[time 759] Re: [time 756] Re: [time 749] Re: [time 748] About L_0: reply to Hitoshi

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 14 Sep 1999 08:27:47 +0300 (EET DST)

On Mon, 13 Sep 1999, Hitoshi Kitada wrote:

> Dear Matti,
> I hope you would respond to the following questions of mine:
> ----- Original Message -----
> From: Hitoshi Kitada <hitoshi@kitada.com>
> To: <time@kitada.com>
> Sent: Sunday, September 12, 1999 12:52 AM
> Subject: [time 749] Re: [time 748] About L_0: reply to Hitoshi
> > Dear Matti,
> >
> > I have simple questions before going further.
> >
> > What is z in your explanation? Is it the third spatial coordinate,
> > constituting the three dimensional spatial coordinates (x,y,z) together with
> > x_T=(x,y)? And what is Psi? Is Psi a wave function of the total universe in
> > your context? If so, it seems not uniquely determined by the equation

Psi is configuration space spinor field describing the state
of the Universe.

x,y,z refer to Minkowski coordinates.
Directions of classical four-momentum and classical spin
associated with 3-surfaces (by classical conservation laws
for Kaehler action) define t,z uniquely as coordinates in
directions of momentum and spin. (x,y) is determined
only modulo rotation. This is enought to construct
p-adic counterpart of real theory in GCI manner.

> >
> > L_0(tot)Psi=0.
> >
> > If Psi is not unique, what universe are you speaking of? Or does this
> equation
> > hold for any state Psi of the universe? Also do you speak of a universe
> having
> > matter inside it? How are the interactions between matters included?
> >
> > Also, are L_0(tot) and L_0 the same or different operators? And * in your
> > equation
> >
> > i(d/dt)*Psi = [p_T^2/P -L_0/P]Psi==L_0* Psi
> >
> > a convolution or something else?

There are two mutually consistent interpretations.

a) L_0(tot)Psi=0 (L_n(tot)Psi=0 more generally)
states Super Virasoro invariance of physical states.

b) L_0(tot) can be regarded as infinite-dimensional counterpart
of 3-dimensional spinor Laplacian, square of Dirac operator.
t in equation is purely group theoretical parameter, it is
not spacetime coordinate. p_k are translation generators
of the algebra of Diff^4 invariant translations.

Of course there is a lot to do before I can say
that I really understands all the details of the construction.

What I have done is to study of Super Virasoro conditions on
*single particle level*: deduced general view about spectrum
of massless particles and calculated the masses of
particles resulting from p-adic thermodynamics.

What should be done is to look how L_(tot)Psi=0 condition
for entire universe relates to corresponding approximate conditions
(which must hold true apart from interaction terms) for subsystems. This
is something, which I have not actually done.


Appendix: Could one construct Schrodinger equation by 'Haniltonizing'
Dirac equation in configuration space

It is not at all obvious whether the proposed manner
to end up with Schrodinger type equation is the only
one. One could also consider the possibility that
Hamiltonian form for Dirac equation in configuration
space might be the correct approach instead of
Hamiltonian form for its square.

Just to clarify my own thoughts about what
is involved I decided to collect some arguments
showing why the reduction to Hamiltonian form of Dirac
does not work,

a) L_0(tot) represents in well defined
sense square of configuration space Dirac operator.

b) This would suggest that one could perhaps consider
the Dirac equation rather than its square. Ordinary
Dirac equation allows transformation to Hamiltonian form

idPsi/dt= H_DPsi

and similar trick might work now.

It seems however that this is not the case. The
reason is that there are *two* possibilities
for the Dirac operator corresponding to Neveu-Scwartz and
Ramond represntations of Super Virasoro. One would
be forced to select between either of these possibilities
and this does not seem to make sense.

To see what would happen it is useful to look what would happen
if one tried this ansatz.

1. Ramond<--> leptons, NS<--> quarks

Ramond representation correspond to super generators G_n, n *integer*
linear in leptonic oscillator operators and
NS representation corresponds to generators G_r, r *half odd integer*
linear in quark oscillator operators (NS ).
One can say that there are two nonequivalent representations
for configuration space gamma matrices: in
terms of quark oscillator operators (NS) and in terms
of leptonic oscillator operators (Ramond).

2. L_0(tot) as square of Ramond G?

a) In case of Ramond representation G_n corresponds
to n=integer and one has

L_0(tot)= G_0(tot)G_0(tot)^dagger

where Super Virasoro generator G_0 is generalization of Dirac

p_kgamma^k --> p_kgamma^k + ....,

where +... acts in vibrational degrees of freedom of 3-surface
and p_k acts in cm degrees of freedom. At pointlike
limit one would obtain just p_kgamma^k, the ordinary Dirac operator.
G_0 is hermitian in string models but in TGD it is non-Hermitian
(being linear in leptonic oscillator operators).

3. L_0(tot) as square of NS G?

In case of NS representation G_r corresponds to r= half odd integer.

L_0(tot)=G_(1/2)G_(-1/2)^dagger+ h.c

4. L_0(tot) as sum of square of NS and Ramond G:s

Actually one can construct L_0(tot) as sum of
Ramond and NS contributions.

5. Could one reduce everything to Dirac equation?

The conditions G_0 Psi=0 and G_1/2Psi=0 are analogous
to Dirac equation. These conditions are satisfied
by single particle states. The first guess would
that these equations are the basic equation for the
time development but it seems that this is not the
case. The problem is that it seems that G_0Psi=0
or G_1/2 Psi=0 can hold true only for asymptotically
and only for single particle states.

The detailed (somewhat out of date) calculations are
in the first chapters of 'TGD' on my homepage.


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