[time 707] Re: [time 706] Re: [time 702] Time operator?

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 7 Sep 1999 18:39:30 +0300 (EET DST)

On Tue, 7 Sep 1999, Stephen P. King wrote:

> Dear Hitoshi and Matti,
>
> >
> > Dear Matti et al.,
> >
> > > Dear Hitoshi et all,
> > >
> > >
> > > On Tue, 7 Sep 1999, Hitoshi Kitada wrote:
> > >
> > > > Dear Stephen,
> > > >
> [snip]
> [SPK]
> > > > > 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."
> [HK]
> > > > 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.
> [MP]
> > > 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?
> [HK]
> > 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?
>
> [MP]
> > > 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.
> [HK]
> > 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?

[MP] In Hilber space in general QM. For instance, models of two level
systems are based on 2-dimensional Hilbert space with Hamiltonian
dictating the time evolution.

>
> [MP]
> > > 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.
> [HK]
> > 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?

[MP] Dynamical laws in sense of standard physics by definition assume
geometric time as an additional dimension. In QM Hamiltonian
is Lie-algebra generator of time translation symmetry and Lie symmetries
require manifold structure and metric.

One can also consider formulations in which one has just a sequence of
events governed by some rule. The proposal of Smolin for spin network
dynamics is good example. The concept of energy becomes however
problematic since energy is symmetry related concept and symmetry assumes
geometric time plus time translation symmetry.
The nondeterministic dynamics of quantum jumps partially dictated by NMP
is second example.

What supports the concept of geometric time is the success of
quantum field theories: Hamiltonian is derived
from action principle as generator of time translation symmetry
and yields S-matrix which describes this world excellently!
Note the beauty and economy of all this: symmetry determines dynamics!

> 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?

>
Best,
MP

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