**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Sun, 10 Oct 1999 04:31:26 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"**Previous message:**Stephen P. King: "[time 926] Re: [time 913] Mind over Matter"**In reply to:**Ben Goertzel: "[time 913] RE: [time 912] Re: [time 911] RE: [time 910] Re: [time 909] About your proof of unitarity"**Next in thread:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"

Dear Matti,

I looked through your "formal" proof. I agree it is correct as far as

*epsilon>0. And it is free of the criticism that you did not treat the inner
*

product <m|n> of the full states <m| and |n> before (i.e. you considered

<m|P|n> before).

I think physicists might say OK since many of them seem not aware of

mathematics.

My mathematical interest is in how the states <m| and |n> behave as

epsilon ->0. This is necessary to be considered because <m|=<m(z)| and

|n>=|n(z)> actually depend on z=i\epsilon and |m(z)> satisfies

(H+z)|m(z)>=(H_0+z)|m_0>=z|m_0> by your equation (2), hence

H|m(z)>=-z(|m(z)>-|m_0>) which is not necessarily 0 when epsilon>0. If one

wants to have equation H|m(i\epsilon)>=0, one needs to let epsilon ->0. If one

goes to the limit epsilon ->0, they would go outside the Hilbert space \HH.

For if the scattering states |m(i0)> and |n(i0)> remain in \HH, they are

genuine eigenstates of H and scattering does NOT occur. Thus it is necessary

for |m(i0)> and |n(i0)> to be outside \HH. Then the inner product <m(z)|n(z)>

goes to infinity as epsilon ->0 and S-matrix diverges. Thus one has to

consider "generalized" eigenfunctions or eigenvectors |m(i0> and |n(i0)> which

are characterized by the conditions

H|m(i0)>=0, and |m(i0)> does not belong to the Hilbert space \HH.

"Generalized" means that they are outside \HH.

In this respect, formal proof does not give the unitarity.

There will be a way to avoid this difficulty. A physicist Prugovecky whom

Stephen referred to was researching scattering theory when I was a graduate

student (probably he might have been a graduate student too or not long later

than that). As someone whom Stephen referred to said, Prugovecky is aware of

the difficulty of this sort, and uses Gel'fand triple to avoid it. This is

understandable recalling his career in scattering theory. As this example

suggests, physicists could be more careful in their treatment of divergences.

Renormalization technique would not give logical solutions but it could be

replaced by more mathematical arguments, which could be the shortest way to

get physicists' dreams. "Blind mathematicians" could give better help to

physicists than mathematical dreamers could, although personally I like to

remain a dreamer myself. This time I made criticisms on your proofs with a

dare to be away from the standpoint of a dreamer. This was because there

seemed a possibility that a rigorous proof without divergence might have been

gotten if some push was given. I hope you would not take my criticisms as

something like attempts to destroy your proofs.

Best wishes,

Hitoshi

----- Original Message -----

From: Matti Pitkanen <matpitka@pcu.helsinki.fi>

To: <time@kitada.com>

Sent: Sunday, October 10, 1999 12:07 AM

Subject: [time 923] Unitarity

*>
*

*>
*

*> 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
*

*> V=L_0(int).
*

*>
*

*>
*

*> a) One has the basic equation
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*>
*

*> |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
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*> 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|)
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*> to get
*

*>
*

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

[Hitoshi]

Here not -iepsilon, but +iepsilon.

*> 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?
*

*>
*

*> Best,
*

*> MP
*

*>
*

*>
*

*>
*

*> P.S
*

*>
*

*> 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.
*

*>
*

*>
*

**Next message:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"**Previous message:**Stephen P. King: "[time 926] Re: [time 913] Mind over Matter"**In reply to:**Ben Goertzel: "[time 913] RE: [time 912] Re: [time 911] RE: [time 910] Re: [time 909] About your proof of unitarity"**Next in thread:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"

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