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

*Thu, 7 Oct 1999 07:34:10 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 891] Re: [time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"**Previous message:**Hitoshi Kitada: "[time 889] Re: [time 888] Re: [time 886] Unitarity finally understood!"**In reply to:**Matti Pitkanen: "[time 888] Re: [time 886] Unitarity finally understood!"**Next in thread:**Hitoshi Kitada: "[time 891] Re: [time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"

On Thu, 7 Oct 1999, Hitoshi Kitada wrote:

*> Dear Matti,
*

*>
*

*> Let me make a question at each posting for the time being.
*

*>
*

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

*>
*

*> Subject: [time 888] Re: [time 886] Unitarity finally understood!
*

*>
*

*>
*

*> >
*

*> >
*

*> > Dear Hitoshi,
*

*> >
*

*> > The posting did not containg any proof of unitarity. I attach a latex file
*

*> > with proof.
*

*> >
*

*> > Best,
*

*> > MP
*

*> >
*

*> >
*

*> > \documentstyle [10pt]{article}
*

*> > \begin{document}
*

*> > \newcommand{\vm}{\vspace{0.2cm}}
*

*> > \newcommand{\vl}{\vspace{0.4cm}}
*

*> > \newcommand{\per}{\hspace{.2cm}}
*

*> > %matti
*

*> >
*

*> >
*

*> > \subsection{Formal proof of unitarity}
*

*> >
*

*> >
*

*> > Consider now the formal proof of the unitarity.
*

*> > Orthogonality condition guaranteing
*

*> > unitarity can be expressed also as the condition
*

*> >
*

*> >
*

*> > \begin{eqnarray}
*

*> > \frac{1}{1+X^{\dagger}} P\frac{1}{1+X}&=&G \per ,\nonumber\\
*

*> > \nonumber\\
*

*> > G(m,n) &=&\delta (m,n) \langle m\vert m\rangle\per .
*

*> > \end{eqnarray}
*

*>
*

*> >From where the projection P comes?
*

*>
*

*> My understanding:
*

*>
*

*> |m> = |m_0> + |m_1> = |m_0> - X|m>,
*

*>
*

*> X=(L_0+iz)^{-1}V, V=L_0(int),
*

*>
*

*> thus
*

*>
*

*> |m> = (1+X)^{-1}|m_0>.
*

*>
*

*> So unitarity
*

*>
*

*> <m|n> = <m_0|(1+X^\dagger)^{-1}(1+X)^{-1}|n_0> = <m_0|n_0>
*

*>
*

This should be written

<m|Pn> = <m_0|n_0>

since it is projections P|m> which form outgoing states.

*> means
*

*>
*

*> (1+X^\dagger)^{-1}(1+X)^{-1} = G.
*

*>
*

Hence

(1+X^\dagger)^{-1}P(1+X)^{-1} = G.

*> Here P in
*

*>
*

*> |m_0> = P|m_0>
*

*>
*

*> cannot come to the center between (1+X^\dagger)^{-1} and (1+X)^{-1}, since P
*

*> does not commute with L_0 and V.
*

The point is that it is on mass shell projections |Pm> of states |m>

to the space of free states |m_0>. which are outgoing states. These states

should form orthogonal basis and this is what I want to prove.

This is precisely the picture also in standard quantum field theories

(LSZ formula).

Best,

MP

**Next message:**Hitoshi Kitada: "[time 891] Re: [time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"**Previous message:**Hitoshi Kitada: "[time 889] Re: [time 888] Re: [time 886] Unitarity finally understood!"**In reply to:**Matti Pitkanen: "[time 888] Re: [time 886] Unitarity finally understood!"**Next in thread:**Hitoshi Kitada: "[time 891] Re: [time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"

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