DIRAC'S TRICK

Abstract. In this work we explore the possibility of using Dirac's trick,

namely, linearizing a dierential equation using a suitable Cliord algebra,
with equations other than those for relativistic spin 1/2 particles.The work-
ing hypothesis is that the resulting solutions could have a physical and/or
geometrical meaning.

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

Our equation can be re-written as

3
X
i~ + i~cei m0 c2 e0 = 0
t i=1
x i

2. Our proposal

(1) start with a linear dierential equation in variables (x, y, z, t).

(2) by Fourier transforming it get the dispersion relation D(k1 , k2 , k3 , ) = 0
(3) solve for one of the variables, say , in a suitable Cliord algebra.
(4) introduce a representation for the algebra in an Ideal (using primitive idem-
potents)
(5) get the dierential equation for the spinors
(6) solve it in simple cases
(7) establish connection, if any, with solutions of the original equation.
(8) see if the spinors have a geometrical and/or physical meaning.

3. Partial linearization of Schrdinger's equation (paraxial

approximation)

The one electron non-relativistic equation for motion in free space is

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 = ue1 + ve2 + vwe

e
then

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

5. Back to the equation.

Our equation can now be written as

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

As in quantum mechanics we seek solutions of the form

= 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

When u=v=0 there are still non-trivial

  solutions.
1
It is easy to see that the solution occurs i u=v=0 and k=w which cor-
0  
0
responds to forward propagation. Similarly the solution occurs i u=v=0
1
and k = w which corresponds to backwards propagation.