THE PHOTON'S WAVEFUNCTION

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instead of matrices and spinors. We use a four-sided diamond to represent the fourdimensional biquatemion gradient 18

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0== - - + L i n - ,

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(3.29)

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and as we know from experiment that tight has spin 1, we let the wavefunction be a pure biquatemion (Le., a vector wavefunction)

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F== Linifn'

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(3.30)

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where i is the ordinary commutative imaginary number, not to be confused with iI, i2, and i 3 , the generalized noncommutative imaginary quantities of quatemion theory. Notice that 2 1 82 (3.31) 0 = 0 0 = \1 - e2 8t 2 = 0,

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where 0 is the D' Alembertian. 19 So if we write (3.24) in the form

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n OOF=memeF,

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it is tempting to try the factorized equation

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(3.32)

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OF=pF,

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where

(3.33) (3.34)

P==r;'

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But (3.33) is not covariant. Under a Lorentz transformation, 0 transforms as 20 a four-vector: that is, U out, where U is a biquatemion such that OU = 1.

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(3.35)

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Now then, to tum OF into 0' fI, we must multiply it on the left by U, on the right by T, and F' must be obtained from F by F' = U* FT, where T is still to be determined: UOFT

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= UO utU* FT

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=O'F'.

---1

(3.36)

ISIn Appendix C, we only talk about the quatemion gradient (i.e., a gradient when the coefficients of the quaternion 'R = Xo + Ln inxn describing a point in four dimensional space are all real). For Minkowsky's space-time. we use biquaternions (last section of Appendix C) (i.e., use complex coefficients, with Xo = ct and Xl = ix, X2 = iy, X3 = iz in the previous expression for'R). Then it is convenient to define the biquaternion gradient as i multiplied by the gradient defined in Appendix C. 19Sometimes the D' Alembertian is written as 0 2 , perhaps because it is composed of square derivatives (i.e., second derivatives). But we feel that the superscript 2 is an unnecessary emphasis, for a square is already a square. 20See Appendix C or [506] or 9 of Lanczos's book on mechanics [386].

EXTREME QUANTUM THEORY OF LIGHT WITH A TWIST

Then we see that regardless of what T is, the right-hand side of (3.33) does not transform properly unless U is a quaternion with real coefficients rather than a biquaternion for U" to be equal to U. But if U* = U, the transformation can only be an ordinary spatial rotation (Le., the time component will be left unchanged).21 If our wave equation is to transform accordingly with respect to a general Lorentz transformation that also transforms the time, we must have another biquaternion multiplying F on the right-hand side of (3.33) (3.37) where

=uxut,

(3.38)

(Le., X must transform like 0). Moreover, if we multiply (3.37) by we must recover (3.24); that is,

0 from the left,

OOF = JLOXF

JL 2 F.

(3.39) (3.40) (3.41)

Then

OX =JL

and X must be given by Let us define A such that F

= OA; then the wave equation we are looking for is

OF = JL2 A,

(3.42) (3.43)

OA =:F.

These are LanclOs's equations. 22 Instead of a single biquaternion wavefunction candidate F, we were forced by relativity to have two: F and A. To find out how they transform, we must now determine T. We want F to remain a pure biquaternion in all Lorentz frames; otherwise, this formulation will not be covariant. The simplest way for this to hold is if our F is what is called a six-vector. For instance, T cannot be U; otherwise, F will transform as the conjugate of a four-vector and will develop a scalar component in some Lorentz frames. What we want is to find a T such that for a given biquaternion 2 = Zo + E~=l inzn, the same biquaternion 21 = U* 2T in a different Lorentz frame will have its scalar component unchanged (Le., z~ = Zo must be a true scalar). The simplest solution (i.e., where F is a six-vector) turns out23 to be T = ut

21 See Appendix C. 22 Actually. Lanczos's equations are more general in that the wavefunction :F does not need to be a pure biquaternion and can transform in a different way. Lanczos's equations can describe all scalar. all vector. and all pseudovector particles. 23This can be seen by noticing that (3.35) also implies that Ua = 1. Taking the complex conjugate of the former equation and of (3.35). we find that U t U 1 and U* ut = 1. Then Zo is not changed by