Canonical--an adjective derived from canon--essentially means "standard" or "generally accepted" or "part of the backstory."

Religion

This word is used by theologians and canon lawyers to refer to the canons of the Eastern Orthodox and Roman Cathlolic churches, adopted by ecumenical councils. It also means "part of the canon", i.e., one of the books composing the Bible, 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 consists of vectors with 1 in one coordinate and 0 elsewhere. But this is not at all "canonical." It is just a choice which makes certain calculations on vectors easy, but most linear operators on Euclidean space only take on an especially simple form when written as a matrix relative to some basis other than the standard one.

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". 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 word canonicality; others use canonicity. In fields other than computer science, canonicity is this word's canonical form.

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. 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 form methods of counting up physical states of an ensemble of systems, typically gases. Thus they can be immediately applied to practical problems in thermodynamics.

See also

• Canonicalization is a transformation to get the canonical form.