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

*Mon, 27 Sep 1999 04:13:05 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 818] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**Previous message:**Matti Pitkanen: "[time 816] Re: [time 815] A summary on [time 814] Still about construction of U"**In reply to:**Hitoshi Kitada: "[time 815] A summary on [time 814] Still about construction of U"**Next in thread:**Matti Pitkanen: "[time 818] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"

Dear Matti,

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

Subject: [time 816] Re: [time 815] A summary on [time 814] Still about

construction ofU

*>
*

*>
*

*>
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*> Thank you for good posting. Your are right in that Hilbert space
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*> is extended. One however obtains S-matrix for which other half
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*> of unitary condition with summation over intermediate states of
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*> extended Hilbert space is satisfied and this makes
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*> S-matrix physical. Other half of unitarity conditions
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*> involving sum over the intermediate states in smaller Hilbert space is
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*> lost.
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*>
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*> See below.
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skip

*> > This is not your expectation. Why this happened? There are two possible
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reasons:

*> >
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*> > 1) The first is that we have assumed that both of \Psi and \Psi_0 are in
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the

*> > Hilbert space \HH. If we assume \Psi_0 is in \HH, then \Psi must be
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outside \HH.

*>
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*> This is certainly the case since Psi contains superposition of
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*> off mass shell states. p^2-L_0(vib)=0 is not satisfied for Psi.
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*> If this were not the case, the entire equation would be nonsensical
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*> since right hand side would be of form (L_0(int)/+ie)Psi.
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*> Thus we have Hilbert spaces which we could call Hilb_0 and Hilb.
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*>
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*>
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*> One the other hand. Psi is image of on mass shell state under Psi_0-->Psi
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*> and S-matrix is defined as matrix elements
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*>
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*> SmM== <Psi_0(m),Psi (M)>.
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*>
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*> One restricts outgoing momenta to on mass shell momenta in inner product.
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*> This means projection of Psi (m) to the space Hilb_0 spanned by Psi_0:s
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*> when one calculates inner products defining S-matrix.
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*>
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*> One obtains unitarity relations
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*>
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*> sum_N SmN (SnN)^* = delta (m,n)
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*>
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*> from completeness in Hilb: sum_N |N> <N|=1
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*>
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*> but NOT
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*>
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*> sum_m smM (SmN)*.
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*>
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*> since Hilb_0 completeness relation sum_m |m><m|=1 are not true in Hilb
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*> but become sum_m |m><m>= P, P projector to Hilb_0.
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*>
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*> But this seems to be enough! One obtains S-matrix with orthogonal
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*> rows: this gives probability conservation plus additional conditions.
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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.

*>
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*>
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Best wishes,

Hitoshi

**Next message:**Matti Pitkanen: "[time 818] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**Previous message:**Matti Pitkanen: "[time 816] Re: [time 815] A summary on [time 814] Still about construction of U"**In reply to:**Hitoshi Kitada: "[time 815] A summary on [time 814] Still about construction of U"**Next in thread:**Matti Pitkanen: "[time 818] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"

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