[Phys-l] points are not vectors

[John Denker <jsd@av8n.com>]

I wrote:

> > pair of position vectors,
> > in which case the bound vector is almost equivalent to a pair of
> > points.
> >
> > In other cases the "bound vector" is a pair of vectors of different
> > type, such as a point-of-application (which is a position vector)


On 09/06/2010 01:56 PM, Jeffrey Schnick wrote:

> Is a point in space really a position vector?  In what sense does a
> point in space have magnitude and direction?  I can see that a position
> vector can be usesful in specifing the location of a point in space
> relative to some other point in space but I don't see the point itself
> actually being that position vector.  

Your point is extremely well taken.  Points are not vectors.

I hate to play lawyer, but you will notice that I said "almost"
equivalent ... for precisely this reason.  The relationship
between points and vectors is tricky, and I didn't want to
derail the previous discussion.


So, here's the deal, as I see it:  Points are tiny round dots,
with no extent in any direction.  Vectors have extent in one
dimension, bivectors have extent in two dimensions, et cetera.
Here's the diagram:
  http://www.av8n.com/physics/clifford-intro.htm#sec-svbt

This idea goes back to Grassmann's (belatedly) celebrated
Ausdehnungslehre (1844) if not before.

Such points exist as things unto themselves, as fundamental
entities in geometry and physics.  They exist independent of
what observers (if any) are looking at them, and independent
of any coordinate systems (if any) are being used.

Given two points A and B, we can construct the vector V(A,B)
that extends from A to B.  Such a vector is another thing
that exists as a fundamental entity in geometry and physics,
independent of observer, coordinate system, basis, et cetera.

If we choose an origin O, we can define the position vectors
P(A) := V(O,A) and P(B) := V(O,B).

There is /some/ kind of correspondence between the point A
and the position vector P(A), but it is not a one-to-one
correspondence.  Position vectors are not quite the same 
as points.  For one thing, they are vectors, not points.  
Secondly, points exist unconditionally, whereas the meaning
of the position vectors is conditional on the choice of 
origin.

The fundamental laws of physics are (so far as we know!) 
independent of the choice of origin.  This invariance is
connected to conservation of momentum via Noether's theorem.
Therefore picking a new origin is a gauge symmetry.  This is 
perhaps the oldest dynamical symmetry in physics;  it is so
old that Galileo took it for granted when he formulated the
principle of relativity, namely invariance with respect to
uniform motion, which also ranks very high on the list of
symmetries.

Since position vectors are not gauge-invariant, they cannot
enter the laws of physics directly, but only in gauge-invariant
combinations such as P(B) - P(A) ... which is equal to V(A,B).