Hitoshi Kitada (email@example.com)
Mon, 11 Oct 1999 14:45:03 +0900
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
Just a few comments below.
----- Original Message -----
From: Matti Pitkanen <firstname.lastname@example.org>
To: Hitoshi Kitada <email@example.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
> > 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.
> TGD however states are classical configuration space spinor fields and no
> second quantization occurs so that difficulties cannot be avoided.
> 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):
> Dear Hitoshi,
> Slight correction to the spontaneous digression.
> On Sun, 10 Oct 1999, Matti Pitkanen wrote:
> > On Sun, 10 Oct 1999, Hitoshi Kitada wrote:
> > [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.
returning to [time 928]:
> > 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
> > goes to the limit epsilon ->0, they would go outside the Hilbert space
> > 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
> > for |m(i0)> and |n(i0)> to be outside \HH. Then the inner product
> > goes to infinity as epsilon ->0 and S-matrix diverges. Thus one has to
> > consider "generalized" eigenfunctions or eigenvectors |m(i0> and |n(i0)>
> > 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.
> 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.
> 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
> 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.
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