In
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
and
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the continuum limit or scaling limit of a
lattice model characterizes its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world processes, such as
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
. Indeed, according to
Donsker's theorem, the discrete
random walk
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space.
An elementary example of a rand ...
would, in the scaling limit, approach the true
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
.
Terminology
The term ''continuum limit'' mostly finds use in the physical sciences, often in reference to models of aspects of
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, while the term ''scaling limit'' is more common in mathematical use.
Application in quantum field theory
A lattice model that approximates a
continuum 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 ...
in the limit as the lattice spacing goes to zero may correspond to finding a
second order phase transition of the model. This is the scaling limit of the model.
See also
*
Universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale-invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite sc ...
References
*H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena'' (1971)
*
H. Kleinert, ''Gauge Fields in Condensed Matter'', Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, Vol. II, "STRESSES AND DEFECTS", pp. 743–1456,
World Scientific (Singapore, 1989) Paperback '' (also available online
an
''
*
H. Kleinert and V. Schulte-Frohlinde, ''Critical Properties of φ
4-Theories''
World Scientific (Singapore, 2001) Paperback '' (also availabl
online''
Lattice models
Lattice field theory
Renormalization group
Critical phenomena
Articles containing video clips
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