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|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
|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 03 3-D Rotation
|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.
|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).
|One way to satisfy Eq.(1) is to represent x and p by the matrices:|
|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
|Figure 06 is highly conjectural just for illustration purpose). However, we can use the algebraic representation to proceed by first making the spacetime noncommutative.|