THE DIRAC EQUATION in .NET

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THE DIRAC EQUATION
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To combine relativistic invariance with quantum mechanics let us return to the correspondence principle. In the usual configuration space representation of quantum mechanics, we associate the operators ih(ojot) and (hji) Vi = (hji)(ojoXi) to the energy E and momentum pi respectively. For a free massive particle, the energy is given in terms of the momentum by
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in the nonrelativistic picture in the relativistic case
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(2-2a) .(2-2b)
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Unless explicitly specified, we shall use the convenient system of units such that h=c=l. In the same way that the correspondence principle transforms Eq. (2-2a) into the Schrodinger eq~ation for the wave function l/J(x, t) = <x, t!l/J):
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i at l/J(x, t) = - 2m l/J(x, t)
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it leads, in the relativistic case, from Eq. (2-2b) to the Klein-Gordon equation:
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2 2) ,o/(x, t) = 0 I,
01/1
(2-3)
Although this equation does not have the Schrodinger-like form (2-1), we may remedy this by casting it in a matrix form. Introducing the notations
1/1-1 ==ml/l
the vector
I/Ii) ==
i = 1,2,3
{I/I.} (ex
-1,0, ... , 3) satisfies i
0% = (mf3+~<x.v)1/I ot
for a suitable set of 5 x 5 hermitian matrices. The reader may explicitly write this set of matrices and obtain an auxiliary condition in order to reproduce a set of equations equivalent to Eq. (2-3).
If we want to interpret l/J as a wave function, we have to find a nonnegative norm, conserved by/the time evolution. There does indeed exist a continuity equation:
where the four-vector j/l == (l
+ div j == a/If' =
(2-4)
p,/) is defined as
2~ (l/J* ~ l/J 21m
tt* l/J)
(2-5)
~ [l/J*Vl/J -
(Vl/J)*l/JJ
QUANTUM FIELD THEORY
In integral form we have equivalently
-~ at
d xp
dS j
which expresses that the change in the total "charge" inside the volume V corresponds to the flux of j through the surface S enclosing V. However, the density p is not positive definite. Therefore, it may well be considered as the density of a conserved quantity (the electric charge, for instance), but not as a positive probability. A second problem arises when we realize the existence of negative energy solutions. Any plane wave function l/J(x, t)
e-i(Et-p'x)
satisfies Eq. (2-3), provided E2 = p2 + m 2 . Thus negative energies E = p2 + m 2 are on the same footing as the physical ones E = p2 + m 2 This is a severe difficulty because the spectrum is no longer bounded from below. It seems that an arbitrarily large amount of energy may be extracted from the system. For a particle initially at rest, this will be the case if an external perturbation allows it to jump over the energy gap I1E = 2m between the positive and negative continuum of states. This is clearly a failure of the concept of stable stationary states. These reasons seemed at a time so overwhelming that they led Dirac to introduce another equation. Although the latter has a positive norm, we shall ultimately have to face the same problems of physical interpretation of the negative energy states. At that stage, we shall come back to the Klein-Gordon equation and recast our relativistic quantum mechanics as a many-body theory, where the negative energy states may be interpreted in terms of antiparticles.
2-1-2 The Dirac Equation
Since the Klein-Gordon equation was found physically unsatisfactory, we shall try to construct a wave equation
(2-6)
where l/J is a vector wave function and Qt, f3 are hermitian matrices to make H hermitian, such that a positive conserved probability density exists. We now insist on the three following points:
1. The components of l/J must satisfy the Klein-Gordon equation, so that a plane wave with E2 = p2 + m 2 is a solution.
2. There exists a four-vector current density which is conserved and whose fourth component is a positive density. 3. The components of l/J do not have to satisfy any auxiliary condition, namely, at a given time they are independent functions of x. We shall also have to verify the relativistic covariance of this formalism.
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