mathematical physics Mathematical physics refers to 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 t ...

and

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 processes, such as

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

. Indeed, according to

Donsker's theorem In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let X_1, X_2, X_3, \ldots be ...

, the discrete

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

would, in the scaling limit, approach the true

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 a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

, while the term ''scaling limit'' is more common in mathematical use.

Application in quantum field theory

A lattice model that approximates a

continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...

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 ...

in the limit as the lattice spacing goes to zero may correspond to finding a

second order phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...

of the model. This is the scaling limit of the model.

References

*H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena'' * 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 φ⁴-Theories''
World Scientific (Singapore, 2001)
Paperback '' (also availabl
online
'' Lattice models Lattice field theory Renormalization group Critical phenomena Articles containing video clips {{lattice-stub

Terminology

Application in quantum field theory

See also

References