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SECOND QUANTIZATION   19

for all k,k, λ,λ, see Equation (1.32), and in executing the double sum overλ and λ it does not matter in which order one does this. The h.c.  term in thesecond sum in Equation (1.62) is also zero, of course, because it is the Hermi-

tian conjugate of the one we just considered apart from the order of the oper-ators, but they commute as well. The symmetry argument (luckily) does nothold for the first sum in Equation (1.62) because that sum involves the prod-uct of  aλ(k)a

†λ

(k) and its h.c.  These do not commute, see Equation (1.31),so the trick employed in Equation (1.63) does not work for the first sum.

Using Equation (1.50) for the dot product of the polarization vectors andinserting the factor 1/(4π ) from Equation (1.58) we get

Pem  =  12

k,λ

aλ(k)a

†λ(k) + a

†λ(k)aλ(k)

k   (1.64)

=   12

k,λ a†λ(k)aλ(k) + 1k   (1.65)

=

k,λ

N λ(k) +

  12

k   (1.66)

=

k,λ

N λ(k)k   (1.67)

where we have used the commutation relations Equation (1.31) in Equation(1.64) and introduced the number operator  N λ(k) from Equation (1.35) in

Equation (1.65). The ‘zero point momentum’ evident in Equation (1.66)sums to zero if  k  has no preferred direction for an electromagnetic field ina vacuum, a reasonable assumption. We evade the zero point momentum,contrary to the case for the energy, see Equation (1.57).

The result in Equation (1.67) is equally remarkable as the result inEquation (1.57) for the same reasons. It shows that the momentum of the electromagnetic field is quantized, confirming that we may speak of ‘photons’. The quantum of momentum is  k   ( k   really) where  k   was firstintroduced in the plane wave solutions of the wave Equation (1.21). The

vector k  derives its physical meaning from that expression:   |k|  is the wavevector with magnitude 2π/λ. We conclude that the momentum    k   of aphoton is equal to   2π/λ  or p = h/λ, the deBroglie relation. We can use theEinstein relation  m2 = E2 − p2 to calculate the mass of the photon. UsingE =  ωk and  p =  k we find m

2 = 0. This is as expected because a ‘particle’can travel with the speed of light (in a vacuum) only if it is massless.

Also here we see that the choice of pre-factor in Equation (1.29) andEquation (1.30) is a good one because it leads again to a result in agreementwith experiment.

1.2.4 Polarization and Spin

The polarization factors ελ(k) and its complex conjugate define the polariza-tion as discussed earlier. A photon with momentum  k  along the z-axis that