Lattice gauge theory
Non-perturbative gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional path integral, which is computationally intractable.
By working on a discrete spacetime, the path integral becomes finite-dimensional, and can be evaluated by stochastic simulation techniques such as the Monte Carlo method.
When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.
Hence, to simulate QCD with Lie group SU(3), a 3×3 unitary matrix is defined on each link.
(a color 3-vector, the space on which the fundamental representation of SU(3) acts), a bispinor (Dirac 4-spinor), an nf vector, and a Grassmann variable.
By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to
Quantities such as particle masses are stochastically calculated using techniques such as the Monte Carlo method.
To reduce the computational burden, the so-called quenched approximation can be used, in which the fermionic fields are treated as non-dynamic "frozen" variables.
[4][5] The results of lattice QCD computations show e.g. that in a meson not only the particles (quarks and antiquarks), but also the "fluxtubes" of the gluon fields are important.
[citation needed] Lattice gauge theory is also important for the study of quantum triviality by the real-space renormalization group.
[7] Triviality has yet to be proven rigorously, but lattice computations have provided strong evidence for this[citation needed].
This fact is important as quantum triviality can be used to bound or even predict parameters such as the mass of Higgs boson.
[8] Originally, solvable two-dimensional lattice gauge theories had already been introduced in 1971 as models with interesting statistical properties by the theorist Franz Wegner, who worked in the field of phase transitions.
[9] When only 1×1 Wilson loops appear in the action, lattice gauge theory can be shown to be exactly dual to spin foam models.
