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

Stephen P. King (stephenk1@home.com)
Mon, 06 Sep 1999 16:36:48 -0400

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."

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. 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

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