Hello:
I made three resolutions for 1998, and here is the fulfillment of one
of them. In this post I will express the Lorentz force using
quaternion potentials and operators. By using potentials and
operators, it may be possible to create other laws like the Lorentz
force, in particular, one for gravity.
What is the Lorentz force? It is the force acting on a moving charge.
F = e E + e b x B (Jackson I.3)
e is the electric charge
E is the Electric field 3-vector
b is the relative velocity, v/c
B the magnetic field 3-vector
If the cross product term is zero, then for a point source, the Lorentz
force becomes Coulomb's law, F = k e e'/r^2. There is a limitation on
the Lorentz force written this way. This version describes only a 3-
vector which is not covariant under a Lorentz transformation.
Jackson gets around to the "manifestly covariant" form only deep
into his tour de force.
dp^alpha/d tau = m dU^alpha/d tau = e/c F^{alpha beta} U_beta
Jackson 11.144!
Translate this into E's and B's, b's and g's (g is gamma = 1/(1 - b^2)^.5)
d Energy/d tau = e gb.E
d P/d tau = e (g E + gbxB)
Form a quaternion from the scalar change in energy term and the 3-
vector d P/d tau (tau = (t^2 - x^2)^.5). I choose to write the electric
field E as a potential, E = -dA/dt - Del phi and the magnetic field B as
B = DelxA. The covariant Lorentz force written as a quaternion
equation is
F = e (-gb.dA/dt - gb.Del phi, -g dA/dt - g Del phi + gbx(DelxA))
This is just a rewrite, containing no new content. It is the next step
that is interesting: finding the combination of operators and
potentials that can generate this manifestly covariant form of the
Lorentz force in the Lorentz gauge.
Part of this hunt must involve creating E and B fields, just as was
done for the Maxwell equations. Here is the same initial step.
< -E > < B >
[d/dt, Del](phi, A) = (d phi/dt - Del.A, dA/dt + Del phi + DelxA)
Another component will be the 4-velocity.
(g, gb)
If these two lines are multiplied together, the result is not the
Lorentz force F from above. That is not surprising since, for example,
it took a little algebraic work to get only the covariant terms of the
Maxwell equations in the Lorentz gauge.
Here is the answer:
e ([d/dt, Del]* (phi, A)* (g, gb)* - (g, gb)* [d/dt, Del](phi, A))/2
= e (-gb.dA/dt - gb.Del phi, -g dA/dt - g Del phi + gbx(DelxA))
Let's justify that assertion. Look at the first term.
e [d/dt, Del]* (phi, A)* (g, gb)*
= e (d phi/dt - Del.A, - dA/dt - Del phi + DelxA)(g, -gb)
= e (g d phi/dt - g Del.A - gb.dA/dt - gb.Del phi + gb.(DelxA),
- gb d phi/dt + gb Del.A - g dA/dt - g Del phi + g DelxA
+ dA/dtxgb + Del phixgb - (DelxA)xgb)
There are 13 terms in all, and only 5 are keepers! The second term
of the operator equation will generate the same 13 terms, but many
signs will get swapped along the way :-)
-e (g, gb)* [d/dt, Del] (phi, A)
= e (-g, gb)(d phi/dt - Del.A, dA/dt + Del phi + DelxA)
= e (-g d phi/dt + g Del.A - gb.dA/dt - gb.Del phi - gb.(DelxA),
-------- ----------
+ gb d phi/dt - gb Del.A - g dA/dt - g Del phi - g DelxA
------- ---------
+ gbxdA/dt + gbxDel phi + (gbx(DelxA))
---------
Only the 5 underlined terms will remain by adding these two
operator terms (note that the cross product flips signs by moving the
gb 3-vector to the other side). These are the 5 terms wanted!
By writing the covariant form of the Lorentz force as an operator
acting on a potential, it may be possible to create other "Lorentz
force-like" laws. For point sources in the classical limit, these new
laws must have the form of Coulomb's law, F = k e e'/r^2. An obvious
candidate is Newton's law of gravity, F = -G m m'/r^2. This would
require a different type of scalar potential, one that always had the
same sign (a non-trival difference :-) I'll get to that by Superbowl
Sunday, but first I must confront my second resolution and write the
Schrodinger equation with quaternions.
Doug
http://world.com/~sweetser
Trying to land the big one in 1998