+ \7 /\ f.

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(C.37)

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Notice that the conditions for left regularity (C.35) and (C.35) only differ from those for right regularity (C.37) and (C.37) through the sign of the curlll of f. So if we differentiate (C.35) or (C.37) with respect to Xo and take the divergency of the

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11 If a quatemion function :F is both left and right regular, the curl of f must vanish (i.e., f must be the gradient of a scalar function).

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QUATERNIONS AND SPECIAL RELATIVITY

corresponding second equation [i.e., either (C.35) or (C.37)], and add the result, we find that

= O. (C.38) n=O n In other words, if :F is left or right regular, the scalar component of :F satisfies the four-dimensional Laplace equation. Similarly, you can easily show [146] that the other components of :F also satisfy a four-dimensional Laplace equation if :F is left or right regular. There is much more that can be done in quaternion calculus. There is, for instance, an analog of the Cauchy integral formula (the basis of the technique of contour integration); however, limitations of space and time do not allow us to delve too deeply into the delights of quaternion calculus here. We refer the interested reader to [146] and [584].

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C.3 BIQUATERNIONS AND LORENTZ TRANSFORMATIONS The use of quaternions in special relativity was pioneered by Conway in 1911 [118] and, independently, by Silberstein in 1912 [566]. For an extended list of references, see for instance, the review paper by Rastall [506]. Here we adopt a similar approach to that of Lanczos in 9 of [386]. Let us regard the quaternion

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'R:=XO+ LXnin n=l

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(C.39)

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as describing a point in four-dimensional space. Now consider another coordinate system obtained from the present one by a general four-dimensional rotation. Let 'R' be the quaternion that represents the same point in the rotated coordinate system. Then 'R and 'R' are related by a linear transformation with six degrees of freedom, which in quaternion notation can be written as [96]

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'R' =A'RB,

(C.40)

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where A and B are two quaternions of unit norm, that is,

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,A.,4 = B8 = 1.

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(C.4l)

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Lorentz transformations are particular four-dimensional rotations that preserve the Minkowskian length. The Minkowskian length can be written in a simple way using the quaternion norm if we let the quaternion coefficients be complex rather than real. Quaternions with complex coefficients are called biquaternions. We represent a point in space time by (C.39) with Xo = ct, Xl = ix, X2 = iy, and X3 = iz, where c is the speed of light, t is the time coordinate, and x, y, z are the three Cartesian spatial coordinates. The square of the invariant Minkowskian length is given then by 1'R12 = c2t 2 _ x2 _ y2 - z2, (C.42)

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BIQUATERNIONS AND LORENTZ TRANSFORMATIONS

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where R has the special property

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Rt =R.

(C.43)

The dagger in (C.43) denotes the operation that Hamilton called biconjugation, which is nowadays called Hermitian conjugation. This operation is defined as

(C.44)

So (C.43) simply says that R is Hermitian. For a four-dimensional rotation to keep the Minkowskian length invariant, it is both necessary and sufficient that it preserve the hermicity of R. Suppose that the Minkowskian length remains invariant, then so does its square and we can write

c2t'2 _ x'2 _ y,2 _ Z'2

= ~t2 _

x 2 _ y2 _ z2

= IRI2

(C.45)

as R = R*. But as the rotation preserves thenormofR, IRI2 = IR'1 2, so that (C.45) implies that c2t'2 _ x,2 _ y,2 _ z,2 = IR'1 2. (C.46) Therefore, R' must be equal to R'* (i.e., if the rotation keeps the Minkowskian length invariant, it must also preserve the hermicity of R). To see that preserving the hermicity ofR is sufficient to keep the Minkowskian length invariant, notice that when the hermicity is preserved, c2t,2 _ xl2 _ y,2 _ z,2 = IR'1 2. (C.47) As the rotation preserves the norm, IR'I 2 implies that