Quantum differential calculus

In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A over a field k means the specification of a space of differential forms over the algebra. The algebra A 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 A-A-bimodule Ω1 over A, i.e. one can multiply elements of Ω1 by elements of A in an associative way: a(ωb)=(aω)b, ∀a,b∈A, ω∈Ω1.
2. A linear map d:A->Ω1 obeying the Leibniz rule d(ab)=a(db)+(da)b, ∀a,b∈A
3. Ω1={a(db) | a,b∈A}
4. (optional connectedness condition) ker⁡ d=k1

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 d are constant functions.

An exterior algebra or differential graded algebra structure over A means a compatible extension of Ω1 to include analogues of higher order differential forms

Ω=⊕nΩn, d:Ωn->Ωn+1

obeying a graded-Leibniz rule with respect to an associative product on Ω and obeying d2=0. Here Ω0=A and it is usually required that Ω is generated by A,Ω1. 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 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.

## Contents

* 1 Note
* 2 Examples
* 3 See also
* 4 Further reading

## Note

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

a(db)=(db)a, ∀a,b∈A

since this would imply that d(ab−ba)=0, ∀a,b∈A, 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 A=ℂ[x] the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by λ∈ℂ and take the form Ω1=ℂ.dx,(dx)f(x)=f(x+λ)(dx),df=f(x+λ)−f(x)λdx This shows how finite differences arise naturally in quantum geometry. Only the limit λ->0 has functions commuting with 1-forms, which is the special case of high school differential calculus.
2. For A=ℂ[t,t−1] the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by q≠0∈ℂ and take the form Ω1=ℂ.dt,(dt)f(t)=f(qt)(dt),df=f(qt)−f(t)q(t−1)dt This shows how q-differentials arise naturally in quantum geometry.
3. For any algebra A one has a universal differential calculus defined by Ω1=ker⁡(m:A⊗A->A),da=1⊗a−a⊗1,∀a∈A where m 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

* v
* t
* e

Quantum mechanics

Background|

* Introduction
* History
* timeline
* Glossary
* Classical mechanics
* Old quantum theory

Fundamentals|

* Bra–ket notation
* Casimir effect
* Complementarity
* Density matrix
* Energy level
* ground state
* excited state
* degenerate levels
* Vacuum state
* Zero-point energy
* QED vacuum
* QCD vacuum
* Hamiltonian
* Operator
* Quantum coherence
* Quantum decoherence
* Measurement
* Quantum
* Quantum realm
* Quantum system
* Quantum state
* Quantum number
* Quantum entanglement
* Quantum superposition
* Quantum nonlocality
* Quantum tunnelling
* Quantum levitation
* Quantum fluctuation
* Quantum annealing
* Quantum foam
* Quantum noise
* Heisenberg uncertainty principle
* Spontaneous parametric down-conversion
* Von Neumann entropy
* Spin
* Scattering theory
* Symmetry in quantum mechanics
* Symmetry breaking
* Spontaneous symmetry breaking
* Wave propagation
* Quantum interference
* Wave function
* Wave function collapse
* Wave–particle duality
* Matter wave
* Qubit
* Qutrit
* Observable
* Probability distribution

Formulations|

* Formulations
* Heisenberg
* Interaction
* Matrix mechanics
* Schrödinger
* Path integral formulation
* Phase space

Equations|

* Dirac
* Klein–Gordon
* Pauli
* Rydberg
* Schrödinger

Interpretations|

* Interpretations
* Bayesian
* Consistent histories
* Copenhagen
* de Broglie–Bohm
* Ensemble
* Hidden variables
* Many worlds
* Objective collapse
* Quantum logic
* Relational
* Stochastic
* Transactional
* Cosmological

Experiments|

* Afshar
* Bell's inequality
* Cold Atom Laboratory
* Davisson–Germer
* Delayed-choice quantum eraser
* Double-slit
* Franck–Hertz experiment
* Leggett–Garg inequality
* Mach-Zehnder inter.
* Elitzur–Vaidman
* Popper
* Quantum eraser
* Schrödinger's cat
* Quantum suicide and immortality
* Stern–Gerlach
* Wheeler's delayed choice

Science|

* Q-analog
* List
* Quantum algebra
* Quantum calculus
* Quantum differential calculus
* Quantum geometry
* Quantum group
* Quantum Bayesianism
* Quantum biology
* Quantum chemistry
* Quantum chaos
* Quantum cognition
* Quantum cosmology
* Quantum dynamics
* Quantum economics
* Quantum evolution
* Quantum finance
* Quantum game theory
* Quantum measurement problem
* Quantum mind
* Quantum probability
* Quantum social science
* Quantum stochastic calculus
* Quantum spacetime

Technology|

* Quantum algorithms
* Quantum amplifier
* Quantum cellular automata
* Quantum finite automata
* Quantum electronics
* Quantum logic gates
* Quantum clock
* Quantum channel
* Quantum bus
* Quantum circuit
* Phase qubit
* Matrix isolation
* Quantum dot
* Quantum dot display
* Quantum dot solar cell
* Quantum dot cellular automaton
* Quantum dot single-photon source
* Quantum dot laser
* Quantum well
* Quantum computing
* Timeline
* Quantum cryptography
* Post-quantum cryptography
* Quantum error correction
* Quantum imaging
* Quantum image processing
* Quantum information
* Quantum key distribution
* Quantum machine
* Quantum machine learning
* Quantum metamaterial
* Quantum metrology
* Quantum network
* Quantum neural network
* Quantum optics
* Quantum programming
* Quantum sensors
* Quantum simulator
* Quantum teleportation
* Quantum levitation
* Time travel
* Quantum complexity theory

Extensions|

* Quantum statistical mechanics
* Relativistic quantum mechanics
* Quantum field theory
* Axiomatic quantum field theory
* Quantum field theory in curved spacetime
* Thermal quantum field theory
* Topological quantum field theory
* Local quantum field theory
* Conformal field theory
* Two-dimensional conformal field theory
* Liouville field theory
* History
* Quantum gravity

* Category
* Physics Portal
* Physics WikiProject
* Commons

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Quantum differential calculus. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Quantum_differential_calculus&oldid=217116"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template