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

*Sat, 9 Oct 1999 21:52:43 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 923] Unitarity"**Previous message:**Matti Pitkanen: "[time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof of unitarity"**In reply to:**Hitoshi Kitada: "[time 920] Re: [time 919] Re: [time 914] Re: [time 909] About your proof of unitarity"**Next in thread:**Matti Pitkanen: "[time 924] Re: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof ofunitarity"

Dear Matti,

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

Subject: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your

proof of unitarity

skip

*> > I am speaking of general context without such a condition. If the limit
*

above

*> > exists, then it follows from it the unitarity.
*

*> >
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*> You argue that you can avoid somehow the assumption about the
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*> existence of the time development operator and get unitarity from
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*> algebraic structure alone. Or that you have unitary time development
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*> operator in case that you have only E=0 states
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*> of Hamiltonian?
*

This is not my argument, but it is an old theory of T. Kato and S. T. Kuroda:

Theory of simple scattering and eigenfunction expansions, Functional Analysis

and Related Topics, Springer-Verlag, 1970, pp. 99-131,

and

The abstract theory of scattering, Rocky Mount. J. Math., Vol. 1 (1971),

127-171.

I am not sure if your interaction term satisfies their assumptions. If it

works with your case, their argument treats the Hamiltonian without assuming

conditions like Virasoro conditions. They get a unitarity (completeness in

their terminology) for general spectra. The spectral projection onto the space

corresponding to E=0 would then give your unitarity.

The problem may be in the interaction term if their method does not work.

A question related with this is if the E=0 states are genuine eigenvectors

or generalized ones. Maybe to see if this is the case or not is included in

your problem?

Best wishes,

Hitoshi

**Next message:**Matti Pitkanen: "[time 923] Unitarity"**Previous message:**Matti Pitkanen: "[time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof of unitarity"**In reply to:**Hitoshi Kitada: "[time 920] Re: [time 919] Re: [time 914] Re: [time 909] About your proof of unitarity"**Next in thread:**Matti Pitkanen: "[time 924] Re: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof ofunitarity"

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