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

*Fri, 8 Oct 1999 09:56:38 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 909] About your proof of unitarity"**Previous message:**Hitoshi Kitada: "[time 907] A proof of unitarity"

I will print the text, read it and respond later.

Best,

MP

On Fri, 8 Oct 1999, Hitoshi Kitada wrote:

*> Dear Matti,
*

*>
*

*> Rethinking about your proof, I found an alternative simple proof of unitarity
*

*> of S-matrix. Of course this is a proof on a formal level.
*

*>
*

*> The proof:
*

*>
*

*> Set
*

*>
*

*> R_0(z)=(H_0-z)^{-1}, z: non-real, H_0=L_0(free), V=L_0(int),
*

*>
*

*> P = projection onto the eigenspace of H_0 with eigenvalue 0.
*

*>
*

*> Let m_0 in P\HH (=eigenspace of H_0 with eigenvalue 0). Then the scattering
*

*> state m(z) satisfies (I omit the bracket notation)
*

*>
*

*> m(z)=m_0+m_1(z)=m_0-R_0(z)Vm(z).
*

*>
*

*> Thus
*

*>
*

*> Pm(z)=Pm_0+Pm_1(z)=m_0+z^{-1}PVm(z). (1)
*

*>
*

*> Therefore
*

*>
*

*> PVm(z)=zPm_1(z). (2)
*

*>
*

*> Your assumption is
*

*>
*

*> VPm_1(z)=0. (A)
*

*>
*

*> Thus (2) and (A) yield
*

*>
*

*> VPVm(z)=zVPm_1(z)=0, which means
*

*>
*

*> <m(z)|VPV|m(z)>=0,
*

*>
*

*> hence
*

*>
*

*> PVm(z)=0.
*

*>
*

*> This and (2) imply
*

*>
*

*> Pm_1(z)=z^{-1}PVm(z)=0.
*

*>
*

*> Thus by (1) we have
*

*>
*

*> Pm(z)=m_0.
*

*>
*

*> Therefore
*

*>
*

*> <m(z)|P|n(z)>=<m_0|n_0>: unitarity. (U)
*

*>
*

*>
*

*> Comment 1. As Im z -> 0,
*

*>
*

*> m(z) = m_0 - R_0(z)Vm(z) = (1+R_0(z)V)^{-1}m_0
*

*>
*

*> would be outside the Hilbert space \HH. Then (U) might lose its meaning as Im
*

*> z -> 0. This would require the introduction of some larger space \HH_-.
*

*>
*

*> Comment 2. The present formulation of yours uses P explicitly. Namely m=m(z)
*

*> may be outside of P\HH. Thus it is free of my criticism in [time 882].
*

*>
*

*>
*

*> Best wishes,
*

*> Hitoshi
*

*>
*

*>
*

**Next message:**Matti Pitkanen: "[time 909] About your proof of unitarity"**Previous message:**Hitoshi Kitada: "[time 907] A proof of unitarity"

*
This archive was generated by hypermail 2.0b3
on Sun Oct 17 1999 - 22:40:47 JST
*