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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

The Early Time Expansion of the Heat Kernel

Stefan Lippoldt November 8, 2016

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

Motivation & Introduction

Heat Kernel

• rigin: 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

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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: PDE: u(x, s ց 0) = u0(x) ∂su(x, s) = −∆u(x, s) the unique solution reads u(x, s) = e−s∆u0(x) =

• y

e−s∆δ(x − y)

• Heat Kernel

u0(y) =

• y

K(x, y; s)u0(y)

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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: PDE: K(x, y; s ց 0) = δ(x − y) ∂sK(x, y; s) = −∆K(x, y; s) using momentum representation of δ(x − y) we get K(x, y; s) = e−s∆

• ddp

(2π)d eip(x−y) =

• ddp

(2π)d e−sp2+ip(x−y)

= e− (x−y)2

4s

(4πs)d/2

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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: ∂kΓk = 1

2 Tr

• (Γ(2)

k

+ Rk)−1∂kRk

• What does “Tr” mean?

⇒ It is the trace over the considered space of functions appearing in the path integral (

• Dϕ):

Tr M =

• x
• y
• n

¯ Φn(x)M(x, y)Φn(y) for example in flat space: Tr f (∆) =

• x
• y
• ddp

(2π)d e−ipxf (∆)δ(x − y)eipy =

• x
• ddp

(2π)d f (p2)

=

• x

∞

• dz

z(d−2)/2 (4π)d/2 Γ(d/2)f (z)

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

Laplace Transform consider a function of a positive argument f : R+ → R Laplace Transform ˜ f : R+ → R: f (z) =

∞

• ds ˜

f (s)e−sz for example:

1 (z+a)x = ∞

• ds sx−1e−as

Γ(x)

e−sz, a, x > 0 generally one can show:

∞

• ds s−x ˜

f (s) =

1 Γ(x) ∞

• dz zx−1f (z)

∞

• ds sn ˜

f (s) = (−1)n lim

z ց 0 f (n)(z)

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

Heat Kernel in FRG simpler form of the trace: Tr M=

• x
• y
• n

¯ Φn(x)M(x, y)Φn(y)=

• x
• y

tr

• M(x, y)
• n

Φn(y)¯ Φn(x)

• =
• x

tr [M(x, y)]y=x for example in flat space: e−s∆δ(x − y) = e−(x−y)2/(4s) (4πs)d/2 Tr f (∆) =

• x

[f (∆)δ(x − y)]y=x =

• x

∞

• ds ˜

f (s)[e−s∆δ(x − y)

• Heat Kernel

]y=x =

• x

∞

• ds

˜ f (s) (4πs)d/2 =

• x

∞

• dz

z(d−2)/2 (4π)d/2 Γ(d/2)f (z)

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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: ∆ = −DµDµ What to do with δ(x − y)? ⇒ δ(x, y) is unit oprator in the considered space of functions:

• y

δ(x, y)Φ(y) = Φ(x) ⇔ δ(x, y) =

• n

Φn(x)¯ Φn(y) for example in flat space: δ(x − y) =

• ddp

(2π)d eipxe−ipy

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

The World Function What to do with (x − y)2? ⇒ geodesic distance d2

g(x, y)

introduce the “world function” σ(x, y) for convenience: σ(x, y) :=

d2

g (x,y)

2

, Dµσ = 0,

1 2gµν(Dµσ)(Dνσ) = σ,

DµDνσ = gµν for example in flat space: σ(x, y) = (x−y)2

2

, ηµρ(x − y)ρ = 0,

1 2ηµν

ηµρ(x − y)ρ ηνκ(x − y)κ = σ(x, y), ∂µηνρ(x − y)ρ = ηµν

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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 S1

the metric gϕϕ = r 2, Laplacean ∆ = − 1

r 2 ∂2 ϕ

eigenfunctions of the Laplacean: fn,l(ϕ) = el·inϕ

√ 2πr ,

n ∈ N0, l ∈ Dn =

• {1}, n = 0

{−1, 1}, n > 0 , ∆fn,l = λn · fn,l, λn = n2

r 2

Heat Kernel: K(ϕ1, ϕ2; s) =

∞

• n=0
• l∈Dn

e−sn2/r 2 el·inϕ1 √ 2πr e−l·inϕ2 √ 2πr = e−σ(ϕ1,ϕ2)/(2s)

√ 4πs

• n∈Z

e−

• σ(n)(ϕ1,ϕ2)−σ(ϕ1,ϕ2)
• /(2s)

σ(n) corresponds to d(n)

g

= n · (2πr) + dg

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

Early Time Expansion

• ne can show that the Heat Kernel allows for an early time

expansion of the form: K(x, y; s) =

e

σ(x,y) 2s

(4πs)d/2 ∞

• n=0

snAn(x, y) the An(x, y) are the expansion coefficients (to be determined) this form misses topological properties of the manifold

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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 use the algebraic relation: eXQ =

∞

• n=0

1 n![X, Q]neX

implying: f (∆)Q =

∞

• n=0

1 n![∆, Q]nf (n)(∆)

traces we are interested in are of the following form: Tr

• Qµ1...µmDµ1 . . . Dµme−s∆

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

Coincidence Limit of the An for the evaluation of the flow equation, all we need are the coincidence limits of derivatives of the Dµ1 . . . DµmAn 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: Dµ1 . . . DµmAn ∼ R(n+m)/2 to linear order in the curvature we get: A0 = ✶, DµA0 = 0, DµDνA0 = Rµν

6 ✶ + 1 2Fµν,

A1 = R

6 ✶

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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

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Motivation & Introduction Heat Kernel in FRG Early Time Expansion

Thank you for your attention!

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