In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure means the specification of a space of differential forms here is regarded as a coordinate ring but it is important that it may be noncommutative and hence not an actual algebra of coordinate functions on any actual space, so this represents a point of view replacing the specification of a differentiable structure for an actual space. In ordinary differential geometry one can multiply differential 1-forms by functions from the left and from the right, and there exists an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:

are constant functions.

An exterior algebra or differential graded algebra to include analogues of higher order differential forms

. The product of differential forms is called the exterior or wedge product. The noncommutative or quantum de Rham cohomology is defined as the cohomology of this complex.

A higher order differential calculus can mean an exterior algebra, or it can mean the partial specification of one, up to some highest degree, and with products that would result in a degree beyond the highest being unspecified.

The above definition lies at the crossroads of two approaches to noncommutative geometry. In the Connes approach a more fundamental object is a replacement for the Dirac operator in the form of a spectral triple, and an exterior algebra can be constructed from this data. In the quantum groups approach to noncommutative geometry one starts with the algebra and a choice of first order calculus but constrained by covariance under a quantum group symmetry.

Note

is commutative or functions on an actual space. This is because we do not demand that

, which would violate axiom 4 when the algebra was noncommutative. As a byproduct, this enlarged definition includes finite difference calculi and quantum differential calculi on finite sets and finite groups (finite group Lie algebra theory).

Examples

1. has functions commuting with 1-forms, which is the special case of high school differential calculus.

2. -differentials arise naturally in quantum geometry.

3. one has a universal differential calculus is the algebra product. By axiom 3., any first order calculus is a quotient of this.

See also

• Quantum geometry

• Noncommutative geometry

• Quantum calculus

• Quantum group

• Quantum spacetime

Further reading