Hitoshi Kitada (hitoshi@kitada.com)
Sun, 26 Sep 1999 01:01:54 +0900
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
> >
> >
> > >
> > >
> > >
> > > 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.
Your U is U(\infty, -\infty) = lim_{t-> +\infty} U(t,-t) ? If so how do you
define it?
>
>
> 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?
And how do you define (or construct) U from this equation?
>
> satisfies Virasoro condition
>
> L_0(tot)Psi=0 <--> (H-E)Psi=0
Did you change E=0 to general eigenvalue E?
>
> L_0(tot)<--> H: both Hermitian.
H is related with H_0 by H = H_0 + V or H = H_0 - V?
>
> 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.
What makes the quantum jumps occur? Is it outside of the realm of U?
>
>
> 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?
>
> No!
>
> Best,
> MP
>
>
Best wishes,
Hitoshi
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