Editing Wick rotation

{{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>

== 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''}}.

== Statics and dynamics ==
Wick rotation relates statics problems in {{mvar|n}} dimensions to dynamics problems in {{math|''n'' − 1}} dimensions, trading one dimension of space for one dimension of time. A simple example where {{math|1=''n'' = 2}} is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve {{math|''y''(''x'')}}. The spring is in equilibrium when the energy associated with this curve is at a critical point (an extremum); this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy, we integrate the energy spatial density over space:
: <math>E = \int_x \left[ k \left(\frac{dy(x)}{dx}\right)^2 + V\big(y(x)\big) \right] dx,</math>
where {{math|''k''}} is the spring constant, and {{math|''V''(''y''(''x''))}} is the gravitational potential.

The corresponding dynamics problem is that of a rock thrown upwards. The path the rock follows is that which extremalizes the [[Action (physics)|action]]; as before, this extremum is typically a minimum, so this is called the "[[principle of least action]]". Action is the time integral of the [[Lagrangian mechanics|Lagrangian]]:
: <math>S = \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V\big(y(t)\big) \right] dt.</math>

We get the solution to the dynamics problem (up to a factor of {{mvar|i}}) from the statics problem by Wick rotation, replacing {{math|''y''(''x'')}} by {{math|''y''(''it'')}} and the spring constant {{mvar|k}} by the mass of the rock {{mvar|m}}:
: <math>iS = \int_t \left[ m \left(\frac{dy(it)}{dt}\right)^2 + V\big(y(it)\big) \right] dt = i \int_t \left[ m \left(\frac{dy(it)}{dit}\right)^2 - V\big(y(it)\big) \right] d(it).</math>

== Both thermal/quantum and static/dynamic ==
Taken together, the previous two examples show how the [[path integral formulation]] of quantum mechanics is related to statistical mechanics. From statistical mechanics, the shape of each spring in a collection at temperature {{mvar|T}} will deviate from the least-energy shape due to thermal fluctuations; the probability of finding a spring with a given shape decreases exponentially with the energy difference from the least-energy shape. Similarly, a quantum particle moving in a potential can be described by a superposition of paths, each with a phase {{math|exp(''iS'')}}: the thermal variations in the shape across the collection have turned into quantum uncertainty in the path of the quantum particle.

== Further details ==
The [[Schrödinger equation]] and the [[heat equation]] are also related by Wick rotation.

Wick rotation also relates a [[quantum field theory]] at a finite [[inverse temperature]] {{math|''β''}} to a statistical-mechanical model over the "tube" {{math|'''R'''<sup>3</sup> × ''S''<sup>1</sup>}} with the imaginary time coordinate {{math|''τ''}} being periodic with period {{math|''β''}}.  However, there is a slight difference. Statistical-mechanical [[N-point function|{{mvar|n}}-point functions]] satisfy positivity, whereas Wick-rotated quantum field theories satisfy [[Schwinger function#Reflection positivity|reflection positivity]].{{Explain|date=August 2016}}

Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the [[inner product]], as in this case the rotation would cancel out and have no effect.

== Rigorous proof ==
{{Needs expansion|date=January 2025}}
Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the [[Osterwalder–Schrader axioms]].<ref>{{Cite journal|last=Schlingemann|first=Dirk|date=1999 |title=From Euclidean Field Theory To Quantum Field Theory|url=https://www.worldscientific.com/doi/abs/10.1142/S0129055X99000362|journal=Reviews in Mathematical Physics|volume=11|issue=9|pages=1151–78|doi=10.1142/S0129055X99000362|arxiv=hep-th/9802035|bibcode=1999RvMaP..11.1151S |s2cid=9851483|issn=0129-055X}}</ref>

== See also ==
* {{Section link|Circular points at infinity#Imaginary transformation}}
* [[Complex spacetime]]
* [[Imaginary time]]
* [[Schwinger function]]

== References ==
{{Reflist}}
* {{cite journal |last=Wick |first=G. C. |date=1954 |title=Properties of Bethe–Salpeter Wave Functions |journal=[[Physical Review]] |doi=10.1103/PhysRev.96.1124 |bibcode=1954PhRv...96.1124W |volume=96 |issue=4 |pages=1124–1134}}

== External links ==
{{wikiquote}}
* [http://math.ucr.edu/home/baez/qg-fall2006/#f06week02a A Spring in Imaginary Time] &ndash; a worksheet in Lagrangian mechanics illustrating how replacing length by imaginary time turns the parabola of a hanging spring into the inverted parabola of a thrown particle
* [https://web.archive.org/web/20050403215217/http://www.mth.kcl.ac.uk/~streater/lostcauses.html#X Euclidean Gravity] &ndash; a short note by [[Ray Streater]] on the "Euclidean Gravity" programme.

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[[Category:Quantum field theory]]
[[Category:Statistical mechanics]]