**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sat, 25 Sep 1999 19:35:07 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"**Previous message:**Hitoshi Kitada: "[time 809] Re: [time 808] Stillabout construction of U"**In reply to:**Matti Pitkanen: "[time 808] Re: [time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**Next in thread:**Hitoshi Kitada: "[time 811] Re: [time 810] Re: [time 809] 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
*

*>
*

*>
*

*> >
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*> >
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*> > On Sat, 25 Sep 1999, Hitoshi Kitada wrote:
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*> >
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*> > > Dear Matti,
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*> > >
*

*> > > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
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*> > >
*

*> > > Subject: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re:
*

*> [time
*

*> > > 799] Stillabout construction of U
*

*> > >
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*> > >
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*> > > >
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*> > > >
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*> > > >
*

*> > > > You might be right in that one can formally introduce
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*> > > > time from S-matrix. Indeed the replacement p_+--> id/dt in
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*> > > > mass squared operator p_kp^k= 2p_+p_--p_T^2
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*> > > > seems to lead to Schrodinger equation if my earlier arguments
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*> > > > are correct.
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*> > > >
*

*> > > > This replacement is however not needed and is completely ad hoc since
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*> > > > the action of p_+ is in any case well defined.
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*> > >
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*> > > By "action of p_+" what do you mean? Does it make your "quantum jump"
*

*> occur?
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*> >
*

*> > I introduce lightcone coordinatse for momentum space which is isomorphic
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*> > o 4-dimensional Minkowski space. p_0+p_= p_+ and p_0-p_z= p-. In
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*> > these coordinates p^2= 2p_+p_--px^2-p_y^2. The idea is that p^2 is
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*> > *linear* in p_+--> id/dt and one one obtains Schrodinger equation
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*> > using the replacement trick.
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*> >
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*> > >
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*> > > > Unless one interprets
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*> > > > the time coordinate conjugate to p_+ as one configuration space
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*> > > > coordinate associated with space of 3-surfaces at light cone boundary
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*> > > > delta M^4_+xCP_2.
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*> > >
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*> > > I do not understand this sentence.
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*> > >
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*> >
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*> >
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*> >
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*> > Diff^4 invariant momentum generators are defined in the following manner.
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*> > Consider Y^3 belonging to delta M^4_+xCP_2 ("lightcone boundary").
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*> > There is unique spacetime surface X^4(Y^3) defined as absolute minimum
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*> > of Kaehler action.
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*> >
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*> > Take 3-surface X^3(a) defined by the intersection of lightcone
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*> > proper time a =constant hyperboloidxCP_2 with X^4(Y^3). Translate it
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*> > infinitesimal amount to X^3(a,new)and find the new absolute minimum
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*> > spacetime surface goinb through X^3(a,new). It intersectors
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*> > lightcone at Y^3(new). Y^3(new) is infinitesimal translate
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*> > of Y^3: it is not simple translate but slightly deformed surface.
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*> >
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*> > In this manner one obtains what I called Diff^4 invariant infinitesimal
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*> > representation of Poincare algebra when one considers also infinitesimal
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*> > Lorentz transformations. These infinitesimal transformations need
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*> > *not* form closed Lie-algebra for finite value a of lightcone proper time
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*> > but at the limit a--> the breaking of Poincare invariance is expected
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*> > to go to zero and one obtains Poincare algebra since the distance to
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*> > the lightcone boundary causing breaking of global Poincare invariance
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*> > becomes infinite. The Diff^4 invariant Poincare algebra p_k(a--> infty)
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*> > defines momentum generators appearing in Virasoro algebra.
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*> >
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*> >
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*> > Returning to the sentence which You did not understand: p_+(a--> infty)
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*> > acts on the set of 3-surfaces belonging to lightcone boundary and
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*> > one can assign to the orbit of 3-surface coordinate. This plays effective
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*> > role of time coordinate since it is conjugate to p_+.
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*> >
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*> >
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*> >
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*> >
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*> >
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*> >
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*> > >
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*> > > [skip]
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*> > >
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*> > > > > > In TGD approach one has
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*> > > > > >
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*> > > > > > L_0(tot) Psi=0 rather than HPsi = EPsi! No energy, no time!!
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*> > > > > >
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*> > > > > > By the way, this condition is analogous to your condition
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*> > > > > > that entire universe has vanishing energy
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*> > > > > >
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*> > > > > > HPsi=0
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*> > > > > >
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*> > > > > > Thus there is something common between our approaches!
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*> > > > >
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*> > > > > Then you agree with that there is no time for the total universe?
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*> > > > >
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*> > > >
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*> > > >
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*> > > > I agree in the sense that there is no need to assign time to U: just
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*> > > > S-matrix describes quantum evolution associated with each quantum
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*> jump.
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*> > >
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*> >
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*> >
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*> > > If the total state \Psi is an eigenstate of the total Hamiltonian
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*> L_0(tot) of
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*> > > yours, how the "quantum jump" occur? See
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*> > >
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*> > > L_0(tot) \Psi = 0,
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*> > >
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*> > > and \Psi is the total state. There is nothing happen. Scattering operator
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*> S
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*> > > of the universe becomes I, the identity operator. No scattering occur.
*

*> How
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*> > > quantum jump can exist?
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*> >
*

*> > No! L_0(tot) is not time development operator! U is not
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*> > 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)

*>
*

*> >
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*> >
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*> > a) The action of U on Psi_0 satisfying Virasoro conditions
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*> > for single particle Virasoro generators is
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*> > defined by the formula
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*> >
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*> > Psi= Psi_0 - [1/L_0(free)+iepsilon ]L(int)Psi
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*>
*

*> 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.

*>
*

*> >
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*> > satisfies Virasoro condition
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*> >
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*> > L_0(tot)Psi=0 <--> (H-E)Psi=0
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*>
*

*> Did you change E=0 to general eigenvalue E?
*

This is just analogy. L_0(tot) corresponds to H-E mathematically.

*>
*

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

*> 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.

*>
*

*> >
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*> > L_0(free) =sum_n L_0(n): L_0(free)<--->H_0: both Hermitian
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*> >
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*> > L_0(n) Psi_0=0 for every n <--> H_0 Psi_0=0
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*> >
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*> > L_0(int) <--> V: both Hermitian.
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*> >
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*> > n labels various particle like 3-surfaces X^3(a-->infty)
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*> > associated with spacetime surface and L_0(n) is
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*> > corresponding Virasoro generator defined
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*> > by regarding X^3(n) as its own universe.
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*> >
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*> > The structure of scattering solution is similar to the
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*> > solution of Schrodinger equation in time dependent perturbation
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*> > theory. This was what I finally discovered.
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*> >
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*> >
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*> > b) The map Psi_0---> Psi=Psi_0 + ..., with latter normalized properly,
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*> > defines by linear extension the unitary time development operator U:
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*> >
*

*> > Psi_i---> UPsi_i is defined by this unitary map.
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*> >
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*> > Here is the quantum dynamics of TGD.
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*> > One can say that U assings to a state corresponding scattering state.
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*> >
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*> > c) In quantum jump Psi_i-->UPsi_i --> Psi_f
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*> > and one indeed obtains nontrivial theory.
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*>
*

*> 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.

*>
*

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

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

not said nothing about this.

*> >
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*> > >
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*> > > > This might be even impossible.
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*> > > >
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*> > > > But there is geometric time associated with imbedding
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*> > > > space and spacetime surfaces: in this respect TGD differs from
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*> > > > GRT where also TGD formalism would lead to a loss of geometric time.
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*> > >
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*> > > Then you agree that also geometric time does not exist?
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*> >
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*> > No!(;-) I hope the preceding argument clarifies this point.
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*> >
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*> > >
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*> > > >
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*> > > > And there is the subjective time associated with
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*> > > > quantum jump sequence (nothing geometrical) and psychological time is
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*> kind
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*> > > > of hybrid of subjective and geometric time.
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*> > >
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*> > > In view of the two observation above, there is no psychological time of
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*> the
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*> > > total universe?
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*> >
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*> > No!
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*> >
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*> > Best,
*

*> > MP
*

*> >
*

Best,

MP

**Next message:**Hitoshi Kitada: "[time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"**Previous message:**Hitoshi Kitada: "[time 809] Re: [time 808] Stillabout construction of U"**In reply to:**Matti Pitkanen: "[time 808] Re: [time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**Next in thread:**Hitoshi Kitada: "[time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"

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