**Hitoshi Kitada** (*hitoshi@kitada.com*)

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 872] Re: [time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**Previous message:**Hitoshi Kitada: "[time 870] Re: [time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Next in thread:**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"

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:
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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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*> >
*

*> > Subject: [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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*> > >
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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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*> > >
*

*> > > <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

*> > of real numbers works or not?
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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.
*

What is "zero modes" and how is it related with quantum jump?

*>
*

*> 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.
*

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

in recent researches.

*>
*

*> Best,
*

*> 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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*>
*

Best wishes,

Hitoshi

**Next message:**Matti Pitkanen: "[time 872] Re: [time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**Previous message:**Hitoshi Kitada: "[time 870] Re: [time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Next in thread:**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"

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