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

*Fri, 8 Oct 1999 23:24:15 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Ben Goertzel: "[time 911] RE: [time 910] Re: [time 909] About your proof of unitarity"**Previous message:**Matti Pitkanen: "[time 909] About your proof of unitarity"**Next in thread:**Ben Goertzel: "[time 911] RE: [time 910] Re: [time 909] About your proof of unitarity"

Dear Matti,

Comments are below.

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

Subject: [time 909] About your proof of unitarity

*>
*

*> Dear Matti,
*

*>
*

*> Rethinking about your proof, I found an alternative simple proof of
*

*> unitarity of S-matrix. Of course this is a proof on a formal level.
*

*>
*

*> [MP] I hardly dared to open your posting(;-) and only wondered what
*

*> kind of new devilish trick proving my poor S-matrix trivial you
*

*> have discovered this time! Have you read Hofstadter's book 'Goedel,
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*> Escher, Bach'? If not you should do it.
*

I am a specialist of Goedel's proof.

*> There are hilarious stories whose
*

*> basic theme is Goedel's theorem: every complete axiom system with
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*> sufficient complexity contains inconsistency. In one story
*

*> the great dream of Achilles is to build a record player which is complete.
*

*> When N:th generation record player is constructed,
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*> Tortoise gives for poor Achilles as a gift a record whose name happens
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*> to be "I am record not playable by record player of generation N". Every
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*> time Achilles manages to construct a new version
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*> of his record player the record brought by Tortoise manages to smash it
*

*> into pieces. So I will look at your proof with horror in my heart(;-).
*

*>
*

*>
*

*> The proof:
*

*>
*

*> Set
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*>
*

*> R_0(z)=(H_0-z)^{-1}, z: non-real, H_0=L_0(free), V=L_0(int),
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*>
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*> P = projection onto the eigenspace of H_0 with eigenvalue 0.
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*>
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*> Let m_0 in P\HH (=eigenspace of H_0 with eigenvalue 0). Then the
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*> scattering
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*> state m(z) satisfies (I omit the bracket notation)
*

*>
*

*> m(z)=m_0+m_1(z)=m_0-R_0(z)Vm(z).
*

*>
*

*> Thus
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*>
*

*> Pm(z)=Pm_0+Pm_1(z)=m_0+z^{-1}PVm(z). (1)
*

*>
*

*>
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*> [MP]
*

*> Isn't it dangerous to talk about P|m(z)> since the action
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*> of P itself is defined as integral over infinitesimal circle around z=0?
*

No, your definition of P by integral does not work.

*> You now how extremely tricky creature residue calculus is.
*

*>
*

*> Since I cannot afford too many record players, I am only willing
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*> to talk about P|m> defined as
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*>
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*> Int_C dz p(z)|m(z)
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*>
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*> with p(z)= (1/2pi)* (1/L_0-iz).
*

*> **************************
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*>
*

*>
*

*> [Hitoshi]
*

*> Therefore
*

*>
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*> PVm(z)=zPm_1(z). (2)
*

*> ***************
*

*>
*

*>
*

*> [MP] This looks ok but I am a somewhat worried about applying P to
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*> |m_1(z)>.
*

*>
*

*> [MP]
*

*> Your assumption is
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*>
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*> VPm_1(z)=0. (A)
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*>
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*> [MP] My assumption is actually weaker. P involves integration
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*> around small circle.
*

Your integral is not well-defined.

*>
*

*> [Hitoshi]
*

*> Thus (2) and (A) yield
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*>
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*> VPVm(z)=zVPm_1(z)=0, which means
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*>
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*> <m(z)|VPV|m(z)>=0,
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*>
*

*> hence
*

*>
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*> PVm(z)=0.
*

*>
*

*> [MP] This states that state PV|m(z)> has zero norm. In real
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*> context this implies PV|m(z)> =0.
*

No, P and V does not commute.

*> In p-adic context the
*

*> fact that state has vanishing p-adic norm does not
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*> however imply that state vanishes. In fact that rows of T
*

*> matrix have zero norm p-adically. So that this step
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*> is not allowed p-adically.
*

*> *******************
*

*>
*

*> [Hitoshi]
*

*> This and (2) imply
*

*>
*

*> Pm_1(z)=z^{-1}PVm(z)=0.
*

*>
*

*>
*

*> [MP] Here there is dangerous 1/z which goes to zero and the
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*> conclusion should be taken with grain of salt. In any case
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*> Pm_1(z)=0 leads to trivial S-matrix.
*

You need to treat 1/z and its limit Im z->0. You seem to assume the existence

of

lim_{Im z->0} (1-R_0(z)V)^{-1}.

But this is the main issue of the problem. The rest that you gave and I gave

are only miscellaneous arguments.

*>
*

*> [Hitoshi]
*

*> Thus by (1) we have
*

*>
*

*> Pm(z)=m_0.
*

*>
*

*> Therefore
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*>
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*> <m(z)|P|n(z)>=<m_0|n_0>: unitarity. (U)
*

*> **********
*

*>
*

*>
*

*> [MP] By the same proof one has
*

*>
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*> <m_0|n(z)> = <m_0|P|n(z)>= <m_0|n_0> so that S-matrix is trivial!
*

*> And my record player would be smashed into pieces again!
*

Maybe you have to begin your proof on the basis of sound mathematics.

The limit Im z->0 is the point. Others are just miscellaneous and trivial

things for mathematicians.

*>
*

*>
*

*> I think that the problem is that one should not include P|n(z)>
*

*> into the vocabulary but talk only about P|n> involving integration
*

*> over z.
*

Your formulae including n(z) in the integrand is different from what you want

to mean.

The p-adically weak point of your proof is that

*> zero norm for state does not imply vanishing of state in
*

*> p-adic context.
*

*> ******************
*

*>
*

*>
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*> Comment 1. As Im z -> 0,
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*>
*

*> m(z) = m_0 - R_0(z)Vm(z) = (1+R_0(z)V)^{-1}m_0
*

*>
*

*> would be outside the Hilbert space \HH. Then (U) might lose its meaning as
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*> Im
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*> z -> 0. This would require the introduction of some larger space \HH_-.
*

*>
*

*> Comment 2. The present formulation of yours uses P explicitly. Namely
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*> m=m(z)
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*> may be outside of P\HH. Thus it is free of my criticism in [time 882].
*

*>
*

*>
*

*> Best,
*

*>
*

*> MP
*

*>
*

*>
*

*>
*

*>
*

Physicists look like to talk much. Mathematicians just talk about what they

knows without making redundant comments.

Best wishes,

Hitoshi

**Next message:**Ben Goertzel: "[time 911] RE: [time 910] Re: [time 909] About your proof of unitarity"**Previous message:**Matti Pitkanen: "[time 909] About your proof of unitarity"**Next in thread:**Ben Goertzel: "[time 911] RE: [time 910] Re: [time 909] About your proof of unitarity"

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