To see the Lorentz invariance, recall that for the Pauli matrices,
σ
1
and
σ
3
are real, and
σ
2
is
imaginary.
σ
1
=
parenleftbigg
0
1
1
0
parenrightbigg
, σ
2
=
parenleftbigg
0
−
i
i
0
parenrightbigg
, σ
3
=
parenleftbigg
1
0
0
−
1
parenrightbigg
(128)
So,
σ
1
⋆
=
σ
1
, σ
2
⋆
=
−
σ
2
, σ
3
⋆
=
σ
3
(129)
σ
1
T
=
σ
1
, σ
2
T
=
−
σ
2
, σ
3
T
=
σ
3
(130)
This implies
σ
1
T
σ
2
=
σ
1
σ
2
=
−
σ
2
σ
1
,
σ
3
T
σ
2
=
σ
3
σ
2
=
−
σ
2
σ
3
and
σ
2
T
σ
2
=
−
σ
2
σ
2
. That is,
σ
j
T
σ
2
=
−
σ
2
σ
j
(131)
and so
δ
(
ψ
R
T
σ
2
) =
1
2
(
iθ
j
+
β
j
)
ψ
R
T
σ
j
T
σ
2
=
1
2
(
−
iθ
j
−
β
j
)(
ψ
R
T
σ
2
)
σ
j
(132)
Combining this with the transformation of
ψ
R
in Eq. (125) we see that
L
Maj
in Eq. (127) is
Lorentz invariant.
Since
σ
2
=
parenleftbigg

i
i
parenrightbigg
the Majorana mass can be expanding out to
ψ
R
T
σ
2
ψ
R
= (
ψ
1
ψ
2
)
parenleftbigg
−
i
i
parenrightbiggparenleftbigg
ψ
1
ψ
2
parenrightbigg
=
−
i
(
ψ
1
ψ
2
−
ψ
2
ψ
1
)
(133)
Thus we have shown that
ψ
1
ψ
2
−
ψ
2
ψ
1
is Lorentz invariant. We often write this as
ψ
1
ψ
2
−
ψ
2
ψ
1
=
ψ
α
ψ
β
ε
αβ
, ε
αβ
=
parenleftbigg
0
1
−
1
0
parenrightbigg
(134)
which avoids picking a
σ
2
.
There’s only one problem: if the fermion components commute
ψ
1
ψ
2
−
ψ
2
ψ
1
= 0
! For Majo
rana masses to be nontrivial, fermion components can’t be regular numbers, they must be anti
commuting numbers. Such things are called
Grassmann
numbers and satisfy a Grassmann
algebra. Further explanation of why spinors must anticommute is given in the Lecture II5, on
the spinstatistics theorem. Grassmann numbers are discussed more in Lecture II7 on the path
integral.
7.2
Notation for Weyl spinors
In QED, we will be mostly interested in Dirac spinors, like the electron. But since Weyl spinors
correspond to irreducible representations of the Lorentz group, it is sometimes helpful to have
concise notation for constructing products and contractions of Weyl spinors only. This notation
is useful in many contexts besides gauge theories, such as supersymmetry. It is also related to
the spinorhelicity formalism we will discuss in Lecture IV3. If you’re just interested in QED,
you can skip this part.
18
Section 7
Let us write
ψ
for lefthanded spinors and
ψ
˜
for righthanded spinors. Sometimes the nota
tion
ψ
is used, especially in the contexts of supersymmetry, but this can be confused with the
bar notation for a Dirac spinor,
ψ
¯
≡
ψ
†
γ
0
, so we will stick with
ψ
˜
. We index the two compo
nents of lefthanded Weyl spinors with Greek indices from the beginning of the alphabet, i.e.
ψ
α
. For righthanded spinors, we use dotted Greek indices, like
ψ
˜
α
˙
. A Dirac spinor is
ψ
=
parenleftBigg
ψ
α
ψ
˜
β
˙
parenrightBigg
(135)
Conventionally, lefthanded spinors (and righthanded antispinors) have upper undotted indices
and righthanded spinors (and lefthanded antispinors) have lower dotted indices.
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 Summer '19
 mechanics, Quantum Field Theory, Lie group, lorentz group