Now, the four-momentum is a three-dimensional volume integral of the quaternion

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THE PHOTON'S WAVEFUNCTION

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(3.98)

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where 33 G(r[r') = Analogously,

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(3.100) If it were not for ytk2 + J.L2 in (3.99), G(r[r') would be a three-dimensional delta function and :Fa('R.) and A",('R.) would depend on Woo and ~a, respectively, at the space-time point 'R. only. But as it stands, the fields depend on the wavefunctions in a nonlocal way. This nonlocal dependency implies that the wavefunction can even vanish at a point in space where the energy density does not. So we cannot interpret the scalar component of the quaternion + J.L2~t~ at a point in space as the probability of finding the photon (the energy quantum) at that point,34 Moreover, as we will see in 5, the interaction with an electron, say, at a given point in space depends on the value of the fields at that point. But due to the nonlocal relation between the fields and the wavefunction, this interaction will be nonlocal in the wavefunction. So the electron will interact with the photon even if it sits at a place where the wavefunction vanishes (see Sec. 25(d) in [480]). This difficulty is not present in momentum space, where the fields depend on the wavefunction in a local way. It is perfectly legitimate to talk about the wavefunction of the photon in momentum space [12]. In fact, the photon wavefunction in momentum space is often used in high-energy physics. We can also see immediately that this problem does not exist in the nonrelativistic limit either. For then, the particle's momentum hk is always much smaller than J.L, so that G (r[ r') reduces to .,jii03 (r- r'), making the fields become local in Wand <P.

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3.4 BACK TO VECTOR NOTATION

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Quaternions are very powerful and we have been able to go quite a long way with them. For some purposes, however, as Cayley once put it [461], "a quaternion formula

33Another way to write (3.98) is to define the operator

{jJ.t2 + k 2 e ik. r , we can rewrite (3.98) as :Fa('R) =

VJ.t2 -

{jJ.t2 -

'\7 2

Then, as

{jJ.t2 -

'\72 e ik . r

'\7 2 Ilt a

('R.). See [387J.

34We will see later that this interpretation holds in a course-grained way [119, 311, 428J. It also holds in the paraxial approximation [151J.

BACK TO VECTOR NOTATION

is like a pocket-map that for use must be unfolded." Before going back to the more familiar vector notation, it is useful to find out which of the vectors in F and A are polar and which are axial. A polar vector remains the same when we change from a right-handed coordinate system to a left-handed one. An axial vector reverses its direction (i.e., goes into minus itself) under such a change of coordinates. To go from our right-handed coordinate system to a left-handed one, we can change Xl into -Xl. Let us denote the new coordinates and all quantities in the new coordinate system in general by adding a prime to their old names. The chainrule tells us that the partial derivatives in the new coordinates relate to those in the old ones by

{)t'

ax~ = -

aXI'

- aX2'

aX3 .

(3.101)

In quatemion notation, this can be written very compactly as

0' =

iIO*i l .

(3.102)

Now (3.42) and (3.43) cannot depend on our particular choice of coordinates, so they must remain valid in the left-handed coordinate system: that is,

O'F'

= ",2 A',

O'A' = F'.

(3.103) (3.104)

Substituting (3.102) into (3.103) and (3.104), multiplying by il on the left and on the right, and taking the complex conjugate, we find that

O(-I)ilF'*il

= ",2 ilA'*il,

(3.105) (3.106)

OiIA'*i 1 = -iIF'*il.

To recover (3.42) and (3.43) from (3.105) and (3.106), F and A must transform according t035

F = (-l)ilF'*il,

(3.107) (3.108)

= iIA'*i l

Now we notice that the imaginary part of36 f in (3.30) must be an axial vector, for it reverses direction when the coordinate system becomes left-handed (its il component remains the same and both its i2 and i3 components change sign; see Figure 3.1). The real part of f is a polar vector, for it remains unchanged when the coordinate system becomes left-handed (its il component changes sign and both its i2 and i3 components remain the same; see Figure 3.2). As to A, we can infer from its anti-Hermiticity that its time component must be a pure imaginary number while the coefficients of its space components must be real. So the vector part of A is a polar vector.

35If the minus sign in (3.105) and (3.106) were grouped with A and we identified -ilA' il instead of ilA' il with A. the time component of A would change sign when the left-handed coordinates are adopted, which does not make any sense. So the minus sign can only be incorporated in the transformation relation for:F as we do in (3.107) and (3.108). 36f is the vector whose Cartesian components a:re the in in (3.30).