕ 2 ; s ) = √ √ 2 π r 2 π r n = 0 l ∈ D n � � � = e − σ ( ϕ 1 ,ϕ 2 ) / ( 2 s ) σ ( n ) ( ϕ 1 ,ϕ 2 ) − σ ( ϕ 1 ,ϕ 2 ) / ( 2 s ) e − √ 4 π s n ∈ Z σ ( n ) corresponds to d ( n ) = n · ( 2 π r ) + d g g 10 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Early Time Expansion one can show that the Heat Kernel allows for an early time expansion of the form: ∞ σ ( x , y ) � e 2 s s n A n ( x , y ) K ( x , y ; s ) = ( 4 π s ) d / 2 n = 0 the A n ( x , y ) are the expansion coefficients (to be determined) this form misses topological properties of the manifold 11 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Field Insertions want to evaluate flow equation on a generic curved background need to expand the effective action in powers of the fields, and need to perform a derivative expansion ∞ � 1 use the algebraic relation: e X Q = n ! [ X , Q ] n e X n = 0 ∞ � n ! [∆ , Q ] n f ( n ) (∆) 1 implying: f (∆) Q = n = 0 traces we are interested in are of the following form: � Q µ 1 ...µ m D µ 1 . . . D µ m e − s ∆ � Tr 12 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Coincidence Limit of the A n for the evaluation of the flow equation, all we need are the coincidence limits of derivatives of the D µ 1 . . . D µ m A n these limits can be calculated recursively, by plugging the ansatz of the heat kernel into its defining equation the coincidence limits then correspond to curvature monomials: D µ 1 . . . D µ m A n ∼ R ( n + m ) / 2 to linear order in the curvature we get: D µ D ν A 0 = R µν 6 ✶ + 1 A 1 = R A 0 = ✶ , D µ A 0 = 0 , 2 F µν , 6 ✶ 13 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Summary heat kernel techniques allow for computations in generic curved spaces (within a curvature expansion) the necessary coefficients can be calculated recursively early time expansion misses topological effects derivative and polynomial expansion of fields is necessary 14 / 15

Motivation & Introduction Heat Kernel in FRG Early Time Expansion Thank you for your attention! 15 / 15