Hitoshi Kitada (hitoshi@kitada.com)
Sun, 26 Sep 1999 01:55:58 +0900
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
> >
> >
> > >
> > >
> > > 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?
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
>
> 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.
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
> > > 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
> not said nothing about this.
>
>
> > >
> > > >
> > > > > 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
>
Best wishes,
Hitoshi
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