Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 26 Sep 1999 08:42:29 +0300 (EET DST)
On Sun, 26 Sep 1999, Hitoshi Kitada wrote:
> 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?
a) L_0(free) and L_0(tot) are Hermitian: I have accustomed to use
self-adjoint and Hermitian in same sense: <m,An> =(Am,n). Please
explain the difference.
b) I answered the question about Hilbert space in previous email.
I glue the answer also here.
The space of configuration space spinor fields defined in
the space of 3-surfaces. For given 3-surface this space reduces
to space of configuration space spinors and is essentially Fock
space spanned by second quantized free imbedding space
spinor fields fields induced to spacetime surface X^4(X^3).
By GCI one can reduce inner product to integration over
3-surfaces belonging to the boundary of imbedding space:
lightcone boundary xCP_2 plus summation over the degenerate
absolute minima of Kaehler action: this is implied by classical
nondeterminism of Kaehler action.
This reduction is crucial simplification: otherwise one would have
difficult time with Diff^4 gauge invariance.
c) What are their spectra in that space?
L_0(n) and L_0(tot) annihilates physical states. Spectrum contains
only zero eigenvalue.
L_0(n)= p^2-L_0(vib,n)
decomposition where p^2 and L_0(vib,n) are Hermitian
implies spectrum for mass squared from L_0(n)Psi_0=0 condition
and this is integer valued
p^2= nm_0^2, n=01,2,3....
The construction of solution so the conditions L_0(n)Psi_0=0
leads to the construction of Super Virasoro representations.
This is horribly rich and complicated mathematical theory. Unitary
representations are well known and are crucially important
for string models and 2-dimensional conformal field theories
describing 2-dimensional critical systems.
Here is one classical reference:
C. Itzykson, H. Saleur,J-B. Zuber (Editors)(1988):
Conformal Invariance and Applications to Statistical Mechanics, Word
Scientific.
> 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?
The spectrum of p^2 is fixed by the condition L_0(n) Psi_0=
p^2= L_0(vib)=m_0^2!
Mass squared quantization results. Mass squared is integer multiple
of (10^(-4) Planck mass) squared. This mass quantization is the
basics of basics of string models and of TGD.
Physically only massless states are important. There is however a small
thermal mixing of m^2=0 states with m^2=n>0 states described by p-adic
thermodynamics for L_0 (rather than energy!). p-Adic thermodynamics
for Super Virasoro representations is extremely predictive since
Bolzman weight exp(-E/T) is replaced with p^(L_0/T) and
T must be 1/inger valued for power of p to exist p-adically.
This leads to predictions of elementary particle masses and accuracy
is better than one percent using the freedom to select
the p-adic prime characterizing particle: p-Adic length scale
hypothesis stating p= about 2^k, k power of prime, however
makes theory extremely predictive.
>And Psi and Psi_0 are genuine eigenstate of H and H_0, i.e. not
> generalized
> eigenfunctions?
As far as Psi_0 is considered everything looks simple. Genuine
eigestates of p^2/L_0(vib).
The construction of the representations of L_0 is essentially
equivalent with construction of states of
two-dimensional conformal quantum field theory
whose states are labelled by integers or even simpler.
Infinite number of oscillator operators generate the states
satisfying Virasoro conditions. There are also additional
quantum numbers: typically there is Kac Moody algebra
associated with some Lie group involved.
Of course, in TGD one has configuration space and it would be naive
to say that everything is simple.
About Psi it is difficult to say. Construction of S-matrix in QFT:s
relies on the same scattering solution. The normalization of
the solution forces the replacement
Psi= Psi_0 /sqrt(Z) + (1(L_0(free)+iepsilon)L_0(int)Psi
In quantum field theories wave function renormalization
constant Z diverges (also bare charge diverges). What
happens in TGD: does Z diverge? Is divergence of
Z in real context the deep reason for the necessity of
p-adic valued S-matrix?
> 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?
I wanted to concentrate to just bare essentials of the magic
formula of time dependent perturbation theory without introducing
any complications unessential for the application of formula.
The beauty of TGD approach is that the decomposition of L_0(tot)
to L_0(free)+L_0(int) is purely topological.
The connected spacetime surface X^4(X^3) characterizing particle reaction
is analogous to string diagram.
a) There are number of outgoing
3-surfaces X^3(n,a--> rightarrow infty) (a is lightcone proper time):
these are like outgoing strings.
b) L_0(tot) is associated with the actual 4-surface X^4(X^3), which
is connected.
c) L_0(free) is sum of L_0(n):s associated with absolute minimum
4-surfaces X^4(X^3(n)) associated with X^3(n)! Same holds true for
p_k(a--> infty). This is the big idea, which makes the definition
of 'free' L_0 completely unique and natural. Note however absolute
minimization of Kaehler action comes into the play.
d) L_0(int) is simply the difference L_0(tot)-sum_nL_0(n)!
Best,
MP
>
> Best wishes,
> Hitoshi
>
> ----- Original Message -----
> From: Hitoshi Kitada <hitoshi@kitada.com>
> To: Time List <time@kitada.com>
> 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
> > > 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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