**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Tue, 7 Sep 1999 07:00:44 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next 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?"**Previous message:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**In reply to:**Stephen P. King: "[time 686] Time operator?"**Next in thread:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"

Hi all,

some comments about time operator below.

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

*> Dear Hitoshi et al,
*

*>
*

*> Hitoshi Kitada wrote:
*

*> 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)
*

*>
*

*> 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."
*

[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

equation is satisfied. For instance, for free Schrodinger equation E=

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

**Next 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?"**Previous message:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**In reply to:**Stephen P. King: "[time 686] Time operator?"**Next in thread:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"

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