Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 4 Oct 1999 07:07:24 +0300 (EET DST)
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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>.
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)>.
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
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