[time 808] Re: [time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 25 Sep 1999 18:03:32 +0300 (EET DST)

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
> >
> >
> >
> > You might be right in that one can formally introduce
> > time from S-matrix. Indeed the replacement p_+--> id/dt in
> > mass squared operator p_kp^k= 2p_+p_--p_T^2
> > seems to lead to Schrodinger equation if my earlier arguments
> > are correct.
> >
> > This replacement is however not needed and is completely ad hoc since
> > the action of p_+ is in any case well defined.
> By "action of p_+" what do you mean? Does it make your "quantum jump" occur?

I introduce lightcone coordinatse for momentum space which is isomorphic
o 4-dimensional Minkowski space. p_0+p_= p_+ and p_0-p_z= p-. In
these coordinates p^2= 2p_+p_--px^2-p_y^2. The idea is that p^2 is
*linear* in p_+--> id/dt and one one obtains Schrodinger equation
using the replacement trick.

> > Unless one interprets
> > the time coordinate conjugate to p_+ as one configuration space
> > coordinate associated with space of 3-surfaces at light cone boundary
> > delta M^4_+xCP_2.
> I do not understand this sentence.

Diff^4 invariant momentum generators are defined in the following manner.
Consider Y^3 belonging to delta M^4_+xCP_2 ("lightcone boundary").
There is unique spacetime surface X^4(Y^3) defined as absolute minimum
of Kaehler action.

Take 3-surface X^3(a) defined by the intersection of lightcone
proper time a =constant hyperboloidxCP_2 with X^4(Y^3). Translate it
infinitesimal amount to X^3(a,new)and find the new absolute minimum
spacetime surface goinb through X^3(a,new). It intersectors
lightcone at Y^3(new). Y^3(new) is infinitesimal translate
of Y^3: it is not simple translate but slightly deformed surface.

In this manner one obtains what I called Diff^4 invariant infinitesimal
representation of Poincare algebra when one considers also infinitesimal
Lorentz transformations. These infinitesimal transformations need
*not* form closed Lie-algebra for finite value a of lightcone proper time
but at the limit a--> the breaking of Poincare invariance is expected
to go to zero and one obtains Poincare algebra since the distance to
the lightcone boundary causing breaking of global Poincare invariance
becomes infinite. The Diff^4 invariant Poincare algebra p_k(a--> infty)
defines momentum generators appearing in Virasoro algebra.

Returning to the sentence which You did not understand: p_+(a--> infty)
acts on the set of 3-surfaces belonging to lightcone boundary and
one can assign to the orbit of 3-surface coordinate. This plays effective
role of time coordinate since it is conjugate to p_+.

> [skip]
> > > > In TGD approach one has
> > > >
> > > > L_0(tot) Psi=0 rather than HPsi = EPsi! No energy, no time!!
> > > >
> > > > By the way, this condition is analogous to your condition
> > > > that entire universe has vanishing energy
> > > >
> > > > HPsi=0
> > > >
> > > > Thus there is something common between our approaches!
> > >
> > > Then you agree with that there is no time for the total universe?
> > >
> >
> >
> > I agree in the sense that there is no need to assign time to U: just
> > S-matrix describes quantum evolution associated with each quantum jump.

> If the total state \Psi is an eigenstate of the total Hamiltonian L_0(tot) of
> yours, how the "quantum jump" occur? See
> L_0(tot) \Psi = 0,
> and \Psi is the total state. There is nothing happen. Scattering operator S
> of the universe becomes I, the identity operator. No scattering occur. How
> quantum jump can exist?

No! L_0(tot) is not time development operator! U is not
exip(iL_0(tot)(t_f-t_i))!! Let me explain.

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

satisfies Virasoro condition

L_0(tot)Psi=0 <--> (H-E)Psi=0

L_0(tot)<--> H: both Hermitian.

L_0(free) =sum_n L_0(n): L_0(free)<--->H_0: both Hermitian

L_0(n) Psi_0=0 for every n <--> H_0 Psi_0=0

L_0(int) <--> V: both Hermitian.

n labels various particle like 3-surfaces X^3(a-->infty)
associated with spacetime surface and L_0(n) is
corresponding Virasoro generator defined
by regarding X^3(n) as its own universe.

The structure of scattering solution is similar to the
solution of Schrodinger equation in time dependent perturbation
theory. This was what I finally discovered.

b) The map Psi_0---> Psi=Psi_0 + ..., with latter normalized properly,
defines by linear extension the unitary time development operator U:

Psi_i---> UPsi_i is defined by this unitary map.

Here is the quantum dynamics of TGD.
One can say that U assings to a state corresponding scattering state.

c) In quantum jump Psi_i-->UPsi_i --> Psi_f
and one indeed obtains nontrivial theory.

The whole point is the possibility to decompose L_0(tot) uniquely
to sum of single particle Virasoro generators L_0(n) plus
interaction term. In GRT one cannot decompose Hamiltonian
representing coordinate condition in this manner.
This decomposition leads to stringy perturbation theory.

> > This might be even impossible.
> >
> > But there is geometric time associated with imbedding
> > space and spacetime surfaces: in this respect TGD differs from
> > GRT where also TGD formalism would lead to a loss of geometric time.
> Then you agree that also geometric time does not exist?

No!(;-) I hope the preceding argument clarifies this point.

> >
> > And there is the subjective time associated with
> > quantum jump sequence (nothing geometrical) and psychological time is kind
> > of hybrid of subjective and geometric time.
> In view of the two observation above, there is no psychological time of the
> total universe?



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