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

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Ben Goertzel: "[time 881] Matti's Theory"**Previous message:**Matti Pitkanen: "[time 879] Re: [time 877] Re: Your assumption"**In reply to:**Hitoshi Kitada: "[time 877] Re: Your assumption"**Next in thread:**Matti Pitkanen: "[time 885] Re: [time 879] Re: [time 877] Re: Your assumption"

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:
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*>
*

*> > Dear Matti,
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*> >
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*> > Comments are below.
*

*> >
*

*> > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
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*> >
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*> > > [time 876] Message for time
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*> > > Sender: owner-time
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*> > > Precedence: bulk
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*> > >
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*> > >
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*> > >
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*> > > Dear Hitoshi,
*

*> > >
*

*> > > The previous version was contained still errors. The following formulas
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*> > > 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)
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*> > > is nothing but representing the projector P in elegant manner.
*

*> > > In any case, I want to represent the correct formulas.
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*> > >
*

*> > > a) Inner product between on mass shell state and scattering state
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*> > > can be defined in the following manner.
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*> > >
*

*> > > I write
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*> > >
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*> > > |m(z)> =|m_0> + |m_1(z)>
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*> > >
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*> > > =|m_0> + (1/(L_0+iz))L_0(int)|m_0>.
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*> >
*

*> > Is L_0 = L_0(free) or L_0(tot)?
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*> >
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*> > >
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*> > > z-->0 limit must be taken in suitable manner.
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*> > >
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*> > >
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*> > > b) Projector P to the on mass shell states is represented as
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*> > >
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*> > > P= (1/2pi)Int_C dz/(L_0+iz).
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*> > >
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*> > >
*

*> > > b) It seems S-matrix can be written
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*> > >
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*> > > S_(m,n)= <m_0|n> = (1/2pi)<m_0|Int_C (dz/(L_0+iz)|n(z)>
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*> > > =<m_0|n_0> + (1/2pi) <m_0|Int_C(1/(L_0+iz)|n_1(z)>.
*

*> >
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*> > 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.
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*> >
*

*> > 2) How is your equality in the second equation above:
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*> >
*

*> > |n> = (1/2pi) Int_C (dz/(L_0+iz)|n(z)>
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*> >
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*> > (= 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)?
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*>
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*>
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*>
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*> >
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*> > >
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*> > >
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*> > > Here one has
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*> > >
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*> > > |n_1(z)> = sum_(n>0) X^n |n_0>,
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*> > >
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*> > > X(z)= (1/L_0+iz) L_0(int).
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*> > >
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*> > > C is small curve encircling origin and Int is integral over this.
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*> > > The integral gives nonvanishing result if there is pole contribution.
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*> > > This requires that the Laurent expansion of inner product <m_0|n_1(z)>
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*> > > contains
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*> > > constant term. This formulation adds nothing to the previous: it only
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*> > > makes
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*> > > it more elegant and rigorous.
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*> > >
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*> > >
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*> > > c) Consider now unitarity conditions.
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*> > >
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*> > > I must find under what conditions one has <m|n>= <m_0|n_0>:
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*> > >
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*> > > <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)
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*> > > <m_1(z^*)|n_1(z)>.
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*> > >
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*> > >
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*> > > The first two terms give opposite results which cancel each
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*> > > other.
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*> > >
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*> > >
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*> > > The third term vanishes if one has
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*> > >
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*> > >
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*> > > (1/2*pi) Int_C dz L_0(int)(1/L_0+iz) |m(z)>=0.
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*> > >
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*> > > Thus the condition says that the residy of the pole of |m(z)>
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*> > > at z=0 is annihilated by L_0(int). This condition is equivalent with
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*> > > the earlier condition so that nothing new is introduced: Int_C...
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*> > > is only an elegant manner to represent projection operator.
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*> > >
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*> > >
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*> > > With Best,
*

*> > > MP
*

*> > >
*

*> > >
*

*> > >
*

*> > >
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*> >
*

*> > 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>.
*

*> >
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*>
*

*>
*

*> No. I am just expressing projector to the space of "free states"
*

*> defined as stats annihilated by L_o(free)==L_0 above. What
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*> 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
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*> only after that it is possible to discuss delicaciess
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*> I tend to believe tha our are right that the condition L_0(int)|m>=0
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*> is too strong and wrong. This looks obvious in more elegant formulation.
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*>
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*>
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*>
*

*> I am just studying a modified condition
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*>
*

*> L_0(int)|m_1>=0
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*>
*

*> where |m_1> is the genuine scattering contribution defined
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*> above. The condition
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*> differs from the earlier one only in that |m>=|m_0>+|m_1>
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*> is replaced by |m_1>. It SEEMS that unitarity is satisfied
*

*> and your counter argument leading to the triviality of S-matrix
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*> does not bite since the condition implies
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*> L_0(int)||m> =L_0(int)|m_0>.
*

*>
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*>
*

*> As a matter fact, I noticed that I must have ended the earlier condition
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*> by just failing to notice that the sum
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*>
*

*> |m_1>= sum_(n=0)X^n |m_0>
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*>
*

*> X= (1/L_0+iz) L_0(int)
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*>
*

*> **starts from n=1, not n=0**!! Rather
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*> stupid error which has generated a lot of swet and heat!
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*>
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*>
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*>
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*>
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*>
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*> Best,
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*> MP
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*>
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*>
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*>
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*>
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*>
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*>
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Best wishes,

Hitoshi

**Next message:**Ben Goertzel: "[time 881] Matti's Theory"**Previous message:**Matti Pitkanen: "[time 879] Re: [time 877] Re: Your assumption"**In reply to:**Hitoshi Kitada: "[time 877] Re: Your assumption"**Next in thread:**Matti Pitkanen: "[time 885] Re: [time 879] Re: [time 877] Re: Your assumption"

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