[time 882] Re: [time 881] Matti's Theory

Hitoshi Kitada (hitoshi@kitada.com)
Tue, 5 Oct 1999 03:13:49 +0900

Dear Ben,

Ben Goertzel <ben@goertzel.org> wrote:

Subject: [time 881] Matti's Theory

> Hitoshi,
> For those of us who are not following all the details of your discussion
> with Matti
> (due to lack of time, in my case), it might be useful if you could briefly
> summarize
> your understanding of Matti's theory at this point, and how it relates to
> your own..

Oh, this would be a good question for all of us!

The following is my understanding; correct me if wrong, Matti.

My interest in Matti's theory began when he started speaking about the
propagator(?) U. According to his definition, he calls the following U a
scattering operator: U Psi_0 = Psi, where Psi and Psi_0 satisfy the relation

Psi = Psi_0 - lim_{z -> 0+i0} R_0(z)VPsi. (1)

Here I use my notation for his:

H = L_0(tot): total Hamiltonian for the universe,

H_0 = L_0(free) = \sum_{n} L_0(n): free Hamiltonian,

L_0(n) = p^2(n) - L_0(vib,n),

V = H - H_0 = L_0(int): interaction,

R_0(z) = (H_0-z)^{-1}, R(z) = (H-z)^{-1}, z: complex number.

Of course these operators must be defined in a suitable Hilbert space \HH, and
H and H_0 must be selfadjoint in \HH.

He assumes in the first place the super Virasoro conditions on the state of
the universe Psi and Psi_0:

H Psi = 0, H_0 Psi_0 = 0. (Vi)

He calls the totalities of such Psi and Psi_0, the scattering space \HH_s and
the free space \HH_0, respectively.

According to my understanding, he assumes the condition (Vi) when he speaks
about the equation (1). I.e. He seems to assume Psi is in \HH_s and Psi_0 is
in \HH_0 in equation (1).

Under this assumption (Vi), one has the following difficulty:

Let us first consider an approximate form of (1):

Psi = Psi_0 - R_0(z)VPsi, z = ie, e > 0, Psi in \HH_s, Psi_0 in \HH_0. (2)

(1) is a limit case of (2).

By a direct calculation, one has

R(z) - R_0(z) = -R(z)VR_0(z) = -R_0(z)VR(z). (3)

If we set

f = (H_0-z)Psi_0,

(3) gives

R(z)f = R_0(z)f - R_0(z)VR(z)f = Psi_0 - R_0(z)VR(z)f.

Thus the solution of (2) is given uniquely by

Psi(z) = R(z)f = R(z)(H_0-z)Psi_0, (4)

if such a Psi(z) exists in \HH_s.

This gives

(H-z)Psi(z) = (H_0-z)Psi_0. (5)

(Vi) yields

-zPsi(z) = -zPsi_0. (6)


Psi(z) = Psi_0.

The solution Psi of (1) may be given by

Psi = lim{e->+0} Psi(z) = Psi_0

with neglecting any topological considerations. Thus

U = I. (7)

There is no scattering.

The difficulty here looks tricky. If one stops thinking that Psi(z) is in
\HH_s (as I suggested in the earlier stage) and assume Psi(z) is in \HH, then
(5) yields just

(H-z)Psi(z) = -zPsi_0,

and taking limit e->+0 might not give (7). But in this case (as well as in the
above) one needs to consider topology in which the convergence occurs. There
is a possibility (actuality in some cases) that Psi(z) goes outside \HH.

In scattering theory, these problems are considered in the following way:

One starts with not assuming any conditions like (Vi) and considers in \HH.

The equation (1) in general context is

Psi = Psi_0 - R_0(z)VPsi, z = E+ie, e>0, E: real (8)

with Psi and Psi_0 hopefully being in \HH when e->+0. The solution is as in
the above

Psi = R(z)(H_0-z)Psi_0

= Psi_0 - R(z)VPsi_0. (9)

The limits R(E+i0), R_0(E+i0) as e->+0 do not belong to the space of bounded
operators in \HH usually. Instead they (in some cases) belong to

B(\HH_+, \HH_-) = {bounded operators from \HH_+ into \HH_-},


\HH_+ \subset \HH \subset \HH_-

with the topological relation

the norm of \HH_+ > the norm of \HH > the norm of \HH_-.

In some of these cases, V acts as a bounded operator from \HH_- into \HH_+.

R(E+i0)V: \HH_- --> \HH_+ --> \HH_-
                V R(E+i0)

is a bounded operator from \HH_- into \HH_-. Thus if Psi_0 is in \HH_-, then
(9) gives a solution Psi in \HH_-.

If this machinery works with Matti's theory, it would give a solution Psi in
\HH_- for Psi_0 in \HH_-. I.e. one can add general Virasoro conditions:
(H-E)Psi=(H_0-E)Psi_0 after one gets a solution for general case.

This method is, however, sometimes difficult and does not work.

An alternative method is to consider time dependent correspondent:

W_+ = lim_{t->\infty} exp(itH) exp(-itH_0).

This is called wave operator usually, but Matti may call this scattering
operator, maybe because he seems to consider a universe with the beginning but
without the end (I am not sure).

This would go to ramifications and I stop technical details here. But I do not
understand why Matti does not like to consider any time dependent formulation.
In principle, these two methods are equivalent. As far as one considers a pair
(H,H_0) of selfadjoint operators in a Hilbert space \HH, there is no reason to
choose a particular method.

The relation of Matti's theory with mine is not clear yet. Maybe I have to
study symmetries in physics in order to include weak and strong forces, but I
could not have found time for my own two years or so.

> ben

Best wishes,

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