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November 7, 2013 129
Before finishing this section, let us introduce the Feynman ‘slash’ notation :
if a µ is a Lorentz vector, we shall mean by /a its contraction with Dirac matrices:
/a ≡ a µ γ µ . (5.11)
The Dirac equation (5.7) can therefore also be written as 9
with the corollary that
/a/b + /b/a = 2 (a · b) ∀ a µ , b ν , (5.12)
/a/a = a 2 . (5.13)
We stress that the vector object a µ and the matrix /a encode exactly the same
information ; further on we shall see how the vector can be recovered once
the matrix is given. A few simple results, which can be checked by repeated
application of the anticommutation rule, are
5.2.2 The Clifford algebra
γ µ /a γ µ = −2 /a ,
γ µ /a /b γ µ = 4(a · b) . (5.14)
By the anticommutation relation (5.7), any product of more than four Dirac
matrices can be reduced to a smaller number. Let us define the enormously
useful object 10 γ 5 ≡ i γ 0 γ 1 γ 2 γ 3 , (5.15)
for which we can immediately derive that
γ 5 γ µ = −γ µ γ 5 , (γ 5 ) 2 = 1 . (5.16)
Also we can define the commutator of Dirac matrices as
σ µν ≡ i 2 [γµ , γ ν ] = i 2 (γµ γ ν − γ ν γ µ ) . (5.17)
Obviously there are 6 independent σ matrices. The most general object that
can be constructed using Dirac matrices is therefore
Γ = S 1 + V µ γ µ + T µν σ µν + A µ γ 5 γ µ + P γ 5 , (5.18)
and these objects form the Clifford algebra. We see that T (p) must be an
element of the Clifford algebra. The various coefficients are called, repectively,
the scalar (S), vector (V µ ), tensor (T µν ), axial-vector (A µ ) and pseudo-scalar
9 It is customary to leave the unit matrix 1 out of the notation. Its presence can always be
inferred where necessary.
10 In some texts the definition of γ 5 is slightly different, for instance it may lack the factor
i. Some care is necessary in comparing results between different texts. The reason why it is
called γ 5 and not γ 4 is that in some older treatments the Minkowski indices were assumed to
run from 1 to 4, with the 4 th index playing the rôle of our 0 th one.