Canonical

For the software company, see Canonical Ltd.

Canonical is an adjective derived from canon. Canon essentially means "rule", "law", "standard", and has come to mean "generally accepted" or "authoritatively correct."

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"

Religion

This word is used by theologians and canon lawyers to refer to the canons of the Roman Catholic, Eastern Orthodox and Anglican Churches adopted by ecumenical councils. It also refers to later law developed by local churches and dioceses of these churches. The function of this collection of various "canons" is somewhat analogous to the precedents established in common law by case law.

In the 20th century, the Roman Catholic Church revised its canon law in 1917 and then again 1981 into the modern Code of Canon Law. 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 law.

Canonical can also mean "part of the canon", i.e., one of the books comprising the biblical canon, as opposed to apocryphal books.

Literature and art

It is used most often when describing bodies of literature or art: those books that all educated people have read make up the "canon" (see also canon (fiction)).

Mathematics

Mathematicians have for perhaps a century or more used the word canonical to refer to concepts that have a kind of uniqueness or naturalness, and are (up to trivial aspects) "independent of coordinates." Examples include the canonical prime factorization of positive integers, the Jordan canonical form of matrices (which is built out of the irreducible factors of the characteristic polynomial of the matrix), and the canonical decomposition of a permutation into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical homomorphism of a group onto any of its quotient groups, or the canonical isomorphism between a finite-dimensional vector space 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, but one could say at a simpler level that "any isomorphism you can think of here depends on choosing a basis.")

Being canonical in mathematics is stronger than being a conventional choice. For instance, the vector space Rⁿ has a standard basis which is canonical in the sense that it is not just a choice which makes certain calculations easy; in fact most linear operators on Euclidean space 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.

Computer science

Some circles in the field of computer science have borrowed this usage 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 is as a hierarchy, with extensions to make it a directed graph". XML Signature defines canonicalization as the process of converting XML content to a canonical form, to take into account changes that can invalidate a signature over that data (from JWSDP 1.6).

For an illuminating story about the word's use among computer scientists, see the Jargon File's entry for the word[1].

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.

The company Canonical Ltd. is named jocularly referencing this usage.

Physics

In theoretical physics, the concept of canonical (or conjugate) variables is of major importance. They always occur in complementary pairs, such as spatial location x and linear momentum p, angle φ and angular momentum L, and energy E and time t. They can be defined as any coordinates whose Poisson brackets 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. Canonical variables are essential in the Hamiltonian formulation of physics, which is particularly important in quantum mechanics. For instance, the Schrödinger equation and the Heisenberg uncertainty relation 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, which states that a symmetry in a variable implies an invariance of the conjugate variable, and vice versa; for instance symmetry under time reversal is followed by energy conservation.

In statistical mechanics, the canonical ensemble, the grand canonical ensemble, and the microcanonical ensemble are archetypal probability distributions for the (unknown) microscopic state of a thermal system, applying respectively in the physical cases of:- a closed system at fixed temperature (able to exchange energy with its environment); an open system at fixed temperature (able to exchange both energy and particles); and a closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in thermodynamics.

See also

• Canon (fiction)
• Canonicalization is a transformation to get the canonical form.

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