Editing Wick rotation (section)

{{Short description|Mathematical trick using imaginary numbers to simplify certain formulas in physics}}
{{primary sources|date=May 2014}}

In [[physics]], '''Wick rotation''', named after Italian physicist [[Gian Carlo Wick]], is a method of finding a solution to a mathematical problem in [[Minkowski space]] from a solution to a related problem in [[Euclidean space]] by means of a transformation that substitutes an imaginary-number variable for a real-number variable.

Wick rotations are useful because of an analogy between two important but seemingly distinct fields of physics: [[statistical mechanics]] and [[quantum mechanics]].  In this analogy, [[inverse temperature]] plays a role in statistical mechanics formally akin to [[imaginary time]] in quantum mechanics: that is, {{math|''it''}}, where {{math|''t''}} is time and {{math|''i''}} is the [[imaginary unit]] ({{math|1=''i''<sup>2</sup> = –1}}).

More precisely, in statistical mechanics, the [[Gibbs measure]] {{math|exp(−''H''/''k''<sub>B</sub>''T'')}} describes the relative probability of the system to be in any given state at temperature {{math|''T''}}, where {{math|''H''}} is a function describing the energy of each state and {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]].  In quantum mechanics, the transformation {{math|exp(−''itH''/''ħ'')}} describes time evolution, where {{math|''H''}} is an operator describing the energy (the [[Hamiltonian (quantum mechanics)|Hamiltonian]]) and {{math|''ħ''}} is the [[reduced Planck constant]].   The former expression resembles the latter when we replace {{math|''it''/''ħ''}} with {{math|1/''k''<sub>B</sub>''T''}}, and this replacement is called Wick rotation.<ref>{{Cite book |last=Zee |first=Anthony |url=https://books.google.com/books?id=n8Mmbjtco78C&dq=zee+wick+rotation&pg=PA289 |title=Quantum Field Theory in a Nutshell |date=2010 |publisher=Princeton University Press |isbn=978-1-4008-3532-4 |edition=2nd |page=289 |language=en}}</ref>

Wick rotation is called a rotation because when we represent [[Complex plane|complex numbers as a plane]], the multiplication of a complex number by the [[imaginary unit]] is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude {{math|''π''/2}} about the origin.<ref>{{Citation |last1=Lancaster |first1=Tom |title=Statistical field theory |date=2014-04-17 |work=Quantum Field Theory for the Gifted Amateur |pages=228–229 |url=http://dx.doi.org/10.1093/acprof:oso/9780199699322.003.0026 |access-date=2023-11-12 |publisher=Oxford University Press |last2=Blundell |first2=Stephen J.|doi=10.1093/acprof:oso/9780199699322.003.0026 |isbn=978-0-19-969932-2 }}</ref>