[time 923] Unitarity

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 9 Oct 1999 18:07:42 +0300 (EET DST)

Dear Hitoshi,

I began to ponder your comment and looked formal scattering theory again
and realized that unitarity proof for S-matrix formally generalizes to
TGD case.

Denote H_0== L_0(free), H==L_0(tot)= L_0(free)+ L_0(int) and

a) One has the basic equation

|m> = |m_0> - 1/(H_0+iepsilon) V |m> (1)

b) One can multiply this equation by H_0+iepsilon and move terms
proportional to |m> to the left hand side to
get (H+iepsilon)|m> right hand side. Left hand side gives
(H_0 +V)|m_0> -V|m_0> by adding and subtracting V|m_0>.
Solving |m> one obtains

|m> = |m_0> -1/(H+ iepsilon) V|m_0> (2)

c) One can also solve |m_0> from the first equation

|m_0> = |m> + 1/(H_0+iepsilon) V|m> (3)

Consider now the matrix element <m|n>: one must show that this
is <m_0|n_0> in order to prove unitarity.

a) Express first <m| in terms of <m_0| using (2)

<m|n> = <m_0|n> +<m_0|V*1/(-H-iepsilon)|n> (4)

b) One can use the fact that H annihilates |n>
to remove 1/(L_0(tot).. term in front of V and replace
the H=0 by -H_0=0 (due to inner product with <m_0|)
 to get

<m|n> = <m_0|n> -<m_0|1/(H_0-iepsilon)V|n>

c) But by equation (3) the state proportional to |n> is in fact |n_0>
and one has

<m|n> =<m_0|n_0>.

Thus one has formal unitarity. The calculation is extremely tricky.

What do you think?



I tend to believe that the condition V|m_1>=0 is correct condition
since it leads to p-adics and is consistent with quantum criticality
even if it would not be needed for unitarity.

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