**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sun, 3 Oct 1999 19:36:06 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 876]"**Previous message:**Matti Pitkanen: "[time 874] Re: [time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 862] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"

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

**Next message:**Matti Pitkanen: "[time 876]"**Previous message:**Matti Pitkanen: "[time 874] Re: [time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**In reply to:**Hitoshi Kitada: "[time 871] Re: [time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re:[time 847]Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 862] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"

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