Lattice field theory on:
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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).
FermiQCD
– A standard library of algorithms for lattice QCD {{quantum-stub
In physics
Physics is the natural science that studies matter, its 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 which ...
, lattice field theory is the study of lattice models
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of co ...
of quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, that is, of field theory on a space or spacetime that has been discretised onto a lattice
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 orna ...
.
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 usingMarkov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
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
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processe ...
is approached.
Just as in all lattice models, numerical simulation gives 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 or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
s. Likewise, non-trivial vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
s can be discovered and probed.
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 (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
. Most quantization methods keep Poincaré invariance
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luc ...
manifest but sacrifice manifest gauge symmetry
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 (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
by requiring gauge fixing
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
. Only after renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
can gauge invariance
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 (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
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, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
. The articles on lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.
Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum ...
and lattice QCD explore these issues in greater detail.
Further 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).
External links
FermiQCD
– A standard library of algorithms for lattice QCD {{quantum-stub