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

*Tue, 7 Sep 1999 11:47:06 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 700] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**Previous message:**Stephen P. King: "[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:**Matti Pitkanen: "[time 701] Re: [time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"

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.

*>
*

*>
*

*> snip
*

*> > > > The local time t of L can be thought as an operator that acts on
*

everything,

*> > > > as it is a numerical multiplication operator. If this t can be
*

canonically

*> > > > conjugate in some sense to H, your expectation would be correct.
*

*> > >
*

*> > > Yes, but this implies that the energy of the LS has some strange
*

*> > > behavior!
*

*> >
*

*> > What is the strange point?
*

*>
*

*> Given the discussion above, we are left wondering how it is that we
*

*> only observe a discrete energy spectra.
*

This is not correct. Not "only," but one "also" observes discrete energy

spectra in addition to the continuous energy spectra.

I think I know why! I will try

*> to explain by using a paraphrase of Schommers' idea, replacing his word
*

*> "reality" with the more accurate, IMHO, word "partially ordered set of
*

*> observations" or "poset of observables" and other notes in [..]
*

*> brackets.
*

*> (ibid. pg. 233.
*

*>
*

*> "Mach's principle requires the elimination of space-time as an active
*

*> cause; space-time cannot give rise to any physically real effects and
*

*> cannot be influenced by any physical condition. This means that there
*

*> can be no interaction between space-time and ...[a given LS's poset of
*

*> observables], in accordance with the fact that the elements x_1, x_2,
*

*> x_3 and t are not accessible to empirical tests. Any change in the
*

*> distance (of [centers of] masses is not due to interactions between
*

*> coordinates or between coordinates and [centers of] masses but is
*

*> entirely caused by the interaction between [centers of] masses. Thus,
*

*> space-time must be considered as an $auxillary element$ for the
*

*> geometrical description of physically real processes. In other words,
*

*> physically real processes are projected on space-time."
*

*>
*

*> (ibid pg. 235)
*

*>
*

*> "It is a typical feature of the Fourier transform that a system
*

*> localized in (r, t)-space must be totally distributed in (p, E)-space [r
*

*> = x_1, x_2, x_3 and p = p_1, p_2, p_3]. That means the momentum p of the
*

*> system is 'uncertain' if it appears as [a] 'point' in (r, t)-space. On
*

*> hte other hand, the position r of the system is uncertain if its
*

*> momentum takes a definite value in (p, E)-space. This property agrees
*

*> qualitatively with Heisenberg's uncertainty relation. ...
*

*> One of the consequences of this picture is, for example, that there can
*

*> be no such concept as the velocity of a particle in the classical sense
*

*> of the word, i.e., the limit to which the difference of the coordinates
*

*> at two instants, divided by the interval \delta t between these
*

*> instants, tends as \delta t tends to zero. Also, within the usual QT
*

*> such a velocity does not exist. Hamilton's equations are not applicable
*

*> if one of the variables r and p is uncertain; Hamilton's equations
*

*> requires that both, the position vector r $and$ the momentum p have
*

*> definite values at any instant - at least in principle. Thus the usual
*

*> QT, we have to use another description in the case of hbar =/= 0."
*

*> ...
*

*>
*

*> pg. 236-7
*

*> "Due to the structure of the Fourier transform, it is not possible to
*

*> give definite values for the coordinates x_1, x_2, x_3and the time t if
*

*> p and E take definite values. And it is not possible to give definite
*

*> values of p and E if r = (x_1, x_2, x_3) and t take definite values.
*

*> Thus, in the analysis of quantum phenomena ... the following question
*

*> arises: How can we express p and E in (r, t)-space, and r and t in (p,
*

*> E)-space? to answer this question consider the following identity...
*

*>
*

*> -i hbar (\partial / \partial r) Phi(r,t)
*

*>
*

*> + \infinity
*

*> = 1/hbar^4 \integral p Phi(p,E) exp{i[p/hbar * r - E/hbar
*

*> t]}dp/(2Pi)^3, (5.47)
*

*> - \infinity
*

*>
*

*> where
*

*>
*

*>
*

*> \partial/\partial r \equivalent i \partial / \partial x_1 + \partial /
*

*> \partial x_2 + \partial / \partial x_3.
*

*>
*

*> Interpretationb of (5.47): Any information given in (r, t)-space can be
*

*> $completely$ transformed into (p, E)-space, and vice versa. Bother
*

*> informations must be physically equivalent; we have $two$
*

*> representations of the $same$ thing. Phi(p, E) is equivalent to Phi(r,
*

*> t), and vice versa. Also, -ihbar \partial/\partial r Phi(r, t) and p
*

*> Phi(p, E) must be equivalent to each other. Thus the operator
*

*>
*

*> p^hat = -i hbar \partial/ \partial r. (5.48)
*

*>
*

*> with the components
*

*>
*

*> p^hat = - i hbar \partial / \partial x_i; i = 1, 2, 3, (5.49)
*

*>
*

*> must be equivalent to the momentump, i.e. the momentum takes the form of
*

*> an operator in (r, t)-space.
*

*> In the same way we can find ... operators for E, r, and t. For example
*

*> we have
*

*>
*

*> -i hbar (\partial / \partial E) Phi(p, E)
*

*>
*

*> + \infinity
*

*> = \integral t Phi(r,t) exp{-i[p/hbar * r - E/hbar t]}dr dt/(2Pi),
*

*> (5.50)
*

*> - \infinity
*

*>
*

*> Thus, the operator
*

*>
*

*> t^hat = -i hbar \partial / \partial E (5.51)
*

*>
*

*> must be equivalent to the time, i.e., the time t takes the form of an
*

*> operator in (p, E)-space."
*

*>
*

*> The idea that I am thinking of is that each LS has associated with it a
*

*> set of such operators and spaces, following the reasoning that each LS
*

*> has its own time! What I am thinking is that LS's can have their own (r,
*

*> t) and (p, E) spaces and can have "agreements" so that they can
*

*> communicate. How they are related to the scattering propagator is the
*

*> subject of future study!
*

*>
*

*> Onward,
*

*>
*

*> Stephen
*

*>
*

Best wishes,

Hitoshi

**Next message:**Matti Pitkanen: "[time 700] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**Previous message:**Stephen P. King: "[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:**Matti Pitkanen: "[time 701] Re: [time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"

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