[time 706] Re: [time 702] Time operator?

Stephen P. King (stephenk1@home.com)
Tue, 07 Sep 1999 10:49:51 -0400

Dear Hitoshi and Matti,

Hitoshi Kitada wrote:
> Dear Matti et al.,
> > Dear Hitoshi et all,
> >
> >
> > On Tue, 7 Sep 1999, Hitoshi Kitada wrote:
> >
> > > Dear Stephen,
> > >
> > > > i.e., Psi_E is also an eigen function of H^hat, but with eigenvalue E +
> > > > \alpha hbar. Since \alpha is arbitrary, the eigenvalues of H^hat would
> > > > take all real values from -\inf. to +\inf, and this is in contradiction
> > > > with the existence of a discrete energy spectra."
> > > Yes, this is right, insofar as it is considered in the usual framework of
> > > QM. I.e. the state vectors are Psi(x), the functions of configuration
> > > variables x only.
> > >
> > > In my case of [time 692], the state vectors are Psi(x,t) =
> > > exp(-itH/h)Psi(x,0), the functions of time and coordinates. By this
> > > difference, one has
> > >
> > > i[t, H] = - i[H, t] = i[T, t] = h.
> > >
> > > Since H is an operator that acts on the functions Psi(x,t) of x and t, and
> > > it
> > > acts as H = -T on them, H naturally has the spectra that is the whole real
> > > line, consistent with the argument you quoted.
> > >
> > > Namely the opeartor H newly defined in [time 692] is different from the
> > > usual
> > > Hamiltonian H. The former acts on the four dimensinal space functions
> > > (exactly, on Psi(x,t) = exp(-itH/h)Psi(x,0)), while the latter on three
> > > dimensional functions. So I reformulated the usual QM to 4 dimesnional
> > > form,
> > > then no contradiction and the canonical conjugateness of t to H follows.
> > There is perhaps problem with the fact that multiplication by t takes
> > you out from the space of energy eigenstates. Or is it a problem?
> Yes, this is a problem. E.g., t exp(-itH/h)Psi(x,t), t^2 exp(-itH/h)Psi(x,t),
> etc. do not belong to a Hilbert space of 4-dimensional space-time, if such a
> Hilbert space could be well-defined at all. I just made an excursion in the
> time continent ;-)

        Forgive my silly question, but is it necessary to define such a Hilbert
space? What would "multiplication by t" mean?

> > Second problem is that localization in time is not possible due
> > to constraints posed by Schrodinger equation: localization would break
> > probability conservation. Thus one question whether time operator
> > is a useful concept.
> I agree. I have the same question. Just made a trip with Stephen's dream :-)

        Thanks, Hitoshi! :-) The problem I see is that we assume that the
Schrodinger equation is a creature that lives in space-time! What is a
localization? Could we not just consider that a measurement defines a
selection from an ensemblen and that such defines a space-time framing?
The "conservation of probability" is a weird thing! Does it work only
for the probabilities of a system "within" a space-time, or does it work
in Hilbert space?

> > The basic point is that Schrodinger equation
> > or any field equation puts constraints on time behaviour and this
> > means that one cannot anymore perform arbitrary operations like
> > time localization affecting the time behaviour without conflict
> > with dynamical law.
> You are right again. I completely agree. This is the same problem if it is
> possible to construct a four dimensional version of the Hilbert space. What I
> proposed is that if the space of states could be thought as the totality of
> the QM orbits exp(-itH/h)Psi(x,t), then the conjugateness of t to H is
> trivial. This is an identical propsoition by nature of positing the problem.

        Matti, are you saying that the dynamical law is a priori to time? How?
I see the "dynamical law" as defining a pattern of behavior of a system
as it evolves in its time. When we say that we localize it in time, we
are refering, to be consistent, to the time of the localizing agent, not
the system in question's time. There is no "time" for all unless we are
merely considering the trivial case when all systems are synchronized...

        Hitoshi, are the QM orbits constructed in a Hilbert space such that
they are strictly orthogonal to each other? This, to me, says that the
LS are independent and thus have independent space-time framings of
their observations. Does this affect your argument?



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