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However, in the discussion of, for instance, photon scattering by an electron in a free space, such an artificial volume should not play any role. If we wish, we could have avoided the V, as was done in § 1 (if, of course, the Dirac delta-function is accepted). The reason why the volume V is introduced in our argument is to show that Planck's formula can be reproduced. We shall now briefly indicate how the limit of an infinite volume can be taken. Recall first the familiar formulae lim 7o V . 89) 33 An Introduction to Field Quantization [Ch.

Hence there are two unit and one zero eigenvalues. 81) / l 0 0\ / = 10 1 0 I. 88) and χ is an arbitrary angle between 0 and 2π. The first and second columns are components of e{1)(k) and ei2)(k) respectively. We have so far considered the radiation field enclosed in a finite volume V. For a system such as the cavity radiation or lattice vibrations of a solid with a finite volume which will be discussed in the next section, the volume V has a realistic physical significance. However, in the discussion of, for instance, photon scattering by an electron in a free space, such an artificial volume should not play any role.

50) t For an alternative formulation of thefieldwith spin 2, see Nath (1965). 51) Here, m is a real non-zero constant. If we define UiXx) = U+(x)gaœQV9 then ΰλν(χ)Λλνισρ(-§) = 0. Since ϋλν(χ) is a tensor, UxAx) - %(*') = a^U^x), under the Lorentz transformation Χμ ""*" ·*μ = (ΙμνΧν. 56) in the form U'ae(x') = δσξδ2η -f ly- SMVt whence with σρξηοωμν υξη{χ), 8μν, σρξη = S ^ σξΟρη + S ^ ρη0σξ δμν, σρ = ~ Κ^μσ^νρ ~ ^μράνσ). 61) 53 An Introduction to Field Quantization [Ch. 60) will be discussed at the end of the next chapter.

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