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

*Sun, 3 Oct 1999 08:47:18 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 863] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Matti Pitkanen: "[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,
*

*>
*

*> My question in the following is that:
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*>
*

*> You stated the scattering space \HH_s is the same as the free space P\HH. This
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*> means
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*>
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*> \HH_s = P\HH,
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*>
*

*> hence
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*>
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*> any free state u=|n_0> in P\HH=\HH_s satisfies
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*>
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*> u = Pu.
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*>
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*> Thus
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*>
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*> 1/(1+X) |n_0> = 1/(1+X) P|n_0> = |n_0>.
*

*>
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*> The last equality here follows from
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*>
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*> 1/(1+X) = 1-\sum_{n=1}^\infty (-X)^n
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*>
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*> and
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*>
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*> X P = 0
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*>
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*> by
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*>
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*> X = X= 1/(L_0 +i*epsilon)*L_0(int)
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*>
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*> and L_0(int) P|n_0> = 0.
*

Yes. Suppose that every state L_0(int)P|n> vanishes and states P|n> span

the space spanned by |n_0>. Then one can express |n_0> as superposition

of P|n>:s and conclude that L_0(int)|n_0> vanishes and S is trivial.

I am really amazed! How it is possible to obtain nontrivial S-matrix

at all in QM? Same kind proof for unitarity should hold also there!

The point is that I "know" that the expansion yields S-matrix. There

is now doubt about that. But how on Earth can I demonstrate the unitarity?

There is something which I do not understand but what is it?

Could this paradox be related to the taking of epsilon-->0 limit?

Condition

L_0(int)* P*(1/(1+X)) |m_0> =0

holds true only at in the sense of limit epsilon-->0 .

Limit of this equation would not be same as equation

obtained putting epsilon=0 from the beginning to get L_0(int|m_0>=0?

This would not be surprising since epsilon prescription can be seen

as a manner to make propagators well defined. I do not know.

Best,

MP

*>
*

*>
*

*> Best wishes,
*

*> Hitoshi
*

*> ----- Original Message -----
*

*> From: Hitoshi Kitada <hitoshi@kitada.com>
*

*> To: Matti Pitkanen <matpitka@pcu.helsinki.fi>
*

*> Cc: Time List <time@kitada.com>; Paul Hanna <phanna@ghs.org>
*

*> Sent: Sunday, October 03, 1999 1:04 PM
*

*> Subject: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of
*

*> S-matrix
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*>
*

*>
*

*> > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
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*> >
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*> > Subject: [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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*> > > On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
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*> > >
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*> > > > Dear Matti,
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*> > > >
*

*> > > > I considered your proof and previous notes. I have a following question
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*> on
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*> > the
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*> > > > present proof:
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*> > > >
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*> > > > If u = Pu for any scattering states u, u satisfies
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*> > > >
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*> > > > Vu = 0
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*> > > >
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*> > > > by your assumption: L_0(int)P|n> = 0. (Here V = L_0(int) and u = |n>.)
*

*> > Then
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*> > > >
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*> > > > (I+X)^{-1}|n> = (I - R(z)V) u = u = |n>.
*

*> > > >
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*> > > > (Here R(z) = (H-z)^{-1}, H=L_0(tot), z= i\epsilon in your notation.)
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*> > > >
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*> > > > This means there is no scattering: S = I.
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*> > >
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*> > >
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*> > > I think that this is not the case.
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*> > >
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*> > >
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*> > > 1/(1+X), X= 1/(L_0 +ie*epsilon))*L_0(int)
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*> > >
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*> > > operates on *"free"* state |n_0> in matrix element of the scattering
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*> > > operator and L_0(int) does not annhilate it.
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*> >
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*> > What is the difference of |n_0> from |n>?
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*> >
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*> >
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*> > 1/(1+X) acts like unity
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*> > > only when acts on *scattering state* |n>.
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*>
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*>
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*>
*

**Next message:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 863] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"

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