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

*Mon, 11 Oct 1999 14:45:03 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 931] Re: [time 928] Re: [time 923] Unitarity"**Previous message:**Matti Pitkanen: "[time 929] Re: [time 928] Re: [time 923] Unitarity"**In reply to:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"**Next in thread:**Matti Pitkanen: "[time 931] Re: [time 928] Re: [time 923] Unitarity"

Dear Matti,

I have been rereading some parts of Prugovecky's book. Given so many

approaches by so various authors, one would well be free in choosing one's

favorite from those many approaches ;-)

As to your approach, if you adopt your definition of S matrix, it would work

for you. This may be sufficient, and it is time for me to shut my mouth (or

hands).

Just a few comments below.

Best wishes,

Hitoshi

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

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

To: Hitoshi Kitada <hitoshi@kitada.com>

Cc: <time@kitada.com>

Sent: Sunday, October 10, 1999 1:46 PM

Subject: [time 928] Re: [time 923] Unitarity

*>
*

*>
*

*> On Sun, 10 Oct 1999, Hitoshi Kitada wrote:
*

*>
*

*> > 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).
*

*>
*

*> Yes, the formal proof is just a modification of the "proof" in formal
*

*> scattering theory. And in case of TGD it fails because there are off mass
*

*> shell contributions in |m>. In second quantized QFT:s off mass shell
*

*> contributions are associated with quantum fields but not with states.
*

Yes, and such a formal proof would not work for QFT.

In

*> TGD however states are classical configuration space spinor fields and no
*

*> second quantization occurs so that difficulties cannot be avoided.
*

Maybe...

*> I am convinced that that the standard approach does not work and
*

*> V|m_1>=0 saves the situation and forces p-adics (sorry, I am so
*

*> enthusiastic that I tend to get dogmatic with my p-adics).
*

*>
*

*>
*

*>
*

The following argument in your digression is obvious (it is unnecessary to use

Green's function expression to get your result):

[time 929]

*>
*

*> Dear Hitoshi,
*

*>
*

*> Slight correction to the spontaneous digression.
*

*>
*

*>
*

*> On Sun, 10 Oct 1999, Matti Pitkanen wrote:
*

*>
*

*> >
*

*> > On Sun, 10 Oct 1999, Hitoshi Kitada wrote:
*

*> >
*

*> snip
*

*>
*

*>
*

*> > [Spontaneuous digression:
*

*> >
*

*> > It just occurred to me that S-matrix should be formulated
*

*> > in terms of the configuration counterpart of Green function G(r,r')
*

*> > at least in zero modes.
*

*> >
*

*> > a) In ordinary QM G(r,r',E=0) satifies
*

*> > H_0G(r,r',0)= -2*pidelta (r,r').
*

*> >
*

*> > b) The counterpart of matrix T in "position
*

*> > representation" seems to be promising object to study.
*

*> >
*

*> > T= X/(1+X) = (1/L_0+iepsilon)V /(1+X).
*

*> >
*

*> > T(x,y), that is T in "position representation" would be what I have
*

*> > earlier considered "scattering kernel".
*

*> >
*

*> >
*

*>
*

*> Here the previous version contained error. Let us try again!
*

*>
*

*> c) By acting to this by PL_0 one obtains
*

*>
*

*> PL_0T|m_0> =0 for any state |m_0>
*

*>
*

*> since the action of PL_0T on any state |m_0> gives
*

*>
*

*> P V/(1+X)|m_0> = PV|m>
*

*>
*

*> by using the definition of T.
*

*>
*

*> This vanishes because PL_0(tot)|m>=0
*

*> and gives PL_0(free)|m>+PV|m> =0 which in turn gives PV|m>=0.
*

*> VP|m_1>=0 IS NOT used as I claimed first.
*

*> Only Virasoro condition for L_0(tot).
*

*>
*

*> >
*

*> >
*

*> > d) PL_0 is analogous to nabla^2 in ordinary scattering theory
*

*> > and has purely geometric interpretation as a square
*

*> > of configuration space Dirac operator.
*

*> >
*

*> > e) In standard acattering theory one would have something like
*

*> >
*

*> > nabla^2 G(r,r',E)= -4*pi delta (r,r').
*

*> >
*

*> > for the Green function.
*

*> >
*

*> > f) The **absence of delta function on the right hand side of PL_0T=0**
*

*> > would reflect the fact that *all rows of T* have vanishing p-adic norm
*

*> > (is like unitary matrix but zero norm rows).
*

*> > PL_0T=0 for two-point function would give my p-adic cohomology
*

*> > geometrical meaning. In real context PL_0T=0 would presumably have no
*

*> > solutions (just like nabla^2 phi=0 has no bounded solutions in E^3.
*

*>
*

*>
*

*> g) Note that nonexistence of bounded solutions to nabla^2 Phi=0
*

*> in real context can be shown using inner product
*

*>
*

*> <Phi_1,Phi_2> = Int nabla Phi_1 *nabla Phi_1 dV
*

*>
*

*> The norm for square integrable solutions of nabla^2 Phi=0 would obviously
*

*> vanish since boundary terms would cancel in partial integration. The
*

*> vanishing of norm in real context shows that there are no
*

*> bounded square integrable solutions of nabla^2 Phi=0. In p-adic context
*

*> situation is different. Similar proof does not work since
*

*> zero norm of state does not imply its vanishing.
*

*>
*

*>
*

*> > h) It should be also noticed that the definition of p-adic delta
*

*> > function migth well be impossible since standard integration does
*

*> > not generalize to p-adic context. Therefore PL_0T=0 instead
*

*> > of equation containing delta function on right hand side is
*

*> > p-adically natural.
*

*> >
*

*> >
*

*> Best,
*

*>
*

*> MP
*

returning to [time 928]:

snip

*> > 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
*

*>
*

*> I understand that you are talking now about standard scattering theory.
*

*> In language of on mass shell states your statement is following:
*

*> S-matrix is trivial if off mass shell states cannot appear
*

*> as intermediate states in expansion of T. Their allowance however
*

*> extends Hilbert space and spoils the basic assumption
*

*> of formal scattering theory.
*

*>
*

*> I understand now quite well your worry about Hlbert spaces and I am
*

*> convinced that the presence of off mass shell contributions kills the
*

*> "proof" in case of TGD. There is no unitarity since scattering states
*

*> contain continuum contribution from off mass shell states.
*

*>
*

*> Already in QFT:s one ends up with diverging renormalization constants
*

*> and there is something sick with quantum field theories: I think
*

*> this sickness is basically this going outside of Hilbert space of
*

*> free states.
*

Only basically. The formalism of QFT is quite different from standard one.

*>
*

*>
*

*> In any case, I realized that my approach based on
*

*> |V|m_1>=0 is actually *not* equivalent with standard approach.
*

I understand if you adopt your definition of S matrix.

*> Notice
*

*> that I consider actually the *projections P|m>* instead of |m> and my
*

*> condition implies that states P|m> form orthonormal basis whereas
*

*> standard "proof" shows the orthonormality for states |m> rather than P|m>
*

*> and this is the origin of difficulties of QFT:s.
*

*>
*

*>
*

*> Of course, the troubles caused by iepsilon -->0 limit
*

*> might be present also in the modified approach dealing only with P|m>.
*

This problem might appear also in your definition.

snip

*> I am grateful for you criticism. It have helped me to see the
*

*> origin of trouble and I am indeed convinced that standard scattering
*

*> theory does not work neither in QFT context and even less in TGD.
*

Due to the difference of formalism of QFT, standard approach does not apply to

QFT, but TGD adopts a similar formalism to standard one and it might work for

TGD.

*> This sounds blasphemous, but I am convinced that V|m_1>=0
*

*> conditions with outgoing states defined as projections P|m> saves from
*

*> the troubles and makes theory rigorous in this respect "but"
*

*> forces p-adics.
*

It is your liberty to adopt your own approach. As well each of us has the

liberty to have our own favorite.

*>
*

*> Best,
*

*> MP
*

**Next message:**Matti Pitkanen: "[time 931] Re: [time 928] Re: [time 923] Unitarity"**Previous message:**Matti Pitkanen: "[time 929] Re: [time 928] Re: [time 923] Unitarity"**In reply to:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"**Next in thread:**Matti Pitkanen: "[time 931] Re: [time 928] Re: [time 923] Unitarity"

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