is discussed in more detail in the Heisenberg group section, below. Informally stated, with certain technical assumptions, every representation of the Heisenberg group is equivalent to the position operators and momentum operators on . Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension . More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation. This was later generalized by Mackey theory – and was the motivation for the introduction of the Heisenberg group in quantum physics. In detail: The continuous Heisenberg group is a central extension of the abelian Lie group by a copy of , the corresponding Heisenberg algebra is a central extension of the abelian Lie algebra (with trivial bracket) by a copy of , the discrete Heisenberg group is a central extension of the free abelian group by a copy of , and the discrete Heisenberg group modulo is a central extension of the free abelian -group by a copy of . In all cases, if one has a representation , where is an algebra and the center maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is Fourier theory. If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to central representations. Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the center of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the scalar matrices. Thus the representation of the center of the Heisenberg group is determined by a scale value, called the quantization value (in physics terms, Planck's constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit). More formally, the group algebra of the Heisenberg group over its field of scalars K, written , has center , so rather than simply thinking of the group algebra as an algebra over the field , one may think of it as an algebra over the commutative algebra . As the center of a matrix algebra or operator algebra is the scalar matrices, a -structure on the matrix algebra is