[time 880] Re: [time 879] Re: [time 877] Re: Your assumption

Mon, 4 Oct 1999 18:40:00 +0900

Dear Matt,

I do not have time to share for inexact formulae and arguments, but I tried to

----- Original Message -----
From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
Sent: Monday, October 04, 1999 6:19 PM
Subject: [time 879] Re: [time 877] Re: Your assumption

>
>
> On Mon, 4 Oct 1999, Hitoshi Kitada wrote:
>
> > Dear Matti,
> >
> >
> > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
> >
> > > [time 876] Message for time
> > > Sender: owner-time
> > > Precedence: bulk
> > >
> > >
> > >
> > > Dear Hitoshi,
> > >
> > > The previous version was contained still errors. The following formulas
> > > provide more correct version. I bet that this is totally trivial for you
> > > and also I realized that the introduction of P=(1/2*pi)Int_cdz(1/L_0+iz)
> > > is nothing but representing the projector P in elegant manner.
> > > In any case, I want to represent the correct formulas.
> > >
> > > a) Inner product between on mass shell state and scattering state
> > > can be defined in the following manner.
> > >
> > > I write
> > >
> > > |m(z)> =|m_0> + |m_1(z)>
> > >
> > > =|m_0> + (1/(L_0+iz))L_0(int)|m_0>.
> >
> > Is L_0 = L_0(free) or L_0(tot)?
> >
> > >
> > > z-->0 limit must be taken in suitable manner.
> > >
> > >
> > > b) Projector P to the on mass shell states is represented as
> > >
> > > P= (1/2pi)Int_C dz/(L_0+iz).
> > >
> > >
> > > b) It seems S-matrix can be written
> > >
> > > S_(m,n)= <m_0|n> = (1/2pi)<m_0|Int_C (dz/(L_0+iz)|n(z)>
> > > =<m_0|n_0> + (1/2pi) <m_0|Int_C(1/(L_0+iz)|n_1(z)>.
> >
> > 1) This (i.e. S_(m,n)= <m_0|n>) is not S-matrix, but wave operator
expressed
> > on energy shell, as I stated several times.
> >
> > 2) How is your equality in the second equation above:
> >
> > |n> = (1/2pi) Int_C (dz/(L_0+iz)|n(z)>
> >
> > (= P|m_0> + (1/2pi) Int_C (dz/(L_0+iz)(1/(L_0+iz))L_0(int)|n_0> (by a))

Ok, this is equal to

P|m_0> + P |n_0>.

> >
> >
> > proved from a)?
>
>
>
> >
> > >
> > >
> > > Here one has
> > >
> > > |n_1(z)> = sum_(n>0) X^n |n_0>,
> > >
> > > X(z)= (1/L_0+iz) L_0(int).
> > >
> > > C is small curve encircling origin and Int is integral over this.
> > > The integral gives nonvanishing result if there is pole contribution.
> > > This requires that the Laurent expansion of inner product <m_0|n_1(z)>
> > > contains
> > > constant term. This formulation adds nothing to the previous: it only
> > > makes
> > > it more elegant and rigorous.
> > >
> > >
> > > c) Consider now unitarity conditions.
> > >
> > > I must find under what conditions one has <m|n>= <m_0|n_0>:
> > >
> > > <m,n> = <m_0|n_0> + (1/2*pi)* Int_C (1/L_0+iz) [<m_0|n(z)>
+<m(z^*)|n_0>]
> > >
> > > + (1/2*pi)^2* Int_C Int_C dz* dz (1/L_0-iz^*) (1/L_0+iz)
> > > <m_1(z^*)|n_1(z)>.
> > >
> > >
> > > The first two terms give opposite results which cancel each
> > > other.
> > >
> > >
> > > The third term vanishes if one has
> > >
> > >
> > > (1/2*pi) Int_C dz L_0(int)(1/L_0+iz) |m(z)>=0.
> > >
> > > Thus the condition says that the residy of the pole of |m(z)>
> > > at z=0 is annihilated by L_0(int). This condition is equivalent with
> > > the earlier condition so that nothing new is introduced: Int_C...
> > > is only an elegant manner to represent projection operator.
> > >
> > >
> > > With Best,
> > > MP
> > >
> > >
> > >
> > >
> >
> > It seems that you start with the space of |n> and the free space of |n_0>.
> > If 1/(L_0+iz) is a resolvent, L_0 (=L_0(free) or L_(tot)?) must be defined
as
> > an operator from \HH into itself. What is your Hilbert space \HH? It must
not
> > be the space of the free space P\HH corresponding to zero energy nor the
space
> > of scattering states |n>.
> >
>
>
> No. I am just expressing projector to the space of "free states"
> defined as stats annihilated by L_o(free)==L_0 above. What
> makes this representation useful is its elegance.

What I meant above is that L_0(free) has continuous spectra if super Virasoro
conditions are not supposed. (Only before making Virasoro assumption, the
operator 1/(L_0(free)+iz) has meaning.) Then 0 is not an isolated spectrum of
L_0(free) and one cannot perform the integration around origin without passing
the spectra of L_0(free). Thus the integral that you asserts to define P is
not well-defined.

>
> The basic problem at this stage is to find proper formulation
> for the condition guaranteing unitarity formally. I think that
> only after that it is possible to discuss delicaciess
> I tend to believe tha our are right that the condition L_0(int)|m>=0
> is too strong and wrong. This looks obvious in more elegant formulation.
>
>
>
> I am just studying a modified condition
>
> L_0(int)|m_1>=0
>
> where |m_1> is the genuine scattering contribution defined
> above. The condition
> differs from the earlier one only in that |m>=|m_0>+|m_1>
> is replaced by |m_1>. It SEEMS that unitarity is satisfied
> does not bite since the condition implies
> L_0(int)||m> =L_0(int)|m_0>.
>
>
> As a matter fact, I noticed that I must have ended the earlier condition
> by just failing to notice that the sum
>
> |m_1>= sum_(n=0)X^n |m_0>
>
> X= (1/L_0+iz) L_0(int)
>
> **starts from n=1, not n=0**!! Rather
> stupid error which has generated a lot of swet and heat!
>
>
>
>
>
> Best,
> MP
>
>
>
>
>
>

Best wishes,
Hitoshi

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