**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sat, 25 Sep 1999 14:39:54 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**Previous message:**Hitoshi Kitada: "[time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"**In reply to:**Matti Pitkanen: "[time 803] Re: [time 801] Re: [time 799] Still about construction of U"**Next in thread:**Hitoshi Kitada: "[time 806] Re: [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. 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

**Next message:**Hitoshi Kitada: "[time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"**Previous message:**Hitoshi Kitada: "[time 804] Re: [time 803] Re: [time 801] Re: [time 799] Still about construction of U"**In reply to:**Matti Pitkanen: "[time 803] Re: [time 801] Re: [time 799] Still about construction of U"**Next in thread:**Hitoshi Kitada: "[time 806] Re: [time 805] Re: [time 804] Re: [time 803] Re: [time 801] Re: [time 799] Stillabout construction of U"

*
This archive was generated by hypermail 2.0b3
on Sat Oct 16 1999 - 00:36:41 JST
*