**Stephen P. King** (*stephenk1@home.com*)

*Mon, 06 Sep 1999 16:36:48 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**Previous message:**Hitoshi Kitada: "[time 697] Re: [time 694] Re: [time 674] Reply to NOW/PAST question"**In reply to:**WDEshleman@aol.com: "[time 694] Re: [time 674] Reply to NOW/PAST question"**Next in thread:**Matti Pitkanen: "[time 700] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"

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;
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*> > ISBN: 9810218516
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*> > 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)

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."

snip

*> > > The local time t of L can be thought as an operator that acts on everything,
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*> > > as it is a numerical multiplication operator. If this t can be canonically
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*> > > conjugate in some sense to H, your expectation would be correct.
*

*> >
*

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

**Next message:**Hitoshi Kitada: "[time 699] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"**Previous message:**Hitoshi Kitada: "[time 697] Re: [time 694] Re: [time 674] Reply to NOW/PAST question"**In reply to:**WDEshleman@aol.com: "[time 694] Re: [time 674] Reply to NOW/PAST question"**Next in thread:**Matti Pitkanen: "[time 700] Re: [time 698] Re: [time 696] Re: [time 695] Re: [time 691] ... Re: [time 686] Time operator?"

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