# [time 810] Re: [time 809] Re: [time 808] Stillabout construction of U

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 25 Sep 1999 19:35:07 +0300 (EET DST)

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
> > >
> > >
> > > >
> > > >
> > > >
> > > > 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?

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?
>
> 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.

>
> >
> > L_0(tot)<--> H: both Hermitian.
>
> H is related with H_0 by H = H_0 + V or H = H_0 - V?

H_0+V: but this is not essential. I wanted only to express
the structural analogies of equations.

>
> >
> > 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?

Quantum jumps just occur. Occurrence of quantum jumps is outside
the realm of U. Strong form of NMP characterizes the dynamics
of qjumps.

>
> >
> >
> > 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.

BTW, this decomposition is important and highly nontrivial point. I have

> >
> > >
> > > > 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,
MP

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