# Identical particles

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In classical physics, it is possible to distinguish individual particles in a system, even if they have the same mechanical properties. One might paint each particle a unique color to distinguish it from the rest, or track the trajectory of each particle.

This does not work for elementary particles of the same species, which are truly identical. The "painting" method fails, as particles are exactly specified by their state vectors. Tracking each particle is equally impossible, as the position of the particle is specified by probablity functions rather than having a particular location.

This has profound implications for thermodynamics. Suppose you have two coins which are not identical. The probability of finding the coins in heads-tails is one in two, while the probability of finding the coints in heads-heads or tails-tails in one in four. If the coins are identical, then the probability of finding the coins in heads-heads, tails-tails, or heads-tails is one in three. Since the thermodynamic properties of a material are determined by the probability of particles being at a certain energy, the assertion of identical particles has some very noticable impacts.

Consider a system with two particles. If the state vector of particle 1 is |ψ> and the state vector of particle 2 is |ψ′>, the state of the combined system is denoted by

|ψ ψ′>

If the particles have the same species, they are identical, and (i) they occupy identical Hilbert spaces, and (ii) |ψψ′> and |ψ′ ψ> have equal probability to collapse to an arbitrary state |φ>:

|<φ|ψψ′>|2 = |<φ|ψ′ψ>|2

This is possible provided the permutation operation simply introduces a phase:

|ψψ′> = eiα |ψ′ψ>

However, two permutations are the identity, so we require e2iα = 1. Then either

|ψψ′> = + |ψ′ψ>

which is called a totally symmetric state, or

|ψψ′> = - |ψ′ψ>

which is called a totally antisymmetric state.

It is an empirical fact that states in Nature are either totally symmetric or totally antisymmetric. There is one caveat: in systems existing in two space dimensions (such as electrons constrained to a surface), a technical loophole in the above argument allows states with mixed symmetry.

Particles that produce totally antisymmetric states are called fermions. The Pauli exclusion principle forbids identical fermions from occupying a single quantum state. Fermions obey Pauli-Dirac statistics.

Particles that produce totally symmetric states are called bosons. Bosons can and do occupy the same quantum state, giving rise to Bose-Einstein statistics. This "clumping" gives rise to such varied phenomena as the laser, Bose-Einstein condensate, and superfluid.

Particles with mixed symmetry are known as anyons, and obey fractional statistics

According to the spin-statistics theorem, bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.