THE DIRAC EQUA nON in .NET framework

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THE DIRAC EQUA nON
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Dirac proposed that the (Xi and 13 matrices be anticommuting matrices of square equal to one: for i # k {(Xi, (Xk} = {(Xi,f3} = (Xl = 13 2 = I
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with the bracket {A, B} of two operators standing for the symmetric combination AB + BA, called the anticommutator. It is easy to verify that condition 1 is fulfilled:
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jp - 8i2l/J = i
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Let us introduce the notation yfl-:
yO =
and the Feynman "slash":
13 = f3(Xi
i = 1,2,3
(2-8)
{yfl-, yV} = 2gfl-V
(1= afl-yfl-
This enables us to rewrite the Dirac equation as
(iyfl- Ofl- - m)l/J
= (i~ -
m)l/J =
(2-9)
The Klein-Gordon equation is then obtained by multiplying by (i~ + m). Four is the smallest dimension in which matrices fulfilling (2-8) can be found.
The matrices CI! and f3 have eigenvalues equal to 1; for i # j, det aiClJ = det - aia i = (_l)d det aiai ; thus their dimension d must be even. Since for d = 2 there exist only three anticommuting hermitian matrices, the Pauli matrices, we have d ;0:: 4.
An explicit representation is provided by
. ( .
-(J'
(2-10)
in terms of the 2 x 2 unit I and Pauli (Ji matrices. This representation is useful when discussing the nonrelativistic limit of the Dirac equation.
Among all possible equivalent representations, obtained by a nonsingular transformation: l' -. U1'U- 1 , the Majorana representation plays a special role. It is designed so as to make the Dirac equation real. This is achieved by interchanging a2 and f3 and changing the sign of al and a3 in the previous
QUANTUM FIELD THEORY
representation:,xl = - (Xl>,x2 =
p, ,x3
= - (X3, /3 = (X2' Then only fJ is imaginary and the Dirac equation:
(:t + a. .
+ i/3 m )
'" = 0
is real; its solutions are linear combinations of real solutions. The matrix U which performs this change of representation and the new form of the y matrices may be easily determined '(see the Appendix).
In the four-dimensional representation (2-10), l/I may be written as a bispinor
l/I =
in terms of two-component spinors cp and X. For reasons that will soon
be clear, cp and X are referred to as the large and small components respectively. They satisfy
ocp i - = mcp
+ -:1
or, explicitly,
= - mx +
(2-11)
v cp
It is interesting to notice the similarity between these equations and two of the four Maxwell equations:
rotE + - = 0
oB at
rot B- - = 0
oE 1 i - = - S ' V(iB) at i
o(iB) 1 i-=-S'V(E)
where (Si)jk == (l/i)eijk' The spin matrices Si play for the spin 1 electromagnetic field the same role as the Pauli matrices (1 for the spin i and (E, iB) is analogous to (<p, X).
The main reason for the construction of the Dirac equation was to obtain a positive probability density p = l, together with a continuity equation 01'P'" = O. Since l/I is a complex spinor, p has to be of the form l/I t[lJtl/l in order to be real and positive. Let us first derive the Dirac equation for l/I t. From (2-9), we deduce
l/It(iytl' 01' + m) = 0
But yl't is easily expressed in terms of yl' :
yOt =
yt =
(f3IX)t
1Xf3
f3(f3IX)f3
yOyyO
Thus, introducing lj/: (2-12) we have
l/I(i it + m) = 0
(2-9a)
THE DIRAC EQUATION
Combining Eqs. (2-9) and (2-9a) leads to l/J( it + it)l/J oJl(l/Ji"l/J) = 0 We have, therefore, a candidate for the current
]Jl = l/JyJll/J
-+-+
. - {l =
~ = liIyol/J = l/Jtl/J = cptcp + xtx _ j = l/Jrl/J = l/Jtrxl/J = cptUX + Xtucp
(2-13)
The density p is positive. Small and large components contribute equally to p whereas j involves cross terms. We shall see below that jJl transforms as a Lorentz four-vector.
2-1-3 Relativistic Covariance
According to the relativity principle, we want to verify that the Dirac equation keeps its form in two frames related by a Poincare transformation. Alternatively, we require that to a system satisfying the equation with certain boundary conditions in a given frame we may associate by Poincare transformations a family of transformed states satisfying the same equation, with transformed boundary conditions. Sticking to the first point of view (independence with respect to the observer), we first remark that translation in variance is obvious. Consider now a Lorentz transformation A Let our system be described by the wave function l/J in the first frame and by l/J' in the transformed frame. Both must satisfy the Dirac equation:
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