192
7. Quantum Field Theory III
2
†
where ω = (m + k
2 )
1/2 . We wish to interpret ˆ
c (k) as the creation operator
s
for a Dirac particle of spin s and momentum k. By analogy with (7.16), we
expect that d ˆ† (k) creates the corresponding antiparticle. Presumably we must
s
define the vacuum by (cf (7.30))
ˆ
c ˆ s (k)|0> = d s (k)|0> = 0
for all k and s = 1, 2.
(7.36)
A two-fermion state is then
†
†
|k 1 , s 1 ; k 2 , s 2 > ∝ c ˆ (k 1 )ˆ c (k 2 )|0>.
(7.37)
s1
s2
But it is here that there must be a difference from the boson case. We require
a state containing two identical fermions to be antisymmetric under the exchange of state labels k 1 ↔ k 2 , s 1 ↔ s 2 , and thus to be forbidden if the two
sets of quantum numbers are the same, in accordance with the Pauli exclusion
principle, responsible for so many well-established features of the structure of
matter.
The solution to this dilemma is simple but radical: for fermions, commutation relations are replaced by anticommutation relations! The anticommutator
of two operators A ˆ and B ˆ is written:
{A, ˆ B ˆ } ≡ A ˆ B ˆ + B ˆ A. ˆ
(7.38)
If two different ˆ
c’s anticommute, then
†
†
†
†
c ˆ (k 1 )ˆ c s2 (k 2 ) + ˆ
c (k 2 )ˆ c s1 (k 1 ) = 0
(7.39)
s1
s2
so that we have the desired antisymmetry
|k 1 , s 1 ; k 2 , s 2 > = −|k 2 , s 2 ; k 1 , s 1 >.
(7.40)
In general we postulate
†
{c ˆ s1 (k 1 ), c ˆ (k 2 )} = (2π)
3 δ
3 (k 1 − k 2 )δ s1 s2
s2
(7.41)
†
†
{c ˆ s1 (k 1 ), c ˆ s2 (k 2 )} = {c ˆ (k 1 ), c ˆ (k 2 )} = 0
s1
s2
and similarly for the d ˆ ’s and d ˆ† ’s. The factor in front of the δ-function depends
on the convention for normalizing Dirac wavefunctions.
We must at once emphasize that in taking this ‘replace commutators by
anticommutators’ step we now depart decisively from the intuitive, quasimechanical, picture of a quantum field given in chapter 5, namely as a system
of quantized harmonic oscillators. Of course, the field expansion (7.35) is
a linear superposition of ‘modes’ (plane-wave solutions), as for the complex
scalar field in (7.16) for example; but the ‘mode operators’ ˆ
c s and d ˆ† are
s
fermionic (obeying anticommutation relations) not bosonic (obeying commutation relations). As mentioned at the end of section 5.1, it does not seem
possible to provide any mechanical model of a system (in three dimensions)
7. Quantum Field Theory III
2
†
where ω = (m + k
2 )
1/2 . We wish to interpret ˆ
c (k) as the creation operator
s
for a Dirac particle of spin s and momentum k. By analogy with (7.16), we
expect that d ˆ† (k) creates the corresponding antiparticle. Presumably we must
s
define the vacuum by (cf (7.30))
ˆ
c ˆ s (k)|0> = d s (k)|0> = 0
for all k and s = 1, 2.
(7.36)
A two-fermion state is then
†
†
|k 1 , s 1 ; k 2 , s 2 > ∝ c ˆ (k 1 )ˆ c (k 2 )|0>.
(7.37)
s1
s2
But it is here that there must be a difference from the boson case. We require
a state containing two identical fermions to be antisymmetric under the exchange of state labels k 1 ↔ k 2 , s 1 ↔ s 2 , and thus to be forbidden if the two
sets of quantum numbers are the same, in accordance with the Pauli exclusion
principle, responsible for so many well-established features of the structure of
matter.
The solution to this dilemma is simple but radical: for fermions, commutation relations are replaced by anticommutation relations! The anticommutator
of two operators A ˆ and B ˆ is written:
{A, ˆ B ˆ } ≡ A ˆ B ˆ + B ˆ A. ˆ
(7.38)
If two different ˆ
c’s anticommute, then
†
†
†
†
c ˆ (k 1 )ˆ c s2 (k 2 ) + ˆ
c (k 2 )ˆ c s1 (k 1 ) = 0
(7.39)
s1
s2
so that we have the desired antisymmetry
|k 1 , s 1 ; k 2 , s 2 > = −|k 2 , s 2 ; k 1 , s 1 >.
(7.40)
In general we postulate
†
{c ˆ s1 (k 1 ), c ˆ (k 2 )} = (2π)
3 δ
3 (k 1 − k 2 )δ s1 s2
s2
(7.41)
†
†
{c ˆ s1 (k 1 ), c ˆ s2 (k 2 )} = {c ˆ (k 1 ), c ˆ (k 2 )} = 0
s1
s2
and similarly for the d ˆ ’s and d ˆ† ’s. The factor in front of the δ-function depends
on the convention for normalizing Dirac wavefunctions.
We must at once emphasize that in taking this ‘replace commutators by
anticommutators’ step we now depart decisively from the intuitive, quasimechanical, picture of a quantum field given in chapter 5, namely as a system
of quantized harmonic oscillators. Of course, the field expansion (7.35) is
a linear superposition of ‘modes’ (plane-wave solutions), as for the complex
scalar field in (7.16) for example; but the ‘mode operators’ ˆ
c s and d ˆ† are
s
fermionic (obeying anticommutation relations) not bosonic (obeying commutation relations). As mentioned at the end of section 5.1, it does not seem
possible to provide any mechanical model of a system (in three dimensions)