Maxwell's equations from Dirac and complex quatenrions

sci.physics.research (moderated) #6351 (10 more) [1]
From: mark@omnifest.uwm.edu (Mark Hopkins)
[1] The Equivalence of the Dirac and Maxwell Equations
Date: Thu Feb 13 12:51:57 EST 1997
Organization: Omnifest
Lines: 138
Originator: anderson@bohr

In this treatment, which employs Hestenes spacetime algebra approach
and explains the correspondence between that approach and Hestenes'
notation to the standard spinor notation used in Quantum Physics, we'll show
the intriguing relation between the Dirac and (classical) Maxwell equations.

This has been demonstrated a couple times before in the recent literature,
but not in a way that is (in my opinion) as accessible as the treatment
given below.

--- -----------------

A spinor, according to the treatment given by mathematicians, is an
element of the even subalgebra of a Clifford algebra. In the context of the
Dirac algebra, spinors take on the form:

psi_H = b + (B1 g01 + B2 g02 + B3 g03) + (E1 g23 + E2 g31 + E3 g12) + e g0123

where { g_0, g_1, g_2, g_3 } are the generators of the Dirac algebra, and
g_ij.. = g_i g_j ... This is the representation of spinors that Hestenes
adopts, thus the subscript. (The actual notation with the B's, E's, etc.,
however, is mine, used to expedite the discussion below. Hestenes writes his
psi_H in the form R exp(g0123 S)).

The relation between this spinor and the one normally used in Quantum
Physics (psi_D, with D for Dirac) can be stated as so:
psi_M |0> = psi_D
where
|0> = / 1 \ and psi_M = psi_H in matrix form
| 0 |
| 0 |
\ 0 /
So what Hestenes did, in effect, is unify the notation used for spinors and
operators by "factoring out the trailing |0>".
Given the restriction of psi_H to the even subalgebra, this relation
defines a one-one correspondence between psi_H and psi_D. The Dirac matrix
representation (stated in block form) is:
g0 = 1 0 g1 = 0 X g2 = 0 Y g3 = 0 Z
0 -1 X 0 Y 0 Z 0
where X = 0 1, Y = 0 -i, Z = 1 0,
1 0 i 0 0 -1
where I'm using scalars k in the blocks to denote scalar multiples kI of the
identity matrix I. In this representation, psi_H is:
psi_M = b - i E B - i e
B - i e b - i E
where B = (B1 X + B2 Y + B3 Z), and E = (E1 X + E2 Y + E3 Z). Finally, this
yields:
psi_D = / b - i E3 \
| E2 - i E1 |
| B3 - i e |
\ B1 + i B2 /

The Dirac equation in the standard notation is:

p-bar psi_D = m c psi_D

where p-bar = h-bar D, D = g0/c d/dt + g1 d/dx + g2 d/dy + g3 d/dz, and the
d's are supposed to denote partial derivatives. This can be written in the
form:
h- bar D psi_M |0> = m c psi_M |0>
or
D psi_M |0> = (mc/h-bar) psi_M |0>

This implies
D psi_M = (mc/h-bar) psi_M F
where F is some factor for which F |0> = |0>. Since D psi_M is in the odd
subalgebra, then F must be too. This pins F down to F = g0. Therefore,

D psi_M = (mc/h-bar) psi_M g0

Since the matrix representation is faithful, the same equation holds in
abstract form, as well:
D psi_H = (mc/h-bar) psi_H g0,
where
D = g0/c d/dt + g1 d/dx + g2 d/dy + g3 d/dz.

So the effect of converting to the Hestenes notation is to stick an
extra factor of g0 on the right. This means that in this form Dirac's
equation has to be stated with a timelike direction explicitly selected.
I think this is directly related to the "problem of time" often mentioned
by people in Quantum Gravity.

Now express this equation in the following representation:
g0 = Q, g1 = Ci, g2 = Cj, g3 = Ck
where
QC = -CQ, Q^2 = 1 = C^2
Q, C commute with the quaternion units i, j and k.
and define I = QC. This also provides a faithful representation of the Dirac
algebra. In this form
g01 = I i, g02 = I j, g03 = I k
g23 = i, g31 = j, g12 = k
g0123 = -I
and
psi_H = (b + E) + I (B - e)
where, now, we write B = B1 i + B2 j + B3 k, E = E1 i + E2 j + E3 k. Dirac's
equation takes on the form:
(Q/c d/dt + C Del) psi_H = mc/h-bar psi_H Q
where Del = i d/dx + j d/dy + k d/dz. Multiplying on the left by Q, this
yields:
(1/c d/dt + I Del) psi_H = mc/h-bar (Q psi_H Q)
and since Q I Q = Q Q C Q = C Q = -Q C = -I we can write:
Q psi_H Q = (b + E) - I (B - e) -- the conjugate of psi_H
Taking the real and imaginary parts, we can then decompose Dirac's equation
into the pair of quaternion equations:
1/c d/dt (b + E) - Del (B - e) = (mc/h-bar) (b + E)
1/c d/dt (B - e) + Del (b + E) = - (mc/h- bar) (B - e)
The quaternions Del B and Del E decompose further into:
Del B = -(div B) + (curl B); Del E = -(div E) + (curl E)
where
div (Ui + Vj + Wk) = dU/dx + dV/dy + dW/dz
curl (Ui + Vj + Wk) = i(dW/dy-dV/dz) + j(dU/dz-dW/dx) + k(dV/dx- dU/dy)
grad S = Del S, for scalars S
corresponding to the original quaternion definitions of these operations made
by Hamilton in the 19th Century. Therefore, taking scalar and vector parts of
each of the two equations above results in the following system of 4
equations:
div E = -1/c (d/dt + w) e
div B = -1/c (d/dt - w) b
curl E + 1/c (d/dt + w) B = -grad b
curl B - 1/c (d/dt - w) E = grad e
where w = mc^2/h-bar is the angular frequency associated with the mass m.
Look familiar? When m = 0 and when psi_H consist only of bivector
components (i.e., e = b = 0), this reduces to the Maxwell equations for the
free field (apart from a factor of c here and there). Therefore, Maxwell's
equations for the free field can be written in the form:
p-bar psi_H = 0
where
psi_H = c (B1 g01 + B2 g02 + B3 g03) + (E1 g23 + E2 g31 + E3 g12)

When m = 0 and when there are sources, e and b are not necessarily both 0 and
one has the following relations:
b = 0, in the absence of magnetic monopole sources
e = - time integral (rho c/epsilon_0)
where rho is the charge density. Therefore, one may identify the occurrence
of e and b in psi_H in the general case (m > 0) as a kind of potential
respectively for "electric" and "magnetic" monopole sources.
The appearance of the non-zero angular frequency is all that really
distinguishes the fermion field from the combined classical (electromagnetic
field + sources).