# [time 700] 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 07:00:44 +0300 (EET DST)

Hi all,

On Mon, 6 Sep 1999, Stephen P. King wrote:

> Dear Hitoshi et al,
>
> snip
>
> "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)
>
>
> -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."

[MP]
The continuity of the energy spectrum follows from the requirement
that states are *localizable in time coordinate*. One can however
give this assumption and as Hitoshi showed one can time operator
in the subspace of energy eigenestates. The multiplication with t
however does *not* leave energy eigenstate to a superposition of
energy eigestates but takes out of this subspace. Time operator
takes you outside the system!

Discrete energy spectrum follows from the assumption that some field
p^2/2m m where p is momentum of planewave ezp(iEt-ipx) (hbar=c=1). For
given momentum p only single energy is allowed.

The physical content of discrete energy spectra is probability
conservation. If there where no constraints, Psi could exist
in finite time interval: system would be created and destroyed.
And also determinism. Once you have linear field equation you have
determinism and constraint on energy spectrum.

[By the way, cognitive spacetime sheets, which exist finite period
of time are indeed reflections of nondeterminism of *nonlinear* field
equations associated with Kaehler action.]

Best,
MP

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