QuantumPhysics.dvi

First of all, a free fermion may be expected to be equivalent to a collection of oscillators, just
as bosonic free fields were. But they cannot quite be the usualharmonic oscillators, because we
have just shown above that harmonic operators produce Bose-Einstein statistics. Instead, thePauli
exclusion principlestates that only a single fermion is allowed to occupy a givenquantum state.
This means that a fermion creation operatorb†for given quantum numbers must square to 0.
Then, its repeated application to any state (which would create a quantum state with more than
one fermion particle with that species) will produce 0.

The smallest set of operators that can characterize a fermion state is given by thefermionic
oscillatorsbandb†, obeying the algebra

{b,b}={b†,b†}= 0 {b,b†}= 1 (19.70)

In analogy with the bosonic oscillator, we consider the following simple Hamiltonian,

H=

ω

2

(

b†b−bb†

)

=ω

(

b†b−

1

2

)

(19.71)

Naturally, the ground state is defined byb| 0 〉= 0 and there are just two quantum states in this
system, namely| 0 〉andb†| 0 〉with energies−^12 ωand +^12 ωrespectively. A simple representation
of this algebra may be found by noticing that it is equivalentto the algebra of Pauli matrices or
Clifford-Dirac algebra in 2 dimensions,

σ^1 =b+b† σ^2 =ib−ib† H=

ω

2

σ^3 (19.72)

Vice-versa, theγ-matrices in 4 dimensions are equivalent to two sets of fermionic oscillatorsbαand
b†α,α= 1,2. The above argumentation demsonstrates that its only irreducible representation is
4-dimensional, and spanned by the states|∅〉,b† 1 |∅〉,b† 2 |∅〉,b† 1 b† 2 |∅〉.

In terms of particles, we should affix the quantum number of momentum~k, so that we now
have oscillatorsbσ(~k) andb†σ(~k), for some possible species indexσ. Postulating anti-commutation
relations, we have

{bσ(~k),bσ′(~k′)}={b†σ(~k),b†σ′(~k′)}= 0

{bσ(~k),b†σ′(~k′)}= 2ωkδσσ′(2π)^3 δ(3)(~k−~k′) (19.73)

where againωk=

√

~k (^2) +m (^2). The Hamiltonian and momentum operators are naturally

H =

∫

d^3 k

(2π)^32 ωk

ωk

∑

σ

b†σ(~k)bσ(~k)

P~ =

∫ d (^3) k
(2π)^32 ωk
~k
∑
σ
b†σ(~k)bσ(~k) (19.74)
The vacuum state|∅〉is defined bybσ(~k)|∅〉= 0 and multiparticle states are obtained by applying
creation operators to the vacuum,
b†σ 1 (~k 1 )···b†σn(~kn)| 0 〉 (19.75)