Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 3 Oct 1999 11:04:52 +0300 (EET DST)
On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
> 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 (;-)
>
>
> > The task would
> > be to find proper formulation or possibly understand why p-adics
> > save the situation.
>
> Before going to p-adics, there is a possibility to be checked: If standpoint
> of real numbers works or not?
Your are right.
There are also mathematical challenges related to the localization
in zero modes occurring for final states of quantum jump.
The final proof would be precise Feynmann rules
yielding S-matrix which is unitary order by order. BTW, I remember
having years ago looked the sketch of the perturbative proof of
unitarity. Analyticity and cuts of scattering amplitudes were
somehow involved.
Best,
MP
>
>
> Best wishes,
> Hitoshi
>
>
>
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