# [time 812] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U

Sun, 26 Sep 1999 12:38:40 +0900

Dear Matti,

Since you seem not answer the following questions, I make other questions:

Are H_0=L_0(free) and H=L_0(tot) not self-adjoint, but only Hermitian? And in
what Hilbert spaces are they defined? What are their spectra in that space?
As you said about L_0(vib), their spectra are discrete? But how about the
part p^2? H and H_0 contains p^2, then their spectra become continuous? And
Psi and Psi_0 are genuine eigenstate of H and H_0, i.e. not generalized
eigenfunctions? I.e., are they in the Hilbert space for H_0 and H? You said
you did not say anything about the decomposition of L_0(tot) into the sum of
L_0(free) and L_0(int), etc. Then does your argument you have been writing
have any meaning at all?

Best wishes,
Hitoshi

----- Original Message -----
Sent: Sunday, September 26, 1999 1:55 AM
Subject: [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
> > >
> > >
> > > >
> > > >
> > > > 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
> >
> >
> > > >
> > > > >
> > > > > > 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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