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Re: [Phys-l] Types of scalars



In order for a discussion to be productive, it is desirable (putting it
mildly) to have first a commonly used definition of a discussed concept.
As I had said before in another discussion, one cannot dispute a definition -
one can either accept it or not. In my previous message, I merely suggested one
about a "scalar" concept. If, as John Denker said, "The whole idea of
scalar=invariant is Dead on Arrival", this only means that it is not acceptable
to John Denker. Then I ask John or anyone else to give another, GENERALLY
ACCEPTED, definition of a "scalar", or, in case it does not exist, just to
suggest something different from "scalar=invariant" and show why it would be
better.
Meanwhile, I want to correct my two statements in the previous message,
remaining within the framework of my original suggestion.
First, there is an exception in rule 1): a component of position vector is, (as
John has noted correctly) not an invariant under translation. Therefore, in the
framework of my definition, it is not a scalar under this transformation, either
(nothing horrible!). I think, this property of position vector under translation
is the only exception in rule 1). This does not invalidate this rule for all
vectors other than position vector (displacement, momentum, velocity, force,
etc.)
Second, my example with parity also needs correction - both eigenvalues P=+1
and P=-1 of the parity operator are scalars under reflection regardless of the
sign (reflection does not change the parity of a function as such).
See also my brief comments below.

Moses Fayngold,
NJIT



________________________________
John Denker wrote:

But that's not the definition. Never has been. Never will be.
Not even the examples given below conform to this "definition".
...
A velocity vector is invariant with respect to translation.
Does that mean velocity is a vector? (??????? - MF)

On the other side of the same coin, the X-component of position is
not invariant with respect to translation. Does that mean it is not
a scalar? (Do you think it is? - MF )

Oh, and what about scalars in thermodynamic state-space. The scalars
exist, even though we don't have any notion of 3-rotation, let alone
4-rotation (Oh, really? So we cannot rotate a container with gas or consider
it from another RF? - MF).

The whole idea of scalar=invariant is Dead on Arrival.

Inventing new "types of scalars" -- in situations where the old types work
fine -- is profoundly misguided. (I wish I knew, what are the old types? -MF)
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