Horace Heffner wrote:
On Jan 8, 2006, at 2:29 PM, Jones Beene wrote:
Horace
What happens to the angular momentum? This seems to deny
conservation of angular momentum.
No problem there. 1 - 1 = 0 All balances before and after pair
formation.
That doesn't look like a proper cross product, or else I don't
understand the full situation.
Conservation of angular momentum is conservation of the vector *sum* of
the constituants of a system.
You have studied this more than me, obviously, but I was under the
impression that as a psuedovector there would be no such additive
cancellation. However, being mildly dyslexic I often get these kinds
of spin things confused, so please correct this line of reasoning.
Angular momentum is a pseudovector. Often, the distinction between
vectors and pseudovectors is overlooked, but it only becomes
important in understanding the effect of symmetry on the solution to
physical system interactions.
A pseudovector can also be described as a vector in which the "head" and
"tail" have been chosen arbitrarily, because there isn't a natural
"frontwards" or "backwards" for it. Thus in certain coordinate
transformations it may misbehave.
HOWEVER, once you have (arbitrarily) chosen a "frontward" and "backward"
orientation, you can certainly find a second pseudovector whose value
will "cancel" the first. Two electrons with flipped spins certainly
have angular momenta which sum to zero, regardless of whether you've
chosen the "right-hand" or "left-hand" orientation on which to base your
angular momentum calculations.
Just about anything which is related to the coordinate system by a
"right hand rule" probably conceals a pseudo-vector; if 95% of the
population were left-handed instead of right-handed the rule would most
likely have been written the other way around. So, for instance, you
might well ask which way the magnetic field around a conducting wire
"points" -- clockwise or counter-clockwise? The choice is arbitrary,
and the magnetic field is also a pseudovector.
An "improper" rotation is one that mirrors one axis. After that, you're
working in a left-hand coordinate system, and that's when you notice
something's gone wrong with the right-hand-rule and all the
pseudovectors which were obtained using it.
Velocity has a "natural frontwards direction" to it, as does an electric
field (er, at least, it does once you've arbitrarily chosen whether to
designate a proton or an electron as "positive"). Neither of those is a
pseudovector.
Here's a thought. If we discover magnetic monopoles, will the magnetic
field still be a pseudovector? Uh........
A pseudovector is a quantity that
transforms like a vector under a proper rotation, but gains an
additional sign flip under an improper rotation (a transformation
that can be expressed as an inversion followed by a proper rotation
which is what we have here with two electrons - an improper
rotation). It follows that any improper rotation multiplies the cross
product by -1 compared to a true vector.
My take on the bottom line of this is that the angular momentum
cancellation you seem to be basing this premise on cannot happen in
normal physics.
...or did I get dyslexic again ?
Jones
You are making the simple complex. Since before and after the proposed
superposition the electrons share the same axis, view the electron
angular momenta along a single axis. You don't need to think in vector
addition terms then, just simple addition. The moments before
superposition are +mu and -mu respectively. +mu - mu = 0 Net momentum
of the system is zero. Afterwards the angular momentum is still zero.
The net momentum of two counter-spinning gyros is zero.
Horace Heffner
