Lattice field theory

In lattice field theory, space or spacetime is discretised onto a lattice. Renormalization and regularization

Renormalization

Renormalization group
On-shell scheme
Minimal subtraction scheme

Regularization

Dimensional regularization
Pauli–Villars regularization
Lattice regularization
Zeta function regularization
Causal perturbation theory
Hadamard regularization
Point-splitting regularization

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In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretized onto a lattice.

## Details

Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer, often using Markov chain Monte Carlo methods. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, one will be able to recover the behaviour of the continuum theory.

Just as in all lattice models, numerical simulation gives access to field configurations that are not accessible to perturbation theory, such as solitons. Likewise, non-trivial vacuum states can be discovered and probed.

The method is particularly appealing for the quantization of a gauge theory. Most quantization methods keep Poincaré invariance manifest but sacrifice manifest gauge symmetry by requiring gauge fixing. Only after renormalization can gauge invariance be recovered. Lattice field theory differs from these in that it keeps manifest gauge invariance, but sacrifices manifest Poincaré invariance-- recovering it only after renormalization. The articles on lattice gauge theory and lattice QCD explore these issues in greater detail.

## References

* M. Creutz, Quarks, gluons and lattices, Cambridge University Press 1985.
* I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press 1997.
* H. Rothe, Lattice Gauge Theories, An Introduction, World Scientific 2005.
* J. Smit, Introduction to Quantum Fields on a Lattice, Cambridge University Press 2002.

## External links

* FermiQCD – A standard library of algorithms for lattice QCD

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