Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 3 Oct 1999 13:40:54 +0300 (EET DST)
On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
> 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?
> >
> > Your are right.
> >
> > 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.
I meant cuts. The imaginary part of scattering amplitude T
which is analytic function of momenta is discontinuous above the
threshold of the process. Unitarity gives
iT- iT^dagger = -TT^dagger imaginary part (the discontinuity)
of forward amplitude, which vanishes only above threshold, can be
related to scattering cross section which in turn corresponds roughly to
TT^dagger.
>
> >
> > Best,
> > MP
> > >
> > >
> > > Best wishes,
> > > Hitoshi
> > >
> > >
> > >
> >
>
> Best wishes,
> Hitoshi
>
>
>
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