# [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof of unitarity

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 9 Oct 1999 13:36:15 +0300 (EET DST)

On Sat, 9 Oct 1999, Hitoshi Kitada wrote:

> Dear Matti,
> ----- Original Message -----
> From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
> Sent: Saturday, October 09, 1999 2:25 PM
> Subject: [time 919] Re: [time 914] Re: [time 909] About your proof of
> unitarity
>
>
> >
> >
> > On Sat, 9 Oct 1999, Hitoshi Kitada wrote:
> >
> > > Dear Matti,
> > >
> > > I understand that you are talking in p-adic context, and as such the
> present
> > > proof does not harm your result.
> > >
> > > I do not want to disturb your satisfaction with your proof. Just I would
> like
> > > to conclude with a comment that the existence of the limit lim
> > > (1+R_0(z)V)^{-1} = lim R(z)(H_0-z) = lm (1-R(z)V): \HH_- -->\HH_- when Im
> z->0
> > > is the main issue, and if this is solved, the unitarity holds also in real
> > > case.
> >
> > But if one has the condition VP|m_1>=0 S matrix is trivial in real
> > context
>
> I am speaking of general context without such a condition. If the limit above
> exists, then it follows from it the unitarity.
>
You argue that you can avoid somehow the assumption about the
existence of the time development operator and get unitarity from
algebraic structure alone. Or that you have unitary time development
operator in case that you have only E=0 states
of Hamiltonian?

This would be very nice. This was my original optimistic belief
but I could not see any manner to achieve unitarity
without the additional assuption. Can one have formal proof
of unitarity in my case?

>
> since T^daggerT=0: this you certaily agree. The limits are
> > certainly delicate but as I said I must try to identify the architecture
> > of unitarity: the condition replacing the representability of S-matrix as
> > time development operator.
> >
> >
> > I hope that you understand that our starting points are different. You
> > have at your use refined scattering theory whereas I am desperately trying
> > to identify basic structural principles leading to "Feynmann rules".
> > Only after that functional analyst can come to my great building and
> > start decoration(-;). You certainly know that even quantum field theories
> > are still unkown territory for mathematicians (say functional integrals).
> > TGD generalize quantum field theories by replacing point like
> > particle with 3-surface: TGD is more interesting for mathematical dreamers
> > than "blind mathematicians" in its recent state.
> >

Best,
MP

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