In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, compactification means changing a theory with respect to one of its
space-time
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be
periodic.
Compactification plays an important part in
thermal field theory
In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.
...
where one compactifies time, in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
where one compactifies the
extra dimensions of the theory, and in two- or one-dimensional
solid state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, where one considers a system which is limited in one of the three usual spatial dimensions.
At the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is
dimensionally reduced.
Compactification in quantum field theory
Any two-dimensional scalar
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 ...
with a generic potential presents a universal feature, first unveiled by Campos Delgado and Dogaru,
namely it is equivalent to a one-dimensional theory of particles, as soon as the parent theory is compactified on a cylinder. In other words, if one starts with a theory of fields described by the action
:
and compactifies the theory on a cylinder of radius
by choosing