-
PREVIOUS LECTURE RECAPDirac equation
(@ + ) = 0
(@@x
+ mc~ ) = 0
p ! p eAcp = i~rA = ( ~A; iA0)
1 =
0 11 0
2 =
0 ii 0
3 =
1 00 1
Thus we take the matrices as (4 4)
1
-
matrices dened as
= 4 =
I 00 I
k =
0 kk 0
k = ik =
0 ikik 0
This equation make sense only if theWave Function is a four
component col-umn matrix.
=
0BB@ 1 2 3 4
1CCADirac equation
2
-
( + ) = 0
using the summation convention can bewritten as
(1@1 + 2@2 + 3@3 + 4@4 + ) = 0
1
@@x + 2
@@y + 3
@@z + 4
@@(ict)
+ =
0Complete form of Dirac Matrices
When written completely by substitut-ing the Pouli Spin Matrices
The gammamatrices actually takes the form
3
-
1 =
0BB@0 0 0 i0 0 i 00 i 0 0i 0 0 0
1CCA 2 =0BB@
0 0 0 10 0 1 00 1 0 01 0 0 0
1CCA
3 =
0BB@0 0 i 00 0 0 ii 0 0 00 i 0 0
1CCA 4 =0BB@1 0 0 00 1 0 00 0 1 00 0 0 1
1CCA
Substituting these matrices in the diracequation we get0BB@
@4 0 i@3 i@1 @20 @4 i@1 + @2 i@3i@3 i@1 + @2 @4 0
i@1 @2 i@3 0 @4
1CCA4
-
0BB@ 1 2 3 4
1CCA + 0BB@ 1 2 3 4
1CCA = 0Thus the Dirac equation gives four cou-pled rst order
dierential equations.
(@4+) 1+ i@3 3+(i@1+@2) 4 = 0(@4+) 2+(i@1@2) 3i@3 4 = 0i@3 1+
(i@1+ @2) 2 (@4 ) 3 = 0(i@1 @2) 1 i@3 2 (@4 ) 4 = 0Conserved
current-Representation in-dependence
1 Conserved current
The main motivation behind the deriva-tion of Dirac equation has
been the factthe Klein Gordon equation is not giv-
5
-
ing a positive denite probability den-sity as required by the
basic postulatesof quantum mechanics. In the follow-ing we show
that Dirac equation actu-ally gives a positive denite
probabilitydensity. An adjoint Dirac spinor ( )in contrast to the
Hermitian conjugatespinor (y) is dened by = y4.They can be
represented as the follow-ing row matrices.
y = (1;2;3;4); = (1;2;3;4)(1)
Normally is considered to be an in-dependent spinor dierent from
. Toobtain its wave equation rst of all theHermitian conjugate of
Dirac equationis taken.@
@xkyk+
@
@xky4+
mc
~y = 0 (2)
6
-
The adjoint Dirac equation is obtainedby multiplying the above
equation fromthe right by 4
@@x
+mc
~ = 0 (3)
As in the case of non-relativistic quan-tum mechanics we
multiply the originalDirac equation from the left by andthe adjoint
equation from the right by and subtract to get
@
@x
( ) = 0 (4)
The ux density four vector in Diractheory can therefore be
written as
s = ic = (cy; icy) (5)
which satises the continuity equation@s@x
= @s = 0: (6)
7
-
It is clear that the quantity
= y =
4X=1
(7)
is positive denite unlike the case of KleinGordon theory. Thus
in Dirac theorythe problem of negative probability den-sity is
eliminated. The probability den-sity is 4 =
y and the quantitysk = ick = c
yk is the uxdensity.
2 Representation independence
Representation independence of Diracequation means that Dirac
equations withdierent form of gamma matrices areequivalent if the
gamma matrices sat-isfy the dening property (ie.) anti-commutation.
Let the Dirac equation
8
-
be written in the form
0
@
@x+mc
~
0 = 0 (8)
In the above equation 0 satises thedening property of
gammamatrices (ie.)theyare 44 matrices satisfying f0; 0g =2 with =
1 : : : 4. By represen-tation independence we mean the as-sertion
that the equation given above
is equivalent to
@@x
+ mc~
= 0
which is the original Dirac equation. Dif-ferent possible sets
of 44 matrices sat-isfying the anti-commutation relation givenabove
are said to be sets of gamma ma-trices in dierent representations.
Therepresentation independence can be provedby using Pauli's
fundamental theoremwhich states that two sets of 44 matri-ces
satisfying the anti-commutation re-
9
-
lations f; g = 2 and f0; 0g =2 with ; = 1 : : : 4 there exists
anonsingular 4 4 matrix S such that
SS1 = 0 (9)
where the matrix S is unique (up toa multiplicative constant).
The Diracequation in the new representation cannow be written
as
SS1 @@x
+mc
~
SS10 = 0
(10)Multiplying from the left by S1 we get
@
@x+mc
~
S10 = 0 (11)
which is same as the original Dirac equa-tion with the solution
given by S0.therefor equation [8] is equivalent to theDirac
equation and the wave functions
10
-
in both representations are related by
0 = S (12)According to Pauli's fundamental theo-rem S can be
chosen to be unitary if 0are Hermitian. The unitarity of the
ma-trix S implies the equivalence of prob-ability density and ux
density in bothrepresentations as seen below.
000 = 0y0400
= ySyS4S1SS1S= (13)
This clearly shows that all the physicalconsequences are same in
both repre-sentations. But the wave functions indierent
representations look dierent.Most common gammamatrices used arethe
following
1. The standard Dirac Pauli represen-11
-
tation used in the present treatment.
2. The Weyl representation in which kand 4 are o diagonal
matrices.
3. The Majorana representation in which
k are purely real and 4 is purelyimaginary.
12
