Noncommutative Geometry: Fuzzy Spaces, the Groenewold-Moyal Plane

In this talk, we review the basics concepts of fuzzy physics and quantum field theory on the Groenwald-Moyal Plane as examples of noncommutative spaces in physics. We introduce the basic ideas, and discuss some important results in these fields. In the end we outline some recent developments in the field.


Introduction
Noncommutative geometry is a branch of mathematics due to Gel'fand, Naimark, Connes, Rieffel and many others [1,2,3]. Physicists in a very short time adopted it and nowadays use this phrase whenever spacetime algebra is noncommutative.
There are two such particularly active fields in physics at present 1. Fuzzy Physics, 2. Quantum Field Theory (QFT) on the Groenewald-Moyal Plane.
Item 1 is evolving into a tool to regulate QFT's, and for numerical work. It is an alternative to lattice methods. Item 2 is more a probe of Planck-scale physics. This introductory talk will discuss both items 1 and 2.

History
The Groenewold-Moyal (G-M) plane is associated with noncommutative spacetime coordinates: It is an example where spacetime coordinates do not commute.
The idea that spatial coordinates may not commute first occurs in a letter from Heisenberg to Peierls [4,5]. Heisenberg suggested that an uncertainty principle such as can provide a short distance cut-off and regulate quantum field theories (qft's). In this letter, he apparently complains about his lack of mathematical skills to study this possibility. Peierls communicated this idea to Pauli, Pauli to Oppenheimer and finally Oppenheimer to Snyder. Snyder wrote the first paper on the subject [6]. This was followed by a paper of Yang [7].
In mid-90's, Doplicher, Fredenhagen and Roberts [8,9] systematically constructed unitary quantum field theories on the G-M plane and its generalizations, even with time-space noncommutativity.
Later string physics encountered these structures.

What is noncommutative geometry
According to Connes [1,2,3], noncommutative geometry is a spectral triple, where A = a C * -algebra, possibly noncommutative, D = a Dirac operator, H = a Hilbert space on which they are represented. If A is a commutative C * -algebra, we can recover a Hausdorff topological space on which A are functions,using theorems of Gel'fand and Naimark. But that is not possible if A is not commutative. But it is still possible to formulate qft's using the spectral triple. A class of examples of noncommutative geometry with A noncommutative is due to Connes and Landi [10].
If some of the strict axioms are not enforced then the examples include SU (2) q , fuzzy spaces, the G-M plane, and many more.
The introduction of noncommutative geometry has introduced a conceptual revolution. Manifolds are being replaced by their "duals", algebras, and these duals are being "quantized", much as in the passage from classical to quantum mechanics.

Fuzzy physics
In what follows, we sketch the contents of "fuzzy physics". Reference [11] contains a detailed survey. For pioneering work on fuzzy physics, see [12,13,14].

What is fuzzy physics [11]
We explain the basic ideas of fuzzy physics by a two-dimensional example: S 2 F . Consider the two-sphere S 2 . We quantize it to regularize by introducing a short distance cut-off. For example in classical mechanics, the number of states in a phase space volume is infinite. But we know since Planck and Bose that on quantization, it becomes This is the idea behind fuzzy regularization.
In detail, this regularization works as follows on S 2 . We have Now consider angular momentum L i : wherex i ∈ Mat 2l+1 ≡ space of (2l+1)×(2l+1) matrices. As l → ∞, they become commutative. They give the fuzzy sphere S 2 F of radius r and dimensions 2l + 1.

Why is this space fuzzy
Asx i ,x j (i = j) do not commute, we cannot sharply localizex i . Roughly in a volume 4πr 2 there are (2l + 1) states.

Field theory on fuzzy sphere
A scalar field on fuzzy sphere is defined as a polynomial inx i , i.e., Differentiation is given by infinitesimal rotation: A simple rotationally invariant scalar field action is given by Simulations have been performed [15,16] on the partition function Z = dΦe −S(Φ) of this model and the major findings include the following: • Continuum limit exists.
Also S 2 F can nicely describe topological features. Hence it seems better suited for preserving symmetries than lattice approximations.

Strings [33]
If N D-branes are close, the transverse coordinates Φ i become N × N matrices with the action given by where f ijk are totally antisymmetric.
The equations of motion give solutions when f ijk are structure constants of a simple compact Lie group. Thus we can have If L i form an irreducible set, then we have L · L = l(l + 1), (2l + 1) = N, and we have one fuzzy sphere. Or we can have a direct sum of irreducible representations: Then we have many fuzzy spheres. Stability analysis of these solutions including numerical studies has been done by many groups.

Quantum gravity and spacetime noncommutativity: heuristics
The following arguments were described by Doplicher, Fredenhagen and Robert in their work in support of the necessity of noncommutative spacetime at Planck scale.

Space-space noncommutativity
In order to probe physics at the Planck scale L, the Compton wavelength M c of the probe must fulfill Such high mass in the small volume L 3 will strongly affect gravity and can cause black holes to form. This suggests a fundamental length limiting spatial localization.

Time-space noncommutativity
Similar arguments can be made about time localization. Observation of very short time scales requires very high energies. They can produce black holes and black hole horizons will then limit spatial resolution suggesting ∆t∆| x| ≥ L 2 , L = a fundamental length.
The G-M plane models above spacetime uncertainties.

What is the G-M plane
The Groenewald-Moyal plane A θ (R d+1 ) consists of functions α, β, . . . on R d+1 with the * -product For spacetime coordinates, this implies, Conversely these coordinate commutators imply the general * -product up to certain equivalencies. The G-M plane also emerges in quantum Hall effect and string physics.

Quantum Hall ef fect (the Landau problem) [34]
Consider an electron in 1-2 plane and an external magnetic field B = (0, 0, B) perpendicular to the plane. Then the Lagrangian for the system is is the electromagnetic potential and x a are the coordinates of the electron. Now if eB → ∞, then This means that on quantization we will have which defines a G-M plane.

Strings [35]
Consider open strings ending on Dp-Branes. If there is a background two-form Neveu-Schwarz field given by the constants B ij = −B ji , then the action is given by As B → ∞ or equivalently g ij → 0, which is just a G-M plane. Fig. 1 indicates different sources wherefrom fuzzy physics and the G-M plane emerge. The question mark is to indicate that the G-M plane may not regularize qft's.

Until 2004/2005, much work was done on
• QFT's on the G-M plane and its renormalization theory, uncovering the phenomenon of UV/IR mixing [36].
This brings into question much of the prehistory-analysis. Examples include the following new results: 1. The Pauli principle can be violated on the G-M plane.
3. There need be no ultraviolet-infrared (UV-IR) mixing in the absence of gauge fields [54].
There is also a striking, clean separation of matter from gauge fields due to the Drinfel'd twist [55], (in the sense that they have to be treated differently) reminiscent of the distinction between particles and waves in the classical theory.
Literature should be consulted for details of these developments.