Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 10 Oct 1999 07:46:18 +0300 (EET DST)
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. In
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).
[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".
c) By acting to this by L_0 one obtains
L_0T=0 when applied to states |m_0>.
since the action of L_0T on any state P|m_1> gives V|m_1>=0.
d) L_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 L_0T=0**
would reflect the fact that *all rows of T* have vanishing p-adic norm
(is like unitary matrix but zero norm rows).
L_0T=0 for two-point function would give my p-adic cohomology
geometrical meaning. In real context L_0T=0 would presumably have no
solutions (just like nabla^2 phi=0 has no bounded solutions in E^3.]
g) 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 L_0T=0 instead
of equation containing delta function on right hand side is
p-adically natural.
>
> I think physicists might say OK since many of them seem not aware of
> mathematics.
>
Physics become aware of mathematics in their own way(;-)!
> 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.
In any case, I realized that my approach based on
|V|m_1>=0 is actually *not* equivalent with standard approach. 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>.
>
> H|m(i0)>=0, and |m(i0)> does not belong to the Hilbert space \HH.
>
> "Generalized" means that they are outside \HH.
>
OK. I understand.
> 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.
>
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.
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.
Best,
MP
>
>
> >
> >
> > 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
> >
> > |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
> > 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|)
> > 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.
> >
> >
>
>
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