Editing Wick rotation (section)

== Overview ==
Wick rotation is motivated by the observation that the [[Minkowski metric]] in natural units (with [[metric signature]] {{math|(− + + +)}} convention)
: <math>ds^2 = -\left(dt^2\right) + dx^2 + dy^2 + dz^2</math>
and the four-dimensional Euclidean metric
: <math>ds^2 = d\tau^2 + dx^2 + dy^2 + dz^2</math>
are equivalent if one permits the coordinate {{mvar|t}} to take on [[imaginary number|imaginary]] values. The Minkowski metric becomes Euclidean when {{math|''t''}} is restricted to the [[imaginary number|imaginary axis]], and vice versa. Taking a problem expressed in Minkowski space with coordinates {{math|''x''}}, {{math|''y''}}, {{math|''z''}}, {{math|''t''}}, and substituting {{math|''t'' {{=}} −''iτ''}}
sometimes yields a problem in real Euclidean coordinates {{math|''x''}}, {{math|''y''}}, {{math|''z''}}, {{math|''τ''}} which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem.

=== Statistical and quantum mechanics ===
Wick rotation connects [[statistical mechanics]] to [[quantum mechanics]] by replacing [[inverse temperature]] with [[imaginary time]], or more precisely replacing {{math|1/''k''<sub>B</sub>''T''}} with {{math|''it''/''ħ''}}, where {{math|''T''}} is temperature, {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]], {{math|''t''}} is time, and {{math|''ħ''}} is the [[reduced Planck constant]].

For example, consider a quantum system whose [[Hamiltonian (quantum mechanics)|Hamiltonian]] {{math|''H''}} has [[eigenvalues]] {{math|''E''<sub>''j''</sub>}}. When this system is in [[thermal equilibrium]] at [[temperature]] {{mvar|T}}, the probability of finding it in its {{math|''j''}}th [[energy]] [[eigenstate]] is proportional to {{math|exp(−''E''<sub>''j''</sub>/''k''<sub>B</sub>''T'')}}. Thus, the expected value of any observable {{math|''Q''}} that commutes with the Hamiltonian is, up to a normalizing constant,
: <math>\sum_j Q_j e^{-\frac{E_j}{k_\text{B} T}},</math>
where {{mvar|j}} runs over all energy eigenstates and {{math|''Q''<sub>''j''</sub>}} is the value of {{math|''Q''}} in the {{math|''j''}}th eigenstate.

Alternatively, consider this system in a [[Quantum superposition|superposition]] of energy [[eigenstates]], evolving for a time {{mvar|t}} under the Hamiltonian {{mvar|H}}. After time {{math|''t''}}, the relative phase change of the {{mvar|j}}th eigenstate is {{math|exp(−''E''<sub>''j''</sub>''it''/''ħ'')}}.  Thus, the [[probability amplitude]] that a uniform (equally weighted) superposition of states
: <math>|\psi\rangle = \sum_j |j\rangle</math>
evolves to an arbitrary superposition
: <math>|Q\rangle = \sum_j Q_j |j\rangle</math>
is, up to a normalizing constant,
: <math>
  \left\langle Q \left| e^{-\frac{iHt}{\hbar}} \right| \psi \right\rangle =
  \sum_j Q_j e^{-\frac{E_j it}{\hbar}} \langle j|j\rangle =
  \sum_j Q_j e^{-\frac{E_j it}{\hbar}}.
</math>
Note that this formula can be obtained from the formula for thermal equilibrium by replacing {{math|1/''k''<sub>B</sub>''T''}} with {{math|''it''/''ħ''}}.