# [time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix

Sun, 3 Oct 1999 16:14:47 +0900

Dear Matti,

Your observation in the following is correct.

Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:

Subject: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time
847]Unitarity of S-matrix

>
>
>
> I noticed what might be the reason for the paradoxal conclusion
> about the triviality of S-matrix.
>
>
> The expression of S-matrix is
>
> <m_0|Sn> = <m_0| P*(1/(1+X)|n_0>
>
> Expand this to geometric series to get
>
> ...= delta (m,n) + sum_n <m_0| X^n|n_0>
>
> = delta (m,n) + (1/i*epsilon) sum_n <m_0| L_0(int) X^(n-1)|n_0>
>
> Here I have used X= (1/L_0(free)+iepsilon)L_0(int) to the first
> X in the expansion in powers of X.
>
> The point is that formula contains 1/epsilon factor!!
>
> Thus the limit is extremely delicate. S-matrix is notrivial
> if L_0(int)|m_0> is of order epsilon and goes to zero at
> the limit epsilon->0.
>
>
> This is dangerously delicate but I think that similar problems
> must be encountered with ordinary time dependent scattering theory
> when one restricts to 'energy shell' E=constant.

Also in time dependent expression, taking the limit t -> \infty requires a
delicate argument and as well dangerous (;-)