Hitoshi Kitada (hitoshi@kitada.com)
Mon, 27 Sep 1999 04:13:05 +0900
Dear Matti,
Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
Subject: [time 816] Re: [time 815] A summary on [time 814] Still about
construction ofU
>
>
>
> Thank you for good posting. Your are right in that Hilbert space
> is extended. One however obtains S-matrix for which other half
> of unitary condition with summation over intermediate states of
> extended Hilbert space is satisfied and this makes
> S-matrix physical. Other half of unitarity conditions
> involving sum over the intermediate states in smaller Hilbert space is
> lost.
>
> See below.
skip
> > This is not your expectation. Why this happened? There are two possible
reasons:
> >
> > 1) The first is that we have assumed that both of \Psi and \Psi_0 are in
the
> > Hilbert space \HH. If we assume \Psi_0 is in \HH, then \Psi must be
outside \HH.
>
> This is certainly the case since Psi contains superposition of
> off mass shell states. p^2-L_0(vib)=0 is not satisfied for Psi.
> If this were not the case, the entire equation would be nonsensical
> since right hand side would be of form (L_0(int)/+ie)Psi.
> Thus we have Hilbert spaces which we could call Hilb_0 and Hilb.
>
>
> One the other hand. Psi is image of on mass shell state under Psi_0-->Psi
> and S-matrix is defined as matrix elements
>
> SmM== <Psi_0(m),Psi (M)>.
>
> One restricts outgoing momenta to on mass shell momenta in inner product.
> This means projection of Psi (m) to the space Hilb_0 spanned by Psi_0:s
> when one calculates inner products defining S-matrix.
>
> One obtains unitarity relations
>
> sum_N SmN (SnN)^* = delta (m,n)
>
> from completeness in Hilb: sum_N |N> <N|=1
>
> but NOT
>
> sum_m smM (SmN)*.
>
> since Hilb_0 completeness relation sum_m |m><m|=1 are not true in Hilb
> but become sum_m |m><m>= P, P projector to Hilb_0.
>
> But this seems to be enough! One obtains S-matrix with orthogonal
> rows: this gives probability conservation plus additional conditions.
The probability conservation (i.e. unitarity of scattering operator) is not so
easy to prove. I just gave an outline. If one would want to get a rigorous
proof, it might require several years.
> Colums are however not orthogonal.
I am not familiar with Dirac notation, but I believe I did not make mistakes
in my formulae, insofar as about its formality.
>
>
Best wishes,
Hitoshi
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