Editing Lattice QCD (section)

===Monte-Carlo simulations===

After [[Wick rotation]], the [[Path integral formulation|path integral]] for the [[Partition function (quantum field theory)|partition function]] of QCD takes the form

<math> Z = \int \mathcal{D} U \, e^{-S[U]} 
= \int \prod_{x, \mu} dU_\mu(x) \, e^{-S[U]} </math>

where the gauge links <math>U_\mu(x) \in \mathrm{SU}(3)</math> range over all the sites <math>x</math> and space-time directions <math>\mu</math> in a 4-dimensional space-time lattice, <math>S[U]</math> denotes the (Euclidean) [[Action (physics)|action]] and <math>dU_\mu(x)</math>  denotes the [[Haar measure]] on <math>\mathrm{SU}(3)</math>. Physical information is obtained by computing observables

<math> \left\langle \mathcal{O} \right\rangle
= \frac{1}{Z} \int \mathcal{D} U \, \mathcal{O}(U)
e^{-S[U]} </math>

For cases where evaluating observables pertubatively is difficult or impossible, a [[Monte Carlo method|Monte Carlo]] approach can be used, computing an observable <math> \mathcal{O} </math> as

<math> \left\langle \mathcal{O} \right\rangle
\approx \sum_{i=1}^{N} \mathcal{O}(U_i)
</math>

where <math>U_1, \dots, U_{N}</math> are [[Independent and identically distributed random variables|i.i.d random variables]] distributed according to the [[Boltzmann distribution|Boltzman distribution]] <math> U_i \sim e^{-S[U_i]}/Z </math>. For practical calculations, the samples <math>\{U_i\}</math> are typically obtained using [[Markov chain Monte Carlo]] methods, in particular [[Hybrid Monte Carlo]], which was invented for this purpose.<ref>{{cite journal | url=https://doi.org/10.1016/0370-2693(87)91197-X | doi=10.1016/0370-2693(87)91197-X | title=Hybrid Monte Carlo | date=1987 | last1=Duane | first1=Simon | last2=Kennedy | first2=A.D. | last3=Pendleton | first3=Brian J. | last4=Roweth | first4=Duncan | journal=Physics Letters B | volume=195 | issue=2 | pages=216–222 }}</ref>