Quantum Fields, Curvature, and Cosmology

General view of QFT Perturbation theory Applications Conclusions

**Quantum** **Fields**, **Curvature**, **and** **Cosmology**

Stefan Holl**and**s

Cardiff University

+ +

10-03-2009

Stefan Holl**and**s 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 Holl**and**s DPG Tagung München, 10 March 2009

General view of QFT Perturbation theory Applications Conclusions

Motivation

Elementary Particles

Exp**and**ing

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 Holl**and**s DPG Tagung München, 10 March 2009

General view of QFT Perturbation theory Applications Conclusions

Motivation

Elementary Particles

Exp**and**ing

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 Holl**and**s 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 Holl**and**s 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 Holl**and**s DPG Tagung München, 10 March 2009

General view of QFT Perturbation theory Applications Conclusions

What scales are we talking about here?

Stefan Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s DPG Tagung München, 10 March 2009

General view of QFT Perturbation theory Applications Conclusions

The PCT theorem

Stefan Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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 Holl**and**s 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], ...)

St**and**ard 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 [Holl**and**s, Leiler to appear].

Stefan Holl**and**s 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 h**and** 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 Holl**and**s 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-**and**rews.ac.uk

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

condensates, Liquid Helium)

Stefan Holl**and**s 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 Holl**and**s DPG Tagung München, 10 March 2009