3 Fermions, bosons and interactions
In this unit we introduce the elementary particles that make up the Standard Model of particle
physics. We define their property using a quantum mechanics and how the interact with each
other by exchanging quanta of information (e.g. other particles).
3.1 Fermions and Bosons statistics
Learning outcome: understand the difference between fermions and bosons, and how they
behave
A multi-particle wave function for non-interacting (e.g. widely separated) particles can be
written as the product of single particle functions. For instance, for a system of three particles
Ψ(1, 2, 3) = 𝑐 Ψ𝐴 (1)Ψ𝐵 (2)Ψ𝐶 (3)
where A, B, C describe the quantum numbers of the state and 1, 2, 3 give the co-ordinates of
the particle. (c is simply a normalisation constant.) Observables are given by the square of
the wave function |Ψ|2 .
If we consider a number of identical, indistinguishable particles, then interchanging a pair of
particles is unobservable.
|Ψ(2, 1, 3)|2 = |Ψ(1, 2, 3)|2
In terms of wave function, this has two possible solutions
Ψ(2, 1, 3) = Ψ(1, 2, 3) (1)
Ψ(2, 1, 3) = −Ψ(1, 2, 3) (2)
These two cases have important physical consequences. E.g. if two particles are in an identical
quantum states
Ψ(1, 2) = 𝑐 Ψ𝐴 (1)Ψ𝐴 (2)
case (1) implies
𝑐1 𝛹𝐴 (1)𝛹𝐴 (2) = 𝑐1 𝛹𝐴 (2)𝛹𝐴 (1)
which is satisfied by any c1, case (2) implies
𝑐2 𝛹𝐴 (1)𝛹𝐴 (2) = −𝑐2 𝛹𝐴 (2)𝛹𝐴 (1)
which is only satisfied if c2 = 0 – the wave function is zero. Particles obeying the two conditions
have completely different behaviours, as they can (or cannot) occupy the same quantum
states.
Bosons have wave functions which are symmetric under the interchange of identical particles.
They obey Bose-Einstein statistics, showing constructive interference of identical single
particle wave-functions. Writing down a wavefunction which is guaranteed to be symmetric
under the interchange of 1 and 2 we have
25
, 1
Ψ= [Ψ𝐴 (1)Ψ𝐵 (2) + Ψ𝐴 (2)Ψ𝐵 (1)]
√2
Bosons have integer spin, i.e. 0, ℏ, 2ℏ, (e.g. photons,𝜋 (pi meson) and other mesons, W±, Z,
gluon, ) They include the quanta of fields, i.e. the carriers of forces. They can be created
and destroyed e.g.
𝑒+ + 𝑒− → 𝑒+ + 𝑒− + 𝛾
Bosons can be their own antiparticles e.g. 𝜋+ ↔ 𝜋−
𝜋0 ↔ 𝜋0
Fermions have wave functions which are antisymmetric under the interchange of identical
particles. They obey Fermi-Dirac statistics, showing destructive interference of identical single
particle wave functions. In particular, no two identical fermions can occupy wave functions
with identical quantum numbers. The antisymmetric combination is
1
Ψ= [Ψ𝐴 (1)Ψ𝐵 (2) − Ψ𝐴 (2)Ψ𝐵 (1)]
√2
1 3 5
Fermions are particles with “half integer” spin, i.e. 2 ℏ, 2 ℏ, 2 ℏ, (e.g. proton, neutron,
electron, neutrino, quarks, ).
They include the constituent particles of matter. For each particle, there is a distinct
antiparticle, e.g.
𝑒− ↔ 𝑒+
𝜈𝑒 ↔ 𝜈̅𝑒
𝑝 ↔ 𝑝̅
𝑛 ↔ 𝑛̅
In particular the neutrino and antineutrino are both neutral, but different.
Fermions obey conservation laws – they are only produced as fermion-antifermion pairs.
3.2 Fundamental particles (fermions)
The fundamental fermions are believed to be the electron-like particles known as leptons and
the quarks. And are grouped in the 3 left-columns in the picture at the front of your
handbook. As we will see later, the proton and neutron – examples of baryons of spin ½ – are
made of quarks. So are mesons of integer spin. Together they make up the hadrons.
Fermions are grouped in 3 families. The first family is made of the electron and neutrino
(leptons) and u and d quarks. The leptons have an associated lepton number 𝐿𝑒 which is (as
far as we know) absolutely conserved.
Leptons e– and 𝜈𝑒 have 𝐿𝑒 = +1
Antileptons e+ and 𝜈̅𝑒 have 𝐿𝑒 = −1
26
In this unit we introduce the elementary particles that make up the Standard Model of particle
physics. We define their property using a quantum mechanics and how the interact with each
other by exchanging quanta of information (e.g. other particles).
3.1 Fermions and Bosons statistics
Learning outcome: understand the difference between fermions and bosons, and how they
behave
A multi-particle wave function for non-interacting (e.g. widely separated) particles can be
written as the product of single particle functions. For instance, for a system of three particles
Ψ(1, 2, 3) = 𝑐 Ψ𝐴 (1)Ψ𝐵 (2)Ψ𝐶 (3)
where A, B, C describe the quantum numbers of the state and 1, 2, 3 give the co-ordinates of
the particle. (c is simply a normalisation constant.) Observables are given by the square of
the wave function |Ψ|2 .
If we consider a number of identical, indistinguishable particles, then interchanging a pair of
particles is unobservable.
|Ψ(2, 1, 3)|2 = |Ψ(1, 2, 3)|2
In terms of wave function, this has two possible solutions
Ψ(2, 1, 3) = Ψ(1, 2, 3) (1)
Ψ(2, 1, 3) = −Ψ(1, 2, 3) (2)
These two cases have important physical consequences. E.g. if two particles are in an identical
quantum states
Ψ(1, 2) = 𝑐 Ψ𝐴 (1)Ψ𝐴 (2)
case (1) implies
𝑐1 𝛹𝐴 (1)𝛹𝐴 (2) = 𝑐1 𝛹𝐴 (2)𝛹𝐴 (1)
which is satisfied by any c1, case (2) implies
𝑐2 𝛹𝐴 (1)𝛹𝐴 (2) = −𝑐2 𝛹𝐴 (2)𝛹𝐴 (1)
which is only satisfied if c2 = 0 – the wave function is zero. Particles obeying the two conditions
have completely different behaviours, as they can (or cannot) occupy the same quantum
states.
Bosons have wave functions which are symmetric under the interchange of identical particles.
They obey Bose-Einstein statistics, showing constructive interference of identical single
particle wave-functions. Writing down a wavefunction which is guaranteed to be symmetric
under the interchange of 1 and 2 we have
25
, 1
Ψ= [Ψ𝐴 (1)Ψ𝐵 (2) + Ψ𝐴 (2)Ψ𝐵 (1)]
√2
Bosons have integer spin, i.e. 0, ℏ, 2ℏ, (e.g. photons,𝜋 (pi meson) and other mesons, W±, Z,
gluon, ) They include the quanta of fields, i.e. the carriers of forces. They can be created
and destroyed e.g.
𝑒+ + 𝑒− → 𝑒+ + 𝑒− + 𝛾
Bosons can be their own antiparticles e.g. 𝜋+ ↔ 𝜋−
𝜋0 ↔ 𝜋0
Fermions have wave functions which are antisymmetric under the interchange of identical
particles. They obey Fermi-Dirac statistics, showing destructive interference of identical single
particle wave functions. In particular, no two identical fermions can occupy wave functions
with identical quantum numbers. The antisymmetric combination is
1
Ψ= [Ψ𝐴 (1)Ψ𝐵 (2) − Ψ𝐴 (2)Ψ𝐵 (1)]
√2
1 3 5
Fermions are particles with “half integer” spin, i.e. 2 ℏ, 2 ℏ, 2 ℏ, (e.g. proton, neutron,
electron, neutrino, quarks, ).
They include the constituent particles of matter. For each particle, there is a distinct
antiparticle, e.g.
𝑒− ↔ 𝑒+
𝜈𝑒 ↔ 𝜈̅𝑒
𝑝 ↔ 𝑝̅
𝑛 ↔ 𝑛̅
In particular the neutrino and antineutrino are both neutral, but different.
Fermions obey conservation laws – they are only produced as fermion-antifermion pairs.
3.2 Fundamental particles (fermions)
The fundamental fermions are believed to be the electron-like particles known as leptons and
the quarks. And are grouped in the 3 left-columns in the picture at the front of your
handbook. As we will see later, the proton and neutron – examples of baryons of spin ½ – are
made of quarks. So are mesons of integer spin. Together they make up the hadrons.
Fermions are grouped in 3 families. The first family is made of the electron and neutrino
(leptons) and u and d quarks. The leptons have an associated lepton number 𝐿𝑒 which is (as
far as we know) absolutely conserved.
Leptons e– and 𝜈𝑒 have 𝐿𝑒 = +1
Antileptons e+ and 𝜈̅𝑒 have 𝐿𝑒 = −1
26
