Quantum differential calculus
Encyclopedia
In quantum geometry
or noncommutative geometry
a quantum differential calculus or noncommtuative differential structure on an algebra
over a field 
means the specification of a space of `differential forms' over the algebra. The algebra 
here is regarded as `coordinate algebra' 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 for an actual space of a differentiable structure. In ordinary differential geometry one can multiply differential 1-forms by functions form the left and the right and has an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:
1. An
-bimodule 
over 
, i.e. one can multiply elements elements of 
by elements of 
in an associative way:
2. A linear map
obeying the Leibniz rule
3.
4. (optional connectedness condition)
The last condition is not always imposed but holds in ordinary geometry when the manifold is connected. It says that the only `functions' killed by
are constant functions.
An exterior algebra or differential graded algebra structure over
means a compatible extension of 
to include analogues of higher order differential forms
obeying a graded-Leibniz rule with respect to an associative product on
and obeying 
. Here 
and it is usually required that 
is generated by 
. The product of differential forms is called the exterior or `wedge' product and often denoted 
. 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 group
s 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.
is commutative or functions on an actual space. This is because we do not demand that
since this would imply 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).
the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by 
and take the form
This shows how finite differences arise naturally in quantum geometry. Only the limit
has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. For
the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by 
and take the form
This shows how
-differentials arise naturally in quantum geometry.
3. For any algebra
one has a universal differential calculus defined by
where
is the algebra product. By axiom 3., any first order calculus is a quotient of this.
Quantum geometry
In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales...
or noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
a quantum differential calculus or noncommtuative differential structure on an algebra
over a field means the specification of a space of `differential forms' over the algebra. The algebra here is regarded as `coordinate algebra' 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 for an actual space of a differentiable structure. In ordinary differential geometry one can multiply differential 1-forms by functions form the left and the right and has an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:1. An
-bimodule over , i.e. one can multiply elements elements of by elements of in an associative way:-
.
2. A linear map
obeying the Leibniz rule
3.
4. (optional connectedness condition)
The last condition is not always imposed but holds in ordinary geometry when the manifold is connected. It says that the only `functions' killed by
are constant functions.An exterior algebra or differential graded algebra structure over
means a compatible extension of to include analogues of higher order differential forms
obeying a graded-Leibniz rule with respect to an associative product on
and obeying . Here and it is usually required that is generated by . The product of differential forms is called the exterior or `wedge' product and often denoted . 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
Spectral triple
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint...
, and an exterior algebra can be constructed from this data. In the quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
s 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
The above definition is minimal and gives something more general than classical differential calculus even when the algebra is commutative or functions on an actual space. This is because we do not demand that
since this would imply 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 algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
theory).
Examples
1. For the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by and take the form
This shows how finite differences arise naturally in quantum geometry. Only the limit
has functions commuting with 1-forms, which is the special case of high school differential calculus.2. For
the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by and take the form
This shows how
-differentials arise naturally in quantum geometry.3. For any algebra
one has a universal differential calculus defined by
where
is the algebra product. By axiom 3., any first order calculus is a quotient of this.See also
- Quantum geometryQuantum geometryIn theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales...
- Noncommutative geometryNoncommutative geometryNoncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
- Quantum calculusQuantum calculusQuantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus". h ostensibly stands for Planck's constant while q stands for quantum...
- Quantum groupQuantum groupIn mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
- Quantum spacetimeQuantum spacetimeIn mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra...