Stephen P. King (firstname.lastname@example.org)
Tue, 06 Apr 1999 23:48:43 -0400
Hitoshi Kitada wrote:
> Dear Stephen,
> Just on technical points...
> > There is also the matter of "equivalence classes".
> Equivalence classes is the notion that the freshmen have to learn at first
> when they enter a university. It is one of indispensable knowledges for anyone
> who wants to learn something about math and nature.
Equivalence classes play a major role in Peter's work. We will, I
suppose, get to them in time. :)
> Prugovecki writes on
> > pg. 447:
> > "...the generic element of \H is not a single function, but rather an
> > equivalence class of almost everywhere (in the Lebesgue sense) equal
> > functions, which are such that one can change the value of any one of
> > these functions \Psi(x) at any given point x without leaving the
> > equivalence class -namely, in physical terms, without changing the
> > quantum state vector. "
> > I had stated in [time 188]:
> > > Yes, Peter and I discussed this for a while. It appears that the
> > > subject-object relation is symmetrical. There is a wonderful thing that
> > > happens when we consider an LS as a subject as a singleton set A and the
> > > other LSs that it is near to as the singleton's complement A^c. If we
> > > think of A^c as a finite number of LSs that can somehow be reduced to a
> > > singleton by some particular observation by A, by symmetry, would we not
> > > expect that A becomes many neighboring yet distinct LSs? As one fuses,
> > > the other fissions, many -> one | one-> many ... Does this make sense?
> > > There exists a mathematical way of saying this but I do not remember it
> > > now. :(
> > It is the fact that I am not stating explicitly the "mathematical way
> > of saying this" that, I think, is the reason I am just making noise
> > here... :( The role of equivalence classes is very important!
> Yes, it is so important that one usually does not refer to the usage of it,
> i.e. people frequently use it without mentioning...
I am only now finding this out. :) One last thing about subject-object
symmetry; an exact symmetry of the pair is a very special case, the case
where uncertainty is zero, I think. I was mistaken in thinking it to be
a generic situation.
I just wish to understand how we formulate the Equivalence Principle
[(EP) : an observer X in free fall in a gravitational field experiences
no differences from that of an unaccelerated observer X'], the caveat is
that this only holds in conventional GR within an infinitesimal
neighborhood of X, due to the tidal forces ("Weyl tensor" in Penrose's
explanation). I think that there is something strange necessary to
quantize this, since infinitesimal ("sharp") quantities are not in
general allowed by QM's uncertainty. But since you define the
uncertainty in a different manner than usual, maybe, just maybe, we can
show that the EP is modified by our thinking.
I need some time to go over my Category theory books to formulate my
thinking better. Lately, I have had little sleep and am making serious
mistakes in my thinking. I apologize... :)
Onward to the Unknown,
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