From Special Relativity to Feynman Diagrams.pdf

L μ φ ν φ l μ φ ν φ ? μν l 1049 where l μ

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L ∂(∂ μ φ) ν φ + L ∂(∂ μ φ ) ν φ η μν L , (10.49) where L ∂(∂ μ φ) = c 2 μ φ ; L ∂∂ μ φ = c 2 μ φ. Substituting in ( 10.49 ) we find: T μν = c (∂ μ φ ν φ + ν φ μ φ) η μν L c . (10.50) In particular we may verify the identity between energy density cT 00 and Hamiltonian density: T 00 = 1 c ( 2 ˙ φ ˙ φ L ) = 1 c ˙ φ ˙ φ + c 2 φ · φ + m 2 c 4 2 | φ | 2 = H c . that is H = c d 3 x T 00 = d 3 x ππ + c 2 φ · φ + m 2 c 4 2 | φ | 2 . (10.51) As far as the momentum of the field is concerned we find P i = d 3 x ˙ φ i φ + ˙ φ∂ i φ P = − d 3 x ( π φ + π φ ) . (10.52) 10.4 The Dirac Equation In the previous sections we have focussed our attention on a scalar field, whose distinctive property is the absence of internal degrees of freedom since it belongs to a trivial representation of the Lorentz group. This means that its intrinsic angular momentum, namely its spin , is zero. We have also studied, both at the classical level and in a second quantized setting, the electromagnetic field which, as a four-vector, transforms in the fundamental representation of the Lorentz group. Its internal degrees of freedom are described by
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10.4 The Dirac Equation 319 the two transverse components of the polarization vector. At the end of Chap.6 we have associated with the photon a unit spin: s = 1 (in units of ). As explained there, by this we really mean that the photon helicity is = 1. Our final purpose is to give an elementary account of the quantum description of electromagnetic interactions. The most important electromagnetic interaction at low energy is the one between matter and radiation. Since the elementary building blocks of matter are electrons and quarks, which have half-integer spin ( s = 1 / 2 ) , such processes will involve the interaction between photons and spin 1/2 particles. It is therefore important to complete our analysis of classical fields by including the fermion fields, that is fields associated with spin 1/2 particles. In this section and in the sequel we discuss the relativistic equation describing particles of spin 1/2, known as the Dirac equation. 10.4.1 The Wave Equation for Spin 1/2 Particles Historically Dirac discovered his equation while attempting to construct a relativistic equation which, unlike Klein–Gordon equation, would allow for a consistent inter- pretation of the modulus squared of the wave function as a probability density. As we shall see in the following, this requirement can be satisfied if, unlike in the Klein– Gordon case, the equation is of first order in the time derivative . On the other hand, the requirement of relativistic invariance implies that the equation ought to be of first order in the space derivatives as well. The resulting equation will be shown to describe particles of spin s = 1 2 .
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