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 ^{-30} cm.; and at the scale of the Planck length space becomes a froth of probabilistic quantum foam (as shown in the diagram) and the notion of a simple, continuous space becomes inconsistent. According to the latest idea in superstring theory, the space at such small scale cannot be described by the Cartesian coordinates, x, y and z; it should be replaced by "noncommutative geometry", where the coordinates are represented by non-diagonal matrix. This is essentially the expression of uncertainty principle in quantum mechanics.
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## Figure 1 Quantum Foam[view large image] |

formulation, the length is not the fundamental attribute. The theory is based on quantized angular momentum, which corresponds to an oriented area element. Thus the area is more fundamental than the length. There is a nonzero absolution minimum volume about 10^{-99} cm^{3}, and it restricts the set of larger volumes to a discrete series of numbers. These quantum states are similar to the energy levels of the hydrogen atom. The idea is similar to the macroscopic and microscopic views of matter, for which the continuous apperance gradually changed to an assembly of discrete atoms at small scale.
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## Figure 2 Quantum Space [view large image] |
## Figure 3 Spin Network [view large image] |

Just as space is defined by a spin network's discrete geometry, time is defined by the sequence of distinct moves that rearrange the network, as shown in Figure 4. Time flows not like a river but like the ticking of a clock, with "ticks" that are about as long as the Planck time: 10^{-43} second. Or, more precisely, time in the universe flows by the ticking of innumerable clocks - in a sense, at every location in the spin network where a quantum "move" takes place, a clock at that location has ticked once. In Figure 4, the lines of the spin network become planes, and the nodes become lines. The result is called a spin foam. Taking a slice through a spin foam at a particular time yields a spin network; taking a series of slices at different times (jumping from one dotted line to another) produces frames of a movie showing the spin network evolving in time. The sequence on the right-hand side of Figure 4 shows a connected group of three volume quanta merge to become a single one.
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## Figure 4 Quantum |
## Spacetime [view large image] |

- Predictions and Tests:
- An important test is whether classical general relativity can be recovered as an approximation to the loop quantum gravity. It has been shown that long-wavelength gravitational waves propagating on otherwise flat space can be described as excitations of specific quantum states in the loop quantum gravity theory. The theory can also reproduce blackhole radiation and the relationship between blackhole's entropy and its surface area.
- The Planck scale is 16 orders of magnitude below the scale probed in the highest-energy particle accelerators currently planned (higher energy is needed to probe shorter distance scales). Thus there seems to be hopeless for the confirmation of quantum gravity theories.
- Another possible effect of discrete spacetime involves very high energy cosmic rays. It was predicted that cosmic-ray protons with an energy greater than 3x10
^{19}ev would scatter off the cosmic microwave background that fills space and should there fore never reach the Earth. However, more than 10 cosmic rays with energy over this limit were detected in an experiment called AGASA. It turns out that the discrete structure of space can raise the energy required for the scattering reaction, allowing higher-energy cosmic-ray protons to reach the Earth. If the ASASA observations hold up, and if no other explanation is found, then it may turn out that the discreteness of space has already been detected. - Loop quantum gravity has opened up a new window to investigate deep cosmological questions such as the origin of the universe. Recent loop quantum gravity calculations indicate that the big bang is actually a big bounce; before the bounce the universe was rapidly contracting. A question of similar profundity concerns the cosmological constant. Recent observations of distant supernovae and the cosmic microwave background strongly indicate that it is associated with a positive energy, which accelerates the universe's expansion. Loop quantum gravity has no trouble incorporating this fact into the theory.
- It remains to be shown that classical general relativity is a good approximate description of the loop quantum gravity theory for distances much larger than the Planck length, in all circumstances; and whether special relativity must be modified at extremely high energies (loop quantum gravity indicates that the universal speed of light is only valid for low energy photons).
- Loop quantum gravity is completely unperturbative and is also background-independent (geometry of spacetime is not fixed), and appears to lead to a pregeometry in which space and time are derived concepts (instead of being a pre-defined entity).

Nevertheless, radiation from distant cosmic explosions called gamma-ray bursts might provide a way to test whether the theory of loop quantum gravity is correct. Gamma-ray bursts occur billions of light-years away and emit a huge amount of gamma rays within a short span. According to loop quantum gravity, each photon occupies a region of lines at each instant as it moves | |

## Figure 5 Test [view large image] |
through the spin network. The discrete nature of space causes higher-energy gamma rays to travel slightly faster than lower- |