Esteemed Prof. Standish,
Thank you for that correction. ;-) But you are missing the point that I
am trying to make here! :_(
----- Original Message -----
From: "Russell Standish" <[EMAIL PROTECTED]>
To: "Stephen Paul King" <[EMAIL PROTECTED]>
Cc: <firstname.lastname@example.org>; "Lee Corbin" <[EMAIL PROTECTED]>
Sent: Wednesday, July 13, 2005 12:02 AM
Subject: Re: The Time Deniers and the idea of time as a "dimension"
On Tue, Jul 12, 2005 at 09:54:55AM -0400, Stephen Paul King wrote:
How familiar are you with the details of quantum mechanics? Did you
happen to know that the notion of an observable in QM has a complex value
and that a real value only obtains after the multiplication of an
observable with its complex conjugate? This operation of conjugation must
involve the selection of some basis.. This makes the problem of a
pre-existing Real value time to be, at least, doubly difficult.
Complex numbers have no natural ordering, as opposed to the Reals,
which do, because in general, complex numbers do not commute with each
other. Only the very special subset of observables can be said to commute
and thus can be mapped to some notion of a "dimension" that one can have
translational transforms as functions.
Tosh! I'm sorry, but you are demonstrating enormous ignorance of QM
with these statements.
1) Observables are Hermitian operators. This means that their
eigenvalues (which are the observed outcomes) are real valued (not
complex valued as you seem to think), and so ordering of observed
values is _not_ the problem you think it is.
Please notice the words "observed outcome"! This is my point! I am not
talking about "after the fact of an observational event" - which is the
intended application of Hermiticity -, I am talking about observables prior
to the specification of the observational context of the particular
observables. There is a big difference between how properties are defined in
QM before and after the specification of a context within which a
measurement and/or observation is made.
BTW, this the what the whole controversy reqarding the "collapse of the
wavefunction"! Prior to the measurement even, the possible properties of an
object of observation are given by a superposition. After the fact, one
obtains a single Boolean representable set of properties.
When we are talking about the notion of Time, we must take this
distiction into account!
2) Complex numbers indeed do not have an ordering (being basically
points on a plane), however they do commute. For any two complex
numbers x and y, xy=yx.
Well, it is the point that complex numbers do not have an ordering that
is my point. I forgot my complex number algebra. ;-)
I will let Wikipedia make my point:
Quantum superposition is the application of the superposition principle to
quantum mechanics. The superposition principle is addition of the amplitudes
of waves from interference. In quantum mechanics it is the amplitudes of
wavefunctions, or state vectors, that add. It occurs when an object
simultaneously "possesses" two or more values for an observable quantity
(e.g. the position or energy of a particle).
More specifically, in quantum mechanics, any observable quantity corresponds
to an eigenstate of a Hermitian linear operator. The linear combination of
two or more eigenstates results in quantum superposition of two or more
values of the quantity. If the quantity is measured, the projection
postulate states that the state will be randomly collapsed onto one of the
values in the superposition (with a probability proportional to the
amplitude of that eigenstate in the linear combination).
The question naturally arose as to why "real" (macroscopic, Newtonian)
objects and events do not seem to display quantum mechanical features such
as superposition. In 1935, Erwin Schrödinger devised a well-known thought
experiment, now known as Schrödinger's cat, which highlighted the dissonance
between quantum mechanics and Newtonian physics.
In fact, quantum superposition does result in many directly observable
effects, such as interference peaks from an electron wave in a double-slit
If two observables correspond to non-commuting operators, they obey an
uncertainty principle and a distinct state of one observable corresponds to
a superposition of many states for the other observable.