Quantum Differential Calculus

In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra [math]\displaystyle{ A }[/math] over a field [math]\displaystyle{ k }[/math] means the specification of a space of differential forms over the algebra. The algebra [math]\displaystyle{ A }[/math] 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:

1. An [math]\displaystyle{ A }[/math]-[math]\displaystyle{ A }[/math]-bimodule [math]\displaystyle{ \Omega^1 }[/math] over [math]\displaystyle{ A }[/math], i.e. one can multiply elements of [math]\displaystyle{ \Omega^1 }[/math] by elements of [math]\displaystyle{ A }[/math] in an associative way: [math]\displaystyle{ a(\omega b)=(a\omega)b,\ \forall a,b\in A,\ \omega\in\Omega^1 . }[/math]
2. A linear map [math]\displaystyle{ {\rm d}:A\to\Omega^1 }[/math] obeying the Leibniz rule [math]\displaystyle{ {\rm d}(ab)=a({\rm d}b) + ({\rm d}a)b,\ \forall a,b\in A }[/math]
3. [math]\displaystyle{ \Omega^1=\{a({\rm d}b)\ |\ a,b\in A\} }[/math]
4. (optional connectedness condition) [math]\displaystyle{ \ker\ {\rm d}=k1 }[/math]

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 [math]\displaystyle{ {\rm d} }[/math] are constant functions.

An exterior algebra or differential graded algebra structure over [math]\displaystyle{ A }[/math] means a compatible extension of [math]\displaystyle{ \Omega^1 }[/math] to include analogues of higher order differential forms

[math]\displaystyle{ \Omega=\oplus_n\Omega^n,\ {\rm d}:\Omega^n\to\Omega^{n+1} }[/math]

obeying a graded-Leibniz rule with respect to an associative product on [math]\displaystyle{ \Omega }[/math] and obeying [math]\displaystyle{ {\rm d}^2=0 }[/math]. Here [math]\displaystyle{ \Omega^0=A }[/math] and it is usually required that [math]\displaystyle{ \Omega }[/math] is generated by [math]\displaystyle{ A,\Omega^1 }[/math]. The product of differential forms is called the exterior or wedge product and often denoted [math]\displaystyle{ \wedge }[/math]. 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

The above definition is minimal and gives something more general than classical differential calculus even when the algebra [math]\displaystyle{ A }[/math] is commutative or functions on an actual space. This is because we do not demand that

[math]\displaystyle{ a({\rm d}b) = ({\rm d}b)a,\ \forall a,b\in A }[/math]

since this would imply that [math]\displaystyle{ {\rm d}(ab-ba)=0,\ \forall a,b\in A }[/math], 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. For [math]\displaystyle{ A={\mathbb C}[x] }[/math] the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by [math]\displaystyle{ \lambda\in \mathbb C }[/math] and take the form [math]\displaystyle{ \Omega^1={\mathbb C}.{\rm d}x,\quad ({\rm d}x)f(x)=f(x+\lambda)({\rm d}x),\quad {\rm d}f={f(x+\lambda)-f(x)\over\lambda}{\rm d}x }[/math] This shows how finite differences arise naturally in quantum geometry. Only the limit [math]\displaystyle{ \lambda\to 0 }[/math] has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. For [math]\displaystyle{ A={\mathbb C}[t,t^{-1}] }[/math] the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by [math]\displaystyle{ q\ne 0\in \mathbb C }[/math] and take the form [math]\displaystyle{ \Omega^1={\mathbb C}.{\rm d}t,\quad ({\rm d}t)f(t)=f(qt)({\rm d}t),\quad {\rm d}f={f(qt)-f(t)\over q(t-1)}\,{\rm dt} }[/math] This shows how [math]\displaystyle{ q }[/math]-differentials arise naturally in quantum geometry.
3. For any algebra [math]\displaystyle{ A }[/math] one has a universal differential calculus defined by [math]\displaystyle{ \Omega^1=\ker(m:A\otimes A\to A),\quad {\rm d}a=1\otimes a-a\otimes 1,\quad\forall a\in A }[/math] where [math]\displaystyle{ m }[/math] 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

• Connes, A. (1994), Noncommutative geometry, Academic Press, ISBN 0-12-185860-X, https://archive.org/details/noncommutativege0000conn 
• Majid, S. (2002), A quantum groups primer, London Mathematical Society Lecture Note Series, 292, Cambridge University Press, doi:10.1017/CBO9780511549892, ISBN 978-0-521-01041-2 

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