# [time 807] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U

Sat, 25 Sep 1999 21:57:59 +0900

Dear Matti,

As you seem not understand the point in the following, I add a comment:

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

Subject: [time 803] Re: [time 801] Re: [time 799] Still about construction of
U

> > >Since ordinary Schr\"odinger equation is consistent with the scattering
> > >matrix formalism avoiding elegantly the difficulties with the
> > >definition of the limit $U(-t,t)$, $t\rightarrow \infty$, it
> > >is natural to take this form of Schr\"odinger equation as starting
> > >point when trying to find Schr\"odinger equation for the 'time'
evolution
> > >operator $U$. One can even forget the assumption
> > >about time evolution and require only
> > >that the basic algebraic information guaranteing
> > >unitarity is preserved. This information boils down to the Hermiticity
> > > of free and interacting Hamiltonians and
> > >to the assumption that the spectra
> > >are identical.
> >
> >
> > It is known that to consider the limit as \epsilon -> 0 in the
Schrodinger
> > equation (1) of your note is equivalent to considering the time limit as
t ->
> > \infty of exp(-itH). So you cannot avoid the difficulty: Below I will try
to
> > show this.
>
> In TGD framework single particle Virasoro generators L_0(n) define
> propagators
>
> 1/(p^2-L_0(vib)+i*epsilon)
>
> appearing in stringy diagrams. L_0(vib) is integer valued and gives rise
> to the universal non-negative integer valued mass squared spectrum of
> string models (in suitable units).
>
> In present case i*epsilon is completely equivalent with
> the presence of i*epsilon in the propagators of relativistic quantum field
> theory: epsilon term guarantees that momentum spacetime integration
> over virtual momenta is performed correctly in case that one
> is forced to integrate over pole of propagator.

Note that the behavior of (p^2-L_0(vib)+i*epsilon)^{-1} when \epsilon goes to
0 is not like that of poles. It is much worth than essential singularities;
the singularities constitute a continuous set in the real line usually. This
is common when considereing any self-adjoint operators that describe physics.
Did you say the above with assuming they constitute a discrete set? Then can
you prove that?

>
> As far as I can understand this has nothing to do with time

This has a relation with time. See my note in [time 804].

Best wishes,
Hitoshi

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