# [time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix

Sun, 3 Oct 1999 18:15:24 +0900

Dear Matti,

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

Subject: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time
855] Re:[time 847]Unitarity of S-matrix

>
>
> 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?
>
>
> There are also mathematical challenges related to the localization
> in zero modes occurring for final states of quantum jump.

What is "zero modes" and how is it related with 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.

In the field theoretical proofs, cut-offs may be necessary. This is seen also
in recent researches.

>
> Best,
> MP
> >
> >
> > Best wishes,
> > Hitoshi
> >
> >
> >
>

Best wishes,
Hitoshi

This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:40:46 JST