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: <everything-list@eskimo.com>; "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.

[RS]
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.

[SPK]

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!

[RS]
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.

[SPK]

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:

http://en.wikipedia.org/wiki/Quantum_superposition
***
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 experiment.

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.

***



Kindest regards,

Stephen

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