# How is perturbation theory applied to the Bunch-Davies state for an interacting quantum field theory?

Feynman diagrams are ordinarily the usual method of perturbative analysis for weakly interacting quantum field theories. However, over a de Sitter background, the total number of particles isn't even conserved in the free field theory. So, how does one apply perturbation theory to the Bunch-Davies state? Feynman diagrams presuppose particle number is conserved in the unperturbed model. This would definitely help us to compute the nongaussian correlations in the cosmic microwave background radiation in inflationary models.

Is the closed time path formalism absolutely necessary?

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If your question is on how to perform perturbation theory in QFT in DeSitter space. Then frankly the question is enormous in scope. It can be done of course -kinda-, it just requires an entire textbook to motivate and explain. And yea, particle number is not a good operator in this business in general! –  Columbia May 11 '11 at 8:06

The Bunch-Davies (BD) vacuum is a "pure state" when it comes to all the creation and annihilation operators in the patch where the BD vacuum is relevant. Inside this patch, we can say what operators annihilate the BD state which uniquely specifies it (well, for massive fields, one has the $\alpha$-vacua as generalizations).

However, there's also a thermal radiation coming from the boundary of this patch. This is the reason for the non-conservation of the particle number. Conceptually, this non-conservation is equivalent to the ordinary thermal ensemble in the Minkowski space - which is a "mixed state". So the two-point functions are essentially thermal two-point functions, and so on.

The correlation functions are computed via Feynman rules that are analogous to those in the Minkowskian thermal state. We don't compute scattering amplitudes because there is no scattering S-matrix in the de Sitter space. That's because an observer's causal diamond misses almost all of the asymptotic regions where global in- and out-states may be defined.

In particular, it is not true that the Feynman diagrams for thermal field theory require the particle numbers to be well-defined (thermal states are always mixtures of different particle number states) or conserved (interactions always change the number of particles, and interactions had to be incorporated to get the right thermal state, anyway). The number of particles is not a good quantum number in violently cosmological backgrounds and one will find nothing interesting if he or she focuses on this bad quantum number, assuming that it carries some wisdom.

At the level of field theory in a curved spacetime, the right rules are known and analogous to thermal QFT; correlation functions and not (unmeasurable) amplitudes may be calculated. When gravity is made dynamical, the calculations only work at the semiclassical level.

Beyond semiclassical QFT in curved spaces

There are subtle things and no one knows the totally rigorous way to describe the de Sitter space. In particular, we don't know whether the relevant Hilbert space is finite-dimensional (as suggested by the finite de Sitter entropy) or infinite-dimensional; whether the degrees of freedom behind the cosmic horizon are "independent" or just reparametrizations of the degrees of freedom inside the visible universe, because of the complementarity principle; whether there can be a fully predictive framework or not, and so on.

Nevertheless, at the same moment, one should realize that all these uncertainties are "very tiny" when the de Sitter radius is large. The thermal wavelength is comparable to the de Sitter radius and the expected number of thermal particles within the whole visible part of the de Sitter space is of order one.

So we're really uncertain about roughly one photon and one graviton that may or may not live somewhere in the whole visible Universe and whose energy is so tiny that the wavelength is comparable to the de Sitter radius, too. In any practical context, even during inflation, this is a very small effect.

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And by the way, the de Sitter space's radius was still much larger than the Planck length even during inflation. That's why the non-uniformities of the CMB temperature today are of order $10^{-5}$ only. These non-uniformities are predicted from the correlation functions. Whether the number of particles in a state is conserved doesn't play any role; the number of particles in the states relevant for inflation can't even be sharply well-defined because they're thermal from all real observers' viewpoint. –  Luboš Motl May 11 '11 at 8:08