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The transformation between functions in the position space x and the momentum space k is effected via the Fourier integral: (k) = (1/2) ^{½}_{}e^{-ikx}_{}(x)dxwhere the momentum p = k. This formula can be readily generalized to the case of transformation between space-time and 4-momentum. | ||

## Figure 23 Curved Momentum Space |
## Figure 24 Space-time Diagram |

- Followings list some peculiar properties of the momentum space, the recent investigations of which may shed some new ideas on quantum gravity.
- Space-time and 4-momentum Symmetry - It was noticed in 1938 by M. Born that many equations are symmetrical (invariant) with the exchange between space-time coordinates and 4-momentum. A modern example is the S-matrix, which yields identical result whether the calculation is performed in coordinate or momentum space. It was not clear what wrapped the momentum space while we know that space-time is distorted by mass-energy (Figure 23).
- Curved Momentum Space - The implication of this coordinates/momentum symmetry and curved momentum space remained unclear until lately in the 20
^{th}century when recent study reveals a startling reality. It is a tenet in both special and general relativity that the space-time interval S is an invariant quantity under the Lorentz transformation in space-time, i.e., S = (c^{2}t^{2}- x^{2})^{½}= (c^{2}t'^{2}- x'^{2})^{½}(see Figure 24). However, it is found that with a curved momentum space the space-time interval S is no longer an invariance although the difference is only 1/10^{18}for two observers separated by a distance of 10^{10}light years. This result implies that there is a certain fuzziness in measuring S. - Noncommutative Space-time - It has been shown in the 1990s that noncommutative space-time is responsible for the curved momentum space. Since fuzziness or quantum uncertainty is associated with noncommutative space-time, these two independent studies seem to reinforce the idea that space-time is granular, and should be treated by quantum theory at the Planck scale.
- Test - A proposed test of the curved momentum space involves the arrival time for high energy photon from gamma-ray burst. It should be detected a little bit later than the lower-energy photon. Observations from telescope in the Canary Islands and NASA's Fermi gamma-ray space telescope have tentatively confirmed such difference. Since the difference could be generated by other mechanisms such as delayed explosions, more data are needed to arrive at a definite conclusion.
- Phase Space - It is suggested that space-time and 4-momentum should be merged into an eight dimensional phase space, in which another invariance replaces S in relativity. Such union demands at least the unit of the two should be compatible. It turns out that the 4-momentum would have the unit of length if it is multiplied by a factor of (G/c
^{3}), which has an extremely small value of 2.47x10^{-39}sec/gm. For the ultimate high energy (the Planck energy for a particle) of 1.22x10^{19}Gev, the combined value would have just the Planck length of 1.6x10^{-33}cm. Thus, it is no wonder that such correction is negligible under normal circumstance. - Entropy - The idea of phase space inevitably brings up the concept of entropy as defined by Boltzmann in 1872. The definition involves an extended phase space containing many particles. The generalized point in such phase space has a natural tendency (statistically) to settle down into a state of equilibrium. Thus the development of the space-time granules can be equated to the evolution of entropy - offering an alternate way to describe the history of the cosmos.

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