Wick rotation

<h2 id="overview">Overview</h2>
Wick rotation is motivated by the observation that the <a href="/facts/Minkowski_metric/vDkfkQJn">Minkowski metric</a> in natural units (with <a href="/facts/Metric_signature/w8d57bQG">metric signature</a> (− + + +) convention)

d
        
          s
          
            2
          
        
        =
        −
        
          (
          
            d
            
              t
              
                2
              
            
          
          )
        
        +
        d
        
          x
          
            2
          
        
        +
        d
        
          y
          
            2
          
        
        +
        d
        
          z
          
            2
          
        
      
    
    {\displaystyle ds^{2}=-\left(dt^{2}\right)+dx^{2}+dy^{2}+dz^{2}}

and the four-dimensional Euclidean metric

d
        
          s
          
            2
          
        
        =
        d
        
          τ
          
            2
          
        
        +
        d
        
          x
          
            2
          
        
        +
        d
        
          y
          
            2
          
        
        +
        d
        
          z
          
            2
          
        
      
    
    {\displaystyle ds^{2}=d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}}

are equivalent if one permits the coordinate t to take on <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary</a> values. The Minkowski metric becomes Euclidean when t is restricted to the <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary axis</a>, and vice versa. Taking a problem expressed in Minkowski space with coordinates x, y, z, t, and substituting t = −iτ
sometimes yields a problem in real Euclidean coordinates x, y, z, τ which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem.

<h3>Statistical and quantum mechanics</h3>
Wick rotation connects <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a> to <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> by replacing <a href="/facts/Inverse_temperature/pEpfRNYb">inverse temperature</a> with <a href="/facts/Imaginary_time/UTTYSUw7">imaginary time</a>, or more precisely replacing 1/kBT with it/ħ, where T is temperature, kB is the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a>, t is time, and ħ is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>.
For example, consider a quantum system whose <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian</a> H has <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> Ej. When this system is in <a href="/facts/Thermal_equilibrium/wHqArIwx">thermal equilibrium</a> at <a href="/facts/Temperature/QyGe4gmj">temperature</a> T, the probability of finding it in its jth <a href="/facts/Energy/wd8PrQM7">energy</a> <a href="/facts/Eigenstate/7jbxd7Dx">eigenstate</a> is proportional to exp(−Ej/kBT). Thus, the expected value of any observable Q that commutes with the Hamiltonian is, up to a normalizing constant,

∑
          
            j
          
        
        
          Q
          
            j
          
        
        
          e
          
            −
            
              
                
                  E
                  
                    j
                  
                
                
                  
                    k
                    
                      B
                    
                  
                  T
                
              
            
          
        
        ,
      
    
    {\displaystyle \sum _{j}Q_{j}e^{-{\frac {E_{j}}{k_{\text{B}}T}}},}

where j runs over all energy eigenstates and Qj is the value of Q in the jth eigenstate.
Alternatively, consider this system in a <a href="/facts/Quantum_superposition/4z1UJg9o">superposition</a> of energy <a href="/facts/Eigenstates/7jbxd7Dx">eigenstates</a>, evolving for a time t under the Hamiltonian H. After time t, the relative phase change of the jth eigenstate is exp(−Ejit/ħ). Thus, the <a href="/facts/Probability_amplitude/ryPgJvs8">probability amplitude</a> that a uniform (equally weighted) superposition of states

|
        
        ψ
        ⟩
        =
        
          ∑
          
            j
          
        
        
          |
        
        j
        ⟩
      
    
    {\displaystyle |\psi \rangle =\sum _{j}|j\rangle }

evolves to an arbitrary superposition

|
        
        Q
        ⟩
        =
        
          ∑
          
            j
          
        
        
          Q
          
            j
          
        
        
          |
        
        j
        ⟩
      
    
    {\displaystyle |Q\rangle =\sum _{j}Q_{j}|j\rangle }

is, up to a normalizing constant,

⟨
          
            Q
            
              |
              
                e
                
                  −
                  
                    
                      
                        i
                        H
                        t
                      
                      ℏ
                    
                  
                
              
              |
            
            ψ
          
          ⟩
        
        =
        
          ∑
          
            j
          
        
        
          Q
          
            j
          
        
        
          e
          
            −
            
              
                
                  
                    E
                    
                      j
                    
                  
                  i
                  t
                
                ℏ
              
            
          
        
        ⟨
        j
        
          |
        
        j
        ⟩
        =
        
          ∑
          
            j
          
        
        
          Q
          
            j
          
        
        
          e
          
            −
            
              
                
                  
                    E
                    
                      j
                    
                  
                  i
                  t
                
                ℏ
              
            
          
        
        .
      
    
    {\displaystyle \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 }}}.}

Note that this formula can be obtained from the formula for thermal equilibrium by replacing 1/kBT with it/ħ.

<h2 id="statics-and-dynamics">Statics and dynamics</h2>
Wick rotation relates statics problems in n dimensions to dynamics problems in n − 1 dimensions, trading one dimension of space for one dimension of time. A simple example where n = 2 is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve 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:

E
        =
        
          ∫
          
            x
          
        
        
          [
          
            k
            
              
                (
                
                  
                    
                      d
                      y
                      (
                      x
                      )
                    
                    
                      d
                      x
                    
                  
                
                )
              
              
                2
              
            
            +
            V
            
              
                (
              
            
            y
            (
            x
            )
            
              
                )
              
            
          
          ]
        
        d
        x
        ,
      
    
    {\displaystyle E=\int _{x}\left[k\left({\frac {dy(x)}{dx}}\right)^{2}+V{\big (}y(x){\big )}\right]dx,}

where k is the spring constant, and 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 <a href="/facts/Action_(physics)/NhWS2Mts">action</a>; as before, this extremum is typically a minimum, so this is called the "<a href="/facts/Principle_of_least_action/QaBTpoKz">principle of least action</a>". Action is the time integral of the <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian</a>:

S
        =
        
          ∫
          
            t
          
        
        
          [
          
            m
            
              
                (
                
                  
                    
                      d
                      y
                      (
                      t
                      )
                    
                    
                      d
                      t
                    
                  
                
                )
              
              
                2
              
            
            −
            V
            
              
                (
              
            
            y
            (
            t
            )
            
              
                )
              
            
          
          ]
        
        d
        t
        .
      
    
    {\displaystyle S=\int _{t}\left[m\left({\frac {dy(t)}{dt}}\right)^{2}-V{\big (}y(t){\big )}\right]dt.}

We get the solution to the dynamics problem (up to a factor of i) from the statics problem by Wick rotation, replacing y(x) by y(it) and the spring constant k by the mass of the rock m:

i
        S
        =
        
          ∫
          
            t
          
        
        
          [
          
            m
            
              
                (
                
                  
                    
                      d
                      y
                      (
                      i
                      t
                      )
                    
                    
                      d
                      t
                    
                  
                
                )
              
              
                2
              
            
            +
            V
            
              
                (
              
            
            y
            (
            i
            t
            )
            
              
                )
              
            
          
          ]
        
        d
        t
        =
        i
        
          ∫
          
            t
          
        
        
          [
          
            m
            
              
                (
                
                  
                    
                      d
                      y
                      (
                      i
                      t
                      )
                    
                    
                      d
                      i
                      t
                    
                  
                
                )
              
              
                2
              
            
            −
            V
            
              
                (
              
            
            y
            (
            i
            t
            )
            
              
                )
              
            
          
          ]
        
        d
        (
        i
        t
        )
        .
      
    
    {\displaystyle 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).}

<h2 id="both-thermalquantum-and-staticdynamic">Both thermal/quantum and static/dynamic</h2>
Taken together, the previous two examples show how the <a href="/facts/Path_integral_formulation/6lt2MHkW">path integral formulation</a> of quantum mechanics is related to statistical mechanics. From statistical mechanics, the shape of each spring in a collection at temperature 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 exp(iS): the thermal variations in the shape across the collection have turned into quantum uncertainty in the path of the quantum particle.

<h2 id="further-details">Further details</h2>
The <a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger equation</a> and the <a href="/facts/Heat_equation/2JPPr5GG">heat equation</a> are also related by Wick rotation.
Wick rotation also relates a <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> at a finite <a href="/facts/Inverse_temperature/pEpfRNYb">inverse temperature</a> β to a statistical-mechanical model over the "tube" R3 × S1 with the imaginary time coordinate τ being periodic with period β. However, there is a slight difference. Statistical-mechanical <a href="/facts/N-point_function/zH7RInAw">n-point functions</a> satisfy positivity, whereas Wick-rotated quantum field theories satisfy <a href="/facts/Schwinger_function/wqVpXznd">reflection positivity</a>.
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 <a href="/facts/Inner_product/HoyjElEy">inner product</a>, as in this case the rotation would cancel out and have no effect.

<h2 id="rigorous-proof">Rigorous proof</h2>

Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the <a href="/facts/Osterwalder%E2%80%93Schrader_axioms/wqVpXznd">Osterwalder–Schrader axioms</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Circular_points_at_infinity/BS6eH0Hn">Circular points at infinity § Imaginary transformation</a></li>
<li><a href="/facts/Complex_spacetime/vaUo5xak">Complex spacetime</a></li>
<li><a href="/facts/Imaginary_time/UTTYSUw7">Imaginary time</a></li>
<li><a href="/facts/Schwinger_function/wqVpXznd">Schwinger function</a></li></ul>

<ul><li>Wick, G. C. (1954). "Properties of Bethe–Salpeter Wave Functions". <a href="/facts/Physical_Review/QxekSF11">Physical Review</a>. 96 (4): 1124–1134. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1954PhRv...96.1124W">1954PhRv...96.1124W</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRev.96.1124">10.1103/PhysRev.96.1124</a>.</li></ul>
<h2 id="external-links">External links</h2>

Wikiquote has quotations related to Wick rotation.

<ul><li><a href="http://math.ucr.edu/home/baez/qg-fall2006/#f06week02a">A Spring in Imaginary Time</a> – 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</li>
<li><a href="https://web.archive.org/web/20050403215217/http://www.mth.kcl.ac.uk/~streater/lostcauses.html#X">Euclidean Gravity</a> – a short note by <a href="/facts/Ray_Streater/GLTqaQ8w">Ray Streater</a> on the "Euclidean Gravity" programme.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. p. 289. ISBN 978-1-4008-3532-4. <a href="978-1-4008-3532-4" target="_blank">978-1-4008-3532-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lancaster, Tom; Blundell, Stephen J. (2014-04-17), "Statistical field theory", Quantum Field Theory for the Gifted Amateur, Oxford University Press, pp. 228–229, doi:10.1093/acprof:oso/9780199699322.003.0026, ISBN 978-0-19-969932-2, retrieved 2023-11-12 <a href="978-0-19-969932-2" target="_blank">978-0-19-969932-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Schlingemann, Dirk (1999). "From Euclidean Field Theory To Quantum Field Theory". Reviews in Mathematical Physics. 11 (9): 1151–78. arXiv:hep-th/9802035. Bibcode:1999RvMaP..11.1151S. doi:10.1142/S0129055X99000362. ISSN 0129-055X. S2CID 9851483. <a href="https://www.worldscientific.com/doi/abs/10.1142/S0129055X99000362" target="_blank">https://www.worldscientific.com/doi/abs/10.1142/S0129055X99000362</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Wick rotation open-in-new

Wick rotation