Quantum infra-red effects in de Sitter space de Sitter空間上の場の理論における量子赤外効果
Quantum infra-red effects in de Sitter space
In cosmic inflation at the early universe and dark energy at the present universe, our universe is exponentially expanding with the respective cosmological constants. To investigate the quantum effects on these universes, we need to understand the quantum field theory in de Sitter (dS) space. Exploring the quantum infra-red (IR) effects specific to dS space, we may better understand inflation and dark energy. In order to investigate the interacting field theories in a time dependent background like dS space, we need to employ the Schwinger-Keldysh formalism. Nonequilibrium physic may play an important role in this regard. In investigating the quantum effects in dS space, we divide the momentum scale into the two regions, inside the cosmological horizon and outside the cosmological horizon. Well inside the cosmological horizon, we have derived a Boltzmann equation in dS space from a Schwinger-Dyson equation, which is a standard tool in nonequilibrium physics. The local physics probed by the Boltzmann equation respects the dS symmetry since the degrees of freedom inside the cosmological horizon are time independent. On the other hand, the degrees of freedom outside the cosmological horizon increase with cosmic expansion. This increase gives rise a growing time dependence to the propagator for a massless and minimally coupled scalar field and gravitational field. It is a direct consequence of their scale invariant fluctuation spectrum. In some field theoretic models on dS space, the dS symmetry is dynamically broken and physical quantities acquire time dependences through such a quantum IR effect. In particular, this IR effect may be relevant to resolve the cosmological constant problem. In the Schwinger-Keldysh perturbation theory, the IR effects at each loop level manifest as a polynomial in the logarithm of the scale factor of the universe. At late times, the leading IR effect comes from the leading logarithm at each loop level. Their growing time dependences mean that the perturbation theory eventually breaks down after a large enough cosmic expansion. In order to understand such a situation, we have to investigate the IR effect nonperturbatively. Remarkably in the models with interaction potentials, the leading IR effects can be evaluated nonperturbatively by the stochastic approach. Furthermore it has been found that the equilibrium solution in the stochastic approach can be rederived in an Euclidean field theory on a 4-dimensional sphere. However in a general model with derivative interactions, we still don't know how to evaluate the nonperturbative IR effects. Especially such a tool is required to understand the quantum IR effects of gravity. It is because the gravitational field contains massless and minimally coupled modes with derivative interactions. As a simple model with derivative interactions, we have investigated the non-linear sigma model. The global symmetry guarantees that it contains massless minimally coupled scalar fields. In addition, we can perform some nonperturbative investigations as it is exactly solvable in the large N limit on an N-sphere. Another point is that there is some similarity to the Einstein action as it consists of the derivative interactions of the metric tensor field. Here we have investigated the contribution to the cosmological constant by evaluating the expectation value of the energy-momentum tensor. From the perturbative investigation, we have found that the coupling constant of the non-linear sigma model becomes time dependent at the one loop level in agreement with power counting of the IR logarithms. In contrast, the leading IR effects to the cosmological constant cancel out each other at the two loop level beyond the power counting. Furthermore, we have shown that the cancellation of the leading IR effects works to all orders on an arbitrary target space. In fact even if we consider the full IR effects, the effective cosmological constant is time independent in the large N limit on an N-sphere. Although the sub-leading IR effects could arise at the three loop level in a generic non-linear sigma model, we have shown that there is a renormalization scheme to cancel it. In this thesis, I summarize the quantum IR effects due to the degrees of freedom at the two regions, inside the cosmological horizon and outside the cosmological horizon.