271. Continuing dS/CFT – correspondence. Part 2

Posted by Dmitry

February 18, 2009

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News: It seems that there are good news for science funding in US. Cosmic Variance points out that science funding in the stimulus package was largely restored: with 3 bill. for NSF, 1.6 bill. for DOE and 1 bill. for NASA. I really hope this is final

In the mean time, when we keep waiting for the final outcome of negotiations, let us continue our small study of dS/CFT. Last time we have concluded that it is hard to realize dS/CFT, since correlations between bulk physics and degrees of freedom living on $I_\pm$ infinities should be absent due to the presence of horizon.

On the other hand, as we have argued, horizon really means absence of correlations only if we prepare (quasi) de Sitter space by some kind of time evolution. So, what kind of correlation between the bulk physics and the physics at infinities $I_\pm$ we might expect? In what follows, we will mostly focus on the simplest case of the three-dimensional de Sitter space dS_3 .

1. How to generally define correlations with boundary?

Formal way to do that is to construct bulk-boundary correlation function $G_{B\partial}(b,x)$ . What do I exactly mean by this Green function?

Suppose that we have a free massive scalar field theory with e.o.m.

$(\Box-m^2)\phi=0$ . (1)

and we have found the general solution of the eq. (1). It is written in terms of mode expansion

$\phi{}(x)=\sum_k{}a_k\phi_k(x)+{\rm c.c.}$ (2)

Every mode $\phi_k(x)$ has an asymptotic behavior

$\phi_k(x)\to{}f_k(b)$ , $x\to{\rm boundary}$ .

In overall, we define the bulk-boundary Green function as

$\phi(x)=\int{}db{}f(b)G_{B\partial}(b,x)$ . (3)

We can write down the explicit expression for the bulk-boundary correlation function as follows. Starting from the e.o.m. (1), we write for the bulk Green function (namely, Feynman propagator)

$(\Box-m^2)G_B(x,x')=-\delta{}(x,x')$ .

Then,

$\phi{}(x')=-\int_V{}dV\phi{}(x)(\Box-m^2)G_B(x,x')$ .

Integrating twice by parts we have

$\phi{}(x')=-\int_{\partial{}V}d\Sigma^\mu\phi{}(x)\partial^\leftrightarrow_\mu{}G_B(x,x')$ .

Taking an appropriate limit (vicinity of boundary for a given spacetime), we can then find the bulk-boundary Green function.

2. The case of

Let us see how these considerations work for the particular case of dS_3 spacetime and calculate bulk-boundary Green function for the $dS_3/{\rm CFT}_2$ .

The metric of dS_3 in the global coordinates is

$ds^2=-d\tau^2+4{\rm cosh}^2\tau\frac{dwd\bar{w}}{(1+w\bar{w})^2}$ ,

where $(w,\bar{w})$ are complex coordinates on the sphere (it is especially convenient to use them, since the topology of $I_\pm$ infinities is the one of a sphere as we discussed).

We again consider massive free scalar field $\phi(x)$ . Near the I_- infinity it has the following asymptotic behavior:

$\phi(t,w,\bar{w})\to\phi_+^{\rm in}(w,\bar{w})e^{h_+\tau}+\phi_-^{\rm in}(w,\bar{w})e^{h_-\tau}$ ,

where

$h_\pm=1\pm{}\sqrt{1-m^2}$ (4)

(we measure mass in units of the Hubble scale) and $\phi_\pm$ is determined from the boundary conditions.

Now, using formula for the boundary-bulk Green function we find that it has the following form for dS_3 (the limit $\tau\to{}-\infty$ is taken):

$\int_{I_-}dwdv\sqrt{h(w)h(v)}[e^{-2(\tau+\tau')}\phi\partial^{\leftrightarrow}_\tau{}G\partial_{\tau{}'}^{\leftrightarrow}\phi]_{\tau=\tau'}$ ,

where

$h(w)=\frac{2}{(1+w\bar{w})^2}$ is invariant measure on the sphere,

or,

after substituting asymptotic behavior and some subsequent algebra,

$\int_{I_-}dwdv\sqrt{h(w)h(v)}\left(c_+\phi_-^{\rm in}\Delta_{h_+}\phi_-^{\rm in}+(+\to{}-)\right)$ ,

where

$\Delta_{h_\pm}=\left(\frac{(1+w\bar{w})(1+v\bar{v})}{|v-w|^2}\right)^{h_\pm}$ . (5)

We are almost ready to formulate the correspondence. Namely, we would like to interpret (5) as the correlation function of a conformal field theory operators ${\cal O}_\phi(w,\bar{w})$ that live on a 2d sphere and have a conformal dimension $h_\pm$ (this is very much analogous to AdS/CFT).

There is one interesting observation to be made, though. The conformal weights (4) become complex if the field $\phi$ is sufficiently heavy (namely, m>H ), so in order to make sense of dS/CFT correspondence we would probably want to stick to the case of light scalars.

On the other hand, heavy scalars are the only ones for which we can even define in- and out- vacuum states (as well as Allen-Mottola vacuum states which are linear combinations of the latter!) in the de Sitter (since the asymptotic behavior at infinities should be oscillatory – we want to separate negative frequency modes from positive frequency ones).

So naively it seems like we either have nice well defined QFT in de Sitter space or the dS/CFT correspondence

To be continued.

Cosmology, Journal club, Lectures on de Sitter spacetime, Master Postdoc, Quantum field theory, String theory

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271. Continuing dS/CFT – correspondence. Part 2

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