Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 3 Oct 1999 19:36:06 +0300 (EET DST)
Dear Hitoshi,
I realized that there must I had not really yet understood the power
and elegance of your approach based on complex variables: I
took i*epsilon only as a dirty trick and did not consider
what projection to on mass shell states actually means.
I had not really formulated properly the inner
product, when one has projection to on mass shell states.
As I noticed earlier, inner product
gives 1/i*epsilon term and this formally diverges and this
suggess that one must define inner product by inclusing
integral around origin and apply residy calculus. The following
is probably more or less what you have been trying to tell me.
a) Inner product between on mass shell state and scattering state
can be defined in the following manner.
I write
|m> =|m_0> + |m(z)>
=|m_0> + (1/(L_0+iz))L_0(int)|m_0>.
z-->0 limit must be taken in suitable manner.
*****
b) It seems S-matrix can be written
S_(m,n)= <m_0|n> = <m_0|n_0> + (1/2pi)Int_C dz <m_0|P|n(z)>.
Here one has
|n(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.
This gives finite result from second term since the first term
since Int_C dz (1/L_0+iz) gives simply 2*pi for L_0 part of the
state.
****************
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 [<m_0|P{n(z)> +<m(z^*)|n_0>]
+ (1/2*pi)^2* Int_C Int_C dz dz* <m(z^*)|n(z)>.
The first two terms give opposite results which cancel each
other.
The third term gives term which vanishes if one has
(1/2*pi) Int_C dz L_0(int) P|m(z)>=0.
Thus the condition says that
** |m(z)> does not have pole at z=0**.
This condition is quite beautiful and purely geometric!
I hope it makes sense. This condition is equivalent with
the earlier conditions.
Best,
MP
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