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84), it will clearly be helpful to expand (x) in a basis of eigenfunctions of hD . 3. There we found that i 0 @0 + i r ; m us (p)e;ip x = 0 so us (p)eip x are eigenfunctions of hD with eigenvalues Ep . Similarly, the functions vs (p)e;ip x (or equivalently, vs (;p)e+ip x) are eigenfunctions of hD with eigenvalues ;Ep . These form a complete set of eigenfunctions, since for any p there are two u's and two v's, giving us four eigenvectors of the 4 4 matrix hD . Expanding in this basis, we obtain Z d3p 1 X s s ap u (p) + bsp vs ( p) (3:87) (x) = (2 )3 p eip x 2Ep s=1 2 where asp and bsp are operator coe cients.

43). In practice it is often convenient to work with speci c spinors . A useful ; choice here would be eigenstates of 3 . For example, if = 10 (spin up along the 3-axis), we get u(p) = pE ; p3;1 pE + p3;01 0 p ;0 ;! 3 Free-Particle Solutions of the Dirac Equation while for = ;0 1 (spin down along the 3-axis) we have pE + p3;0 pE ; p3;10 ;! p 47 ;0 2E 01 : (3:53) 1 In the limit ! 1 the states degenerate into the two-component p spinors of a massless particle. 53) are eigenstates of the helicity operator, i (3:54) h p^ S = 21 p^i 0 0i : A particle with h = +1=2 is called right-handed, while one with h = ;1=2 is called left-handed.

Of course we are being a bit premature in talking about particles and spin. ) Now that we have the general form of u(p) in the rest frame, we can obtain u(p) in any other frame by boosting. Consider a boost along the 3-direction. First we should remind ourselves of what the boost does to the 4-momentum vector. In in nitesimal form, E m 0 1 p3 = 1 + 1 0 0 46 Chapter 3 The Dirac Field where is some in nitesimal parameter. For nite we must write E = exp m 0 1 3 1 0 p 0 m = cosh 10 01 + sinh 01 10 (3:48) 0 cosh = m m sinh : The parameter is called the rapidity.

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A Critical Experiment on the Statistical Interpretation of Quantum Mechanics by Ruark A.E.


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