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Noncommutative Geometry and Quantum Gravity


Algebraic Representation of Geometry
Noncommutative Characteristic
Uncertainty Principle
Quantum Gravity and Qunatization of Time

Algebraic Representation of Geometry

Circle 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
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Noncommutative Characteristic

2-D Rotation 3-D Rotation 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
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Differentiation 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

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


Quantum Gravity and Quantization of Time

Quantum Foam 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
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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:
xixj - xjxi = Kxij ,
where xi can be any one of the coordinates (x,y,z,ct), K is an unspecified function of xi, In particular, one application suggests that the spatial part of the quantum space-time is a continuum, only the time component becomes granular :

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

where = (G/c5)1/2 = 5.4x10-44 sec is the Planck time, and other similar formulas with y, z in place of x. The uncertainty principle is now in the form : xt |< x >|, where < x > denotes the average value of x. Even for < x > 1028 cm, i.e., about the size of the universe, the uncertainty becomes xt 5.4x10-16 cm-sec.

The velocity of light is c = x/t. By applying the commutative relation in Eq.(2), it becomes c' = (1 - /t)(x/t). Therefore, the uncertainty in c is
c = c - c' = (/t)c. We can identify x to the wave-length , and 1/t to the frequency to re-write (at gamma-ray frequency of ~ 1020 Hz) :

c = c - c' = ()c ~ 10-24c ---------- (3) .

Thus, the velocity of light becomes dependent on frequency in such a way that higher frequency photons could travel slower than the lower frequency ones. Current technology can detect a difference in arrival time of about 10-3 sec from gamma-ray bursts more than 10 billion light years away. The GLAST satellite will look for such events as part of its mission protocol. Meanwhile a report from the MAGIC Gamma-ray Telescope indicates a 4 minutes difference in arrival time of a gamma-ray burst signal (at two different frequencies). Whereas the following formula :

t2 - t1 = (D/c)(2 - 1) ---------- (4),

predicts a difference in arrival time (of two different gamma-ray signals) about 10-6 sec for a distance D of about 10 billion light years.