THE DIRAC EQUATION in .NET framework

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THE DIRAC EQUATION
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discarding the Klein-Gordon 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 many-body 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.
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2-4-2 Charge Conjugation
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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
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m)!/J = 0 (i + ej - m)!/Jc = 0
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(i - ej -
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(2-96)
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We demand that this transformation be local and that its square amount at most to mUltiplying !/J by an unobservable phase. To construct !/Jc we conjugate and transpose the first equation and get
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[yIlT( - iO Il - eA Il ) - m]f!7 = 0
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with lfJT = yOT !/J*. In any representation of the y algebra there must exist a matrix C which satisfies .
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Cy:C- 1
-Yll
(2-97)
For instance, in the representation (2-10), C may be taken as
(2-97a) (2-97b)
We then identify
!/Jc as
(2-98)
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 spin-down negative energy
QUANTUM FIELD THEORY
electron at rest. In the absence of external field
Therefore, the charge conjugate of a negative energy spin-down electron is indeed equivalent to a positive energy spin-up 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~)*
(2-99)
(-e~: m)C \Y5~)t/JC
t/JC is described by the same four-vectors p and n, but the sign of the energy has been reversed. Using the notations (2-48), we have
u(p, n) = lJ(p, n)vC(p, n)
(2-100)
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,
(2-98a)
the Dirac equation (2-96) remains unchanged.
The transformation law of the four-vector 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 (Fermi-Dirac 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.
2-4-3 Zero-Mass Particles
In Sec. 2-2-1, 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 (2-101) 5yO= - iy 1ly3 yields Multiplying this equation by y
L pl/J = y5 pOl/J
(2-102)
since, for instance, (y5 yO)y1 = iy2 y3 = (}"23 == The chirality operator Y5 anticommutes with p. For a positive energy solution
l/J(x) = e-ik,xl/J(k)
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