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THE DIRAC EQUATION in .NET framework
THE DIRAC EQUATION PDF 417 Decoder In .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. PDF417 2d Barcode Encoder In Visual Studio .NET Using Barcode maker for Visual Studio .NET Control to generate, create PDF 417 image in Visual Studio .NET applications. discarding the KleinGordon equation no longer hold. As we shall see, it actually describes spinless particles, such as pions. However, the hole interpretation is not satisfactory for bosons, since Fermi statistics plays a crucial role in Dirac's argument. Even for fermions, the concept of an infinitely charged unobservable sea looks rather queer. We have instead to construct a true manybody theory to accommodate particles and antiparticles in a consistent way. This will be achieved by the "second quantization," i.e., the introduction of quantized fields capable of creating or annihilating particles. Recognizing PDF417 2d Barcode In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Bar Code Generation In .NET Framework Using Barcode maker for VS .NET Control to generate, create barcode image in .NET applications. 242 Charge Conjugation
Scan Barcode In VS .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications. PDF 417 Generation In C# Using Barcode creation for .NET framework Control to generate, create PDF417 image in VS .NET applications. Hole theory implies the existence of electrons and positrons with the same mass and opposite charges which obey the same equation. The Dirac equation must therefore admit a new symmetry corresponding to the interchange particle ~ antiparticle. We thus seek a transformation !/J + !/Jc reversing the charge, i.e., such that Draw PDF 417 In .NET Framework Using Barcode printer for ASP.NET Control to generate, create PDF 417 image in ASP.NET applications. Draw PDF 417 In VB.NET Using Barcode generator for .NET framework Control to generate, create PDF 417 image in .NET framework applications. m)!/J = 0 (i + ej  m)!/Jc = 0
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(297) For instance, in the representation (210), C may be taken as
(297a) (297b) We then identify
!/Jc as
(298) with 1'/c an arbitrary unobservable phase, generally taken as being equal to unity. In the present framework charge conjugation is an anti linear transformation. This is consistent with the hole interpretation, since when computing a transition probability the presence of a particle in a certain state will be represented by !/J and its absence by !/J*. Let us examine more closely the properties of this charge conjugation. We compute !/Jc for !/J describing a spindown negative energy QUANTUM FIELD THEORY
electron at rest. In the absence of external field
Therefore, the charge conjugate of a negative energy spindown electron is indeed equivalent to a positive energy spinup electron. For an arbitrary solution t/J of energy momentum p polarized along n, we know that t/J = ep + m 1 + Y5~ t/J 2m 2
where e = 1 denotes the sign of the energy. Since C commutes with in the Dirac representation, Y5 = t/JC = clIfT = Cyo
C t/J* +2Y5~)* (299) (e~: m)C \Y5~)t/JC
t/JC is described by the same fourvectors p and n, but the sign of the energy has been reversed. Using the notations (248), we have u(p, n) = lJ(p, n)vC(p, n) (2100) v(p, n) = lJ(p, n)uC(p, n) where the phase IJ(P, n) may depend on p and n. We recall that the projector (1 + Y5~)/2 projects onto spin states i along n according to the sign of the energy. Thus the spin is reversed by charge conjugation. We note, furthermore, that under a common transformation on the spinor t/J and the potential A, (298a) the Dirac equation (296) remains unchanged.
The transformation law of the fourvector current under charge conjugation is
j~ = lj/y"tjI> j~ = lj/'y"tjI' VCy"cli
Vy;lj/T
We could naively conclude that J~ = lj/y~tjI = jw However, we shall see in the next chapter that tjI and Ij/ have to be considered as anticommuting operators (FermiDirac statistics). Therefore charge conjugation will reverse the sign ofj" and leave ej' A unchanged. THE DIRAC EQUATION
We shall explore the intricacies of the last discrete symmetry, namely time reversal, in Chap. 3, when we have a satisfactory formulation of particles and antiparticles. 243 ZeroMass Particles
In Sec. 221, while constructing the spinor solutions of the free Dirac equation, we have discarded the massless case m = O. However, the neutrinos are massless spin i particles. In addition, we expect that at very high energy massive particles behave as massless. Therefore we reexamine this case. We start from the massless Dirac equation (2101) 5yO=  iy 1ly3 yields Multiplying this equation by y L pl/J = y5 pOl/J
(2102) since, for instance, (y5 yO)y1 = iy2 y3 = (}"23 == The chirality operator Y5 anticommutes with p. For a positive energy solution l/J(x) = eik,xl/J(k)

