The Spin Chronicles (Part 17): When The Field appear

 We continue our discussion, from Part 14, Part 15, and Part 16, of actions of the Clifford group G on the Clifford geometric algebra of space Cl(V). In Part 14 we have listed four different actions:

How Leonardo da Vinci pictured the Last Supper

1). g·u = g u τ(g),

2). g·u = g u ν(g),

3). g·u = π(g) u τ(g),

4). g·u  = π(g) u ν(g).

We have already seen that 1) leads to Lorentz transformations of coordinates in a complex Minkowski space of Special Relativity. Today we will discuss the second action. Let us recall that G consists of all invertible elements of Cl(V). It is defined by the condition gν(g) = ν(g)g =1.  Action 2) can thus be written as

g·u = g u g^-1,

and so it consists of automorphisms of the algebra. In particular g·i = i. The scalar part of u does not transform at all under this action, only the vector part transforms! What can be the realtion, if any, to special relativity in this case? For action 1) the scalar part behaved as the time component of a Minkowski space event. Now the scalar part does not transform at all. The mystery will be solved immediately after, anticipating the result, I introduce the notation for this discussion. The vector part consists of three real and three imaginary numbers. What can it be? A physicist guesses immediately: write u as u = E+iB,

and see what happens!

Writing g = exp(tX), we find that X+ν(X)=0. Again we will consider two cases: X odd and X even. Let us start with X even, thus π(X) = X. As in the case 1) we set X = in, where n is a real unit vector. In this case, for g(t) = exp(itn) we have ν(g(t))=g(-t)=τ(g(t)). The action is the same as in the case 1) - E and B transform as vectors (or covectors?) under 3D rotations in the plane perpendicular to n. The same formulas as for x in the case 1).

Now we cansider the case of X odd. Thus we set X=n. Then exp(tn) is rather easily calculated:

exp(tn) = cosh(t)1 + sinh(t)n.

Exercise 1. Verify the last formula.

Then calculating

(E'+iB') ≡ exp(tn)(E+iB)exp(tn) = (cosh(t)1 + sinh(t)n) (E+iB) (cosh(t)1 - sinh(t)n)

is a matter of a straightforward algebra using the bi-quaternion multiplication formula

(p₀,p)(q₀,q) = (p₀q₀ + p_·q,  p₀q + q₀p ₊ i p⨯q). We obtain

E' = ....

B' = ...

I will not give the results now. I will not take  the pleasure from the reader to find it himself/herself. To find and then compare with what can be found on internet.

Anyway, it looks like Cl(V) contains not only spacetime, but also "the field strength". If so, then a field would be a function on Cl(V) (treated as complexified space-time) with values in Cl(V) (treated this time as a receptacle for field strengths). What kind of a function? V.V. Kassandrov suggested the answer: it should be "analytic" function, "analytic" in a special way. Then analyticity would automatically lead to something similar to Maxwell equations. But this is still in the future - I am only learning this stuff.  So far we are dealing with kinematics, no dynamics yet. Yet there is something else still lacking in this picture - the "conformal infinity", the "absolutes" of projective geometry. Points where parallel lines meet.

Morrice Kline, Projective Geometry, Scientific American, 192, p. 80-86, 1966