**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sun, 3 Oct 1999 13:40:54 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 875] Still about the unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 873] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 866] Re: [time 864] 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,
*

*>
*

*> 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
*

*>
*

*>
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*> >
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*> >
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*> > On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
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*> >
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*> > > Dear Matti,
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*> > >
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*> > > Your observation in the following is correct.
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*> > >
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*> > > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
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*> > >
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*> > > Subject: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time
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*> > > 847]Unitarity of S-matrix
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*> > >
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*> > >
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*> > > >
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*> > > >
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*> > > >
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*> > > > I noticed what might be the reason for the paradoxal conclusion
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*> > > > about the triviality of S-matrix.
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*> > > >
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*> > > >
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*> > > > The expression of S-matrix is
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*> > > >
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*> > > > <m_0|Sn> = <m_0| P*(1/(1+X)|n_0>
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*> > > >
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*> > > > Expand this to geometric series to get
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*> > > >
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*> > > > ...= delta (m,n) + sum_n <m_0| X^n|n_0>
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*> > > >
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*> > > > = delta (m,n) + (1/i*epsilon) sum_n <m_0| L_0(int) X^(n-1)|n_0>
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*> > > >
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*> > > > Here I have used X= (1/L_0(free)+iepsilon)L_0(int) to the first
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*> > > > X in the expansion in powers of X.
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*> > > >
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*> > > > The point is that formula contains 1/epsilon factor!!
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*> > > >
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*> > > > Thus the limit is extremely delicate. S-matrix is notrivial
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*> > > > if L_0(int)|m_0> is of order epsilon and goes to zero at
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*> > > > the limit epsilon->0.
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*> > > >
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*> > > >
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*> > > > This is dangerously delicate but I think that similar problems
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*> > > > must be encountered with ordinary time dependent scattering theory
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*> > > > when one restricts to 'energy shell' E=constant.
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*> > >
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*> > > Also in time dependent expression, taking the limit t -> \infty requires a
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*> > > delicate argument and as well dangerous (;-)
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*> > >
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*> > >
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*> > > > The task would
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*> > > > be to find proper formulation or possibly understand why p-adics
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*> > > > save the situation.
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*> > >
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*> > > Before going to p-adics, there is a possibility to be checked: If
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*> standpoint
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*> > > of real numbers works or not?
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*> >
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*> > Your are right.
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*> >
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*> > There are also mathematical challenges related to the localization
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*> > in zero modes occurring for final states of quantum jump.
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*>
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*> What is "zero modes" and how is it related with quantum jump?
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*>
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*> >
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*> > The final proof would be precise Feynmann rules
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*> > yielding S-matrix which is unitary order by order. BTW, I remember
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*> > having years ago looked the sketch of the perturbative proof of
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*> > unitarity. Analyticity and cuts of scattering amplitudes were
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*> > somehow involved.
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*>
*

*> In the field theoretical proofs, cut-offs may be necessary. This is seen also
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*> 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.

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

*>
*

*>
*

*>
*

**Next message:**Matti Pitkanen: "[time 875] Still about the unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 873] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 866] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"

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