[time 885] Re: [time 879] Re: [time 877] Re: Your assumption


Matti Pitkanen (matpitka@pcu.helsinki.fi)
Wed, 6 Oct 1999 16:50:44 +0300 (EET DST)


Dear Hitoshi,

apologies for slow response: computers were down for days.
Wonderful be free of the curse of 60 emails at the morning!
I have been working with unitarity last days and
dare claim that I understand it now.
More about this in another posting.

Best
MP

Below short comment about your objection.

On Mon, 4 Oct 1999, Hitoshi Kitada wrote:

> Dear Matt,
>
> I do not have time to share for inexact formulae and arguments, but I tried to
> make comments as far as I understand your notation:
>
> ----- Original Message -----
> From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
> To: Hitoshi Kitada <hitoshi@kitada.com>
> Cc: <time@kitada.com>
> 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,
> > >
> > > Comments are below.
> > >
> > > 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.

L_0(free) indeed has discrete on mass shell spectrum. The on mass
shell states having continuous spectrum are a source of
possible problems. Continuous spectrum but since integration
goes to point there are good reason to believe that at
the limit of infinitisemal circle contribution vanishes.
This point deserves certainly thought but is
at this stage luxury.

>
>
> >
> > 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
> > and your counter argument leading to the triviality of S-matrix
> > 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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