Canonical is an adjective
In grammar, an adjective is a 'describing' word; the main syntactic role of which is to qualify a noun or noun phrase, giving more information about the object signified....

 derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" (perhaps originally from kanna "reed", cognate
In linguistics, cognates are words that have a common etymological origin. This learned term derives from the Latin cognatus . Cognates within the same language are called doublets. Strictly speaking, loanwords from another language are usually not meant by the term, e.g...

 to cane), and is used in various meanings.

Basic, canonic, canonical: reduced to the simplest and most significant form possible without loss of generality, e.g., "a basic story line"; "a canonical syllable pattern."

The word is also often used when describing bodies of literature or art: those books that all educated people have supposedly read, or are advised to read, make up the "canon", for example the Western canon
Western canon
The term Western canon denotes a canon of books and, more broadly, music and art that have been the most important and influential in shaping Western culture. As such, it includes the "greatest works of artistic merit." Such a canon is important to the theory of educational perennialism and the...

. (See also canon (fiction)
Canon (fiction)
In the context of a work of fiction, the term canon denotes the material accepted as "official" in a fictional universe's fan base. It is often contrasted with, or used as the basis for, works of fan fiction, which are not considered canonical...



This word is used by theologians and canon law
Canon law
Canon law is the body of laws & regulations made or adopted by ecclesiastical authority, for the government of the Christian organization and its members. It is the internal ecclesiastical law governing the Catholic Church , the Eastern and Oriental Orthodox churches, and the Anglican Communion of...

yers to refer to the canon
Canon law
Canon law is the body of laws & regulations made or adopted by ecclesiastical authority, for the government of the Christian organization and its members. It is the internal ecclesiastical law governing the Catholic Church , the Eastern and Oriental Orthodox churches, and the Anglican Communion of...

s of the Roman Catholic, Eastern Orthodox and Anglican Churches adopted by ecumenical council
Ecumenical council
An ecumenical council is a conference of ecclesiastical dignitaries and theological experts convened to discuss and settle matters of Church doctrine and practice....

s. It also refers to later law developed by local churches and dioceses of these churches. The function of this collection is somewhat analogous to the precedents established in common law
Common law
Common law is law developed by judges through decisions of courts and similar tribunals rather than through legislative statutes or executive branch action...

 by case law
Case law
In law, case law is the set of reported judicial decisions of selected appellate courts and other courts of first instance which make new interpretations of the law and, therefore, can be cited as precedents in a process known as stare decisis...


In the 20th century, the Roman Catholic Church revised its canon law in 1917 and then again in 1983, into the modern Code of Canon Law
Canon law (Catholic Church)
The canon law of the Catholic Church, is a fully developed legal system, with all the necessary elements: courts, lawyers, judges, a fully articulated legal code and principles of legal interpretation. It lacks the necessary binding force present in most modern day legal systems. The academic...

. This code is no longer merely a compilation of papal decrees and conciliar legislation, but a more completely developed body of international church law. It is analogous to the English system of statute
A statute is a formal written enactment of a legislative authority that governs a state, city, or county. Typically, statutes command or prohibit something, or declare policy. The word is often used to distinguish law made by legislative bodies from case law, decided by courts, and regulations...


Canonical can also mean "part of the canon", i.e., one of the books comprising a biblical canon
Biblical canon
A biblical canon, or canon of scripture, is a list of books considered to be authoritative as scripture by a particular religious community. The term itself was first coined by Christians, but the idea is found in Jewish sources. The internal wording of the text can also be specified, for example...

, (e.g. the Gospel of Matthew
Gospel of Matthew
The Gospel According to Matthew is one of the four canonical gospels, one of the three synoptic gospels, and the first book of the New Testament. It tells of the life, ministry, death, and resurrection of Jesus of Nazareth...

 or the Gospel of Mark
Gospel of Mark
The Gospel According to Mark , commonly shortened to the Gospel of Mark or simply Mark, is the second book of the New Testament. This canonical account of the life of Jesus of Nazareth is one of the three synoptic gospels. It was thought to be an epitome, which accounts for its place as the second...

) as opposed to apocrypha
The term apocrypha is used with various meanings, including "hidden", "esoteric", "spurious", "of questionable authenticity", ancient Chinese "revealed texts and objects" and "Christian texts that are not canonical"....

l books (e.g. the Gospel according to the Hebrews).

The term is also applied by Westerners to other religions, but in inconsistent ways: for example, in the case of Buddhism one authority refers to "scriptures and other canonical texts", while another says that scriptures can be categorized into canonical, commentarial and pseudo-canonical.

Canonization is the act by which a Christian church declares a deceased person to be a saint, upon which declaration the person is included in the canon, or list, of recognized saints. Originally, individuals were recognized as saints without any formal process...

 is the process by which a person becomes recognized as a saint
A saint is a holy person. In various religions, saints are people who are believed to have exceptional holiness.In Christian usage, "saint" refers to any believer who is "in Christ", and in whom Christ dwells, whether in heaven or in earth...



In the molecular life sciences, canonical is used to refer to biochemical pathways or other molecular mechanisms that are the classic stereotypical examples of a general type. One example is the Wnt
Wnt signaling pathway
The Wnt signaling pathway is a network of proteins best known for their roles in embryogenesis and cancer, but also involved in normal physiological processes in adult animals.-Discovery:...

/β-catenin signaling pathway, also known as the canonical Wnt pathway. In this pathway, the secreted signaling protein Wnt regulates gene expression by binding to Frizzled receptors and hence preventing the degradation of β-catenin, which acts as a transcriptional activator. The designation "canonical" also distinguishes this signaling pathway from others activated by Wnt ligands, the Wnt/Calcium and Wnt/Planar Cell Polarity pathways, which are referred to as non-canonical Wnt signaling pathways.

Computer science

Some circles in the field of computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

 have borrowed the usage of canonical from mathematicians. It has come to mean "the usual or standard state or manner of something"; for example, "the canonical way to organize a file system
File system
A file system is a means to organize data expected to be retained after a program terminates by providing procedures to store, retrieve and update data, as well as manage the available space on the device which contain it. A file system organizes data in an efficient manner and is tuned to the...

 is as a hierarchy
A hierarchy is an arrangement of items in which the items are represented as being "above," "below," or "at the same level as" one another...

, with extensions to make it a directed graph". XML Signature
XML Signature
XML Signature defines an XML syntax for digital signatures and is defined in the W3C recommendation . Functionally, it has much in common with PKCS#7 but is more extensible and geared towards signing XML documents...

 defines canonicalization as the process of converting XML
Extensible Markup Language is a set of rules for encoding documents in machine-readable form. It is defined in the XML 1.0 Specification produced by the W3C, and several other related specifications, all gratis open standards....

 content to a canonical form, to take into account changes that can invalidate a signature over that data (from JWSDP 1.6).

In enterprise application integration
Enterprise application integration
Enterprise Application Integration is defined as the use of software and computer systems architectural principles to integrate a set of enterprise computer applications.- Overview :...

, the Canonical Model
Canonical Model
Canonical Model is a design pattern used to communicate between different data formats. A form of Enterprise Application Integration, it is intended to reduce costs and standardize on agreed data definitions associated with integrating business systems. A Canonical Model is any model that is...

 is a design pattern
Design pattern
A design pattern in architecture and computer science is a formal way of documenting a solution to a design problem in a particular field of expertise. The idea was introduced by the architect Christopher Alexander in the field of architecture and has been adapted for various other disciplines,...

 used to communicate between different data formats. It introduces an additional format, called the "canonical format", "canonical document type" or "canonical data model". Instead of writing translators between each and every format (with potential for a combinatorial explosion
Combinatorial explosion
In administration and computing, a combinatorial explosion is the rapidly accelerating increase in lines of communication as organizations are added in a process...

), it is sufficient just to write a translator between each format and the canonical format. OASIS
OASIS (organization)
The Organization for the Advancement of Structured Information Standards is a global consortium that drives the development, convergence and adoption of e-business and web service standards...

 is an example of an integration architecture that is based on a canonical data model.

Some people have been known to use the noun canonicality; others use canonicity. In fields other than computer science, canonicity is this word's canonical form.

In computer science, a canonical name record (or CNAME record
CNAME record
A CNAME record or Canonical Name record is a type of resource record in the Domain Name System that specifies that the domain name is an alias of another, canonical domain name. This helps when running multiple services from a single IP address...

) is a type of DNS
Domain name system
The Domain Name System is a hierarchical distributed naming system for computers, services, or any resource connected to the Internet or a private network. It associates various information with domain names assigned to each of the participating entities...

 record, while "canonical number" is the old designation for a MAC address
MAC address
A Media Access Control address is a unique identifier assigned to network interfaces for communications on the physical network segment. MAC addresses are used for numerous network technologies and most IEEE 802 network technologies, including Ethernet...

 on routers and servers.


Mathematicians have for perhaps a century or more used the word canonical to refer to concepts that have a kind of uniqueness or naturalness. Examples include the canonical prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

 of positive integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, the Jordan canonical form of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 (that is built out of the irreducible factors of the characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 of the matrix), and the canonical decomposition of a permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

 into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

 of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 onto any of its quotient groups
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

, or the canonical isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 between a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism. This lack of a canonical isomorphism can be made precise in terms of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

; see natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

. But at a simpler level one could say that "any isomorphism you can think of here depends on choosing a basis." As stated by Joseph Goguen
Joseph Goguen
Joseph Amadee Goguen was a computer science professor in the Department of Computer Science and Engineering at the University of California, San Diego, USA, who helped develop the OBJ family of programming languages. He was author of A Categorical Manifesto and founder and Editor-in-Chief of the...

, "To any canonical construction from one species of structure to another corresponds an adjunction
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...

 between the corresponding categories."

Being canonical in mathematics is stronger than being a conventional choice. For instance, the vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 Rn has a standard basis
Standard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...

, which is canonical in the sense that it is not just a choice that makes certain calculations easy; in fact most linear operators on Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 take on a simpler form when written as a matrix relative to some basis other than the standard one (see Jordan form). In contrast, an abstract n-dimensional real vector space V would not have a canonical basis; it is isomorphic to Rn of course, but the choice of isomorphism is not canonical.

The word canonical is also used for a preferred way of writing something, see the main article canonical form
Canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....


In set theory, the term "canonical" identifies an element as representative of a set. If a set is partitioned
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

 into equivalence classes, then one member can be chosen from each equivalence class to represent that class. That representative member is the canonical member. If you have a canonicalizing function, f(x), that maps x to the canonical member of the equivalence class which contains it, then testing whether two items, a and b, are equivalent is the same as testing whether f(a) is identical to f(b).


In theoretical
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the concept of canonical (or conjugate, or canonically conjugate) variables is of major importance. They always occur in complementary pairs, such as spatial location x and linear momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 p, angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

 φ and angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

 L, and energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 E and time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

 t. They can be defined as any coordinates whose Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...

s give a Kronecker delta (or a Dirac delta in the case of continuous variables). The existence of such coordinates is guaranteed under broad circumstances as a consequence of Darboux's theorem
Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...

. Canonical variables are essential in the Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

 formulation of physics, which is particularly important in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. For instance, the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

 and the Heisenberg uncertainty relation
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

 always incorporate canonical variables. Canonical variables in physics are based on the aforementioned mathematical structure and therefore bear a deeper meaning than being just convenient variables. One facet of this underlying structure is expressed by Noether's theorem
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...

, which states that a (continuous) symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 in a variable implies an invariance
Invariant (physics)
In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...

 of the conjugate variable, and vice versa; for instance symmetry under spatial displacement leads to
conservation of momentum, and time-independence implies energy conservation
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...


In statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, the grand canonical ensemble
Grand canonical ensemble
In statistical mechanics, a grand canonical ensemble is a theoretical collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state...

, canonical ensemble
Canonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...

, and the microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...

 are archetypal probability distributions for the (unknown) microscopic state
Microstate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...

 of a thermal system, applying respectively in the physical cases of:
  1. An open system at fixed temperature (able to exchange both energy and particles with the environment);
  2. A closed system at fixed temperature (able to exchange energy with its environment);
  3. A closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in thermodynamics
    Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...


Occasionally, certain derivations of important ideas from more basic facts are known as "canonical derivations", and can be contrasted with "alternative derivations". One example from statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

 is the canonical derivation of the Boltzmann factors appearing in the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...


Canonical Theory

The Canonical Theory (also named the "canonical form") was developed by Joel E. Keizer and coworkers. They showed that his molecular theory explains many physical, chemical, and biological processes in an unified and canonical way, unlike the other theories cited in previous sections. Ronald F. Fox and Keizer showed the application of the canonical theory to chaos.

Prof. Keizer used the canonical form for the first formulation of statistical thermodynamics valid in far from equilibrium regimes, where the Onsager reciprocal relations
Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists....

 and the Einstein formula for the fluctuations do not work. Keizer also provided fluctuating generalizations of the Boltzmann equation and of hidrodynamics (fluctuating hydrodynamics). The applications of his work to biology are the reason that he was considered as one of the pioneers in the field of computational biology. Cosma Shalizi
Cosma Shalizi
Cosma Rohilla Shalizi is an assistant professor in the Department of Statistics at Carnegie Mellon University in Pittsburgh....

 wrote: "Chapter five applies the canonical theory to various chemical and electrochemical processes. There is a detailed comparison of a model based on the formalism to actual experimental data for a calcium-regulated potassium channel in muscle cells, yielding remarkably close agreement (especially since the channel is really just a single molecule!)... Keizer was, until his premature death in May, 1999, an active and talented scientist who played a significant role not merely in the development of the formal structure of far from equilibrium thermodynamics, but also in its application to experiment, especially in biology. Unlike a number of others who have attempted such cross-overs, he made it work."

Besides the unification of disparate topics as chemical reactions, hydrodynamics, or heat transport in solids, the canonical theory has been applied to solving the problems of traditional disciplines as statistical mechanics.

See also

  • Canonicalization
    In computer science, canonicalization , is a process for converting data that has more than one possible representation into a "standard", "normal", or canonical form...

     is a transformation to get the canonical form.
  • Literary canons
    Western canon
    The term Western canon denotes a canon of books and, more broadly, music and art that have been the most important and influential in shaping Western culture. As such, it includes the "greatest works of artistic merit." Such a canon is important to the theory of educational perennialism and the...

     and Canon (fiction)
    Canon (fiction)
    In the context of a work of fiction, the term canon denotes the material accepted as "official" in a fictional universe's fan base. It is often contrasted with, or used as the basis for, works of fan fiction, which are not considered canonical...

  • Canonical, Ltd.: an open source
    Open source
    The term open source describes practices in production and development that promote access to the end product's source materials. Some consider open source a philosophy, others consider it a pragmatic methodology...

    computer company
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