Editing Spin–statistics theorem (section)

===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''.