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

*Sun, 3 Oct 1999 11:04:52 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next 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"**Previous message:**Matti Pitkanen: "[time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 866] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Next in thread:**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"

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
*

*>
*

*>
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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 standpoint
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*> 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

*>
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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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**Next 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"**Previous message:**Matti Pitkanen: "[time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 866] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Next in thread:**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"

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