Lattice field theory

Quantum field theory on a lattice

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. This involves studying field theory on a space or spacetime that has been discretised onto a lattice.

## Details

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Although most lattice field theories are not exactly solvable, they are immensely appealing due to their feasibility for computer simulation, 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 behavior of the continuum theory as the continuum limit is approached.

Just as in all lattice models, numerical simulation provides access to field configurations that are not accessible to perturbation theory, such as solitons. Similarly, non-trivial vacuum states can be identified and examined.

The method is particularly appealing for the quantization of a gauge theory using the Wilson action. Most quantization approaches maintain Poincaré invariance manifest but sacrifice manifest gauge symmetry by requiring gauge fixing. It's only after renormalization that gauge invariance can 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.

## See also

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* Fermion doubling

## Further reading

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* Creutz, M., Quarks, gluons and lattices, Cambridge University Press, Cambridge, (1985). ISBN 978-0521315357 (renewed version: (2023) ISBN 978-1009290395)
* DeGrand, T., DeTar, C., Lattice Methods for Quantum Chromodynamics, World Scientific, Singapore, (2006). ISBN 978-9812567277
* Gattringer, C., Lang, C. B., Quantum Chromodynamics on the Lattice, Springer, (2010). ISBN 978-3642018497
* Knechtli, F., Günther, M., Peardon, M., Lattice Quantum Chromodynamics: Practical Essentials, Springer, (2016). ISBN 978-9402409970
* Lin, H., Meyer, H.B., Lattice QCD for Nuclear Physics, Springer, (2014). ISBN 978-3319080215
* Makeenko, Y., Methods of contemporary gauge theory, Cambridge University Press, Cambridge, (2002). ISBN 0-521-80911-8.
* Montvay, I., Münster, G., Quantum Fields on a Lattice, Cambridge University Press, Cambridge, (1997). ISBN 978-0521599177
* Rothe, H., Lattice Gauge Theories, An Introduction, World Scientific, Singapore, (2005). ISBN 978-9814365857
* Smit, J., Introduction to Quantum Fields on a Lattice, Cambridge University Press, Cambridge, (2002). ISBN 978-0521890519

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