Noncommutative geometry, or
NCG, is a branch of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
concerned with the possible spatial interpretations of algebraic structures for which the commutative law fails, that is, for which
xy does not always equal
yx. For example; 3 steps of 4 units and 4 steps of 3 units length might be different in noncommutative spaces. Although one could technically construct geometries by simply removing this condition (commutativity), the results are typically trivial or uninteresting. The most common usage of the term, therefore, refers to what is properly called differential noncommutative geometry, a subject which was developed extensively by French mathematician
Alain ConnesAlain Connes is a French mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University.-Work:...
. The challenge of NCG theory is to get around the lack of commutative multiplication, which is a requirement of previous geometric theories of algebraic structures. The purpose of noncommutative geometry is as a key mathematical
tool for describing
Planck scaleIn particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1028 eV at which quantum effects of gravity become strong...
geometry, such as in the field of
quantum gravityQuantum gravity is the field of theoretical physics attempting to unify quantum mechanics with general relativity in a self-consistent manner, or more precisely, to formulate a self-consistent theory which reduces to ordinary quantum mechanics in the limit of weak gravity and which reduces to...
,
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
, or any NC
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
including the first successful QFT,
quantum electrodynamicsQuantum electrodynamics is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons...
.
History
The original algebra that motivated noncommutative geometry is the quantized phase-space of nonrelativistic
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
. According to his 1926 papers,
Paul DiracPaul Adrien Maurice Dirac, OM, FRS was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
was aware of describing phase-space in terms of the quantum analog of the
algebra of functionsIn mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
(noncommutative
quantum algebraQuantum algebra is a set of algebraic methods that allows performing quantum calculations using noncommutative algebra. In noncommutative calculations, the outcome of equations differs according to the order in which they are performed. The order of the computation changes the outcome...
). He was also aware of the absence of localization (Heisenberg uncertainty principle) in the these geometries. His work had much in common with noncommutative geometry. Later in 1986
Alain ConnesAlain Connes is a French mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University.-Work:...
introduced a comparable idea to the notion of an
exterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a form of degree zero, to differential forms of higher degree...
and generalized the
De Rham cohomologyIn mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
of
compact manifoldsIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
to noncommutative geometry.
Some of the theory developed by Alain Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular in
ergodic theoryErgodic theory is a branch of mathematics that studies dynamical systemswith an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
. The proposal of
George MackeyGeorge Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
to create a
virtual subgroup theory, with respect to which ergodic
group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
s would become
homogeneous spaceIn mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s of an extended kind, has by now been subsumed. But, in general, Connes work in NCG has predominantly been to emphasize the C*-algebras.
Motivation
In mathematics, there is a close relationship between
spaces, which are geometric in nature, and the numerical
functionsIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
on them. In general, such functions will form a
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. For instance, one may take the ring
C(
X) of
continuousIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
complexA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
-valued functions on a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X. In many important cases (
e.g., if
X is a
compactIn mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space...
Hausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T
2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most...
), we can recover
X from
C(
X), and therefore it makes some sense to say that
X has
commutative geometry.
For other cases and applications, including mathematical
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
and
functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
, noncommutative
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
s arise as the natural candidates for a ring of functions on some noncommutative "space". "Noncommutative spaces", however defined, cannot be too similar to ordinary
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s, as these are known to correspond to commutative rings in many important cases. For this reason, the field is also called
noncommutative topologyNoncommutative topology in mathematics is a term applied to the strictly C*-algebraic part of the noncommutative geometry program. The program has its origins in the Gel'fand duality between the topology of locally compact spaces and the algebraic structure of commutative C*-algebras.Several...
— some of the motivating questions of the theory are concerned with extending known topological invariants to these new spaces.
The problem is that on the microscopic level our traditional (non-quantum mechanical) notion of distance is no longer sufficient. For example, it is not possible to determine both the length and height of an object at the same time. A mathematician describes this situation by saying that the space coordinates of length and height do not commute with each other. This is good motivation to develop new mathematical tools, such as noncommutative geometry.
Motivation for NCG comes from the discovery of a mathematical formulation of
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
by Heisenberg in 1925. From a mathematical point of view, transition from classical mechanics to quantum mechanics amounts to passing from the commutative algebra of classical observables to the noncommutative algebra of quantum mechanical observables.
Definition
A noncommutative geometry of dimension n can be defined as a real spectral triple (A,H,D,J,Γ) or (A,H,D,J) depending whether dimension is even or odd (or equivalently, a
spectral tripleIn mathematical physics, a spectral triple , also known as an unbounded K-cycle, consists of an algebra A, a Hilbert space H and a Dirac operator D and defines a noncommutative space or geometry. The spectral triple, among other things, captures metric aspects of noncommutative spaces...
(A,H,D) with real structure J), if the following seven axioms are satisfied.
- 1 (dimension). There is an integer n, the dimension of the K-cycle
In mathematics, K-homology is a homology theory on the category of compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space...
, such that the length element ds := |D|-1 is an infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
of order 1=n.
- 2 (order one condition). For all a, b ∈ A, the commutation relation holds: [ [D, a], Jb*J†] = 0.
- 3 (smoothness of algebra). For any a ∈ A, [D, a] is bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
on H, and both a and [D, a] belong to the domain of smoothness Dom(δk) of the derivation δ on L(H) given by δ(T):=[|D|, T].
- 4 (orientability, Hochschild cycles and orientation). There is a Hochschild cycle
In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. It is named after Gerhard Hochschild.-Definition of Hochschild homology of algebras:...
c ∈ Zn(A, AA0) whose representative on H is π(c) = Γ if n is even or 1 if n is odd.
- 5 (finiteness of the K-cycle
In mathematics, K-homology is a homology theory on the category of compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space...
). The space of smooth vectors H∞:= Dom(Dk), is a finite projectiveIn mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
left A-moduleIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...
with a Hermitian structure (.|.) given by ∫(ξ|η)dsn:=<ξ|η>.
- 6 (Poincare duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
). The Fredholm index of the operator D yields a nondegenerate intersection form on the K-theoryIn mathematics, K-theory is a tool used in several disciplines. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It also has some applications in operator algebras...
ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
of the algebra (A, AA0).
- 7 (reality). There is an antilinear isometry
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent....
J:H → H such that the representation π0(b):= Jπ(b*)J† commutes with π(A), satisfying J2 = ±1, JD = ±DJ, JΓ = ±ΓJ where the signs are given as follows.
| n mod 8 |
n even: |
0 |
2 |
4 |
6 |
n odd: |
1 |
3 |
5 |
7 |
| J2 = ± 1 |
|
+ |
- |
- |
+ |
|
+ |
- |
- |
+ |
| JD = ± DJ |
|
+ |
+ |
+ |
+ |
|
- |
+ |
- |
+ |
| JΓ = ± ΓJ |
|
+ |
- |
+ |
- |
Noncommutative C*-algebras, von Neumann algebras
Noncommutative
C*-algebraC*-algebras are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional properties:* A is a topologically closed set in the norm topology of...
s are often now called noncommutative spaces. This is by analogy with the
Gelfand representationIn mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
, which shows that commutative C*-algebras are
dualIn mathematics, duality has numerous meanings, and although it is “a very pervasive and important concept in mathematics” and “an important general theme that has manifestations in almost every area of mathematics”, there is no single definition that unifies all concepts of duality.Generally...
to locally compact
Hausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T
2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most...
s. In general, one can associate to any C*-algebra
S a topological space
Ŝ; see
spectrum of a C*-algebraThe spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A...
.
For the
dualityIn mathematics, duality has numerous meanings, and although it is “a very pervasive and important concept in mathematics” and “an important general theme that has manifestations in almost every area of mathematics”, there is no single definition that unifies all concepts of duality.Generally...
between σ-finite measure spaces and commutative
von Neumann algebraIn mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group...
s, noncommutative
von Neumann algebraIn mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group...
s are called
noncommutative measure spaces.
Noncommutative differentiable manifolds
A smooth
Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...
M is a topological space with a lot of extra structure. From its algebra of continuous functions
C(M) we only recover
M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. It is constructed from a smooth vector bundle
E over
M, e.g. the exterior algebra bundle. The Hilbert space
L²(M,E) of square integrable sections of
E carries a representation of
C(M) by multiplication operators, and we consider an unbounded operator
D in
L²(M,E) with compact resolvent (e.g the
signature operatorIn mathematics, the signature operator is an elliptic differential operator defined on a subspace of the space of differential forms on a 4k-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold.-Preliminaries:Let be a compact...
), such that the commutators
[D,f] are bounded whenever
f is smooth. A recent deep theorem states that
M as a Riemannian manifold can be recovered from this data.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple
(A,H,D), consisting of a representation of a
C*-algebra
A on a Hilbert space
H, together with an unbounded operator
D on
H, with compact resolvent, such that
[D,a] is bounded for all
a in some dense subalgebra of
A. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.
The theory of characteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operator
K-theoryIn mathematics, K-theory is a tool used in several disciplines. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It also has some applications in operator algebras...
and
cyclic cohomologyIn homological algebra, cyclic homology and cyclic cohomology are homology theories for associative algebras introduced by Alain Connes around 1980, which play an important role in his noncommutative geometry...
. Several generalizations of now classical index theorems allow for effective extraction of numerical invariants from spectral triples.
Noncommutative affine schemes
In analogy to the
dualityDuality may refer to:In philosophy, logic, and psychology:* Dualism, a twofold division in several spiritual, religious, and philosophical doctrines* Dualism , where the body and mind are considered to be irreducibly distinct...
between affine schemes and
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s, we can also have noncommutative affine schemes.
For example, there exist an analog of the celebrated
Serre dualityIn algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
for noncommutative projective schemes. This can be shown for the noncommutative projective
schemeIn mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
when it is a commutative
noetherianIn mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian ring, a ring that satisfies the ascending chain condition on ideals....
Gorenstein ringIn commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
.
Examples of noncommutative spaces
- In Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" distribution in phase space invertibly. A synonym is phase-space quantization...
, the symplecticIn mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed, nondegenerate, 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists...
of classical mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
is deformed into a noncommutative phase space generated by the position and momentum operatorsIn mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the formor its generalizations under the operation of matrix multiplication...
.
- The standard model
The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
of particle physics is another example of a noncommutative geometry, cf noncommutative standard modelIn theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise is an extension of the Standard Model to include a modified form of general relativity. This unification implies a few constraints on...
.
- The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively, such as in quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...
, and still functions as a test case for more complicated situations.
- Nonncommutative algebras arising from foliations
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
.
- Examples related to dynamical systems arising from number theory
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, such as the Gauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.
See also
- Commutativity
In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
- Fuzzy sphere
In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is...
- JLO cocycle
In noncommutative geometry, the JLO cocycle is a cocycle in entire cyclic cohomology. It is a non-commutative version of the classic Chern character of the conventional differential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra of...
- Moyal product
In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℜ2n, equipped with its Poisson bracket .This particular star product is also sometimes...
- Noncommutative quantum field theory
In mathematical physics, a branch of mathematics and physics, noncommutative quantum field theory is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the spatial coordinates are...
- Noncommutative standard model
In theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise is an extension of the Standard Model to include a modified form of general relativity. This unification implies a few constraints on...
- Noncommutative topology
Noncommutative topology in mathematics is a term applied to the strictly C*-algebraic part of the noncommutative geometry program. The program has its origins in the Gel'fand duality between the topology of locally compact spaces and the algebraic structure of commutative C*-algebras.Several...
- Connection (algebraic framework)
Geometry of quantum systems is mainlyphrased in algebraic terms of modules andalgebras. Connections on modules are...
- Supergeometry
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
- von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group...
External links