Path Integrals, asymptotic expansions and Zeta determinants - PowerPoint PPT Presentation

Path Integrals, asymptotic expansions and Zeta determinants Matthias Ludewig Universität Potsdam February 11, 2016 Matthias Ludewig (Uni Potsdam) 11.02.16 1 / 24

The heat equation Let L = ∇ ∗ ∇ be a Laplace type operator, acting on sections of a vector bundle V on a compact n -dimensional Riemannian manifold M . The heat or diffusion equation ∂ ∂tu ( t, x ) + Lu ( t, x ) = 0 , u (0 , x ) = u 0 ( x ) has a unique solution for given initial data u 0 ∈ L 2 ( M, V ) , and it is given by ˆ p L u ( t, x ) = t ( x, y ) u ( y )d y, t > 0 M where p L t ∈ C ∞ ( M × M, V ⊠ V ∗ ) is the heat kernel of L . Matthias Ludewig (Uni Potsdam) 11.02.16 2 / 24

The heat kernel as a path integral By physicist’s reasoning, the heat kernel can be written as the "sum over all histories", weighted with their probability. ˆ 1 � − 1 � formally � � 0 ] − 1 D γ � 2 d s p L (4 πt ) − n/ 2 [ γ � 1 t ( x, y ) = exp � ˙ γ ( s ) 4 t 0 paths x �→ y Matthias Ludewig (Uni Potsdam) 11.02.16 3 / 24

The heat kernel as a path integral By physicist’s reasoning, the heat kernel can be written as the "sum over all histories", weighted with their probability. ˆ 1 � − 1 � formally � � 0 ] − 1 D γ � 2 d s p L (4 πt ) − n/ 2 [ γ � 1 t ( x, y ) = exp � ˙ γ ( s ) 4 t 0 � �� � paths x �→ y Classical action Matthias Ludewig (Uni Potsdam) 11.02.16 3 / 24

The heat kernel as a path integral By physicist’s reasoning, the heat kernel can be written as the "sum over all histories", weighted with their probability. ˆ 1 � − 1 � formally � � 0 ] − 1 D γ � 2 d s p L (4 πt ) − n/ 2 [ γ � 1 t ( x, y ) = exp � ˙ γ ( s ) 4 t 0 � �� � paths x �→ y Classical action The slash in the integral sign denotes division by the normalization Z = (4 πt ) N/ 2 , N = dimension of path space D γ denotes the Riemannian volume measure of the space of paths. Matthias Ludewig (Uni Potsdam) 11.02.16 3 / 24

The heat kernel as a path integral formally e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ p L (4 πt ) − n/ 2 t ( x, y ) = paths x �→ y Matthias Ludewig (Uni Potsdam) 11.02.16 4 / 24

The heat kernel as a path integral formally e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ p L (4 πt ) − n/ 2 t ( x, y ) = paths x �→ y Theorem We have ˆ 0 ] − 1 d W xy ; t ( γ ) . p L [ γ � 1 t ( x, y ) = C xy ( M ) 0 ] − 1 is the stochastic parallel transport in V , W xy ; t is a In the theorem, [ γ � 1 conditional Wiener measure and C xy ; t ( M ) = { γ ∈ C ([0 , t ] , M ) | γ (0) = x, γ (1) = y } Matthias Ludewig (Uni Potsdam) 11.02.16 4 / 24

The heat kernel as a path integral formally e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ p L (4 πt ) − n/ 2 t ( x, y ) = paths x �→ y Theorem (L. ’15) We have e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 d Σ - H 1 γ p L t ( x, y ) = lim | τ |→ 0 H xy ; τ ( M ) where the limit goes over any sequence of partitions τ = { 0 = τ 0 < τ 1 < · · · < τ N = 1 } of the interval [0 , 1] , the mesh of which tends to zero. In the theorem, � � H xy ; τ ( M ) = γ ∈ H xy ( M ) | γ | [ τ j − 1 ,τ j ] is a geodesic ∀ j Matthias Ludewig (Uni Potsdam) 11.02.16 5 / 24

The space of finite energy paths An important path space is the space of finite energy paths � � γ ∈ H 1 ([0 , t ] , M ) | γ (0) = x, γ (1) = y H xy ( M ) := . This is an infinite-dimensional Hilbert manifold with the Riemannian metric ˆ 1 � � ( X, Y ) H 1 := ∇ s X, ∇ s Y d s, X, Y ∈ T γ H xy ( M ) . 0 It is the "Cameron-Martin-manifold" corresponding to the Wiener measure. Matthias Ludewig (Uni Potsdam) 11.02.16 6 / 24

formally p L (4 πt ) − n/ 2 e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ t ( x, y ) = H xy ( M ) Matthias Ludewig (Uni Potsdam) 11.02.16 7 / 24

formally p L (4 πt ) − n/ 2 e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ t ( x, y ) = H xy ( M ) • The heat kernel has an asymptotic expansion ∞ e t ( x, y ) = e − d ( x,y ) 2 / 4 t � p L t j Φ j ( x, y ) , t ( x, y ) ∼ e t ( x, y ) (4 πt ) n/ 2 j =0 Matthias Ludewig (Uni Potsdam) 11.02.16 7 / 24

formally p L (4 πt ) − n/ 2 e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ t ( x, y ) = H xy ( M ) • The heat kernel has an asymptotic expansion ∞ e t ( x, y ) = e − d ( x,y ) 2 / 4 t � p L t j Φ j ( x, y ) , t ( x, y ) ∼ e t ( x, y ) (4 πt ) n/ 2 j =0 • The path integral has a formal Laplace expansion. Matthias Ludewig (Uni Potsdam) 11.02.16 7 / 24

formally p L (4 πt ) − n/ 2 e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ t ( x, y ) = H xy ( M ) • The heat kernel has an asymptotic expansion ∞ e t ( x, y ) = e − d ( x,y ) 2 / 4 t � p L t j Φ j ( x, y ) , t ( x, y ) ∼ e t ( x, y ) (4 πt ) n/ 2 j =0 • The path integral has a formal Laplace expansion. Goal Compare the two asymptotic expansions. In this talk: The lowest order term. Matthias Ludewig (Uni Potsdam) 11.02.16 7 / 24

formally p L (4 πt ) − n/ 2 e − E ( γ ) / 2 t [ γ � 1 0 ] − 1 D γ t ( x, y ) = H xy ( M ) • The heat kernel has an asymptotic expansion ∞ e t ( x, y ) = e − d ( x,y ) 2 / 4 t � p L t j Φ j ( x, y ) , t ( x, y ) ∼ e t ( x, y ) (4 πt ) n/ 2 j =0 • The path integral has a formal Laplace expansion. Goal Compare the two asymptotic expansions. In this talk: The lowest order term. Matthias Ludewig (Uni Potsdam) 11.02.16 7 / 24

Laplace’s Expansion Let Ω be a finite-dimensional Riemannian manifold. If φ : Ω − → R has the unique non-degenerate minimum x 0 , then a ( x 0 ) e − φ ( x ) / 2 t a ( x ) d x ∼ e − φ ( x 0 ) / 2 t � � 1 / 2 det ∇ 2 φ | x 0 Ω Matthias Ludewig (Uni Potsdam) 11.02.16 8 / 24

Laplace’s Expansion Let Ω be a finite-dimensional Riemannian manifold. If φ : Ω − → R has the unique non-degenerate minimum x 0 , then a ( x 0 ) e − φ ( x ) / 2 t a ( x ) d x ∼ e − φ ( x 0 ) / 2 t � � 1 / 2 det ∇ 2 φ | x 0 Ω Formal conclusion If there is a unique shortest geodesic γ xy connecting x to y , then formally [ γ xy � 1 0 ] − 1 0 ] − 1 D γ ∼ e − d ( x,y ) 2 / 4 t e − E ( γ ) / 2 t [ γ � t � � 1 / 2 ∇ 2 E | γxy det H xy ( M ) since E ( γ xy ) = d ( x, y ) 2 / 2 . Matthias Ludewig (Uni Potsdam) 11.02.16 8 / 24

A formal proof Taylor expand E ( γ ) = E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) , where X is the vector field with exp γ xy ( X ) = γ . Matthias Ludewig (Uni Potsdam) 11.02.16 9 / 24

A formal proof Taylor expand E ( γ ) = E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) , where X is the vector field with exp γ xy ( X ) = γ . Then � � − E ( γ ) exp D γ 2 t � � �� − 1 E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) = exp D X 2 t Matthias Ludewig (Uni Potsdam) 11.02.16 9 / 24

A formal proof Taylor expand E ( γ ) = E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) , where X is the vector field with exp γ xy ( X ) = γ . Then � � − E ( γ ) exp D γ 2 t � � �� − 1 E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) = exp D X 2 t substitute X �→ t 1 / 2 X gives � � − 1 = e − E ( γ xy ) / 2 t 4 ∇ 2 E | γ xy [ X, X ] + O ( t 1 / 2 | X | 3 ) exp D X Matthias Ludewig (Uni Potsdam) 11.02.16 9 / 24

A formal proof Taylor expand E ( γ ) = E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) , where X is the vector field with exp γ xy ( X ) = γ . Then � � − E ( γ ) exp D γ 2 t � � �� − 1 E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) = exp D X 2 t substitute X �→ t 1 / 2 X gives � � − 1 = e − E ( γ xy ) / 2 t 4 ∇ 2 E | γ xy [ X, X ] + O ( t 1 / 2 | X | 3 ) exp D X and in the limit t → 0 � � − 1 ∼ e − E ( γ xy ) / 2 t 4 ∇ 2 E | γ xy [ X, X ] exp D X Matthias Ludewig (Uni Potsdam) 11.02.16 9 / 24

A formal proof Taylor expand E ( γ ) = E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) , where X is the vector field with exp γ xy ( X ) = γ . Then � � − E ( γ ) exp D γ 2 t � � �� − 1 E ( γ xy ) + 1 2 ∇ 2 E | γ xy [ X, X ] + O ( | X | 3 ) = exp D X 2 t substitute X �→ t 1 / 2 X gives � � − 1 = e − E ( γ xy ) / 2 t 4 ∇ 2 E | γ xy [ X, X ] + O ( t 1 / 2 | X | 3 ) exp D X and in the limit t → 0 � � − 1 ∼ e − E ( γ xy ) / 2 t 4 ∇ 2 E | γ xy [ X, X ] exp D X ∼ e − E ( γ xy ) / 2 t det � � − 1 / 2 ∇ 2 E | γ xy Matthias Ludewig (Uni Potsdam) 11.02.16 9 / 24

Remark At a geodesic γ , the Hessian of the energy is given by ˆ 1 ˆ 1 � � � � � � ∇ 2 E | γ [ X, Y ] = ∇ s X, ∇ s Y d s + R γ ( s ) , X ( s ) ˙ γ ( s ) , Y ( s ) ˙ d s 0 0 Matthias Ludewig (Uni Potsdam) 11.02.16 10 / 24

Remark At a geodesic γ , the Hessian of the energy is given by ˆ 1 ˆ 1 � � � � � � ∇ 2 E | γ [ X, Y ] = ∇ s X, ∇ s Y d s + R γ ( s ) , X ( s ) ˙ γ ( s ) ˙ , Y ( s ) d s 0 0 � �� � := R γ ( s ) X ( s ) Matthias Ludewig (Uni Potsdam) 11.02.16 10 / 24