**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Tue, 7 Sep 1999 13:40:33 +0900*

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Dear Matti et al.,

*> Dear Hitoshi et all,
*

*>
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*>
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*> On Tue, 7 Sep 1999, Hitoshi Kitada wrote:
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*>
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*> > Dear Stephen,
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*> >
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[snip]

*> > >
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*> > > i.e., Psi_E is also an eigen function of H^hat, but with eigenvalue E +
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*> > > \alpha hbar. Since \alpha is arbitrary, the eigenvalues of H^hat would
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*> > > take all real values from -\inf. to +\inf, and this is in contradiction
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*> > > with the existence of a discrete energy spectra."
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*> >
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*> > Yes, this is right, insofar as it is considered in the usual framework of
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QM.

*> > I.e. the state vectors are Psi(x), the functions of configuration
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variables x

*> > only.
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*> >
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*> > In my case of [time 692], the state vectors are Psi(x,t) =
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*> > exp(-itH/h)Psi(x,0), the functions of time and coordinates. By this
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*> > difference, one has
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*> >
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*> > i[t, H] = - i[H, t] = i[T, t] = h.
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*> >
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*> > Since H is an operator that acts on the functions Psi(x,t) of x and t, and
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it

*> > acts as H = -T on them, H naturally has the spectra that is the whole real
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*> > line, consistent with the argument you quoted.
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*> >
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*> > Namely the opeartor H newly defined in [time 692] is different from the
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usual

*> > Hamiltonian H. The former acts on the four dimensinal space functions
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*> > (exactly, on Psi(x,t) = exp(-itH/h)Psi(x,0)), while the latter on three
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*> > dimensional functions. So I reformulated the usual QM to 4 dimesnional
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form,

*> > then no contradiction and the canonical conjugateness of t to H follows.
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*>
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*> There is perhaps problem with the fact that multiplication by t takes
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*> 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 ;-)

*> Second problem is that localization in time is not possible due
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*> to constraints posed by Schrodinger equation: localization would break
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*> probability conservation. Thus one question whether time operator
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*> is a useful concept.
*

I agree. I have the same question. Just made a trip with Stephen's dream :-)

The basic point is that Schrodinger equation

*> or any field equation puts constraints on time behaviour and this
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*> means that one cannot anymore perform arbitrary operations like
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*> time localization affecting the time behaviour without conflict
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*> 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.

*>
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*>
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*>
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*> Best,
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*> MP
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*>
*

Best wishes,

Hitoshi

**Next message:**WDEshleman@aol.com: "[time 703] Re: [time 694] Re: [time 674] Reply to NOW/PAST question"**Previous message:**Matti Pitkanen: "[time 701] Re: [time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**In reply to:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**Next in thread:**Stephen P. King: "[time 706] Re: [time 702] Time operator?"

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