**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Sun, 26 Sep 1999 01:55:58 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 812] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"**Previous message:**Matti Pitkanen: "[time 810] Re: [time 809] Re: [time 808] Stillabout construction of U"**In reply to:**Hitoshi Kitada: "[time 809] Re: [time 808] Stillabout construction of U"**Next in thread:**Matti Pitkanen: "[time 813] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"

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
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*> >
*

*> >
*

*> > >
*

*> > >
*

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

*> > > > 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.
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*> > > >
*

*> > > > By "action of p_+" what do you mean? Does it make your "quantum jump"
*

*> > occur?
*

*> > >
*

*> > > I introduce lightcone coordinatse for momentum space which is
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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
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*> > > *linear* in p_+--> id/dt and one one obtains Schrodinger equation
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*> > > using the replacement trick.
*

*> > >
*

*> > > >
*

*> > > > > 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
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boundary

*> > > > > delta M^4_+xCP_2.
*

*> > > >
*

*> > > > I do not understand this sentence.
*

*> > > >
*

*> > >
*

*> > >
*

*> > >
*

*> > > Diff^4 invariant momentum generators are defined in the following
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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
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*> > > of Kaehler action.
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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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*> > >
*

*> > > In this manner one obtains what I called Diff^4 invariant infinitesimal
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*> > > representation of Poincare algebra when one considers also
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infinitesimal

*> > > Lorentz transformations. These infinitesimal transformations need
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*> > > *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
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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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*> > >
*

*> > >
*

*> > > Returning to the sentence which You did not understand: p_+(a--> infty)
*

*> > > 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
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effective

*> > > 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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*> > > > [skip]
*

*> > > >
*

*> > > > > > > In TGD approach one has
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*> > > > > > >
*

*> > > > > > > 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
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*> > jump.
*

*> > > >
*

*> > >
*

*> > >
*

*> > > > 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
*

*> > > >
*

*> > > > L_0(tot) \Psi = 0,
*

*> > > >
*

*> > > > and \Psi is the total state. There is nothing happen. Scattering
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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
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*> 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
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*> of TGD, p-Adic TGD, and consciousness book all these (-infty,infty):ies
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*> 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
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*>
*

*> 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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*>
*

*> >
*

*> > >
*

*> > >
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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
*

*> > > defined by the formula
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*> > >
*

*> > > 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?

*> >
*

*> > 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
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*>
*

*> between two state basis as
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*>
*

*> U_m,N = <Psi_0(m), Psi(N)>
*

*>
*

*> This matrix is unitary as an overlap matrix between two orthonormalized
*

*> state basis.
*

*>
*

*>
*

*>
*

*> >
*

*> > >
*

*> > > satisfies Virasoro condition
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*> > >
*

*> > > 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?

*>
*

*>
*

*> >
*

*> > >
*

*> > > 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
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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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*> > >
*

*> > > The structure of scattering solution is similar to the
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*> > > solution of Schrodinger equation in time dependent perturbation
*

*> > > theory. This was what I finally discovered.
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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.
*

*> > >
*

*> > > 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
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*> > > 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
*

*> not said nothing about this.
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*>
*

*>
*

*> > >
*

*> > > >
*

*> > > > > This might be even impossible.
*

*> > > > >
*

*> > > > > 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.

*> > > >
*

*> > > > 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
*

*>
*

Best wishes,

Hitoshi

**Next message:**Hitoshi Kitada: "[time 812] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"**Previous message:**Matti Pitkanen: "[time 810] Re: [time 809] Re: [time 808] Stillabout construction of U"**In reply to:**Hitoshi Kitada: "[time 809] Re: [time 808] Stillabout construction of U"**Next in thread:**Matti Pitkanen: "[time 813] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"

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