PHY304 Particle Physics Dr C N Booth

Fermions and Bosons

A multi-particle wave function for non-interacting (e.g. widely separated) particles can be written as the product of single particle functions.
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 clearly interchanging a pair of particles is quite unobservable.

|Ψ(2,1,3)|2 = |Ψ(1,2,3)|2

This has two possible solutions

Ψ(2,1,3) = +Ψ(1,2,3)
Ψ(2,1,3) = −Ψ(1,2,3)

These two cases have important physical consequences. E.g. if two particles are in identical quantum states

Ψ(1,2,) = c ψA(1) ψA(2)
The first case implies c1 ψA(2) ψA(1) = c1 ψA(1) ψA(2) which is clearly satisfied by any c1,
the second case implies c1 ψA(2) ψA(1) = −c1 ψA(1) ψA(2)
which is only satisfied if c2 = 0 - the wave function is zero.

Particles obeying the two conditions have completely different behaviours.

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 we have

Ψ = 1/√2A(1)ψB(2) + ψA(2)ψB(1)).

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

Ψ = 1/√2A(1)ψB(2) − ψA(2)ψB(1)).

Fermions are particles with "half integer" spin, i.e. ħ,  ħ,  ħ, ... (e.g. proton, neutron, electron, neutrino, quarks, ...).
They include the constituents particles of matter. Foreach particle, there is a distinct antiparticle

 e ↔  e+
ν ↔ ν̅ both neutral, but different.
They obey conservation laws - they are only produced as fermion-antifermion pairs.

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
They are their own antiparticles e.g. 

The fundamental fermions are believed to be the electron-like particles known as leptons and the quarks.  (As we will see later, the proton and neutron - examples of baryons - are made of quarks.  So are mesons, and together they make up the hadrons.)

The basic constituents are the electron and neutrino (leptons) and u and d quarks.  The leptons have an associated lepton number L which is (as far as we know) absolutely conserved.

Leptons:  e and ν have  L = +1
Antileptons:  e+ and ν̅ have  L = −1

In fact, though the above 4 particles are all that is required to make the present day universe, the pattern is repeated, occurring 3 times, with heavier, unstable versions of the ordinary particles..  So, in the lepton sector we have:

extremely small
<2 eV/c2
<2 eV/c2
<2 eV/c2
and their antiparticles.

[The above neutrino mass limits are the result of oscillation experiments; direct measurements in decays give much poorer limits for the mass of νμ and ντ, 0.19 MeV/c2 and 18 MeV/c2 respectively. In all kinematic calulations we will treat the three species of neutrino as massless. Recent evidence indicates that, though these masses are certainly very small, they are not zero. This may have important consequences for our theories of the particle families. For more information on these recent results, please see some of the web pages below - but note this is definitely non-examinable!]

Each generation has its own distinct lepton number, Le, Lμ, Lτ. This is what makes the different neutrinos distinct, and forbids μ →  e γ.

As we will see later, the quark types can be changed by the weak interaction, so there is only a global baryon number B.  Quarks have B = +, antiquarks have B = −.

e.g. proton (uud) has B = +1; neutron (udd) has B = +1;
antiproton p̅ (u̅u̅d̅) has B = −1;  π+ (ud̅) has B = 0.
We will discuss the heavier quarks, and the allowed combinations of quarks which form hadrons, later.

The leptons and quarks, and the interactions between them, are parts of the standard model of particle physics, to which we will return many times during the course.  Supplementary material on the standard model, mainly of a popular or non-technical nature, can be obtained from a number of sources.  You may wish to consult some of the following information on the Web (though some of this material is really aimed at a younger or less specialised audience!):

More information on neutrinos, and their possible masses, is available from the following sources.

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