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

*Fri, 8 Oct 1999 16:22:28 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 910] Re: [time 909] About your proof of unitarity"**Previous message:**Matti Pitkanen: "[time 908] Re: [time 907] A proof of unitarity"**Next in thread:**Hitoshi Kitada: "[time 910] Re: [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,

Escher, Bach'? If not you should do it. There are hilarious stories whose

basic theme is Goedel's theorem: every complete axiom system with

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,

Tortoise gives for poor Achilles as a gift a record whose name happens

to be "I am record not playable by record player of generation N". Every

time Achilles manages to construct a new version

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

R_0(z)=(H_0-z)^{-1}, z: non-real, H_0=L_0(free), V=L_0(int),

P = projection onto the eigenspace of H_0 with eigenvalue 0.

Let m_0 in P\HH (=eigenspace of H_0 with eigenvalue 0). Then the

scattering

state m(z) satisfies (I omit the bracket notation)

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

Thus

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

[MP]

Isn't it dangerous to talk about P|m(z)> since the action

of P itself is defined as integral over infinitesimal circle around z=0?

You now how extremely tricky creature residue calculus is.

Since I cannot afford too many record players, I am only willing

to talk about P|m> defined as

Int_C dz p(z)|m(z)

with p(z)= (1/2pi)* (1/L_0-iz).

**************************

[Hitoshi]

Therefore

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

***************

[MP] This looks ok but I am a somewhat worried about applying P to

|m_1(z)>.

[MP]

Your assumption is

VPm_1(z)=0. (A)

[MP] My assumption is actually weaker. P involves integration

around small circle.

[Hitoshi]

Thus (2) and (A) yield

VPVm(z)=zVPm_1(z)=0, which means

<m(z)|VPV|m(z)>=0,

hence

PVm(z)=0.

[MP] This states that state PV|m(z)> has zero norm. In real

context this implies PV|m(z)> =0. In p-adic context the

fact that state has vanishing p-adic norm does not

however imply that state vanishes. In fact that rows of T

matrix have zero norm p-adically. So that this step

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

conclusion should be taken with grain of salt. In any case

Pm_1(z)=0 leads to trivial S-matrix.

[Hitoshi]

Thus by (1) we have

Pm(z)=m_0.

Therefore

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

**********

[MP] By the same proof one has

<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!

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. The p-adically weak point of your proof is that

zero norm for state does not imply vanishing of state in

p-adic context.

******************

Comment 1. As Im z -> 0,

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

Im

z -> 0. This would require the introduction of some larger space \HH_-.

Comment 2. The present formulation of yours uses P explicitly. Namely

m=m(z)

may be outside of P\HH. Thus it is free of my criticism in [time 882].

Best,

MP

**Next message:**Hitoshi Kitada: "[time 910] Re: [time 909] About your proof of unitarity"**Previous message:**Matti Pitkanen: "[time 908] Re: [time 907] A proof of unitarity"**Next in thread:**Hitoshi Kitada: "[time 910] Re: [time 909] About your proof of unitarity"

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