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