# [time 701] Re: [time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 7 Sep 1999 07:06:36 +0300 (EET DST)

Dear Hitoshi et all,

On Tue, 7 Sep 1999, Hitoshi Kitada wrote:

> Dear Stephen,
>
>
> > Dear Hitoshi et al,
> >
> > Hitoshi Kitada wrote:
> > snip
> > [SPK]
> > > > Space and Time, Matter and Mind : The Relationship Between Reality and
> > > > Space-Time by W. Schommers, (October 1994) World Scientific Pub Co;
> > > > ISBN: 9810218516
> >
> > > > I will write up a relevant quote as soon as possible...
> > [HK]
> > > Thanks for the quotation.
> >
> >
> > I don't know how to write the equations correctly in ascii, but would
> > it not be similar to Schommers' notion?
> >
> > Quantum Theory and Picture of Reality, W. Schommers (ed.)
> > Springer-Verlag (April 1989) pg. 220-1
>
> I read this book before.
>
> >
> > "It was argued by Schrodinger and Pauli that such a time-energy relation
> > should be a straightforward consequence of a commutation rule with the
> > structure...
> >
> > [T^hat, H^hat] = i hbar I^hat (5.8)
> >
> > which is the commutation rule between the time operator T^hat and the
> > Hamiltonian H^hat representing the variables t and E. Does there exist
> > such a time-operator T^hat within the usual QT? The answer is no, for
> > the following reason... [given the position-momentum relation [x^hat_i,
> > p^hat_i] = i hbar I^hat, i = 1,2,3] ... it follows that there should
> > exist the relation
> >
> > i hbar * (\partial f(T^hat))/(\patial T^hat) = [f(T^hat), H^hat] (5.9)
> >
> > which is completely equivalent to (5.6). [i hbar * (\partial
> > f(x^hat_i))/(\patial x^hat_i) = [f(x^hat_i), p^hat] , i = 1, 2, 3 (5.6)]
> > The application of (5.9) to the unitary operator
> >
> > f(T^hat) = exp{i \alpha T^hat} (5.10)
> >
> > leads to
> >
> > -hbar \alpha f(T^hat) Phi_E = Ef(T^hat)Phi_E - H^hat f(T^hat) Phi_E,
> > (5.11)
> >
> > where \alpha is a real number and Phi_E an eigenfunction of H^hat with
> > eigenvalues E:
> >
> > H^hat Phi_E = E Phi_E . (5.12)
> >
> > With
> >
> > Psi_E = f(T^hat)Phi_E, (5.13)
> >
> > we have
> >
> > H(hat) Psi_E = (E + hbar\alpha)Psi_E
> >
> > 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?
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. 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.

Best,
MP

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