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

*Sun, 3 Oct 1999 13:26:04 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 863] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Previous message:**Hitoshi Kitada: "[time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Matti Pitkanen: "[time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"

Dear Matti,

My question in the following is that:

You stated the scattering space \HH_s is the same as the free space P\HH. This

means

\HH_s = P\HH,

hence

any free state u=|n_0> in P\HH=\HH_s satisfies

u = Pu.

Thus

1/(1+X) |n_0> = 1/(1+X) P|n_0> = |n_0>.

The last equality here follows from

1/(1+X) = 1-\sum_{n=1}^\infty (-X)^n

and

X P = 0

by

X = X= 1/(L_0 +ie*epsilon))*L_0(int)

and L_0(int) P|n_0> = 0.

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

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

*>
*

*> Subject: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix
*

*>
*

*>
*

*> >
*

*> >
*

*> > On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
*

*> >
*

*> > > Dear Matti,
*

*> > >
*

*> > > I considered your proof and previous notes. I have a following question
*

on

*> the
*

*> > > present proof:
*

*> > >
*

*> > > If u = Pu for any scattering states u, u satisfies
*

*> > >
*

*> > > Vu = 0
*

*> > >
*

*> > > by your assumption: L_0(int)P|n> = 0. (Here V = L_0(int) and u = |n>.)
*

*> Then
*

*> > >
*

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

*> > >
*

*> > > (Here R(z) = (H-z)^{-1}, H=L_0(tot), z= i\epsilon in your notation.)
*

*> > >
*

*> > > This means there is no scattering: S = I.
*

*> >
*

*> >
*

*> > I think that this is not the case.
*

*> >
*

*> >
*

*> > 1/(1+X), X= 1/(L_0 +ie*epsilon))*L_0(int)
*

*> >
*

*> > operates on *"free"* state |n_0> in matrix element of the scattering
*

*> > operator and L_0(int) does not annhilate it.
*

*>
*

*> What is the difference of |n_0> from |n>?
*

*>
*

*>
*

*> 1/(1+X) acts like unity
*

*> > only when acts on *scattering state* |n>.
*

**Next message:**Matti Pitkanen: "[time 863] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Previous message:**Hitoshi Kitada: "[time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Matti Pitkanen: "[time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"

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