quantum field theory in de sitter space - kekresearch.kek.jp/group/...introduction • quantum field...
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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
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. Baym
L.V. Keldysh
T. Kita
A. 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 parts
by 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
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