You figured out everything yourself while I was posting my answer.

Cheers

> Your equation (4) has contains a single tau derivative (d/d > tau)y_L. I?m not sure why it is symmetric in the
> transformation tau -> -tau ? could you explain?

Indeed, (4) as I wrote it is not invariant w.r.t. \tau->-\tau. This is because I forgot to add \tanh \tau in the second term Fixed now and thanks for finding misprint.

> I?d also like to ask you: I?ve heard before that de Sitter > spacetime does not admit a S-matrix. Is this true? And is
> it related to this divergence at tau = 0 that you?re
> discussing here?

You probably had in mind AdS, didn’t you? Quantum gravity in AdS does not have a description in terms of S-matrix, instead, one has to describe it in terms of boundary degrees of freedom (AdS/CFT).

Actually, dS admits in-out S-matrix for odd dimensions; what I calculate is exactly S-matrix elements between in- and out-states. When number of dimensions is even, de Sitter admits in-in S-matrix (singularity at \tau=0 basically means that there is particle production and in-out S-matrix is of no good).

Your comments and questions are very much welcome.
Cheers.

Your equation (4) has contains a single tau derivative (d/d tau)y_L. I’m not sure why it is symmetric in the transformation tau -> -tau — could you explain?

I’ve found instead, if I first substitute y_L(tau) = exp(-tau(d-1)/2) z_L(tau), then the equation takes the form

z_L”(tau) – ( (1/4)(d(d – 2) + 1) – m^2 – L (d + L – 2) sech^2(t) ) z_L(tau) = 0,

which can also be solved in terms of hypergeometric functions (probably similar/same answer as yours? I’ve not checked…) and is symmetric under tau -> – tau.

I’d also like to ask you: I’ve heard before that de Sitter spacetime does not admit a S-matrix. Is this true? And is it related to this divergence at tau = 0 that you’re discussing here?

Thank you!

P.S. I should say I “discovered” your blog a week or two ago and I hope to read it on a regular basis and also try to learn some QFT from your older posts.