Editing Spin–statistics theorem (section)

==Background==
The statistics of [[indistinguishable particles]] is among the most fundamental of physical effects. The [[Pauli exclusion principle]]{{snd}} that every occupied [[quantum state]] contains at most one fermion{{snd}} controls the formation of matter. The basic building blocks of matter such as [[proton]]s, [[neutron]]s, and [[electron]]s are all fermions. Conversely, [[photon]] and other particles which mediate forces between matter particles, are bosons. A spin–statistics theorem attempts to explain the origin of this fundamental dichotomy.<ref name="DuckSudarshanBook" />{{rp|4}}

Naively, spin, an angular momentum property intrinsic to a particle, would be unrelated to fundamental properties of a collection of such particles. However, these are indistinguishable particles: any physical prediction relating multiple indistinguishable particles must not change when the particles are exchanged. 

===Quantum states and indistinguishable particles===
In a quantum system, a physical state is described by a [[quantum state|state vector]]. A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions. A pair of indistinguishable particles such as this have only one state. This means that if the positions of the particles are exchanged (i.e., they undergo a permutation), this does not identify a new physical state, but rather one matching the original physical state. In fact, one cannot tell which particle is in which position.

While the physical state does not change under the exchange of the particles' positions, it is possible for the state vector to change sign as a result of an exchange. Since this sign change is just an overall phase, this does not affect the physical state.

The essential ingredient in proving the spin-statistics relation is relativity, that the physical laws do not change under [[Lorentz transformation]]s. The field operators transform under [[Lorentz transformation]]s according to the spin of the particle that they create, by definition.

Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time direction. Otherwise, the notion of being spacelike is meaningless. However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.

[[Lorentz transformations]] include 3-dimensional rotations and [[Lorentz Boost|boosts]]. A boost transfers to a [[frame of reference]] with a different velocity and is mathematically like a rotation into time. By [[analytic continuation]] of the correlation functions of a quantum field theory, the time coordinate may become [[imaginary number|imaginary]], and then boosts become rotations. The new "spacetime" has only spatial directions and is termed ''Euclidean''.

===Exchange symmetry or permutation symmetry===

[[Boson]]s are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles, the wavefunction does not change. [[Fermion]]s are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the [[Pauli exclusion principle]]: two identical fermions cannot occupy the same state. This rule does not hold for bosons.

In quantum field theory, a state or a wavefunction is described by [[field operator]]s operating on some basic state called the [[Vacuum state|''vacuum'']]. In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator

:<math>
\iint \psi(x,y) \phi(x)\phi(y)\,dx\,dy
</math>

(with <math>\phi</math> an operator and <math>\psi(x,y)</math> a numerical function with complex values) creates a two-particle state with wavefunction <math>\psi(x,y)</math>, and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter.

Let us assume that <math>x \ne y</math> and the two operators take place at the same time; more generally, they may have [[spacelike]] separation, as is explained hereafter.

If the fields '''commute''', meaning that the following holds:

:<math>\phi(x)\phi(y)=\phi(y)\phi(x),</math>

then only the symmetric part of <math>\psi</math> contributes, so that <math>\psi(x,y) = \psi(y,x)</math>, and the field will create bosonic particles.

On the other hand, if the fields '''anti-commute''', meaning that <math>\phi</math> has the property that

:<math>\phi(x)\phi(y)=-\phi(y)\phi(x),</math>

then only the antisymmetric part of <math>\psi</math> contributes, so that <math>\psi(x,y) = -\psi(y,x)</math>, and the particles will be fermionic.