Dirac Strick 2
Dirac Strick 2
Dirac Strick 2
1. Diracs trick
In a modern language, not the one Dirac himself used, the starting point is the
relativistic dispersion relationship for an electron:
E 2 = p2 c2 + m20 c4
Next we decide to work on the complex Cliord algebra Cl4 .1 This algebra is
generated by four vectors e0 , e1 , e2 , e3 subject to the relations
ei ej + ej ei
= ij
2
We dene a vector in Cl4 by means of
3
X
=
E pi cei + m0 c2 e0
i=1
Then it is trivial to deduce that
E
E = p2 c2 + m20 c4
=E 2
so
E is a square root of E2 in Cl4 .
The following step is to identify
E = i~
t
p = i~
3
X
E = i~ = pc + m0 c2 e0 = pi cei + m0 c2
t i=1
3
X
= i~cei + m0 c2 e0
i=1
xi
In the case of Dirac's equation this equations are then modied so that they are
expressed in an algebra Cl1,3 in order to end up with a relativistically invariant for-
mulation. We will not repeat that here. In the original formulation the members of
the Cliord algebra are replaced by a matrix representation and the wavefunctions
become spinors (single column matrices).
1
We are aware that fans of Geometric algebra would use a real Cliord algebra.
1
DIRAC'S TRICK 2
2. Our proposal
2mE
2 + =0
~2
where E is the electron energy.
Fourier transforming this equation we get
2m
| 2i k |2 ( k ) + 2 E( k)=0
~
where denotes a Fourier transform and
k = (u, v, w)
is the wave-vector. Calling
2mE
k2 =
h2
we have
| k |2 ( k ) + k 2 ( k ) = 0
so only waves for which
| k |2 +k 2 = 0
will be propagated. This means that
u2 + v 2 + w2 = k 2
This is the dispersion relation for the free electron. From the dispersion relation
p
w= k 2 u2 v 2
and expanding in Taylor series to rst order (high energy approximation, paraxial
approximation)
u2 + v 2
w=k
2k
The dierential equation with this dispersion relation is
1 1 2 2
( + 2 ) = k
2i z 8k 2 x2 y
DIRAC'S TRICK 3
or
i 2 2
( + 2 ) = (2i)k
z 4k x2 y
4. Linearization a la Dirac
Since
k 2 = w2 + u2 + v 2
we start with the Cliord algebra Cl3,0 with generators e1 , e2 , e3 that satisfy the
relations
e21 = 1
e22 = 1
e23 = 1
and the usual anti commutation relations for a Cliord algebra.
Dene
k 2 = u2 + v 2 + w2
e
so the Cliord number k
e is a root of k 2 = w2 + u2 + v 2 .
Next, as in Dirac's approach we make the substitutions
1
e
w
2i z
1
u
2i x
1
v
2i y
and the completely linearized equation becomes
1 e1 1 e2 1e3
k = + +
2i x 2i y 2i z
or
And, what does it mean to multiply by i?
As explained by many authors what we have to do is to move to a matrix
representation:
4.1. matrix representation. But it is well known that Cl3,0 is isomorphic to the
algebra of complex 22 matrices and a convenient representation is given by the
Pauli matrices, so without further ado we identify
0 1
e1 =
1 0
0 i
2 =
i 0
1 0
3 =
0 1
DIRAC'S TRICK 4
1 e1 1 e2 1e3
k = + +
2i x 2i y 2i z
or
1 0 1 1 0 i 1 1 0
k = + +
2i 1 0 x 2i i 0 y 2i 0 1 z
but now is to be understood as a spinor
1
=
2
6. The solutions
= e2i(ux+vy+wz)
for a xed spinor so
1 0 0 1 0 i 1 0
k =u +v +w
0 1 1 0 i 0 0 1
or
1 0 0 1 0 i 1 0
k u v w =0
0 1 1 0 i 0 0 1
that can be simplied to
k w u + iv
=0
u vi k + w
This equation can have non trivial solutions i
k w u + iv
det =0
u vi k + w
k 2 u2 v 2 w 2 = 0
in agreement with the dispersion relation.
The generic element of the null space of
k w u + iv
u vi k + w
is
(u iv, k w)
or, alternatively,
(k + w, u + iv)
and
u iv
= e2i(ux+vy+wz)
kw
(except for normalization factors). For the special cases u=v=0 and k = w see
below.
It can be shown that the components of these solutions are also solutions of the
original equation (but the converse is not true). The proof will be omitted but is
the same as the proof that the components of the solutions of Dirac's equation are
also solutions to the Klein-Gordon equation.
DIRAC'S TRICK 5
7. Speculation