# [time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 7 Oct 1999 07:34:10 +0300 (EET DST)

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

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