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

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

Dear Stephen,

> Dear Hitoshi et al,
>
> 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

>
> "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."

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

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