Quantum Field Theory in de Sitter space

Hiroyuki Kitamoto (Sokendai)

with

Yoshihisa Kitazawa (KEK,Sokendai)

based on arXiv:1004.2451 [hep-th]

Introduction

• Quantum field theory in de Sitter space concerns deep mysteries: inflation in the early universe and dark energy of the present universe

• In a time dependent background like dS space, there is no stable vacuum

• In such a background, Feynman-Dyson perturbation theory breaks down and we need to use Schwinger-Keldysh formalism N.C. Tsamis, R.P. Woodard

S. Weinberg

A.M. Polyakov

• Our problem is related to non-equilibrium physics, for example, Boltzmann equation A.M. Polyakov

• We derive a Boltzmann equation in dS space from a Schwinger-Dyson equation

• We investigate the energy-momentum tensor of an interacting field theory to estimate the effective cosmological constant

Scalar field theory in dS space

Poincare coordinate

We rescale the field

Infra-red divergence

If we apply Feynman rules to deal with the interaction, the integrations over time give rise to IR divergences at the infinite future

Feynman-Dyson perturbation theory breaks down in dS space

Premise for Feynman rule

in-out formalism

In the Feynman-Dyson formalism, the vacuum expectation value is given by the transition amplitude from to

This is because due to the time translational invariance

Schwinger-Keldysh formalism

in-in formalism

There is no time translational symmetry in dS space,and so we can’t prefix In this case, we can evaluate the vev only with respect to

Because there are two time indices （＋ ,－） , the propagator has 4 components

At time vertexes, we sum ( ＋ , － ) indices like products of matrices

The integration over time is manifestly finite due to causality

Boltzmann equation

In a time dependent background, we need to consider excited states in general

It is possible that the distribution function has time dependence due to the interaction

Intuitively, we can derive a Boltzmann equation from transition amplitude

in Minkowski space

in dS space

If ,

This indicates the instability of dS space?

A.M. Polyakov

because energy doesn’t conserve

• Transition amplitude is based on in-out formalism

• We should derive the Boltzmann equation on dS background in in-in formalism

Schwinger-Dyson equation

∑

=

+

We focus on the （－＋） component of the propagator

There is explicit time dependence at the integration

We consider the following identity from Schwinger-Dyson equation

Boltzmann equation can be derived from this identity

L.P. Kadanoff, G. BaymL.V. KeldyshT. KitaA. Hohenegger, A. Kartavtsev, M. Lindner

Full propagator is

We investigate propagators well inside the cosmological

horizon:

Assumption

Second term is treated perturbatively

The left hand side: Time derivative

The right hand side: Collision term

At the vertexes,

Because there is no time translational symmetry in dS space, the collision term has the on-shell part and the off-shell part

In Minkowski space,

The on-shell term

The off-shell term

Infra-red effect

The on-shell and off-shell parts have IR divergences at

So we redefine the on-shell and off-shell partsby transferring the contribution of within the energy resolution to

When , IR divergences cancel out in this procedure

When , IR divergence remains

We focus on the case that the initial distribution function is thermal

Thermal distribution case

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Spectral weight

The on-shell state weight is reduced to compensate the weight of off-shell states

Change of distribution function

In the case, logarithmic divergent term

remains but this term has a cut-off

It leads to the change of distribution function

Here we adopt a fixed physical UV cut-off

Mass renormalization

counter term:

virtuality:

We represent these results by physical quantities

Explicit time dependence disappears when it is expressed by physical quantities

Effective cosmological constant

Einstein equation:

Is it possible that has time dependence?

Contribution from free field

Conformal anomaly:

From :

• Contribution from the interaction gives growing time dependence to ?

• Such effects screen the cosmological constant?

Contribution from inside the cosmological horizon

We substitute the results of the Boltzmann equation to

Although this effect reduces the cosmological constant, it vanishes as the universe cools down with time

Contribution from outside the cosmological horizon

Inside the cosmological horizon, the degrees of freedom are constant because we adopt a fixed physical UV cut-off

Outside the cosmological horizon, the degrees of freedom increase as time goes on

: IR cut-off

constant increase

We estimate this effect from outside the cosmological horizon in the case

The leading contribution is log order in massless case

Increase in the degrees of freedom screens the cosmological constant is screened in theory

The cosmological constant is also screened in theory

N.C. Tsamis, R.P. Woodard

Conclusion

• We have investigated an interacting scalar field theory in dS space in Schwinger-Keldysh formalism

• We have investigated the time dependence of the propagator well inside the cosmological horizon by Boltzmann equation

• We have found the nontrivial change of matter distribution function and spectral weight

• However, explicit time dependence disappears when it is expressed by physical quantities

• Contribution from inside the cosmological horizon doesn’t give time dependence to the cosmological constant except for cooling down

• Increase in the degree of freedom outside the cosmological horizon screens the cosmological constant, and this effect grows as time goes on

Future work

• Non-thermal distribution case

• Physics around and beyond the cosmological horizon

• Quantum effects of gravity

• Non-perturbative effects