# [time 813] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 26 Sep 1999 08:05:07 +0300 (EET DST)

On Sun, 26 Sep 1999, Hitoshi Kitada wrote:

> Dear Matti,
>
> You have not answered completely to my former questions:
>
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
>
> Subject: [time 810] Re: [time 809] Re: [time 808] Stillabout construction of
> U
>
>
> >
> >
> > On Sun, 26 Sep 1999, Hitoshi Kitada wrote:
> >
> > > Dear Matti,
> > >
> > > I have trivial (notational) questions first. I hope you would write
> exactly
> > > (;-) After these points are made clear, I have further questions.
> > >
> > > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
> > >
> > > Subject: [time 808] Re: [time 806] Re: [time 805] Re: [time 804] Re:
> [time
> > > 803] Re:[time 801] Re: [time 799] Stillabout construction of U
> > >
> > >
> > > >
> > > >
> > > > On Sat, 25 Sep 1999, Hitoshi Kitada wrote:
> > > >
> > > > > Dear Matti,
> > > > >
> > > > > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
> > > > >
> > > > > Subject: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re:
> > > [time
> > > > > 799] Stillabout construction of U
> > > > >
> > > > >
> > > > > [skip]

> >
> > U is *counterpart* of U(-infty,infty) of ordinary QM. I do not
> > however want anymore to ad these infinities as arguments of U!
> > They are not needed.
> >
> > [I made considerable amount of work by deleting from chapters
> > of TGD, p-Adic TGD, and consciousness book all these (-infty,infty):ies
> > and $t\rightarrow \infty$:ies. I hope that I need not add them
> > back!(;-)]
> >
> > I define U below: U maps state Psi_0 satisfying single
> > particle Virasoro conditions
> >
> > L_0(n)Psi_0 =0
> >
> > to corresponding scattering state
> >
> > Psi= Psi_0 + (1/sum_nL_0(n)+iepsilon)*L_0(int) Psi
> >
> > (this state must be normalized so that it has unit norm)
> >
> >
> >
> >
> > >
> > > >
> > > >
> > > > a) The action of U on Psi_0 satisfying Virasoro conditions
> > > > for single particle Virasoro generators is
> > > > defined by the formula
> > > >
> > > > Psi= Psi_0 - [1/L_0(free)+iepsilon ]L(int)Psi
> > >
> > > To which Hilbert spaces, do Psi and Psi_0 belong?
>
> What Hilbert spaces do you think for Psi and Psi_0 to belong to?

The space of configuration space spinor fields defined in
the space of 3-surfaces. For given 3-surface this space reduces
to space of configuration space spinors and is essentially Fock
space spanned by second quantized free imbedding space
spinor fields fields induced to spacetime surface X^4(X^3).

By GCI one can reduce inner product to integration over
3-surfaces belonging to the boundary of imbedding space:
lightcone boundary xCP_2 plus summation over the degenerate
absolute minima of Kaehler action: this is implied by classical
nondeterminism of Kaehler action.

This reduction is crucial simplification: otherwise one would have
difficult time with Diff^4 gauge invariance.

>
> > >
> > > And how do you define (or construct) U from this equation?
> >
> > Just as S-matrix is constructed from the scattering solution
> > in ordinary QM. I solve the equation iteratively by subsituting
> > to the right hand side first Psi=Psi_0; calculat Psi_1 and
> > substitute it to right hand side; etc.. U get perturbative
> > expansion for Psi.
> >
> > I normalize in and define the matrix elements of U
> >
> > between two state basis as
> >
> > U_m,N = <Psi_0(m), Psi(N)>
> >
> > This matrix is unitary as an overlap matrix between two orthonormalized
> > state basis.
> >
> >
> >
> > >
> > > >
> > > > satisfies Virasoro condition
> > > >
> > > > L_0(tot)Psi=0 <--> (H-E)Psi=0
> > >
> > > Did you change E=0 to general eigenvalue E?
> >
> > This is just analogy. L_0(tot) corresponds to H-E mathematically.
>
> I questioned this in relation with your equation below:
>
> H_0 Psi_0=0.
>
> Is the eigenvalue for Psi_0 in this equation different from that for Psi in
>
> (H-E)Psi=0
>
> in the above?
>

I think you are taking the analogy with Hamiltonian too far by assuming
that their IS time development associated with L_0(tot) and
corresponding eigenvalue. The point is that there is NO such time
development and you question does not make sense. I just abstract the
general structure of scattering solution of Schrodinger equation.
Hermitian operators sum_nL_0(n)<-->H_0, L_0(tot)<-->H, L_0(int)<-->V,
and iterative solution of the scattering solution form of
Schrodinger equation. It is this algebraic structure which gives rise
to unitary S-matrix: one does not need time.

Of course, I *could* be wrong and certainly this is purely formal
solution. In any case, it leads to similar string diagrammatics as
encountered in string models: this is what convinces me that I am on
correct track.

By the way, there is also possible formal connection with Bethe-Salpeter
equation describing relativistic bound station formation in QFT. BS
contains time coordinate for each particle separately.

a) *Physically* the momenta p_+ of various particles associated with
3-surface X^3(n) correspond to energy in Schroedinger equation and off
mass shell particles (using QFT terminology) appear in the scattering
part of solution since otherwise denominator would give a pole.

b) One could also consider the possibility of assigning to each single
particle not only momentum p_+(n) but also time coordinate
x_+(n) with each energy: kind of Bethe-Salpeter equation with several
times with each time approaching to infinity might be alternative attempt
to assing time in rigorous manner to these equations. I want to
however emphasize that time or times are not needed for physical
interpretation in TGD framework. They only generate
problems with covariance. If the introduction of time serves
some purpose, this purpose must be proof of unitarity of the resulting
S-matrix.

Best,
MP

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