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

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 808] Re: [time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**Previous message:**Hitoshi Kitada: "[time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**In reply to:**Matti Pitkanen: "[time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"**Next in thread:**Hitoshi Kitada: "[time 802] Re: [time 799] Still about construction of U"

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
*

*> > >non-bound states for free and interacting Hamiltonians
*

*> > >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

**Next message:**Matti Pitkanen: "[time 808] Re: [time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**Previous message:**Hitoshi Kitada: "[time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**In reply to:**Matti Pitkanen: "[time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"**Next in thread:**Hitoshi Kitada: "[time 802] Re: [time 799] Still about construction of U"

*
This archive was generated by hypermail 2.0b3
on Sat Oct 16 1999 - 00:36:41 JST
*