Noncommutative Geometry and Quantum Gravity

Contents

Algebraic Representation of Geometry

Geometry is defined as the mathematics of shapes: the branch of mathematics that is concerned with the properties and relationships of points, lines, angles, curves, surfaces, and solids. It implies the existence of objects in a continuous medium such as the circle in Figure 01. Such objects can be transformed into another level of abstraction with a formula. For example, the circle can be described by the formula: x2 + y2 = 1, where x and y are the coordinates of a point P(x,y) on the circle of radius 1. This algebraic representation is very useful if we wish to study objects beyond our visualization such as something in 5-dimensional spacetime, or when the concept of continuous medium breaks down (as will be discussed presently for the case of quantum gravity).

Figure 01 Circle [view large image]

Noncommutative Characteristic

		Commutativity in mathematics means the same result in an operation irrespective of the order in which two or more objects are inter-changed. Addition and multiplication are commutative processes, while subtraction and division are not. Other examples involve rotations. While two consecutive rotations in 2 dimension is commutative (Figure 02) such that ab = ba = c, rotations in 3 dimension is noncommutative as shown in Figure 03 - R1 R2 R2 R1.
Figure 02 2-D Rotation	Figure 03 3-D Rotation [view large image]

Thus the transition from a commutative operation (such as multiplication) xy = yx to a noncommutative operation can be achieved by making xy = qyx or xy - yx = (q -1)yx with q 1. Such change of rule alters all the mathematical manipulation subsequently. For example, consider taking the derivative of the curve y = x2 with the noncommutative rule x(x) = q(x)x, where x = x'-x (Figure 04): y/x = [(x+x)2 - x2]/x = [(1+q)(x)x + (x)2]/x, at the limit x 0, dy/dx = (1+q)x, which equals to the usual form only when q = 1.

Figure 04 Differentiation

Uncertainty Principle

Another way to express commutative relation is to follow the process of quantization in quantum mechanics: xp - px = i ---------- (1), where x is the position of the particle, p is the x-component of its momentum, and = h/2 = 1.054x10-27 erg-sec is the Planck's constant divided by 2. It can be shown that taking the average of Eq.(1) leads to the expression for the uncertainty principle: xp (Figure 05, see "Schwartz's Inequality" for proof).

Figure 05 Uncertainty Principle

One way to satisfy Eq.(1) is to represent x and p by the matrices:

Quantum Gravity and Quantization of Time

By combining the laws of quantum mechanics and general relativity, it is deduced that in a region the size of the Planck length (10-33 cm.), the vacuum fluctuations are so huge (relatively) that space as we know it "boils" and becomes a froth of quantum foam. In such a scenario, the space appears completely smooth at the scale of 10-12 cm.; a certain roughness starts to show up at the scale of 10-20 cm.; and at the scale of the Planck length space becomes a froth of probabilistic quantum foam (as shown in Figure 06) and the notion of a simple, continuous space becomes inconsistent. It implies that spacetime has to be quantized at the Planck scale even though we don't know what it looks like (BTW, the quantum foam in

Figure 06 Quantum Foam [view large image]

Figure 06 is highly conjectural just for illustration purpose). However, we can use the algebraic representation to proceed by first making the spacetime noncommutative.