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15) 2 2 6(kn)ô(k*+m*) = β(-*ο) 5(Ä; +/w ). 16) It is now clear that if φ(χ) is constructed as an arbitrary superposition of plane waves, the associated fields φ(+\χ) and φ{"\χ) consist of the positive and negative frequency parts respectively. As this is an invariant decomposition we refer to φ^+\χ) and φ^~\χ) as the positive and the negative frequency part respectively. 17) 49 An Introduction to Field Quantization and A™(x) = (2π)~3 fd* with [Ch. 18) ί 1 ko > 0 = \ [ — 1 ko < 0. 22) 2 ζ1 (χ) ^ 0 for x > 0, doA(x)\Xo=0 = -d(x).

60) will be discussed at the end of the next chapter. 4. 63) with is known as the Duffin-Kemmer(-Petiau) equation, where ψ(χ) has sixteen components and the βμ are four 16X16 hermitian matrices. The algebra of the βμ matrices has three irreducible representations and these are one-, five- and ten-dimensional representations respectively. Consequently the field ψ(χ) consists of three irreducible fields, the first of which is trivial (ip(x) = 0), while the other two are spin 0 and spin 1 fields respectively.

In other words, if f(x) and g(x) are arbitrary functions f(x)d,g(x)^[dMf(X)]g(x), f(x)(eM+dM) g(x) = eMWx) g(x)]. 4> 45 [Ch. 6) will often appear. A space-like surface in Minkowski space is denoted by σ, or, if we wish to emphasize that the surface passes through a specific point x9 by a(x). It will prove convenient to define & four-vector differential surface area at the point x by άαμ{χ) = (dx2 dxs dxo, dx± dxz dx0, dx\ dx2 dxo, — idx\ dx2 dx3) . 7) — ι 3! \e/ivXQ\dxvdxxdx{ e> where εμνηρ is the Levi-Civita symbol, b MvXQ — 1 —1 0 if (μνλρ) = even permutation of (1234), if (μνλρ) = odd permutation of (1234), otherwise.