Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 25 Sep 1999 14:39:54 +0300 (EET DST)
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. 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.
Below are still some comments.
Best,
MP
On Sat, 25 Sep 1999, Hitoshi Kitada wrote:
> Dear Matti,
>
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
>
> Subject: [time 803] Re: [time 801] Re: [time 799] Still about construction of
> U
>
>
> >
> >
> > On Sat, 25 Sep 1999, Hitoshi Kitada wrote:
> >
> > > Dear Matti,
> > >
> > > I have several questions on your construction of S-matrix.
> > >
> > > 1.
> > >
> > > > Contrary to earlier expectations, it seems that one cannot assign
> > > >explicit Schr\"odinger equation with S-matrix although the
> > > >general structure of the solutions of the Virasoro conditions
> > > >is same as that associated with time dependent perturbation theory
> > > >and S-matrix is completely analogous to that obtained as
> > > >time evolution operator $U(-t,t)$, $t\rightarrow \infty$ in
> > > >the perturbation theory for Schr\"odinger equation.
> > >
> > > Does this mean that your former equation
> > >
> > > >\begin{eqnarray}
> > > >i\frac{d}{dt}\Psi&=& H\Psi\per , \nonumber\\
> > > >H &\equiv& k\left[-P_T^2 - L_0(vib)-L_0(int)\right]\Psi\per .
> > > >\end{eqnarray}
> > >
> > > is wrong or cannot be derived by the former note?
> >
> > As such the equation is probably wrong. The point is
> > that I was forced to make *ad hoc* replacement
> > of Diff^4 invariant translation generator p_+ with id/dt
> >
> > p_+--> id/dt
> >
> >
> > in order to obtain Schrodinger equation. The introduction
> > of time t leads to potential problems with Poincare invariance,
> > which however could be avoided. But this is the main
> > ad hoc element of construction.
> >
> >
> > Starting directly from Super Virasoro conditions and just
> > writing "scattering solution" for them one avoids
> > all ad hoc hypothesis and manifest Poincare invariance is achieved.
> > One however loses Schrodinger equation but this is not needed
> > since informational "time evolution" is
> > totally characterized by S-matrix. Thus I am tending to believe that
> > Heisenberg was right: S-matrix has nothing to do with time evolution with
> > respect to geometric time.
>
> I will show at the bottom that there is a relation between S-matrix
> formulation and the time-dependent formulation. Of course, this is possible
> when there are scattering states associated with the Hamiltonian H.
>
> On the point that the total energy is zero is equivalent to non-existence of
> time, consider the case:
>
> H\Psi = E\Psi with E not = 0,
>
> where \Psi belongs to the total Hilbert space. That is, \Psi is an eigenstate
> of H with non-zero energy.
>
> Then time evolution is
>
> exp(-itH)\Psi = exp(-itE)\Psi.
>
> This means there is no QM motion, i.e. there is no time. That is, even if
> there is non zero energy state, it happens that the universe has no time.
>
> >
> >
> >
> > >
> > > 2.
> > >
> > > >\begin{eqnarray}
> > > >\Psi&=&\Psi_0 + \frac{V}{E-H_0-V+i\epsilon} \Psi \per .
> > > >\end{eqnarray}
> > >
> > > (This is equation (1) of your note.)
> > >
> > Sorry: this equation is mistyped:
> >
> > \Psi&=&\Psi_0 + \frac{V}{E-H_0+i\epsilon} \Psi \per .
> >
> > The presence of V in denominator would make it to diverge.
> > The ordering is also important: V is to the right.
>
> Either equation leads to a correct time dependent formulation. Just by
> exchanging roles of some factors.
>
> >
> >
> > >
> > > >Since ordinary Schr\"odinger equation is consistent with the scattering
> > > >matrix formalism avoiding elegantly the difficulties with the
> > > >definition of the limit $U(-t,t)$, $t\rightarrow \infty$, it
> > > >is natural to take this form of Schr\"odinger equation as starting
> > > >point when trying to find Schr\"odinger equation for the 'time'
> evolution
> > > >operator $U$. One can even forget the assumption
> > > >about time evolution and require only
> > > >that the basic algebraic information guaranteing
> > > >unitarity is preserved. This information boils down to the Hermiticity
> > > > of free and interacting Hamiltonians and
> > > >to the assumption that the spectra
> > > >non-bound states for free and interacting Hamiltonians
> > > >are identical.
> > >
> > >
> > > It is known that to consider the limit as \epsilon -> 0 in the
> Schrodinger
> > > equation (1) of your note is equivalent to considering the time limit as
> t ->
> > > \infty of exp(-itH). So you cannot avoid the difficulty: Below I will try
> to
> > > show this.
> >
> > In TGD framework single particle Virasoro generators L_0(n) define
> > propagators
> >
> > 1/(p^2-L_0(vib)+i*epsilon)
> >
> > appearing in stringy diagrams. L_0(vib) is integer valued and gives rise
> > to the universal non-negative integer valued mass squared spectrum of
> > string models (in suitable units).
> >
> > In present case i*epsilon is completely equivalent with
> > the presence of i*epsilon in the propagators of relativistic quantum field
> > theory: epsilon term guarantees that momentum spacetime integration
> > over virtual momenta is performed correctly in case that one
> > is forced to integrate over pole of propagator.
> >
> > As far as I can understand this has nothing to do with time but I
> > could of course be wrong.
>
> That time does not exist follows from the eigenequation
>
> H\Psi = E\Psi.
>
> But it does not follow from the form of propagators. If H has a continuous
> spectral subspace (this space is sometimes called scattering space of H),
> then H can have time.
>
> In fact in my formulation,
>
> H \Psi = 0
>
> implies non-existence of time of the universe. But if we want to consider a
> scattering state \Phi as the total satte that is orthogonal to eigenspace,
> hence to \Psi, then one can recover time.
>
> The local time of a local system arises in the same way. A state \psi of a
> local system L is considered a kind of a part of the total state \Psi. Then
> it can be shown that \psi can be a scattering state of the local Hamiltonian
> H_L. This gives the local time t_L of the system L, but at the same time
> there is no time of the total universe.
>
> >
> >
> > [In string models similar expansion is derived from
> > stringy perturbation theory in which time parameter corresponds
> > to string coordinate. The problem in string models is
> > that this gives only first quantized theory and one should
> > second quantize. In TGD this problem is not encountered. Super
> > Virasoro generators are of second quantized form from
> > beginning. Also topological description of particle reactions
> > emerges automatically via the decomposition of L_0(tot) to
> > a sum of single particle Virasoro generators L_0(n). This
> > decomposition was described in later posting.]
> >
> >
> >
> >
> >
> > >
> > > The Schrodinger equation (1) in your note should be written as:
> > >
> > > \Psi = \Psi_0 - V ((E+i \epsilon)-H)^{-1} \Psi, (1)
> >
> > Yes with H replaced with free Hamiltonian: I am sorry for my typo.
> > I hope that it does not affect you conclusions.
>
> No, it does not affect my conclusion at all.
>
> >
> >
> > >
> > > or
> > >
> > > \Psi = \Psi_0 - ((E+i \epsilon)-H)^{-1} V \Psi. (2)
> > >
> > > The reason we consider two equations is that the fractional expression in
> > > your note has two interpretations, since the relevant factors are
> operators,
> > > hence they are noncommutative in general.
> >
> > Yes. In my equation 1/(E-H_0 +iepsilon)*V is meant to be the
> > correct ordering. This is easy to see: by operating with E-H_0 to both
> > sides of the equation one obtains (E-H_0) Psi= (E-H_0)Psi+ VPsi =VPsi
> > which is just Schrodinger equation. Provided that order is
> > correct.
> >
> >
> > > Also the sign on the RHS (right
> > > hand side) should be minus if the definition of H is H = H_0 + V.
> > >
> >
> >
> > > We write z = E+i\ep, where \ep = \epsilon > 0 and E is any real number.
> > >
> > > In case (1), the equation is equivalent to
> > >
> > > (I - V (H-z)^{-1} ) \Psi = \Psi_0.
> > >
> > > This is rewritten as
> > >
> > > (H - z - V) (H - z)^{-1} \Psi = \Psi_0,
> > >
> > > which is equivalent to
> > >
> > > (H_0-z) (H-z)^{-1}\Psi = \Psi_0 (1)'
> > >
> > > or
> > >
> > > (H-z)^{-1}\Psi = (H_0-z)^{-1}\Psi_0 (1)"
> > >
> > > or
> > >
> > > \Psi = (H-z)(H_0-z)^{-1}\Psi_0
> > >
> > > = \Psi_0 + V(H_0-z)^{-1}\Psi_0 (1)'''
> > >
> > > by H = H_0 + V.
> > >
> > > Thus if \Psi_0 is a given fixed state function, \Psi depends on z =
> E+i\ep,
> > > and it should be written as a function of z:
> > >
> > > \Psi = \Psi(z).
> > >
> > > Note that these hold only when \ep > 0, NOT for an infinitesimal number
> \ep
> > > because the inverse (H-(E+i\ep))^{-1} does not exist for \ep = 0 in
> general.
> > >
> > > In case (2), the equation is equivalent to
> > >
> > > \Psi_0
> > >
> > > = (I - (H-z)^{-1} V)\Psi (2)'
> > >
> > > = (H-z)^{-1} (H-z-V)\Psi
> > >
> > > = (H-z)^{-1} (H_0 - z)\Psi.
> > >
> > > This is rewritten:
> > >
> > > \Psi(z) = (H_0-z)^{-1} (H-z) \Psi_0. (2)"
> > >
> > > If we transform \Psi_0 and \Psi to
> > >
> > > \Phi_0 = (H-z) \Psi_0,
> > >
> > > \Phi = (H-z) \Psi,
> > >
> > > the equation (2)" becomes
> > >
> > > \Phi(z) = (H-z) (H_0-z)^{-1} \Phi_0. (2)'''
> > >
> >
> >
> >
> > Certainly one cannot get rid of time in ordinary wave mechanics.
>
> As I stated, it is possible to have time locally without the total time.
>
> > If you have energy, you have also time!
>
> Even when an energy is not zero, if that energy is eigenenergy, then there is
> no time.
>
> >
> >
> > 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.
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.
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.
>
> >
> >
> > Also in general Relativity Hamiltonian vanishes as a constraint.
> > In TGD However General Coordinate invariant
> > as gauge invariance is replaced by Super Virasoro invariance
> > which operates at lightcone boundary and H is replaced with L_0.
> >
> >
> > Best,
> >
> > MP
> >
> >
>
Thank you for your note.
With Best,
MP
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