Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 4 Oct 1999 12:19:24 +0300 (EET DST)
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))
>
>
> 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.
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
This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:40:46 JST