Quantum Fields, Curvature, and Cosmology

etp.physik.uni.muenchen.de

Quantum Fields, Curvature, and Cosmology

General view of QFT Perturbation theory Applications Conclusions

Quantum Fields, Curvature, and Cosmology

Stefan Hollands

Cardiff University

+ +

10-03-2009

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Outline

Introduction/Motivation

What is QFT?

Expansions

Applications in Cosmology

Outlook

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Motivation

Elementary Particles

Expanding

Universe

Quantum fields on

curved spacetime

QFT

Quantum Field Theory

General

Relativity

QFT on manifolds is relevant formalism to describe

quantized matter at large spacetime curvature (→ early

Universe).

Interesting physical effects: primordial fluctuations (→

structure formation, Cosmic Microwave Background,

Baryon/Anti-Baryon asymmetry, Hawking/Unruh effect, ...)

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Motivation

Elementary Particles

Expanding

Universe

Quantum fields on

curved spacetime

QFT

Quantum Field Theory

General

Relativity

QFT on manifolds is relevant formalism to describe

quantized matter at large spacetime curvature (→ early

Universe).

Interesting physical effects: primordial fluctuations (→

structure formation, Cosmic Microwave Background,

Baryon/Anti-Baryon asymmetry, Hawking/Unruh effect, ...)

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Introduction

1 Quantum Field Theory was invented

in physics to describe the behavior

of elementary particles (Electron e − ,

Photon γ, ...) in particle

accelerators.

2 It has made successful predictions

of incredible precision.

3 Not fully understood. (→

Mathematics: One of the “Clay

problems”)

Famous ”g − 2” for electron

[Kinoshita et al., Dehmelt et al.]:

Stefan Hollands DPG Tagung München, 10 March 2009

a e − = 1159652175.86(8.84) × 10 −12

∆a e − ≤ 12.4(9.5) × 10 −12


General view of QFT Perturbation theory Applications Conclusions

Quantum Fluctuations and Structure of Universe

Today

Early

Universe

Time

BIG

BANG

Macroscipic

Density

Fluctuations

LARGE!

(CMB)

Microscope

..

φ

Quantum Field

Fluctuations

SMALL!

φ .

φ

2

+ 3H

_k

2

a

+ = 0

k k k

Image: WMAP, Springel et al.

Consider quantized field

eqn. on curved manifold

Example: gφ = 0

g: Lorentzian metric, e.g.

g = −dt 2 + a(t) 2 ds 2

R 3.

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

What scales are we talking about here?

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Why is QFT in curved space so different from flat

space?

No S-matrix

No natural particle interpretation, no vacuum state

No spacetime symmetries (no Poincare-symmetry)

No Hamiltonian/conserved energy (Stability?

Thermodynamics?)

=⇒ Quantum field in the presences of spacetime curvature

does require a new way of thinking and a new formalism, as I

will explain in this talk.

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Why is QFT in curved space so different from flat

space?

No S-matrix

No natural particle interpretation, no vacuum state

No spacetime symmetries (no Poincare-symmetry)

No Hamiltonian/conserved energy (Stability?

Thermodynamics?)

=⇒ Quantum field in the presences of spacetime curvature

does require a new way of thinking and a new formalism, as I

will explain in this talk.

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Why is QFT in curved space so different from flat

space?

No S-matrix

No natural particle interpretation, no vacuum state

No spacetime symmetries (no Poincare-symmetry)

No Hamiltonian/conserved energy (Stability?

Thermodynamics?)

=⇒ Quantum field in the presences of spacetime curvature

does require a new way of thinking and a new formalism, as I

will explain in this talk.

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Why is QFT in curved space so different from flat

space?

No S-matrix

No natural particle interpretation, no vacuum state

No spacetime symmetries (no Poincare-symmetry)

No Hamiltonian/conserved energy (Stability?

Thermodynamics?)

=⇒ Quantum field in the presences of spacetime curvature

does require a new way of thinking and a new formalism, as I

will explain in this talk.

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

What is QFT?

“Equations” ↔ Algebraic relations

(+ bracket between quantum

structure) fields (OPE)

“Solutions” ↔ Quantum states

OPE:

Expansion of two quantum fields

O1(x1)O2(x2) in terms of certain

coefficients C depending on

spacetime.

OPE coefficients have covariant

behavior under embedding

[S.H. & Wald, Brunetti et al., Verch]:

(M’,g’)

Functor

States: Collections of n-point

functions wn = 〈O1 ...On〉Ψ in which States do not have such a

OPE holds.

behavior under embedding!

ψ

ψ

(M,g)

CM’,g’ CM,g Stefan Hollands DPG Tagung München, 10 March 2009

Functor


General view of QFT Perturbation theory Applications Conclusions

What is the OPE?

General formula: [Wilson], [Zimmermann], ...,[S.H.]

〈Oj1 (x1) · · · Ojn(xn)〉Ψ

x 4

x 3

y x1

x2

∼ C i j1...jn (x1,... ,xn;y)


OPE−coefficients ↔ structure“constants ′′

〈Oi(y)〉Ψ

M

Points are scaled

towards y

Physical idea: Separate the short distance regime of

theory (large ”energies”) from the energy scale of the state

(small) E 4 ∼ 〈ρ〉Ψ.

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application: In Early Universe have different scales:

Temperature E ∼ T(t) ∼ a(t) −1 .

Curvature radius R(t) ∼ H(t) −1 = a(t)/˙a(t).

OPE-coefficients can be calculated.

Can derive fundamental theorems about QFT on curved

spacetimes: Spin-Statistics [Verch, S.H. & Wald],

Parity-Time-Charge [S.H.]:

Flat space:

Θ|in,momenta〉 =

|out, −momenta〉 C .

Cuved space: No “same universe”

formulation of PCT: Relates an “in” state to “out”

state in universe with opposite time-orientation!

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

The PCT theorem

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Curvature expansion

C(x1,... ,xn;y)

= structureconstants

= Q[∇ k R(y),couplings]

× Lorentzinv.Minkowskidistributions

v(ξ1,...,ξn)

ξi—Normal coordinates

M

R αβγδ(y)

x 1

ξ1

y

T M

y

ξ3

x3

Curvature expansion: departures from flat spacetime.

“Minkowski part” v(ξ1,... ,ξn) can be calculated in terms of

“ordinary” Feynman diagrams.

Stefan Hollands DPG Tagung München, 10 March 2009

ξ2

x2


General view of QFT Perturbation theory Applications Conclusions

Expansion techniques: Diagrams

v(ξ 1 , ... , ξ n−1 )

Σ Feynman

Diagrams

massless

propagators

ξ

1

ξ

2

ξ 4

=

ξ

3

1 Need to address

“renormalization” issues.

2 Diagrams “live” in

tangent space TyM.

3 E.g. dimensional

regularization possible

at this stage.

M

ξ 1

y

Ty M

=⇒ Renormalization possible to arbitrary orders!

[S.H. & Wald]

Stefan Hollands DPG Tagung München, 10 March 2009

ξ 4

ξ 3

ξ

2


General view of QFT Perturbation theory Applications Conclusions

Expansion techniques: Diagrams

v(ξ 1 , ... , ξ n−1 )

Σ Feynman

Diagrams

massless

propagators

ξ

1

ξ

2

ξ 4

=

ξ

3

1 Need to address

“renormalization” issues.

2 Diagrams “live” in

tangent space TyM.

3 E.g. dimensional

regularization possible

at this stage.

M

ξ 1

y

Ty M

=⇒ Renormalization possible to arbitrary orders!

[S.H. & Wald]

Stefan Hollands DPG Tagung München, 10 March 2009

ξ 4

ξ 3

ξ

2


General view of QFT Perturbation theory Applications Conclusions

Expansion techniques: Diagrams

v(ξ 1 , ... , ξ n−1 )

Σ Feynman

Diagrams

massless

propagators

ξ

1

ξ

2

ξ 4

=

ξ

3

1 Need to address

“renormalization” issues.

2 Diagrams “live” in

tangent space TyM.

3 E.g. dimensional

regularization possible

at this stage.

M

ξ 1

y

Ty M

=⇒ Renormalization possible to arbitrary orders!

[S.H. & Wald]

Stefan Hollands DPG Tagung München, 10 March 2009

ξ 4

ξ 3

ξ

2


General view of QFT Perturbation theory Applications Conclusions

Example: 3-point OPE

To leading order in perturbation theory, and leading order in

deviation from flat space, 3-point OPE in scalar λφ 4 -theory has

structure

φ(x1)φ(x2)φ(x3) ∼

[ D

+

σij

λ

Cl2(αi) + ...] φ(y)

a


OPE−coefficient C(x1,x2,x3;y)

(+other operators)

Cl2(z)—Clausen function

σij—geodesic distance

a—curved space area of triangle

D—geometrical determinant

3−Point Operator Product

a

α

x 1

1

M

α2

x2 Stefan Hollands DPG Tagung München, 10 March 2009

y

α3

x 3

σ 23


General view of QFT Perturbation theory Applications Conclusions

Application 1): Quantum field theory fluctuations in early

universe, where curvature cannot be neglected.

Example: Consider w3 = 〈φφφ〉Ψ where φ suitable field

parametrizing density contrast δρ/ρ.

Step 1: Compute OPE-coefficients from perturbation

theory (reliable in asymptotically free theories).

Step 2: Write w3 ∼

i Ci φφφ 〈Oi〉Ψ.

Step 3: Get form factors 〈Oi〉Ψ e.g. from (a) “AdS-CFT”,

(b) view as input parameters.

→ Non-Gaussianities in CMB, bispectrum (→ fNL = w3/w 3/2

2

[Shellard,Maldacena,Spergel,...], [Eriksen et al., Bartolo et al., Cabella et al., Gaztanaga et al. (constraints from

WMAP data),...]), ...

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application 1): Quantum field theory fluctuations in early

universe, where curvature cannot be neglected.

Example: Consider w3 = 〈φφφ〉Ψ where φ suitable field

parametrizing density contrast δρ/ρ.

Step 1: Compute OPE-coefficients from perturbation

theory (reliable in asymptotically free theories).

Step 2: Write w3 ∼

i Ci φφφ 〈Oi〉Ψ.

Step 3: Get form factors 〈Oi〉Ψ e.g. from (a) “AdS-CFT”,

(b) view as input parameters.

→ Non-Gaussianities in CMB, bispectrum (→ fNL = w3/w 3/2

2

[Shellard,Maldacena,Spergel,...], [Eriksen et al., Bartolo et al., Cabella et al., Gaztanaga et al. (constraints from

WMAP data),...]), ...

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application 1): Quantum field theory fluctuations in early

universe, where curvature cannot be neglected.

Example: Consider w3 = 〈φφφ〉Ψ where φ suitable field

parametrizing density contrast δρ/ρ.

Step 1: Compute OPE-coefficients from perturbation

theory (reliable in asymptotically free theories).

Step 2: Write w3 ∼

i Ci φφφ 〈Oi〉Ψ.

Step 3: Get form factors 〈Oi〉Ψ e.g. from (a) “AdS-CFT”,

(b) view as input parameters.

→ Non-Gaussianities in CMB, bispectrum (→ fNL = w3/w 3/2

2

[Shellard,Maldacena,Spergel,...], [Eriksen et al., Bartolo et al., Cabella et al., Gaztanaga et al. (constraints from

WMAP data),...]), ...

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application 2): “Quantum inequalities” [Fewster et al., Ford, Flanagan, Roman, Teo,

Verch, Visser et al....]

Classically: Energy density ρ = 1

2 ˙ φ2 + 1

always non-negative. But in QFT 〈ρ(x)〉Ψ < 0 for suitably

chosen states Ψ. Physical issues:

2 ( ∂φ) 2 + V (φ) ≥ 0 is

Exotic spacetimes (“warp-drive”, “time-machine” ...)

Violations of the second law of thermodynamics

Physically desirable to obtain lower bounds on negative

energies to rule out such pathologies → “Quantum

Inequalitites”. Can be proven via OPE [Bostelmann, Fewster], e.g.


〈ρ(t)〉Ψ|s(t)| 2 dt ≥ − 1

16π 3


ω

m

4 |ˆs(ω)| 2 dω .

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

How a wormhole would look like on

Times Square Image: [SCIENTIFIC AMERICAN

2000]

Plot of Energy = 〈ρ(t)〉Ψ in a state

with negative energy: (a) −-tive

energy must be “repaid” (c) with

“interest”, (b) cannot last for long.

Taken from [Ford et al.]

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application 3): “Boltzmann-type collision equations”. (Rigorous

results: [Hugenholz], [Chen], [Yau, H.Z.], [Salmhofer], [Erdos], ...)

Standard Boltzmann equation states (schematic):

=

˙nk(t) + 3H(t)nk(t)


d(momentapi)δ 4

2

(momenta) |amplitude|

particles

npi (t)

Very important e.g. in cosmology to track number distributions

in non-equilibrium situations (→ Baryon/Anti-baryon

asymmetry... [Kolb & Turner]). How to derive/justify/correct this

equation in the framework of QFT on manifolds?

When can particle numbers nk = 〈Nk(t)〉Ψ be defined?

What are the curvature corrections to this equation?

When does the equation hold/fail?

Such questions can be treated systematically using QFT in

curved spacetime [Hollands, Leiler to appear].

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application 4): “Accelerated expansion of Universe/inflation”

[Dappiaggi et al.], [Starobinsky], [Shapiro & Sola],[Schützhold], [Yokoyama],...

There is growing experimental evidence that our universe is

undergoing accelerated expansion. If we take quantum field

theory effects seriously, we should consider

Gµν = (radiation , dust) + 8πGN 〈Tµν〉Ψ .

The right hand side can have an important effect in

cosmological situations. One finds for a scalar field with

suitable coupling [Dappiaggi et al.]:

e 4t



H(t) + A1/A


H(t)

− A

,

so the solution asymptotes to a deSitter space in the future

t → ∞.

One can also argue [Schützhold] that the QCD “trace anomaly” gives

rise to a contribution to dark energy of order

〈ρ〉Ψ ∼ (100MeV) 3 H(t).

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Application 5): (Analogue) Hawking effect [Unruh], [Schützhold], [Geray et al.],

[Volovic], [Visser], ,...

Hawking effect says that black holes emit particles at a

characteristic temperature. It is very tiny for large black holes.

But analogous effects in other systems (“dumb holes”).

Pair creation Image:

www.st-andrews.ac.uk

Analogue of Hawking effect in fluid. (Other systems: Wave guides, Bose-Einstein

condensates, Liquid Helium)

Stefan Hollands DPG Tagung München, 10 March 2009


General view of QFT Perturbation theory Applications Conclusions

Conclusions

Thanks to recent developments QFT in curved spacetime

is a well-developed formalism capable of treating physically

interesting interacting models

Wide set of computational techniques available

Applications in Early Universe/Cosmology, ...

Interesting mathematical structure

Able to address some fundamental questions about nature

Stefan Hollands DPG Tagung München, 10 March 2009

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