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

*Thu, 7 Oct 1999 13:20:44 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"**Previous message:**Matti Pitkanen: "[time 888] Re: [time 886] Unitarity finally understood!"**In reply to:**Hitoshi Kitada: "[time 887] Re: [time 886] Unitarity finally understood!"**Next in thread:**Matti Pitkanen: "[time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"

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!

*>
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*>
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*> Dear Hitoshi,
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*>
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*> The posting did not containg any proof of unitarity. I attach a latex file
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*> with proof.
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*>
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*> Best,
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*> MP
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*>
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*>
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*> \documentstyle [10pt]{article}
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*> \begin{document}
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*> \newcommand{\vm}{\vspace{0.2cm}}
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*> \newcommand{\vl}{\vspace{0.4cm}}
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*> \newcommand{\per}{\hspace{.2cm}}
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*> %matti
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*>
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*>
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*> \subsection{Formal proof of unitarity}
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*>
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*>
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*> Consider now the formal proof of the unitarity.
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*> Orthogonality condition guaranteing
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*> unitarity can be expressed also as the condition
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*>
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*>
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*> \begin{eqnarray}
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*> \frac{1}{1+X^{\dagger}} P\frac{1}{1+X}&=&G \per ,\nonumber\\
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*> \nonumber\\
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*> G(m,n) &=&\delta (m,n) \langle m\vert m\rangle\per .
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*> \end{eqnarray}
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*>From where the projection P comes?
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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>

means

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

Best wishes,

Hitoshi

**Next message:**Matti Pitkanen: "[time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"**Previous message:**Matti Pitkanen: "[time 888] Re: [time 886] Unitarity finally understood!"**In reply to:**Hitoshi Kitada: "[time 887] Re: [time 886] Unitarity finally understood!"**Next in thread:**Matti Pitkanen: "[time 890] Re: [time 888] Re: [time 886] Unitarity finally understood!"

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