## Noncommutative Geometry and Quantum Gravity

### Contents

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

### 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).

### 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 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.

### 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: x p  (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 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.

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 : x t  |< 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 x t 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.