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

*Sun, 3 Oct 1999 10:05:18 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 866] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 862] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"

I noticed what might be the reason for the paradoxal conclusion

about the triviality of S-matrix.

The expression of S-matrix is

<m_0|Sn> = <m_0| P*(1/(1+X)|n_0>

Expand this to geometric series to get

...= delta (m,n) + sum_n <m_0| X^n|n_0>

= delta (m,n) + (1/i*epsilon) sum_n <m_0| L_0(int) X^(n-1)|n_0>

Here I have used X= (1/L_0(free)+iepsilon)L_0(int) to the first

X in the expansion in powers of X.

The point is that formula contains 1/epsilon factor!!

Thus the limit is extremely delicate. S-matrix is notrivial

if L_0(int)|m_0> is of order epsilon and goes to zero at

the limit epsilon->0.

This is dangerously delicate but I think that similar problems

must be encountered with ordinary time dependent scattering theory

when one restricts to 'energy shell' E=constant. The task would

be to find proper formulation or possibly understand why p-adics

save the situation.

On Sun, 3 Oct 1999, Matti Pitkanen wrote:

*>
*

*>
*

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

*>
*

*> > 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 +i*epsilon)*L_0(int)
*

*> >
*

*> > 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
*

*> >
*

*> >
*

*> > > 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:**Hitoshi Kitada: "[time 866] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 862] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"

*
This archive was generated by hypermail 2.0b3
on Sun Oct 17 1999 - 22:40:46 JST
*