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In
Lattice Methods for Quantum Chromodynamics
', World Scientific, Singapore, (2006). * Gattringer, C., Lang, C. B., ''Quantum Chromodynamics on the Lattice'', Springer, (2010). * Knechtli, F., Günther, M., Peardon, M., ''Lattice Quantum Chromodynamics: Practical Essentials'', Springer, (2016). * Lin, H., Meyer, H.B., ''Lattice QCD for Nuclear Physics'', Springer, (2014). * Makeenko, Y., ''Methods of contemporary gauge theory'', Cambridge University Press, Cambridge, (2002). . * Montvay, I., Münster, G.,
Quantum Fields on a Lattice
', Cambridge University Press, Cambridge, (1997). * Rothe, H., ''Lattice Gauge Theories, An Introduction'', World Scientific, Singapore, (2005). * Smit, J., ''Introduction to Quantum Fields on a Lattice'', Cambridge University Press, Cambridge, (2002). {{quantum-stub
In physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, lattice field theory is the study of lattice models
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ...
of quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. This involves studying field theory on a space or spacetime that has been discretised onto a lattice.
Details
Although most lattice field theories are not exactly solvable, they are immensely appealing due to their feasibility for computer simulation, often usingMarkov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that ...
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
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
, such as soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s. Similarly, non-trivial vacuum states can be identified and examined.
The method is particularly appealing for the quantization of a gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
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
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
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
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
. The articles on lattice gauge theory and lattice QCD explore these issues in greater detail.
See also
* Fermion doublingFurther reading
* Creutz, M., ''Quarks, gluons and lattices'', Cambridge University Press, Cambridge, (1985). (renewed version: (2023) ) * DeGrand, T., DeTar, C.,Lattice Methods for Quantum Chromodynamics
', World Scientific, Singapore, (2006). * Gattringer, C., Lang, C. B., ''Quantum Chromodynamics on the Lattice'', Springer, (2010). * Knechtli, F., Günther, M., Peardon, M., ''Lattice Quantum Chromodynamics: Practical Essentials'', Springer, (2016). * Lin, H., Meyer, H.B., ''Lattice QCD for Nuclear Physics'', Springer, (2014). * Makeenko, Y., ''Methods of contemporary gauge theory'', Cambridge University Press, Cambridge, (2002). . * Montvay, I., Münster, G.,
Quantum Fields on a Lattice
', Cambridge University Press, Cambridge, (1997). * Rothe, H., ''Lattice Gauge Theories, An Introduction'', World Scientific, Singapore, (2005). * Smit, J., ''Introduction to Quantum Fields on a Lattice'', Cambridge University Press, Cambridge, (2002). {{quantum-stub