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Noncommutative Characteristic

Uncertainty Principle

Application to Quantum Gravity

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: x ^{2} + y^{2} = 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).
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## 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 02 2-D Rotation |
## Figure 03 3-D Rotation |

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).
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## Figure 05 Uncertainty Principle |
One way to satisfy Eq.(1) is to represent x and p by the matrices: |

The replacement of ordinary number by matrix can be interpreted as turning a definite entity into a blurry one.

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
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## 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. |

For example: xy = qyx, which can be re-written in a form similar to Eq.(1): xy - yx = (q-1)yx. It is then straight forward to interpret x and y as fuzzy entities. This simple example has been generalized to:

x

where x

tx - xt = ix, ---------- (2)

where (G/c

xt |< x >|

where < x > denotes the average value of x. Even for < x > 10

xt 5.4x10