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

*Mon, 6 Sep 1999 18:58:30 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 693] Correction to [time 692] re [time 686] Time operator?"**Previous message:**Hitoshi Kitada: "[time 691] Re: [time 690] Re: [time 689] Re: [time 688] Re: [time 687] Re: [time 686] Time operator?"**In reply to:**Stephen P. King: "[time 686] Time operator?"**Next in thread:**Hitoshi Kitada: "[time 693] Correction to [time 692] re [time 686] Time operator?"

Dear Stephen,

The local time t can be thought as a canonical conjugate to H in the following

sense:

For the state vector Psi(t) of an LS, say L, the Scroedinger equation holds

identically:

h d

- -- Psi(t) + H Psi(t) = 0. (h being the Planck constant/(2 pi) )

i dt

(Recall that the local time t is defined so that this equation becomes the

identity. I.e. the local time t is defined as the exponent t of exp(-itH).

Thus the state vector

Psi(t) = exp(-itH) Psi(0)

of L (with the initial state Psi(0) ) automatically satisfies the above

equation.)

Define an operator T equal to

h d

- --

i dt

Then the Schroedinger equation (identity) becomes

T Psi(t) + H Psi(t) = 0.

Thus every state vector Psi(t) of L is identically a solution of the

Scroedinger equation with Hamiltonian H, and we have on such states

T = -H. (*)

This T clearly satisfies

i[T, t] = h.

In this sense, T is a canonical conjugate to t. H is related with T by the

above relation (*), which holds identically on the state vectors of L. This

means H (restricted to the space of the state vectors of L) is a canonical

conjugate to t.

Best wishes,

Hitoshi

**Next message:**Hitoshi Kitada: "[time 693] Correction to [time 692] re [time 686] Time operator?"**Previous message:**Hitoshi Kitada: "[time 691] Re: [time 690] Re: [time 689] Re: [time 688] Re: [time 687] Re: [time 686] Time operator?"**In reply to:**Stephen P. King: "[time 686] Time operator?"**Next in thread:**Hitoshi Kitada: "[time 693] Correction to [time 692] re [time 686] Time operator?"

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