Calculation of Heat Determinant Coefficients for ... - Semantic Scholar

Calculation of Heat Determinant Coefficients for Scalar Laplace type Operators by Benjamin Jerome Buckman

Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics with Specialty in Analysis

New Mexico Institute of Mining and Technology Socorro, New Mexico August, 2014

ABSTRACT In spectral geometry, one would like to know how much we can distinguish two manifolds by the spectrum of a differential operator. Heat invariants, such as the heat trace and heat content have been studied and isospectral manifolds have been found. We introduce a new heat invariant called the heat determinant which is a step foreword in the study of non-spectral invariants on manifolds. The heat determinant is not a pure spectral invariant in that it does not depend only on the eigenvalues of an operator, but also the eigenfunctions of that operator as well. We find the asymptotic expansion in t of this new invariant and calculate the first 3 terms of the heat determinant for the scalar Laplacian on manifolds without boundary. Keywords: Heat invariants; Heat kernel; Asymptotic expansions; Spectral geometry; Heat determinant

CONTENTS

1. INTRODUCTION

1

2. BACKGROUND INFORMATION 2.1 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7

Basic Definitions . . Riemannian Metric . Tensors . . . . . . . . Covariant Derivative Curvature . . . . . . Geodesics . . . . . . Synge Function . . .

. . . . . . .

3 5 6 9 9 10 10

2.1.8

Covariant Taylor Series . . . . . . . . . . . . . . . . . . . . .

13

Laplace Type Operators . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.1 Heat Kernel . . . . . . . . . Heat Invariants . . . . . . . . . . . 2.3.1 Heat Trace . . . . . . . . . . 2.3.2 Heat Content . . . . . . . . 2.3.3 Modified Heat Determinant

. . . . .

16 16 17 17 18

. . . .

20 20 22 22 25

3.3

Heat Determinant Asymptotics for Scalar Operators . . . . . . . . .

27

3.4

Coefficients of Heat Determinant Asymptotic Expansion . . . . . .

33

2.2 2.3

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3. HEAT DETERMINANT 3.1 Definition of Heat Determinant . . . . . 3.2 Calculation of Heat Determinant . . . . 3.2.1 Mixed Derivative of Heat Kernel 3.2.2 Heat Determinant Asymptotics .

4. CONCLUSION

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36

ii

5. APPENDIX 5.1 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2

Evaluation of Coincidence Limits . . . . . . . . . . . . . . . . . . . .

REFERENCES

37 37 38 44

iii

This thesis is accepted on behalf of the faculty of the Institute by the following committee:

Ivan Avramidi, Advisor

I release this document to the New Mexico Institute of Mining and Technology.

Benjamin Jerome Buckman

Date

CHAPTER 1 INTRODUCTION In 1966, Marc Kac popularized the question “Can you hear the sound of a drum?” [14]. In other words, can one determine the shape of a drum based on the frequencies one can hear? The area of spectral geometry was created to answer this question. We take a manifold and find the spectrum of some differential operator, such as the Laplacian, on this manifold. It is obvious that if you have a large drum and a small drum, the frequencies will be different. In spectral geometry, we want to answer the question “how much does the spectrum of an operator determine the geometry of our manifold”. About the same time Kac popularized the question, two different manifolds (2 distinct 16-dimensional tori) were found to have the same eigenvalues [15]. The problem in 2-dimensions was not answered till 1992 [12]. Thus one cannot hear the shape of a drum. Spectral geometry then became the question of how much we can infer of the geometry of an object from the spectrum of an operator. Mathematicians then turned to spectral invariants, objects that are invariant on a manifold that depend on the spectrum of an operator. In practice, it is rather difficult to calculate the eigenvalues of a differential operator on a manifold, thus we study asymptotic regimes. One of the most important operators on a manifold is the Laplacian, ∆. It is the most “natural” second order elliptic partial differential operator on a manifold and it is seen in many important partial differential equations such as the wave and diffusion equations. The wave equations describes how waves propagate through space, while the diffusion equation describes how some material diffuses through space, such as particles or energy. For this paper, we will focus on the heat equation which describes how heat diffuses through a system. The heat equation is given by   ∂ − ∆ f (t, x ) = 0, (1.1) ∂t which we must give an initial condition f (0, x ) = g( x ),

(1.2)

and appropriate boundary conditions if present. The fundamental solution of the heat equation is called the heat kernel. The heat kernel, itself, has many applications to physics and mathematics, more specifically quantum gravity and spectral geometry. 1

From the heat kernel, we can define many invariants. Some spectral invariants that have been studied already include the heat trace and the heat content. In this paper we introduce a non-spectral invariant called the heat determinant. This new invariant is non-spectral since it depends on the eigenvalues and the eigenfunctions of our operator. We calculate the first 3 terms in the asymptotic expansion of the heat determinant. This paper is organized as follows: Chapter 2 contains all the background information we need, focusing on differential geometry. It will introduce all the notation and objects we will be using throughout this paper. We also mention other heat invariants. In chapter 3, we define the heat determinant on vector bundles. We then go through calculation of the heat determinant, then calculating its asymptotics. In section 3.4, we calculate the first 3 coefficients in our asymptotic expansion. In chapter 4, we have our conclusion where we summarize our results. In chapter 5, we have our appendix where we have put many of our technical calculations.

2

CHAPTER 2 BACKGROUND INFORMATION 2.1

Differential Geometry

In this section, we will have a brief overview of differential geometry and the tools which we will be using. For a more in depth introduction, see [3], [8], or [4].

2.1.1

Basic Definitions

In this section, we will give the standard definitions of the mathematical objects we will be using throughout. Definition. An n-dimensional manifold (without bounday), M, is a Hausdorff, second countable, topological space such that for any point p in M, there exists a homeomorphism ϕU : U → V, where U is an open neighborhood of p and V is an open subset of Rn . We call the ordered pair (U, ϕU ) a coordinate patch and we define the local coordinate of a point p ∈ M as ϕU ( p) ∈ V ⊂ Rn . A manifold is called a differential manifold if for every pair of coordinate patches, (U1 , ϕU1 ) and (U2 , ϕU2 ) with −1 U1 ∩ U2 6= ∅ , the new map φ = ϕU2 ◦ ϕU : V1 ⊂ Rn → Rn is differentiable. We 1 will only deal smooth manifolds, that is, φ is smooth (i.e. infinitely differentiable) for all pairs of coordinate patches with non-empty intersections. All manifolds herein are smooth manifolds of dimension n. Definition. A vector (also called tangent vector or contravariant vector) at a point p in a coordinate patch (U1 , xU1 ) on a manifold M is an element of Rn denoted as X p = µ 1 , · · · , X n ) such that for any other coordinate patch (U , x ) with p ∈ ( XU1 ) = ( XU 2 U2 U1 1 U2 , then # " µ n ∂xU2 µ ν ( p) XU . (2.1) XU2 = ∑ ν 1 ∂xU1 ν =1

3

A vector can also be represented as the differential operator # n ∂ ν X p = ∑ XU , ν ∂xU ν =1

(2.2)

p

ν is the coordinate basis of U in local coordinates. where ∂/∂xU

Definition. The Jacobian matrix between two coordinate patches given by (U1 , xU1 ) and (U2 , xU2 ), is the matrix of partial derivatives µ ! ∂xU2 (2.3) ν ∂xU 1 and the Jacobian is µ

∂xU2

J = det

ν ∂xU 1

! .

(2.4)

Suppose we have another smooth manifold, B , and a smooth mapping π : B → M. Then we define the triple (B , π, M) as a bundle where B is the bundle space and π is the projection. The we define a fiber as the image of π −1 ( p), where p ∈ M. A section of a bundle is a map s : M → B such that s( p) ∈ π −1 ( p). If we let B be a vector space, then we call our bundle a vector bundle. We define the tangent space of M at a point p as the vector space containing all tangent vectors to M at p and is denoted as Tp M. We define tangent bundle as the set of all tangent spaces at every point of M and it is denoted as T M. Then we define a vector field as a smooth mapping X : M → T M which can also be represented in local coordinates as n

X ( p) =

∂

∑ X ν ( p) ∂xν .

(2.5)

ν =1

Definition. A covector (also called a cotangent or covariant vector) is a linear functional α p : Tp M → R and is referred to as the dual of a vector. Thus the dual of Tp M is the set of all linear functionals on Tp M and is called the cotangent space at the point p and is denoted as Tp∗ M. Similarly, the cotangent bundle is the set of all cotangent spaces at every point of M and is denoted as T ∗ M. Then we define a covector field (also called a 1-form) as a smooth mapping α : M → T ∗ M. We also have a covector bundle which is the dual to the vector bundle. We will henceforth use the notation ∂µ = ∂x∂µ . The dual to the coordinate basis ∂µ is denoted as dx µ and satisfies the condition µ

dx µ (∂ν ) = ∂ν (dx µ ) = δν , 4

(2.6)

where µ δν



=

if µ = ν if µ 6= ν

1 0

(2.7)

is the Kronecker delta. We will be using Einstein summation notation throughout this paper, thus we sum over repeated indices (i.e. βµ αµ = ∑µ βµ αµ ).

2.1.2

Riemannian Metric

A Riemannian metric is the smooth assignment of a positive definite inner product in Tp M to p ∈ M. Definition. An inner product on a vector space v is defined as the mapping h·, ·i : V × V → C that satisfies the properties that for all x, y, z ∈ V and α, β ∈ C, 1. h x, yi = hy, x i, 2. hαx + βy, zi = α h x, zi + β hy, zi , 3. h x, x i ≥ 0, 4. h x, x i = 0 implies x = 0. Then inner product defines a norm denoted as kvk =

p

hv, vi.

The Riemannian metric is denoted as gµν and it is a function of x on M. The metric is a positive definite matrix. The inverse of the metric is denoted as ( gµν )−1 = gµν . We will also denote the determinant of the metric as g = det gµν . Thus the inner product defined on Tp M is given by the metric as

hu, vi = gµν uµ vν .

(2.8)

The inverse defines an inner product on Tp∗ M. Using gµν and gµν , we are able to raise and lower indices, thus each vector has a corresponding covector and vise-versa given by vµ = gµν vν or αµ = gµν αν . (2.9) Definition. Suppose we have a smooth curve on a
manifold M parametrized by a variable τ ∈ [0, t]. Then x (τ ) = x1 (τ ), · · · , x n (τ ) is the trajectory of the curve in M. Then the µth component of the tangent vector of the curve is clength of the curve from τ = 0 to τ = t is s=

Zt 0

dx µ (τ ) dτ

r

t µ ν

dx (τ ) Z

= dτ gµν dx dx . dτ

dτ dτ dτ 0

5

. Therefore the ar-

(2.10)

The standard Euclidean volume form denoted as dx = dx1 · · · dx n is not invariant under coordinate transformation. Thus we introduce the Riemannian volume form which is invariant. Definition. The Riemannian volume element is 1

d vol( x ) = g 2 ( x )dx.

(2.11)

.

2.1.3

Tensors

Definition. A tensor of type (p,q) of a vector space V is a multilinear map ∗ · · × V }∗ → R T:V · · × V} × V | × ·{z | × ·{z p times

and is denoted as

(2.12)

q times

µ µ ···µ p

Tν11ν22···νq ,

(2.13)

where each index goes from 1 to n. A tensor is invariant under coordinate changes with every upper index transforming as a vector and every lower index transforming as a covector. Then a tensor field is the smooth assignment of tensor to every point of M and is denoted as µ µ ···µ p Tν11ν22···νq ( x ). (2.14) Suppose T is a tensor of type (2,0) or (1,1) or (0,2), then we call T a matrix. Then we denote the trace of the matrix T as tr T = gµν Tµν = T µ µ .

(2.15)

Definition. Let Sq be the group of permutations of the set Zq = {1, 2, · · · , q}. A permutation ϕ : Zq → Zq is represented by   1 ··· q . (2.16) ϕ (1) · · · ϕ ( q ) An elementary permutation is a permutation that exchanges the order of only two elements. We say a permutation is even (odd) if it is the product of an even (odd) number of elementary permutations. We define the function  1 if ϕ is even sign( ϕ) = . (2.17) −1 if ϕ is odd 6

Using the above definition, we define the anti-symmetrization and symmetrization of tensors: Definition. Let T be a tensor of type (0,q). the symmetrization of the indices of tensor T is given by 1 T(ν1 ν2 ···νq ) = Tνϕ(1) νϕ(2) ···νϕ(q) . (2.18) q! ϕ∑ ∈S q

Secondly, we define the anti-symmetrization of tensor T as T[ν1 ν2 ···νq ] =

1 q!

∑

ϕ ∈ Sq

sign( ϕ) Tνϕ(1) νϕ(2) ···νϕ(q) .

(2.19)

We will use vertical lines to denote omission of an index from symmetrization and antisymmetrization, i.e. T[ν1 ν2 ···νj−1 |νj |νj+1 ···νq ] =

1 ( q − 1) !

∑

ϕ ∈ S q −1

sign( ϕ) Tνϕ(1) νϕ(2) ···νϕ( j−1) νj νϕ( j+1) ···νϕ(q−1) . (2.20)

The Levi-Civita symbol is not a tensor, but is what we call a tensor density and it will be used often. The Levi-Civita symbol is εµ1 µ2 ···µn = ε µ1 µ2 ···µn  if {µ1 , µ2 , · · · , µn } is an even perm. of {1, 2, · · · , n}  1 −1 if {µ1 , µ2 , · · · , µn } is an odd perm. of {1, 2, · · · , n} . =  0 otherwise (2.21) Throughout this paper, we will use the Levi-Civita density, which is just the LeviCivita symbol scaled by the metric. The density is invariant and indices can be raised or lowered by the metric. The density will be denoted as 1

Eµ1 µ2 ···µn = g 2 ε µ1 µ2 ···µn

and

1

E µ1 µ2 ···µn = g− 2 εµ1 µ2 ···µn .

(2.22)

The Levi-Civita symbol is most helpful in writing the determinant of a matrix: det( Aµ ν ) = εµ1 ···µn A1 µ1 · · · An µn

= ε ν1 ···νn Aν1 1 · · · Aνn n 1 = εµ1 ···µn ε ν1 ···νn Aν1 µ1 · · · Aνn µn . n!

(2.23)

We note that the Levi-Civita symbol satisfies εµ1 ···µn Aν1 µ1 · · · Aνn µn = εν1 ···νn det ( Aµ ν ) , 7

(2.24)

and µ ···µ

εµ1 ···µk α1 ···αn−k ε ν1 ···νk α1 αn−k = (n − k )! δν11···νkk ,

(2.25)

where µ ···µ

µ

µ

δν11···νkk = k! δ[ν1 · · · δν k] . 1

(2.26)

k

These will be used in the following lemma. Lemma 1. Suppose we have a matrix Aµ ν with its inverse ( Aµ ν )−1 denoted as Bν µ , and we define the object F µ1 ···µk ν1 ···νk = then

1 εµ1 ···µk α1 ···αn−k ε ν1 ···νk β1 ··· β n−k A β1 α1 · · · A β n−k αn−k , (n − k)! F µ1 ···µk ν1 ···νk = k!B[µ1 ν1 · · · Bµk ] νk detA.

(2.27)

(2.28)

Proof. Let us multiply (2.27) by k A’s, then we obtain Aν1 λ1 · · · Aνk λk F µ1 ···µk ν1 ···νk =

1 εµ1 ···µk α1 ···αn−k ε ν1 ···νk β1 ··· β n−k (n − k)!

× A λ1 · · · A λ k A α1 · · · A αn−k 1 = εµ1 ···µk α1 ···αn−k ε λ1 ···λk α1 ···αn−k detA (n − k)! µ ···µ =δλ11 ···λkk detA. ν1

νk

β1

(2.29)

β n−k

(2.30) (2.31)

Now, by multiplying by k B’s, we have Bλ1 κ1 · · · Bλk κk Aν1 λ1 · · · Aνk λk F µ1 ···µk ν1 ···νk =δκν11 · · · δκkk F µ1 ···µk ν1 ···νk ν

= F µ1 ···µk κ1 ···κk .

(2.32)

Therefore µ ···µ

F µ1 ···µk κ1 ···κk = Bλ1 κ1 · · · Bλk κk δλ11 ···λk detA k

= k!B

[ µ1

κ1

8

···B

µk ]

κk detA.

(2.33)

2.1.4

Covariant Derivative

Partial derivatives are not invariant under a change of coordinates, therefore we must define an “invariant derivative”. First we need the affine connection which relates the tangent vectors at two different points on a manifold. The affine connection that is torsion-free and compatible with the metric is called the LeviCivita connection and is denoted by the Christoffel symbols which are given by Γα µν =

1 αβ g (∂µ g βν + ∂ν gµβ − ∂ β gµν ). 2

(2.34)

Then the covariant derivatives of a vector field v and a covector field α are

∇ν vµ = ∂ν vµ + Γµ βν v β and ∇ν αµ = ∂ν αµ − Γ β µν α β .

(2.35)

We also have covariant derivative of a tensor: µ µ ···µ

µ µ ···µ

βµ ···µ

µ µ ··· β

∇α Tν11ν22···νq p =∂α Tν11ν22···νq p + Γµ1 βα Tν1 ν22 ···νqp + · · · + Γµ p βα Tν11ν22···νq µ µ ···µ

µ µ ···µ

− Γ β ν1 α Tβν12 ···2 νq p − · · · − Γ β νq α Tν11ν22··· β p .

(2.36)

We also consider a vector bundle, V , on our manifold, M. Let Aµ be a connection on this vector bundle, then the covariant derivative of a section ϕ of the bundle V is given by ∇µ ϕ = ∂µ ϕ + Aµ ϕ. (2.37)

2.1.5

Curvature

The Riemann curvature tensor Rµ ναβ is found by finding the commutator of covariant derivatives evaluated on a vector. So we get   ∇α , ∇ β vµ = Rµ ναβ vν (2.38) with

Rµ νγδ = ∂γ Γµ νδ − ∂δ Γµ νγ + Γµ βγ Γ β νδ − Γµ βδ Γ β νγ .

(2.39)

We can also evaluate the commutator of covariant derivatives on a covector and we have (2.40) [∇γ , ∇δ ] αµ = − Rν µγδ αν . get

Lastly, we can evaluate the commutator on a tensor of type ( p, q) and we µ µ ···µ

[∇γ , ∇δ ] Tν11ν22···νq p =

p

∑

µ ···µ

Rµa βγδ Tν11ν2 ···aν−q 1

a =1

βµ a+1 ···µ p

q

−

b =1

9

µ µ ···µ p

∑ Rβ νb γδ Tν11···2νb−1 βνb+1 ···νq . (2.41)

We can raise and lower indices on the Riemann tensor just like any tensor. The Riemann tensor is anti symmetric in the first two indices (i.e. Rµνγδ = − Rνµγδ ), anti symmetric in the last two indices (i.e. Rµνγδ = − Rµνδγ ), and is symmetric in the exchanging of these pairs (i.e. Rµνγδ = Rγδµν ). By contracting over indices, we can define new tensors. Definition. The Ricci tensor is given by Rµν = Rγ µγν

(2.42)

R = gµν Rµν = Rµ µ .

(2.43)

and the scalar curvature is given by

The commutator of covariant derivatives of a section, ϕ, of a vector bundle, V , defines the curvature 2-form, R, of the bundle connection, A,   ∇µ , ∇ν ϕ = Rµν ϕ, (2.44) where

  Rµν = ∂µ Aν − ∂ν Aµ + Aµ , Aν .

2.1.6

(2.45)

Geodesics

Definition. A geodesic is a curve x (τ ) on a manifold M such that the following equation holds:  µ γ ν dx d2 x µ dx ν µ dx dx ∇ν = =0 (2.46) + Γ νγ dτ dτ dτ dτ dτ 2 for some parametrization τ which we call the affine parameter. A geodesic is a curve for which the tangent vector to the curve transported along the curve remains the tangent vector and is referred to as the shortest curve between two points. Any two points on a manifold M that are sufficiently close can be connected by a single geodesic. The distance along the geodesic between two points x and x 0 is called the geodesic distance and is denoted as d( x, x 0 ).

2.1.7

Synge Function

Before we define the Synge function, we need to introduce the injectivity radius of a manifold.

10

Definition. Let x 0 ∈ M, then we define the ball of radius r as  Br ( x 0 ) = x ∈ M | d( x, x 0 ) < r .

(2.47)

Definition. Let x, x 0 ∈ U ⊂ M. We say x and x 0 are conjugate in U if they are connected by more than one geodesic. Definition. Let rinj ( x 0 ) be the largest radius of the ball Br ( x 0 ) such ∀ x ∈ Br ( x 0 ), x and x 0 are not conjugate in Br ( x 0 ). In other words, rinj ( x 0 ) is the largest ball around x 0 such that all points are connected to x 0 by a single geodesic. We call rinj ( x 0 ) the injectivity radius of the point x 0 . Definition. The injectivity radius of the manifold, M, is given by Γinj (M) = inf rinj ( x 0 ).

(2.48)

x 0 ∈M

Thus if r < Γinj , then ∀ x 0 ∈ M, Br ( x 0 ) will not contain conjugate points. Definition. The Synge function (also called a world function) is a bi-scalar function and is defined as half the square geodesic distance between points x and x 0 on M. Thus 1 1 σ( x, x ) = d2 ( x, x 0 ) = t 2 2 0

Zt 0

dτ gµν

dx µ dx ν . dτ dτ

(2.49)

We note σ is multivalued for conjugate points. We will consider σ as single valued when, along the shortest geodesic between x and x 0 , d( x, x 0 ) < Γinj . We will then use this distance along this shortest geodesic as d( x, x 0 ) in (2.49). Thus σ is only defined “locally”. Before we continue further, we will mention some of the notation we will be using. The coincidence limit of a bi-scalar function is denoted as   f ( x, x 0 ) = lim f ( x, x 0 ). (2.50) x→ x0

For the Synge function, we will use indices to denote covariant derivatives: σµ = ∇µ σ( x, x 0 ) and σµ0 = ∇µ0 σ( x, x 0 ),

(2.51)

where ∇µ is the covariant derivative with respect to the x coordinate and ∇µ0 is the covariant derivative with respect to the x 0 coordinate. Thus we can have multiple indices indicating multiple derivatives (i.e. σµ0 ν = σµν0 = ∇µ0 ∇ν σ = ∇ν ∇µ0 σ).

11

The Synge function is very important in that it completely describes the local geometry near x and x 0 . The Synge function has the following properties: 0

σµ σµ =2σ, σµ σ σν0 σ

µ

ν0

ν

µ

σν0 σν =2σ,

=σν ,

σµ σ

=σµ ,

σµ0 σ

µ

µ0

(2.52)

ν0

=σν0 ,

(2.53)

ν0

=σν0 .

(2.54)

We will denote the inverse matrix of σµν as γµν , the inverse matrix of σµν0 as 0 0 0 γµν , and the inverse matrix of σµ0 ν0 as γµ ν . Thus it is easy to see that γ matrices satisfy 0

µ

γµν σνα = δα , σµ γµν = σν , σν0 γ

ν0 µ

µ

γµν σν0 α = δα , σµ γ

= σµ ,

σµ0 γ

µν0

µ0 ν0

ν0

=σ , ν0

=σ .

(2.55) (2.56) (2.57)

See Section 5.2 for more on the Synge function, its derivatives and their coincidence limits which will be used extensively. We introduce the Van Fleck-Morette determinant which is defined as 1

1

∆( x, x 0 ) = g− 2 ( x ) g− 2 ( x 0 )det (−σµν0 ).

(2.58)

Again, see Section 5.2 for information on the derivatives and the coincidence limits of ∆. We will use the following change of variables later: 0

0 σµ σν ξ = √ and ξ ν = √ . t t

µ

(2.59)

These change of coordinates satisfy the following: 0

| ξ |2 = ξ µ ξ µ = ξ ν ξ ν 0 = σν0 µ ξ µ = ξ ν0

and

2σ , t 0

σµν0 ξ ν = ξ µ .

(2.60) (2.61)

Then the standard partial derivatives transform as 0

σν µ ∂ ∂ √ = 0 ∂x µ t ∂ξ ν

and

√ µ ∂ ∂ = tγ ν0 µ . 0 ∂x ∂ξ ν

(2.62)

Thus we have σµ

0 ∂ ∂ = ξ ν ν0 , µ ∂x ∂ξ

12

(2.63)

and the volume element becomes n

1

1

dvol( x ) = g 2 ( x ) dx = t 2 ∆−1 ( x, x 0 ) g 2 ( x 0 ) dξ.

(2.64)

We also introduce the Gaussian average over the variable ξ

h f (ξ )i a =

 a  n2 Z π

  1 dξ g 2 ( x 0 ) exp − aξ 2 f (ξ ),

(2.65)

Rn

where a is a positive constant. In particular, we have

hξ µ1 ξ µ2 · · · ξ µ2k+1 i a = 0, hξ µ1 ξ µ2 · · · ξ µ2k i a =

(2k)! k ( 22 a )

k!

g(µ1 µ2 · · · gµ2k−1 µ2k ) .

(2.66) (2.67)

For more on the Gaussian integral and Gaussian average, see section 5.1.

2.1.8

Covariant Taylor Series

We consider a section, ϕ, of a vector bundle, V , on the manifold M. To expand ϕ in covariant Taylor series, we will transport ϕ from x to x 0 , then expand it in a Taylor series along the geodesic, then transport it back to x. We first introduce the parallel transport operator P ( x, x 0 ), which parallel transports a field from the point x 0 to x along the connecting geodesic. Thus, the parallel transport operator is defined as σµ ∇µ P = 0, (2.68) with [P ] = 1. By taking derivatives of the above equation and symmetrizing, it is easy to see that h i ∇(µ1 · · · ∇µk ) P = 0, (2.69) with the same properties holding for primed derivatives and inverse of P given by P −1 ( x, x 0 ) = P ( x 0 , x ). Therefore, let us define ϕ¯ = P −1 ϕ.

(2.70)

We expand this in a Taylor series along the geodesic using our affine parameter, τ. We find ! ∞ 1 k dk ¯ ϕ¯ = ∑ τ ϕ . (2.71) k k! dτ k =0 τ =0

13

µ

d ¯ = ∇µ ϕ, ¯ and that We know that dτ = U µ ∂µ , where U µ = dx dτ . Noting that ∂µ ϕ µ ν U ∇µ U = 0 since it is just the geodesic equation, we have ∞

ϕ¯ =

 1 k  µ1 µk −1 U · · · U ∇ · · · ∇ P ϕ . τ ∑ k! ( µ1 µk ) τ =0 k =0

(2.72)

0

Noting that −τU µ (0) = σµ (as derived in Section 5.2), and that evaluating P −1 and its derivatives at τ = 0 is the same as taking the coincidence limit, we find ∞

h i (−1)k µ10 µ0k ϕ¯ = ∑ σ ···σ ∇ ( µ1 · · · ∇ µ k ) ϕ . k! k =0

(2.73)

Therefore, we have the covariant Taylor expansion of a field ϕ given by ∞

ϕ=P

2.2

h i (−1)k µ10 µ0k ∇ · · · ∇ ϕ . σ · · · σ ∑ k! ( µ1 µk ) k =0

(2.74)

Laplace Type Operators

For this paper, we will only be looking at second-order elliptic partial differential operators, more specifically, the scalar Laplacian acting on manifolds without boundary. However, the heat determinant we introduce later can be applied any second-order elliptic partial differential operator. Definition. Let H = L2 (V ) be the Hilbert space (complete inner product space) of square integrable sections of our vector bundle V . The inner-product on H is defined as

( ϕ, ψ) =

Z

dvol( x ) h ϕ, ψi ,

(2.75)

M

where h·, ·i is the inner product in the fiber of our bundle V . Suppose we have a second order partial differential operator given by L = − aµν ∇µ ∇ν + bµ ∇µ + c,

(2.76)

where a, b, and c could all be functions of x. Then we denote the leading symbol as σL (ξ, x ) = aµν ξ µ ξ ν . The operator L is elliptic if ∀ξ 6= 0 and ∀ x ∈ M, σL (ξ, x ) ≥ C |ξ |2 for some C > 0. i.e. aµν ( x ) ξ µ ξ ν ≥ C |ξ |2

∀ξ 6= 0 and ∀ x ∈ M.

where |ξ |2 = δµν ξ µ ξ ν . Let C ∞ (V ) be the space of smooth functions on V . 14

(2.77)

Definition. A Laplace type operator is an elliptic operator L : C ∞ (V ) → C ∞ (V ) of the form L = −∆ + Q (2.78) where Q is a smooth endomorphism of the bundle V and ∆ is the standard Laplacian ∆ = gµν ∇µ ∇ν .

(2.79)

The adjoint of an operator L is denoted as L∗ which satisfies

h f , Lgi = h L∗ f , gi

(2.80)

for all f and g in H. An operator is self-adjoint if L∗ = L and the domains of L∗ and L are identical. The Laplace type operators are symmetric (i.e. h f , Lgi = h L f , gi). Laplace type operators have self-adjoint extensions, but they, themselves, are not selfadjoint. We will not distinguish between L and its self-adjoint extension. We only deal with self-adjoint elliptic operators on compact manifolds without boundary, since they have a countable set of eigenvalues and eigenfunctions. Recall that if λi is an eigenvalue with the corresponding eigenfunction ϕi , then they satisfy the equation Lϕi = λi ϕi .

(2.81)

If the set of eigenfunctions forms a complete orthonormal basis on our Hilbert space, then we can spectrally decompose a function of the operator L: ∞

f ( L)φ =

∑

f (λk ) ( ϕk , φ) ϕk .

(2.82)

k =0

where φ is a section of our vector bundle. This makes life easy when dealing with elliptic operators. We define the trace of a function of an operator as the sum of the function evaluated at our eigenvalues, ∞

Tr f ( L) =

∑

f ( λ k ).

(2.83)

k =0

For a more indepth discussion on operators and spectral decomposition, see [6].

15

2.2.1

Heat Kernel

As we have already mentioned, we will only study compact manifolds without boundary. The heat kernel of an operator L is denoted as U (t, x, x 0 ) and satisfies the heat equation

(∂t + L) U (t, x, x 0 ) = 0,

(2.84)

U (0, x, x 0 ) = δ( x, x 0 ),

(2.85)

with the initial condition

1

1

where δ( x, x 0 ) = g 4 ( x ) g 4 ( x 0 )δ( x − x 0 ) is the covariant δ function. The operator L acts on unprimed coordinates. The heat kernel gives us the general solution to to an arbitrary initial condition. That is, the solution of the initial value problem

(∂t + L) f (t, x ) = 0, f (0, x ) = h( x ),

(2.86) (2.87)

is given by f (t, x ) =

Z

dvol( x 0 ) U (t, x, x 0 ) h( x 0 ).

(2.88)

M

We can spectrally decompose the heat kernel to 0

U (t, x, x ) =

∞

∑ exp (−tλk ) ϕk (x) ϕ∗k (x0 ).

(2.89)

k =0

2.3

Heat Invariants

In this section, we discuss a few heat invariants. We will assume L is the general scalar Laplace type operator L = −∆ + Q, where Q is a non-constant smooth endomorphism.

16

(2.90)

2.3.1

Heat Trace

One of the most important heat invariants is the heat trace. It has been studied extensively [11]. The heat trace is given by the trace of the heat semigroup exp(−tL), Θ(t) = Tr exp (−tL) =

∞

∑ e−tλk .

(2.91)

k =1

This is a pure spectral invariant, the asymptotics of which play an important role in spectral geometry and mathematical physics. It is well known that as t → 0, Θ(t) ∼ (4πt)

− n2

∞

(−1)k ∑ k! Ak tk , k =0

(2.92)

where Ak are called the heat trace invariants. They are given by integral of local invariants, more specifically, the first two are A0 = vol(M),   Z 1 A1 = dvol Q − R . 6

(2.93) (2.94)

M

2.3.2

Heat Content

Another heat invariant is the heat content and is given by the integral of the heat kernel of the manifold. For scalar operators, the heat content is defined by Z

Π(t) =

dvol( x ) dvol( x 0 ) U (t, x, x 0 ) =

∞

∑ e−tλk |Φk |2 ,

(2.95)

k =1

M×M

where Φk =

Z

dvol ϕk .

(2.96)

M

From (2.89), we find that Z

dvol( x 0 ) U (t; x, x 0 ) = (exp(−tL) · 1) ( x ).

(2.97)

M

Thus expanding the heat content as t → ∞ gives us Π(t) ∼

∞

(−1)k ∑ k! Πk tk , k =0 17

(2.98)

where Π0 = vol(M), Π1 =

Z

(2.99)

dvol Q,

(2.100)

M

Πk =

Z

dvol Q (−∆ + Q)k−2 Q,

k ≥ 2.

(2.101)

M

We note that if Q = 0, then Π(t) = vol(M).

2.3.3

Modified Heat Determinant

We introduce a modified heat determinant which we will not use since it vanishes for manifolds without boundary. We define a matrix P˜ as P˜µν0 (t; x, x 0 ) = ∇µ ∇ν0 U (t; x, x 0 ) ∞

=

∑ e−tλl ∇µ ϕl (x)∇ν0 ϕl (x0 ).

(2.102)

l =1

Thus we can define the modified heat determinant as K˜ (t) =

Z

dx dx 0 det P˜µν0 (t; x, x 0 )

M×M

=

1 n!

Z

0

0

dx dx 0 εµ1 ···µn εν1 ···νn P˜µ1 ν0 · · · P˜µn νn0 .

(2.103)

1

M×M

We define a new object E˜ as E˜ k1 ···kn =

Z

dx εµ1 ···µn ∇µ1 ϕk1 · · · ∇µn ϕkn

M

=

Z

(2.104) dϕk1 ∧ · · · ∧ dϕkn .

M

We note that dϕk1 ∧ · · · ∧ dϕkn = d( ϕk1 ∧ dϕk2 ∧ · · · ∧ dϕkn ), thus using Stokes’ theorem, we have E˜ k1 ···kn =

Z

ϕk1 ∧ dϕk2 · · · ∧ dϕkn .

∂M

Thus E˜ = 0 if our manifold has no boundary. 18

(2.105)

We can write K˜ as 1 K˜ (t) = n!

Z

0

0

dx dx 0 εµ1 ···µn εν1 ···νn

M× M ∞

(

∑

×

! e−tλk1 ∇µ1 ϕk1 ( x )∇ν0 ϕk1 ( x 0 ) 1

k 1 =1

(2.106)

.. . ∞

×

∑

!) e−tλkn ∇µn ϕkn ( x )∇νn0 ϕkn ( x 0 )

,

k n =1

then 1 K˜ (t) = n! k

∞

∑

1 ,··· ,k n =1

e−t(λk1 +···+λkn ) E˜ k1 ···kn E˜ k0 1 ···kn

∞

=

∑

e

(2.107) −t(λk1 +···+λkn )

1≤k1 ≤···≤k n

E˜ k1 ···kn E˜ k0 1 ···kn .

From (2.105), it is clear that for manifolds without boundary K˜ = 0. However if the boundary of our manifold is not empty, then the modified heat determinant is non-zero and we can study this new heat invariant that depends on the eigenvalues and eigenfunctions of our operator L. However, this only works for a scalar operator, as it is not invariant for elliptic second-order differential operators on vector bundles.

19

CHAPTER 3 HEAT DETERMINANT In this section, we will define the heat determinant of an elliptic second order partial differential operator on a vector bundle, then look more specifically at the heat determinant for scalar Laplacian.

3.1

Definition of Heat Determinant Define the tensor P as Pµν0 (t; x, x 0 ) = tr U ∗ (t; x, x 0 )∇µ ∇ν0 U (t; x, x 0 ),

(3.1)

where U (t; x, x 0 ) is the heat kernel of an operator L. We note that the trace here is over the fiber of our vector bundle, thus Pµν0 is a matrix with scalar entries. We also not that if L is self-adjoint, then U ∗ (t, x, x 0 ) = U (t, x 0 , x ). Then the heat determinant is defined as K (t) =

Z

dx dx 0 detPµν0 (t, x, x 0 ),

(3.2)

M×M

which can also be written as 1 K (t) = n!

Z

dx dx 0 εµ1 ···µn εν1 ···νn Pµ1 ν0 (t, x, x 0 ) · · · Pµn νn0 (t, x, x 0 ). 1

(3.3)

M×M

Let ϕk be the eigenfunction corresponding to the eigenvalue λk of the operator L. Then we define f kl ( x ) = h ϕk , ϕl i , (3.4) then we have

( ϕk , ϕl ) =

Z

dvol( x ) f kl ( x ) = δkl .

(3.5)

M

Now define

 Bkl µ = ϕk , ∇µ ϕl , 20

(3.6)

which then defines the one-forms

 Bkl = Bkl µ dx µ = ϕk , ∇µ ϕl dx µ = h ϕk , Dϕl i .

(3.7)

Lastly, we define ln Ekl1 ··· 1 ··· k n

Z

=



 dx εµ1 ···µn ϕk1 , ∇µ1 ϕl1 · · · ϕkn , ∇µn ϕln

M

Z

=

(3.8) Bk1 l1 ∧ · · · ∧ Bkn ln .

M

Thus we can write the tensor P as ( ∞

∑ e−tλk ϕk (x0 ) ϕk (x)

Pµν0 (t; x, x 0 ) = tr

!

=

∑

e −t(λk

∑

∗ 0 e−t(λk +λl ) Bkl µ ( x ) Bkl ν 0 ( x ).

k,l =1 ∞

=

∑ e−tλl ∇µ ϕl (x)∇ν0 ϕl (x0 )

+λ ) l

!)

l =1

k =1 ∞

∞

ϕk ( x ), ∇µ ϕl ( x ) h ϕk ( x 0 ), ∇ν0 ϕl ( x 0 )i 

k,l =1

(3.9) Then K (t) =

1 n!

Z

dx dx 0 εµ1 ···µn εν1 ···νn

M×M ∞



∑

×

e

∗ 0 Bkl µ1 ( x ) Bkl ν10 ( x )

e

−t(λk +λl )

∗ 0 Bkl µn ( x ) Bkl νn0 ( x )

k,l =1

.. . 

∞

∑

×

k,l =1

=

1 n! k

∞

Z

∑

× e .. . 

× e 1 n! k

(3.10)



dx dx 0 εµ1 ···µn εν1 ···νn

1 l1 ··· k n ln =1M×M



=



−t(λk +λl )

∞

∑

1 l1 ··· k n ln =1

− t ( λ k 1 + λ l1 )

Bk1 l1 µ1 ( x ) Bk∗1 l1 ν0 ( x 0 ) 1

− t ( λ k n + λ ln )

 (3.11)

Bkn ln µn ( x ) Bk∗n ln νn0 ( x 0 )



ln ∗ l1 ···ln e−t(λk1 +λl1 +···+λkn +λln ) Ekl1 ··· ···k n E k ···k n . 1

21

1

(3.12)

3.2

Calculation of Heat Determinant

Let’s fix a point x 0 . Then in a ball Br ( x 0 ) of radius r < Γinj , the heat kernel for an arbitrary elliptic second order differential operator is given by   1 1 σ ( x, x 0 ) 0 2 ( x, x 0 )P ( x, x 0 ) Ω ( t; x, x 0 ), U (t; x, x ) = exp − ∆ (3.13) n/2 2t (4πt) where ∆( x, x 0 ) is the Van-Fleck-Morette determinant (2.58), P ( x, x 0 ) is the parallel transport operator, and Ω(t, x, x 0 ) is the transport function. It is well known that at t → 0 , Ω as the asymptotic expansion given by ∞

(−t)k ak ( x, x 0 ), Ω(t, x, x ) ∼ ∑ k! k =0 0

(3.14)

where an ( x, x 0 ) are the heat kernel coefficients which have been calculated (see [2] and sources therein). In this section, we calculate the mixed derivative of the heat kernel and find a general formula for the det P.

3.2.1

Mixed Derivative of Heat Kernel

We look at the mixed derivative of the heat kernel first for vector valued operators. For an arbitrary vector-valued second order differential operator, let us define 1 Ψ(t; x, x 0 ) = ∆ 2 ( x, x 0 )P ( x, x 0 )Ω(t; x, x 0 ). (3.15) Then 0

U (t; x, x ) =

(4πt)

σ( x, x 0 ) exp − 2t 

1 n/2



Ψ(t; x, x 0 ).

(3.16)

Taking two covariant derivatives of this, we have

∇µ ∇ν0 U =

1

 σ exp − 2t (4πt)n/2    1 1  σ 0 Ψ + σµ Ψν0 + σν0 Ψµ + Ψµν0 . × σµ σν0 Ψ − 2t µν 4t2

(3.17)

Then we get Pµν0

 σ 1 1 = tr U (t, x , x )∇µ ∇ν0 U (t, x, x ) = exp − Y 0, t 2t µν (4πt)n ∗

0

0

22

(3.18)

where Yµν0

  1 ∗ ∗ ∗ ∗ = σµ σν0 tr Ψ Ψ − σµν0 tr Ψ Ψ + σµ tr Ψ Ψν0 + σν0 tr Ψ Ψµ 2t + 2t tr Ψ∗ Ψµν0 .

(3.19)

So if we define the following: Sµν0 = σµ σν0 tr Ψ∗ Ψ,

(3.20)

∗

Vµν0 = −σµ tr Ψ Ψν0 ,

(3.21)

Wµν0 = −σν0 tr Ψ∗ Ψµ ,

(3.22)

Zµν0 = −σµν0 tr Ψ∗ Ψ + 2t tr Ψ∗ Ψµν0 ,

(3.23)

then

1 S 0. (3.24) 2t µν Since S, V, W, and Z are defined this way, they satisfy the following relations: Yµν0 = Zµν0 + Wµν0 + Vµν0 +

S [ µ1 [ ν 0 S µ2 ] ν 0 ] = V [ µ1 [ ν 0 V µ2 ] ν 0 ] = W [ µ1 [ ν 0 W µ2 ] ν 0 ] 2

1

=S Thus

2

1

 det Pµν0 =

[ µ1

[ν10

V

µ2 ]

ν20 ]

2

1

=S

[ µ1

[ν10

 σ 1 1 exp − t (4πt)n 2t

W

µ2 ]

ν20 ]

= 0.

(3.25)

n det Yµν0 ,

(3.26)

where det Yµν0

 1 µ1 ···µn ν0 ···νn0 = ε ε1 Zµ1 ν0 · · · Zµn νn0 1 n!   1 + n Wµ1 ν10 + Vµ1 ν10 + Sµ1 ν10 Zµ2 ν20 · · · Zµn νn0 2t  + n (n − 1) Wµ1 ν0 Vµ2 ν0 Zµ3 ν0 · · · Zµn νn0 . 2

1

(3.27)

3

So 0

1 2

1 2

det Pµν0 ( x, x ) = g ( x ) g ( x ) where

1

0

(2t)n (4πt)n 1

1

2

 n  0 exp − σ( x, x ) H (t; x, x 0 ), (3.28) t

H (t; x, x 0 ) = g− 2 ( x ) g− 2 ( x 0 ) det Yµν0 (t; x, x 0 ).

23

(3.29)

Therefore, our heat determinant is given by (3.2), which in all of its glory is K (t) =

1

Z

(2t)n (4πt)n

 n  0 dx dx exp − σ( x, x ) det Yµν0 (t; x, x 0 ). t 0

2

M×M

(3.30)

Let r be a real number less than the injectivity radius of the manifold, Γinj (M). Then we note that K ( t ) = K1 ( t ) + K2 ( t ) ,

(3.31)

where K1 ( t ) = K2 ( t ) =

1

Z

(2t)n (4πt)n Z M

dx 0

2

M

Z

dx

Z

0 Br

(x0 )

 n  0 dx exp − σ ( x, x ) det Yµν0 (t; x, x 0 ), t

dx det Pµν0 (t; x, x 0 ).

(3.32) (3.33)

M− Br ( x 0 )

By standard elliptic estimates, it has been shown [13] that ∀ x ∈ M − Br ( x 0 ) and 0 < t < 1 that the heat kernel has the estimate  2 U (t; x, x 0 ) ≤ C1 t− n2 exp − r , (3.34) 4t where C1 is a constant. From the heat equation, we note that each spatial deriva1 tive has the dimension of t− 2 . Thus we have an estimate for (3.1) given by  2 r −(n+2) 0 exp − , Pµν0 (t; x, x ) ≤ C2 t 2t

(3.35)

where C2 is a constant. Thus we have the estimate   nr2 0 − n ( n +2) exp − . det Pµν0 (t; x, x ) ≤ C3 t 2t

(3.36)

where C3 is a constant. Thus as t → 0, K2 (t) ∼ 0,

(3.37)

K ( t ) ∼ K1 ( t ) .

(3.38)

therefore

24

3.2.2

Heat Determinant Asymptotics

From above, we have as t → 0, K (t) ∼

1

Z

(2t)n (4πt)n

2

dvol( x )

Z Br ( x 0 )

M

 n  0 dvol( x ) exp − σ ( x, x ) H (t; x, x 0 ). t 0

(3.39) Now using the coordinate transformation given by (2.60), the integral over Br ( x 0 ) becomes the integral over the Euclidean ball Br/√t (ξ ). As t → 0, √r → ∞, theret fore we integrate ξ over all Rn . Thus 1

t−n(n+ 2 ) K (t) ∼ 2 2n (4π )n

Z

0

dvol( x )

Z

Rn

M

 n  1 dξ g 2 ( x 0 ) exp − ξ 2 ∆−1 (ξ, x 0 ) H (t; ξ, x 0 ). 2 (3.40)

Then using the notation for Gaussian average (2.65), we have  n Z 1 D E t−n(n+ 2 ) 2π 2 0 −1 dvol( x ) ∆ H n , K (t) ∼ 2 n 2 2n (4π )n M

 where ∆−1 H n is a function of x 0 .

(3.41)

2

By using (3.14) and (3.15), we see that Ψ(t; x, x 0 ) has an asymptotic expansion as t → 0 and is given by Ψ(t; x, x 0 ) ∼

∞

∑ tk ψk (x, x0 ),

(3.42)

k =0

where, in general,

(−1)k 12 ∆ ( x, x 0 )P ( x, x 0 ) ak ( x, x 0 ). (3.43) k! Thus Z, V, W, and S all have asymptotic expansions in non-negative integer powers of t since Ψ can be expanded in non-negative integer powers of t. Therefore we can expand H (3.29) in integer powers of t greater than −1. Thus we get ψk ( x, x 0 ) =

H (t; x, x 0 ) ∼ ∆( x, x 0 )

∞

∑

tk hk ( x, x 0 ),

(3.44)

k=−1

where ∆( x, x 0 ) was introduced for convenience. The sum begins from k = −1 since there is a 1t term in H which can be seen in the S term in the det Y (3.27) . Then 1 Z

 t−n(n+ 2 )  π  n2 ∞ k K (t) ∼ dvol( x 0 ) hk ( x, x 0 ) n . t (3.45) ∑ 2 2 2n k=−1 (4π )n M

25

If we define Hk =

Z

dvol( x 0 ) hhk i n ,

(3.46)

2

M

then we have

1 t−n(n+ 2 )  π  n2 K (t) ∼ 2 2n (4π )n

∞

∑

tk Hk .

(3.47)

k=−1

Next, we want to calculate hhk i n2 . We can expand hk ( x, x 0 ) in a covariant Taylor series about the point x 0 . This gives us ∞

∑

0

hk,µ0 ···µ0m ( x 0 ) := 1

0

(3.48)

i (−1)m h ∇(µ1 · · · ∇µm ) hk ( x, x 0 ) . m!

(3.49)

m =0

where

0

σµ1 · · · σµm hk,µ0 ···µ0m ( x 0 ).

hk ( x, x ) =

1

Then we have ∞

h hk i n = 2

∑

m

t2

D

0

0

ξ µ1 · · · ξ µ m

E

m =0

n 2

hk,µ0 ···µ0m . 1

(3.50)

Evaluating the Gaussian integral and relabeling indices, we obtain ∞

h hk i n = 2

∑

m =0

tm

0 0 (2m)! (µ10 µ20 0 . g · · · gµ2m−1 µ2m ) hk,µ0 ···µ2m m i (2n) m!

(3.51)

Let us define hk,m ( x 0 ) =

0 0 (2m)! (µ10 µ20 0 . g · · · gµ2m−1 µ2m ) hk,µ10 ···µ2m m (2n) m!

This gives us

∞

h hk i n = 2

If we define Hk,m =

Z

∑

tm hk,m .

(3.52)

(3.53)

m =0

dvol( x 0 ) hk,m ( x 0 ),

(3.54)

M

then

∞

Hk =

∑

m =0

26

tm Hk,m .

(3.55)

When we plug this all into (3.47), we get 1 t−n(n+ 2 )  π  n2 K (t) ∼ 2 2n (4π )n

∞

∞

∑ ∑

tk+m Hk,m .

(3.56)

k=−1 m=0

When we combine powers of t and re-index the sum, we find 1 t−n(n+ 2 )  π  n2 K (t) ∼ 2 2n (4π )n

where

∞

∑

tk Bk ,

(3.57)

k =−1

k

Bk =

∑

Hm,k−m .

(3.58)

m=−1

Note that we can express Bk =

Z

dvol( x 0 ) bk ( x 0 ),

(3.59)

M

where bk ( x 0 ) =

k

∑

hm,k−m ( x 0 ).

(3.60)

m=−1

It is the coefficients Bk that we wish to calculate. In summary, to calculate Bk , we need to find bk (3.60) or Hk,m (3.54), which are found from hk,m (3.52). To find hk,m , we compute the covariant Taylor expansion coefficients hk,µ1 ···µm (3.49) of hk (3.44). The coefficients hk are calculated from H (3.29) which is calculated from Yµν0 (3.19). To compute Yµν0 , we only need the heat kernel and the Synge function (2.49).

3.3

Heat Determinant Asymptotics for Scalar Operators

For scalar operators, we do not have a trace in (3.1) (i.e. tr Ψ∗ Ψ = Ψ2 ). Therefore (3.20) - (3.23) simplify to Sµν0 = σµ σν0 Ψ2 ,

(3.61)

Vµν0 = −σµ ΨΨν0 ,

(3.62)

Wµν0 = −σν0 ΨΨµ ,

(3.63)

Zµν0 = −σµν0 Ψ + 2tΨΨµν0 .

(3.64)

2

27

Since we have expanded the heat determinant in powers of t, let us also expand detY (thus H) in powers of t. First we need to expand S, V, W, and Z in powers of t. Thus from (3.61) - (3.64) and (3.42), we have   Sµν0 =σµ σν0 ψ02 + 2σµ σν0 ψ1 ψ0 t + σµ σν0 2ψ2 ψ0 + ψ12 t2 (3.65) + σµ σν0 (2ψ3 ψ0 + 2ψ2 ψ1 ) t3 + · · · , Vµν0 = − σµ ψ0 ψ0,ν0 − σµ (ψ1,ν0 ψ0 + ψ0,ν0 ψ1 ) t

− σµ (ψ2,ν0 ψ0 + ψ1,ν0 ψ1 ) t2 + · · · ,
Wµν0 = − σν0 ψ0 ψ0,µ − σν0 ψ1,µ ψ0 + ψ0,µ ψ1 t
− σν0 ψ2,µ ψ0 + ψ1,µ ψ1 t2 + · · · ,

(3.66)

(3.67)

and   Zµν0 = − σµν0 ψ02 + 2ψ0 ψ0,µν0 − σµν0 ψ1 t   2 0 0 0 + 2(ψ1,µν ψ0 + ψ0,µν ψ1 ) − σµν (2ψ2 ψ0 + ψ1 ) t2   + 2(ψ2,µν0 ψ0 + ψ1,µν ψ1 + ψ0,µν ψ2 ) − σµν0 (2ψ3 ψ0 + 2ψ2 ψ1 ) t3

(3.68)

+··· , where each additional index denotes the covariant derivative with respect to x and x 0 respectively. Thus ψk,···µ = ∇µ ψk,··· (3.69) and

ψk,···ν0 = ∇ν0 ψk,···

(3.70)

where ψk,··· is ψk with an arbitrary number of indices. Expanding H (3.29) in powers of t, we have the first three terms in (3.44): o 1 µ1 ···µn ν0 ···νn0 n n n −1 2n 1 E E σµ1 σν0 σµ2 ν0 · · · σµn νn0 ψ0 , h −1 = ∆ (−1) 2 1 n! 2 0 0 1 h0 = ∆−1 E µ1 ···µn E ν1 ···νn (−1)n σµ1 ν0 · · · σµn νn0 ψ02n 1 n!   + n (−1)n σµ1 ψ0,ν10 + σν10 ψ0,µ1 σµ2 ν20 · · · σµn νn0 ψ02n−1 n + (−1)n−1 σµ1 σν10 σµ2 ν20 · · · σµn νn0 2ψ1 ψ02n−1 2 −1

28

(3.71)

(3.72)

  n ( n − 1) n −2 σµ1 σν0 ψ0,µ2 ν0 − σµ2 ν0 ψ1 σµ3 ν0 · · · σµn νn0 2ψ02n−1 + (−1) 3 2 2 1 2 

+ n(n − 1) (−1)n σν10 ψ0,µ1 σµ2 ψ0,ν20 σµ3 ν30 · · · σµn νn0 ψ02n−2 ,    n −1 −1 1 µ1 ···µn ν10 ···νn0 h1 = ∆ E E ψ0,µ1 ν0 − σµ1 ν0 ψ1 σµ2 ν0 σµ3 ν0 2ψ02 n (−1) 2 3 1 1 n!
+ n (−1)n σν10 ψ1,µ1 ψ0 + ψ0,µ1 ψ1 σµ2 ν20 σµ3 ν30 ψ0
+ n (−1)n σµ1 ψ1,ν10 ψ0 + ψ0,ν10 ψ1 σµ2 ν20 σµ3 ν30 ψ0    + n(n − 1) (−1)n−1 σµ1 ψ0,ν10 + σν10 ψ0,µ10 ψ0,µ2 ν0 − σµ2 ν0 ψ1 σµ3 ν0 2ψ0 2 2 3   n n −1 σµ1 σν0 σµ2 ν0 σµ3 ν0 2ψ2 ψ0 + ψ12 ψ0 + (−1) 2 3 1 2   n ( n − 1) + (−1)n−2 σµ1 σν10 ψ0,µ2 ν20 − σµ2 ν20 ψ1 σµ3 ν30 4ψ1 ψ0 2   n ( n − 1) n −2 + σµ1 σν0 2 ψ1,µ2 ν0 ψ0 + ψ0,µ2 ν0 ψ1 σµ3 ν0 ψ0 (−1) 2 2 3 1 2   n ( n − 1) − (−1)n−2 σµ1 σν10 σµ2 ν20 2ψ2 ψ0 + ψ12 σµ3 ν30 ψ0 2   n(n − 1)(n − 2) n −3 + σµ1 σν0 ψ0,µ2 ν0 ψ0,µ3 ν0 − σµ3 ν0 ψ1 4ψ0 (−1) 2 3 3 1 4   n(n − 1)(n − 2) − (−1)n−3 σµ1 σν10 σµ2 ν20 ψ1 ψ0,µ3 ν30 − σµ3 ν30 ψ1 4ψ0 4
+ n(n − 1) (−1)n σν10 ψ1,µ1 ψ0 + ψ0,µ1 ψ1 σµ2 ψ0,ν20 σµ3 ν30   n + n(n − 1) (−1) σν10 ψ0,µ1 σµ2 ψ1,ν20 ψ0 + ψ0,ν20 ψ1 σµ3 ν30   n −1 + 2n(n − 1)(n − 2) (−1) σν0 ψ0,µ1 σµ2 ψ0,ν0 ψ0,µ3 ν0 − σµ3 ν0 ψ1 2 3 3 1 × σµ4 ν40 · · · σµn νn0 ψ02n−3 . (3.73) For simplification, we define a matrix F given by 0

F µν = ∆−1 (−1)n−1

0 0 0 1 E µα1 ···αn−1 E ν β1 ··· β n−1 σα1 β01 · · · σαn−1 β0n−1 . ( n − 1) !

(3.74)

Then by using Lemma 1 and letting Aµν = −σµν and Bµν = −γµν , then 0

0

F µν = −γµν . We note that

0

σµ σν0 F µν = −2σ. 29

(3.75) (3.76)

Introducing 0 0

F µ1 µ2 ν1 ν2 = ∆−1

(−1)n−2 µ1 µ2 α1 ···αn−2 ν10 ν20 β01 ··· β0n−2 E E σα1 β0 · · · σαn−2 β0 . n −2 1 ( n − 2) !

(3.77)

Then by using Lemma 1 and letting Aµν = −σµν and Bµν = −γµν , then 0 0

0

0

0

µ µ

0

0

0

F µ1 µ2 ν1 ν2 = γλ1 ν1 γλ2 ν2 δλ11 λ22 = γµ1 ν1 γµ2 ν2 − γµ2 ν1 γµ1 ν2 .

(3.78)

We note that 0 0

0

0

σµ σν0 F αµβ ν = 2σγαβ − σα σ β , 0 0

(3.79)

0

0

−σµν0 F αµβ ν = (n − 1) F αβ = −(n − 1)γαβ , and

(3.80)

0 0

σµ1 σν0 σµ2 ν0 F µ1 µ2 ν1 ν2 = 2(n − 1)σ.

(3.81)

2

1

Lastly, we introduce 0 0 0

F µ1 µ2 µ3 ν1 ν2 ν3 = ∆−1

(−1)n−3 µ1 µ2 µ3 α1 ···αn−3 ν10 ν20 ν30 β01 ··· β0n−3 E E σα1 β0 · · · σαn−3 β0 . (3.82) n −3 1 ( n − 3) !

Then by using Lemma 1 and letting Aµν = −σµν and Bµν = −γµν , then 0 0 0

0

0

0

0

0

0

F µ1 µ2 µ3 ν1 ν2 ν3 = −γλ1 ν1 γλ2 ν2 γλ3 ν3 δλ11 λ22 λ33 = −3!γ[µ1 |ν1 | γµ2 |ν2 | γµ3 ]ν3 We note that

0

µ µ µ

0 0

0

(3.83)

0

−σµν0 F α1 α2 µβ1 β2 ν = (n − 2) F α1 α2 β1 β2 . If we let

0

0

0

(3.84)

0 0

G α1 α2 β1 β2 = σµ σν0 F α1 α2 µβ1 β2 ν ,

(3.85)

then 0

0

0

0

0

0

0

0

0

0

G α1 α2 β1 β2 = − 2σγα1 β1 γα2 β2 + 2σγα2 β1 γα1 β2 − σ β1 σα2 γα1 β2 − σ β2 σα1 γα2 β1 0

0

0

0

+ σ β 2 σ α2 γ α1 β 1 + σ α1 σ β 1 γ α2 β 2 .

(3.86)

We will now simplify (3.71) - (3.73). We find that h −1 =

0 1 σµ σν0 F µν ψ02n . 2

(3.87)

Hence, using (3.76), h−1 = −σψ02n . 30

(3.88)

Simplifying h0 , we have
0 0 h0 =ψ02n − σµ ψ0,ν0 + σν0 ψ0,µ F µν ψ02n−1 + σµ σν0 F µν ψ1 ψ02n−1   0 0 + σµ1 σν0 ψ0,µ2 ν0 − σµ2 ν0 ψ1 F µ1 µ2 ν1 ν2 ψ02n−1 2

1

+ σν10 ψ0,µ1 σµ2 ψ0,ν20 F

2 µ1 µ2 ν10 ν20

(3.89)

ψ02n−2 .

Plugging in (3.75), (3.76), (3.80), and (3.79), we have
0 h0 =ψ02n + σµ ψ0,ν0 + σν0 ψ0,µ γµν ψ02n−1 − 2σψ1 ψ02n−1    µν0 µ ν0 + ψ0,µν0 − σµν0 ψ1 2σγ − σ σ ψ02n−1   0 0 − ψ0,µ ψ0,ν0 2σγµν − σµ σν ψ02n−2 .

(3.90)

Thus, we finally have 

ν0

µν0



+ σ ψ0,ν0 + σ ψ0,µ + 2σψ0,µν0 γ ψ02n−1   0 − σµ σν ψ0,µν0 + 2nσψ1 ψ02n−1   0 0 + σµ σν ψ0,µ ψ0,ν0 − 2σψ0,µ ψ0,ν0 γµν ψ02n−2 .

h0 =ψ02n

µ

(3.91)

Simplifying h1 , we have   0 h1 = ψ0,µν0 − σµν0 ψ1 F µν 2ψ02n−1
0 − σν0 (ψ1,µ ψ0 + ψ0,µ ψ1 ) + σµ (ψ1,ν0 ψ0 + ψ0,ν0 ψ1 ) F µν ψ02n−2    0 0 − σµ1 ψ0,ν10 + σν10 ψ0,µ1 ψ0,µ2 ν20 − σµ2 ν20 ψ1 F µ1 µ2 ν1 ν2 2ψ02n−2   1 µν0 2 + σµ σν0 F 2ψ2 ψ0 + ψ1 ψ02n−2 2   0 0 + σµ1 σν10 ψ0,µ2 ν20 − σµ2 ν20 ψ1 F µ1 µ2 ν1 ν2 2ψ1 ψ02n−2   0 0 + σµ1 σν10 ψ1,µ2 ν20 ψ0 + ψ0,µ2 ν20 ψ1 F µ1 µ2 ν1 ν2 ψ02n−2   0 0 1 2 0 0 − σµ1 σν1 σµ2 ν2 2ψ2 ψ0 + ψ1 F µ1 µ2 ν1 ν2 ψ02n−2 2    0 0 0 + σµ1 σν10 ψ0,µ2 ν20 − σµ2 ν20 ψ1 ψ0,µ3 ν30 − σµ3 ν30 ψ1 F µ1 µ2 µ3 ν1 ν2 ν3 ψ02n−2
0 0 + σν10 ψ1,µ1 ψ0 + ψ0,µ1 ψ1 σµ2 ψ0,ν20 F µ1 µ2 ν1 ν2 ψ02n−3   0 0 + σν10 ψ0,µ1 σµ2 ψ1,ν20 ψ0 + ψ0,ν20 ψ1 F µ1 µ2 ν1 ν2 ψ02n−3   0 0 0 + σν10 ψ0,µ1 σµ2 ψ0,ν20 ψ0,µ3 ν30 − σµ3 ν30 ψ1 F µ1 µ2 µ3 ν1 ν2 ν3 2ψ02n−3 . 31

(3.92)

Plugging in (3.75), (3.76), (3.78), (3.79), (3.80), and (3.84), we find   0 σµν0 ψ1 − ψ0,µν0 γµν 2ψ02 h1 =

0 + σν0 ψ1,µ ψ0 + ψ0,µ ψ1 + σµ (ψ1,ν0 ψ0 + ψ0,ν0 ψ1 ) γµν ψ0
0 + σµ ψ0,ν0 + σν0 ψ0,µ γµν 2(n − 1)ψ1 ψ0     µ1 ν10 µ2 ν20 µ2 ν10 µ1 ν20 −γ γ 2ψ0 − σµ1 ψ0,ν10 + σν10 ψ0,µ1 ψ0,µ2 ν20 γ γ   − σ 2ψ2 ψ0 + ψ12 ψ0   µν0 µ ν0 0 + ψ0,µν 2σγ − σ σ 2ψ1 ψ0

− 4 (n − 1) σψ12 ψ0    µν0 µ ν0 + ψ1,µν0 ψ0 + ψ0,µν0 ψ1 2σγ − σ σ ψ0   − (n − 1) σ 2ψ2 ψ0 + ψ12 ψ0

(3.93)

0 0 0

+ σµ1 σν10 ψ0,µ2 ν20 ψ0,µ3 ν30 F µ1 µ2 µ3 ν1 ν2 ν3 ψ0   0 0 + 2 (n − 2) ψ0,µν0 2σγµν − σµ σν ψ1 ψ0 − 2 (n − 1) (n − 2) σψ12 ψ0  
0 0 − ψ1,µ ψ0 + ψ0,µ ψ1 ψ0,ν0 2σγµν − σµ σν   µν0 µ ν0 − ψ0,µ (ψ1,ν0 ψ0 + ψ0,ν0 ψ1 ) 2σγ − σ σ   0 0 − ψ0,µ ψ0,ν0 2σγµν − σµ σν 2 (n − 2) ψ1  µ1 µ2 µ3 ν10 ν20 ν30 − 2σµ1 σν10 ψ0,µ2 ψ0,ν20 ψ0,µ3 ν30 F ψ02n−3 . Then simplifying further, we obtain  0 h1 = 2n (ψ1 − σψ2 ) ψ02 − 2ψ0,µν0 γµν ψ02   0 0 + ψ1,µν0 2σγµν − σµ σν ψ02  
0 + σµ ψ1,µ ψ0 + ψ0,µ ψ1 + σν (ψ1,ν0 ψ0 + ψ0,ν0 ψ1 ) ψ0   0 ν µ + 2 (n − 1) σ ψ0,ν0 + σ ψ0,µ ψ1 ψ0  0  0 − 2 σν ψ0,ν0 + σµ ψ0,µ ψ0,αβ0 γαβ ψ0

32

(3.94)

 0  0 0 + 2 σν ψ0,β0 γµβ + σµ ψ0,α γαν ψ0,µν0 ψ0

− (2n − 1) nσψ12 ψ0   0 0 + (2n − 1) ψ0,µν0 2σγµν − σµ σν ψ1 ψ0  
0 0 − ψ1,µ ψ0 + ψ0,µ ψ1 ψ0,ν0 2σγµν − σµ σν   µν0 µ ν0 − ψ0,µ (ψ1,ν0 ψ0 + ψ0,ν0 ψ1 ) 2σγ − σ σ   0 0 − 2 (n − 2) ψ0,µ ψ0,ν0 2σγµν − σµ σν ψ1    µ1 µ2 ν10 ν20 + ψ0,µ1 ν10 ψ0 − 2ψ0,µ1 ψ0,ν10 ψ0,µ2 ν20 G ψ02n−3 .

3.4

Coefficients of Heat Determinant Asymptotic Expansion

We want to find the coefficients Bk for scalar operators, for which we need the coefficients bk (3.60). We will be using the coincidence limits derived in Section 5.2 throughout this section. For our first coefficent, we have b−1 = h−1,0 . From (3.52) and (3.88), and the fact that [σ] = 0, h i h−1,0 = [h−1 ] = −σψ02n = 0. Thus

(3.95)

(3.96)

b−1 = 0.

(3.97)

b0 = h−1,1 + h0,0 .

(3.98)

Next, we want to calculate

We will first calculate h0,0 . From (3.52), we have h0,0 = [h0 ] .

(3.99)

From (3.91), and noting the coincidence limits of σ and its single derivatives are 0, then h i h0,0 = [h0 ] = ψ02n = 1. (3.100) Next, we will calculate h−1,1 . From equation (3.52), h−1,1 =

i 1 µ1 µ2 h g ∇ ( µ1 ∇ µ2 ) h −1 . 2n 33

(3.101)

Plugging in h−1 from (3.88), we find h−1,1

i  1 µ1 µ2 h 2n . = ∇(µ1 ∇µ2 ) −σψ0 g 2n

(3.102)

We only have non-zero terms when both derivatives are on σ, thus since  will  σµν = gµν , we have 1 h−1,1 = − . (3.103) 2 Therefore, we have 1 b0 = . (3.104) 2 Next, we want to calculate b1 = h−1,2 + h0,1 + h1,0 .

(3.105)

First, we will look at h1,0 . From (3.52), h1,0 = [h1 ] .

(3.106)

Since coincidence limits of σ and its first derivatives are 0, from (3.94), we are left with only   h1,0 = 2n [ψ1 ] + 2 ψ0, µ µ0 . (3.107) Next, looking at h0,1 , from (3.52), h0,1 =

 1 µ1 µ2  g ∇ µ1 ∇ µ2 h 0 . 2n

(3.108)

Thus from (3.91), h0,1

  0 1 µ1 µ2 = g ∇µ1 ∇µ2 ψ02n + σ β ψ0,β0 ψ02n−1 + σα ψ0,α ψ02n−1 2n 0

0

+ 2σψ0,αβ0 γαβ ψ02n−1 − σα σ β ψ0,αβ0 ψ02n−1 − 2nσψ1 ψ02n−1  α β0 2n−2 µν0 2n−2 + σ σ ψ0,α ψ0,β0 ψ0 − 2σψ0,µ ψ0,ν0 γ ψ0 .

(3.109)

For the coincidence limit to not be 0, there must be two derivatives on each σ and no derivatives or two on each ψ0 . Thus  0 1 2nψ0, ν ν ψ02n−1 + 2σ β ν ψ0,β0 ν ψ02n−1 + 2σαν ψ0,αν ψ02n−1 h0,1 = 2n  (3.110) 2n−1 αν β0 2n−1 ν ν αβ0 2n−1 . − 2σ σ ν ψ0,αβ0 ψ0 − 2nσ ν ψ1 ψ0 + 2σ ν ψ0,αβ0 γ ψ0 34

Now taking the actual coincidence limit,     1 h0,1 = 1 + [ψ0, ν ν ] − ψ0, µ µ0 − n [ψ1 ] . n

(3.111)

Lastly, we need h−1,2 . h−1,2 =

1

(2n)2 2!

  g ( µ1 µ2 g µ3 µ4 ) ∇ µ1 ∇ µ2 ∇ µ3 ∇ µ4 h −1 .

(3.112)

Plugging in h−1 , we find h−1,2 = −

i  1 ( µ1 µ2 µ3 µ4 ) h 2n . ∇ ∇ ∇ ∇ σψ g g µ µ µ µ 0 2 3 1 4 8n 2

(3.113)

The only derivatives that are not zero are the ones with 2 derivatives on σ and 2 on ψ02n since the symmetrization of 4 indices on σ is 0. Thus we have i h 1 µ1 µ2 µ3 µ4 µ1 µ3 µ2 µ4 µ1 µ4 µ2 µ3 2n h−1,2 = − . g g + g g + g g σ ∇ ∇ ψ ( ) µ µ µ µ 3 1 2 4 0 24n 2 (3.114) Taking the derivatives and coincidence limits, we have h−1,2 = −

n+2 [ψ0, ν ν ] . 12n

(3.115)

Therefore, we find that 

b1 = n [ψ1 ] + ψ0,

µ

µ0





+

5 11 + 12 6n



[ψ0, ν ν ] .

(3.116)

We have calculated b−1 , b0 , and b1 . Plugging in the coincidence limits from Section 5.2, we have b−1 = 0, 1 b0 = , 2  1  2 b1 = 12n − n + 10 R − nQ. 72n

(3.117) (3.118) (3.119)

Therefore, from (3.59), we obtain the first 3 global invariants B−1 = 0, 1 B0 = vol(M), 2  B1 =

Z

M

dvol

(3.120) (3.121) 

 1  2 12n − n + 10 R − nQ . 72n

35

(3.122)

CHAPTER 4 CONCLUSION We have studied the asymptotic expansion of a new heat invariant, the heat determinant. The heat determinant seems like the next logical step in the study of heat invariants on a manifold. We have calculated the first three coefficients in the asymptotic t-expansion, thus we have 1

t−n(n+ 2 ) K (t) ∼ (4π )n2



π 2n

 n2 n

B−1 t−1 + B0 + B1 t + · · ·

o

.

(4.1)

We note that the first term is zero, the second term depends on the volume of the manifold, and the the third term depends on the dimension and the integral of the curvature over the entire manifold. For future work, obviously, we can calculate more coefficients in our expansion, however they do get much more complicated rather quickly. The most important work to be done is to see what this new invariant can tell us about these manifolds, whether we can distinguish between any two manifolds, or if not, how much information about the manifold we can extract from the asymptotic expansion. One could also do more work with the modified heat determinant (2.103), more specifically, looking at manifolds with a boundary and seeing what kind of information we can extract from this other invariant. One could also study the diagonal of the heat kernel given by Kdiag (t) :=

Z

dx detPµν0 (t, x, x ).

(4.2)

M

This invariant contains much less information than the actual heat determinant, however, it is much easier to calculate higher t terms in our asymptotic expansion. The heat determinant is a first step in the study of non-spectral heat invariants. There is still a lot of work to be done in this area.

36

CHAPTER 5 APPENDIX 5.1

Gaussian Integrals

For a real positive symmetric matrix A = ( Aij ), then for any vector B = ( Bi ), we have  D Z E 1 n 1 − − 1 dx exp (− h x, Ax i + h B, x i) = π 2 (detA) 2 exp B, A B , (5.1) 4 Rn

where A−1 = ( Aij ) is the inverse of A. Now we shall differentiate the above equation with respect to Bi on both sides, for which we obtain Z

dx exp (− h x, Ax i + h B, x i) xi

Rn

 D n E   π2 1 − 12 −1 ik B, A B . A Bk exp = (detA) 2 4

(5.2)

If we differentiate again with respect to Bj , we obtain Z

dx exp (− h x, Ax i + h B, x i) xi x j

Rn n

1 π2 = (detA)− 2 2



1 A + Aik Bk A jl Bl 2 ij



 D E 1 −1 B, A B . exp 4

(5.3)

If we continue this process and set B = 0, we get the coefficients to the Taylor expansion of (5.1) in Bi and we find that Z

dx exp (− h x, Ax i) xi1 · · · xi2k+1 = 0

(5.4)

Rn

and Z

n

1

dx exp (− h x, Ax i) xi1 · · · xi2k = π 2 (detA)− 2

Rn

37

(2k)! (i1 i2 A · · · Ai2k−1 i2k ) . 22k k!

(5.5)

We define the Gaussian average as

h f ( x )i = π

− n2

(detA)

1 2

Z

dx exp (− h x, Ax i) f ( x ).

(5.6)

Rn

Thus we have

D

i1

x ···x

and D

5.2

xi1 · · · xi2k

E

=

i2k+1

E

= 0,

(5.7)

(2k)! (i1 i2 A · · · Ai2k−1 i2k ) . 22k k!

(5.8)

Evaluation of Coincidence Limits

The Synge function is defined in section 2.1.7. For more on the Synge function, see [16]. Here, we will evaluate the coincidence limits of the Synge function and its derivatives. We have σ=

0 1 1 σµ σµ = σν0 σν . 2 2

From this, and noting that [σ ] = 0, we find that   σµ = 0.

(5.9)

(5.10)

By differentiating (5.9), we get  σµ = ∇µ  σµ0 = ∇ν0

1 σα σα 2



1 σα σα 2

= σαµ σα ,

(5.11)

= σαµ0 σα .

(5.12)



Before taking more derivatives, we shall find the coincidence limit of σµν . From the definition of σ in (2.49), we have 1 σ= t 2

Zt

dx µ dx ν . dτ dτ

(5.13)

dx µ dx ν ∇α . dτ dτ

(5.14)

dτ gµν

0

Taking a covariant derivative, we have σα = t

Zt

dτ gµν

0

38

Evaluating the covariant derivative, and integrating by parts, we have σα =t

Zt 0

dx µ dτ gµν dτ

dx µ µ =tgµα δα − t dτ

=tUα − t

Zt



Zt

dx ν dx γ ∂α + Γν γα dτ dτ  dτ gµν

dτ

0

2 xν

dx γ dx µ − Γ γα dτ dτ dτ 2

µd δα

0





γ d2 x α ν dx dxν − Γ γα dτ dτ dτ 2

ν

 (5.15)

 .

α where Uα = dx dτ . By definition, Uα is the vector tangent to the geodesic connecting 0 x and x . We note that the second term in the above equation is just the integral of (2.46) multiplied by gµα . Thus we have

σα = tUα .

(5.16)

Uµ = Uα σα µ .

(5.17)

Thus we have from (5.11),

When we take the coincidence limit, we want the limit to not depend on direction, therefore µ (5.18) [σµ ν ] = δν and   σµν = gµν . (5.19) Now differentiating again (5.11) and (5.12), we have
σµν = ∇ν σαµ σα = σαµν σα + σαµ σα ν ,   α σµ0 ν = ∇ν σαµ0 σ = σαµ0 ν σα + σαµ0 σα ν .

(5.20) (5.21)

We note that the derivatives with respect to x always commute with derivatives with respect to x 0 . We also note that we can always commute the first two derivatives since σ is a scalar. Taking a derivative with respect to x of (5.20), we have σµνβ = σαµνβ σα + σαµν σα β + σαµβ σα ν + σαµ σα νβ .

(5.22)

Taking the coincidence limit, we have     σνµβ = − σβµν .

(5.23)

Since σ is symmetric in the first two indices, we have from (5.23),   σµνβ = 0.

(5.24)

39

We also note that by commuting the last two indices in σµνβ , we have σµνβ − σµβν = Rα µνβ σα .

(5.25)

Taking another derivative of (5.22), we have σµνβγ =σαµνβγ σα + σαµνβ σα γ + σαµνγ σα β + σαµν σα βγ

+ σαµβγ σα ν + σαµβ σα νγ + σαµγ σα νβ + σαµ σα νβγ . Taking the coincidence limit gives us       σγµνβ + σβµνγ + σνµβγ = 0.

(5.26)

(5.27)

By commutation of derivatives, we have that σµνβγ − σµνγβ = Rα µβγ σαν + Rα νβγ σµα . Due to the symmetry of the Riemann tensor, we have       σµνβγ = σµνγβ = σνµβγ .

(5.28)

(5.29)

Taking a derivative of (5.25), we have σµνβγ − σµβνγ = Rα µνβγ σα + Rα µνβ σαγ .

(5.30)

Taking this coincidence limit gives us     σµνβγ − σµβνγ = Rγµνβ .

(5.31)

If we switch β and γ, then add this to the above equation, we find       2 σµνβγ − σµβνγ − σµγνβ = Rγµνβ + R βµνγ .

(5.32)

Using (5.27) and (5.29), the above equations simplifies to 


1 σµνβγ = − Rµγνβ + Rµβνγ . 3

(5.33)

To calculate the coincidence limits for primed indices, we just use the formula

[ f ··· ]α = [ f ···α ] + [ f ···α0 ] ,

(5.34)

which is true for any function of x and x 0 [16]. More specifically, we have h i     f µν0 = ∇ν f µ − f µν , (5.35) 40

In summary, we have calculated the coincidence limits of σ and its derivatives to be

[σ] = 0, [σµ ] = 0, [σν0 ] = 0, [σµν ] = −[σµν0 ] = [σµ0 ν0 ] = gµν , [σµνα ] = [σµ0 να ] = [σµ0 ν0 α ] = [σµ0 ν0 α0 ] = 0,

(5.36) (5.37) (5.38) (5.39) (5.40)

and
1 Rµγνβ + Rµβνγ , 3
1 Rµγνβ + Rµβνγ , = 3
1 = − Rµγνβ + Rµβνγ , 3
1 Rµνγβ + Rµβγν , = 3
1 = − Rµγνβ + Rµβνγ . 3

[σµνβγ ] = − [σµνβγ0 ] [σµνβ0 γ0 ] [σµν0 β0 γ0 ] [σµ0 ν0 β0 γ0 ]

(5.41)

We will now calculate the coincidence limits of the Van-Fleck determinant and its derivatives. By definition, 1

1

Thus

∆ = g− 2 ( x ) g− 2 ( x 0 ) det(−σµν0 ).

(5.42)

[∆] = 1.

(5.43)

Taking one derivative, we have ∆µ =

0 0 1 (−1)n E α1 ···αn E β1 ··· β n σα1 β01 µ σα2 β02 · · · σαn β0n . ( n − 1) !

(5.44)

Taking the coincidence limit, since there is a third derivative of σ, we have   ∆µ = 0. (5.45) Taking another derivative of (5.44), we have ∆µν =

0 0 1 (−1)n E α1 ···αn E β1 ··· β n σα1 β01 µν σα2 β02 · · · σαn β0n ( n − 1) ! 0 0 1 + (−1)n−2 E α1 ···αn E β1 ··· β n σα1 β01 µ σα2 β02 ν σα3 β03 · · · σαn β0n . ( n − 2) !

41

(5.46)

Using (3.75) and (3.78), we have   0 0 0 0 0 ∆µν = σαβ0 µν γαβ ∆ + σα1 β0 µ σα2 β0 ν γα1 β1 γα2 β2 − γα2 β1 γα1 β2 ∆. 2

1

(5.47)

Taking the coincidence limit and using (5.41) gives us 

(3.43),

 1 ∆µν = −[σα α0 µν ] = Rµν . 3

(5.48)

Next, we calculate the coincidence limits of ψk and its derivatives. From

(−1)k 12 ∆ ak , (5.49) k! where ak can be calculated using the process given in [2]. We will only need the first couple coincidence limits which were originally calculated by [7], [5], and [9] and are given by ψk =

[ a0 ] =1,

(5.50)

1 [ a1 ] = Q − R, 6

(5.51) (5.52)

for our operator L = −∆ + Q. Thus we have 1

ψ0 = ∆ 2 ,

(5.53)

[ψ0 ] = 1.

(5.54)

and from (5.43), we have Taking a single derivative yields

ψ0,µ =

1 1 ∆µ ∆− 2 , 2

(5.55)

then using (5.45), we have 

 ψ0,µ = 0.

(5.56)

Taking one more derivative of ψ0 , we end up with ψ0,µν =

1 3 1 1 ∆µν ∆− 2 − ∆µ ∆ν ∆− 2 . 2 4

(5.57)

Therefore, using (5.48), we have 

 1 ψ0,µν = Rµν . 6 42

(5.58)

To find the coincidence limits of the primed derivatives, we us the formula (5.34). Next, 1 ψ1 = −∆ 2 a1 . (5.59) Thus

1 [ψ1 ] = − [ a1 ] = R − Q. 6

(5.60)

In summary, we have the coincidence limits

[∆] =1,  ∆µ = [∆ν0 ] = 0,   1 ∆µν =[∆µ0 ν0 ] = Rµν , 3 1 [∆µν0 ] = − Rµν , 3 [ψ0 ] =1, [ψ0,µ ] =[ψ0,ν0 ] = 0, 1 [ψ0,µν ] =[ψ0,µ0 ν0 ] = Rµν , 6 1 [ψ0,µν0 ] = − Rµν , 6 1 [ψ1 ] = R − Q. 6 

43

(5.61) (5.62) (5.63) (5.64) (5.65) (5.66) (5.67) (5.68) (5.69)

REFERENCES [1] I. G. Avramidi. A covariant technique for the calculation of the one-loop effective action. Nuclear Physics B, 355(3):712–754, 1991. [2] I. G. Avramidi. Heat Kernel and Quantum Gravity. Springer, 2000. [3] I. G. Avramidi. Heat Kernel: with Applications to Finance. World Scientific, 2014. Under review. [4] M. Berger. A Panoramic View of Riemannian Geometry. Springer, 2003. [5] S. M. Christensen. Vacuum expectation of the stress tensor in an arbitrary curved background: the covariant point separation method. Physical Review D, 14(10):2490–2501, 1976. [6] L. Debnath and P. Mikusinski. Introduction to Hilbert Spaces with Application. Elsevier, 3rd edition, 2005. [7] B. S. DeWitt. Dynamical Theory of Groups and Fields. Gordon and Breach, 1965. [8] T. Frankel. The Geometry of Physics An Introduction. Cambridge University Press, second edition, 2004. [9] P. B. Gilkey. The spectral geometry of a riemannian manifold. Journal of Differential Geometry, 10(4):608–618, 1975. [10] P. B. Gilkey. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Baco-Raton: Chemical Rubber Company, 1995. [11] P. B. Gilkey. Spectral geometry. In D. Krupka and D. Saunders, editors, Global Analysis, chapter 5, pages 289–326. Elsevier, 2008. [12] C. Gordon, D. Webb, and S. Wolpert. Isospectral plane domains and surfaces via riemannian orbifolds. Inventiones mathematicae, 110(1):1–22, 1992. [13] A. Grigor’yan. Heat Kernel and Analysis on Manifolds. AMS, International Press, 1995. [14] M. Kac. Can one hear the shape of a drum? American Mathematical Monthly, 73:1–23, 1966. [15] J. Milnor. Eigenvalues of the Laplace operator on certain manifolds. Proceedings of the National Academy of Sciences of the United States of America, 51(4):542, April 1964. [16] J. L. Synge. Relativity: The General Theory. North-Holland, 1966. 44

Calculation of Heat Determinant Coefficients for Scalar Laplace type Operators by Benjamin Jerome Buckman

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