Noncommutative geometry

Noncommutative geometry, or NCG, is a branch of mathematics 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 Connes. 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 scale geometry, such as in the field of quantum gravity, string theory, or any NC quantum field theory including the first successful QFT, quantum electrodynamics.

History

The original algebra that motivated noncommutative geometry is the quantized phase-space of nonrelativistic quantum mechanics. According to his 1926 papers, Paul Dirac was aware of describing phase-space in terms of the quantum analog of the algebra of functions (noncommutative quantum algebra). He was also aware of the absence of localization (Heisenberg uncertainty principle) in these geometries. His work had much in common with noncommutative geometry. Later in 1986 Alain Connes introduced a comparable idea to the notion of an exterior derivative and generalized the De Rham cohomology of compact manifolds to noncommutative geometry. ^[1]

Some of the theory developed by Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular in ergodic theory. The proposal of George Mackey to create a virtual subgroup theory, with respect to which ergodic group actions would become homogeneous spaces 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 functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. In many important cases (e.g., if X is a compact Hausdorff space), 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 physics^[2] and functional analysis, noncommutative rings 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 spaces, as these are known to correspond to commutative rings in many important cases. For this reason, the field is also called noncommutative topology — 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 mechanics 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 triple (A,H,D) with real structure J), if the following seven axioms are satisfied.^[3]

• 1 (dimension). There is an integer n, the dimension of the K-cycle, such that the length element ds := |D|^-1 is an infinitesimal 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 on H, and both a and [D, a] belong to the domain of smoothness Dom(δ^k) of the derivation δ on L(H) given by δ(T):=[|D|, T].
• 4 (orientability, Hochschild cycles and orientation). There is a Hochschild cycle c ∈ Z_n(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). The space of smooth vectors H_∞:= Dom(D^k), is a finite projective left A-module with a Hermitian structure (^.|^.) given by ∫(ξ|η)dsⁿ:=<ξ|η>.
• 6 (Poincare duality). The Fredholm index of the operator D yields a nondegenerate intersection form on the K-theory ring of the algebra (A, AA⁰).
• 7 (reality). There is an antilinear isometry 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.

Noncommutative C*-algebras, von Neumann algebras

Noncommutative C*-algebras are often now called noncommutative spaces. This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual to locally compact Hausdorff spaces. In general, one can associate to any C*-algebra S a topological space Ŝ; see spectrum of a C*-algebra.

For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called noncommutative measure spaces.

Noncommutative differentiable manifolds

A smooth Riemannian manifold 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 operator), 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-theory and cyclic cohomology. Several generalizations of now classical index theorems allow for effective extraction of numerical invariants from spectral triples.

Noncommutative affine schemes

In analogy to the duality between affine schemes and commutative rings, we can also have noncommutative affine schemes. For example, there exist an analog of the celebrated Serre duality for noncommutative projective schemes. This can be shown for the noncommutative projective scheme when it is a commutative noetherian Gorenstein ring. ^[4]

Examples of noncommutative spaces

• In Weyl quantization, the symplectic phase space of classical mechanics is deformed into a noncommutative phase space generated by the position and momentum operators.

• The standard model of particle physics is another example of a noncommutative geometry, cf noncommutative standard model.

• 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, and still functions as a test case for more complicated situations.

• Nonncommutative algebras arising from foliations.

• Examples related to dynamical systems arising from number theory, such as the Gauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.

See also

• Commutativity
• Fuzzy sphere
• JLO cocycle
• Moyal product
• Noncommutative quantum field theory
• Noncommutative standard model
• Noncommutative topology
• Connection (algebraic framework)
• Supergeometry
• von Neumann algebra

References

1. ^ Madore, J. (1999) An Introduction to Noncommutative Differential Geometry and its Physical Applications, 2nd ed., London Mathematical Society Lecture Note Series, No. 257. Cambridge Univ. Press.
2. ^ As for applications to particle physics, see noncommutative standard model and noncommutative quantum field theory.
3. ^ Connes, A. (1996) "Gravity coupled with matter and the foundation of noncommutative geometry."
4. ^ Yekutieli, A., and Zhang, James (1997) "Serre Duality for Nonprojective Schemes," Proc. of the A.M.S. 125(3): 697-707.

External links

• Elements of noncommutative geometry by Gracia-Bondia, Varilly and Figueroa
• Introduction to Quantum Geometry by Micho Đurđevich
• Noncommutative Geometry by Alain Connes
• Lectures on Noncommutative Geometry by Victor Ginzburg
• Very Basic Noncommutative Geometry by Masoud Khalkhali
• A walk in the noncommutative garden by Alain Connes and Matilde Marcolli
• Lectures on Arithmetic Noncommutative Geometry by Matilde Marcolli
• An Introduction to Noncommutative Spaces and their Geometry by Giovanni Landi
• Noncommutative Geometry for Pedestrians by J. Madore
• An informal introduction to the ideas and concepts of noncommutative geometry by Thierry Masson (an easier introduction that is still rather technical)
• Noncommutative geometry on arxiv.org
• Noncommutative Geometry, Quantum Fields and Motives by Alain Connes and Matilde Marcolli
• Noncommutative Geometry Blog
• Noncommutative Geometry Resources