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Elementary Particles and the World of Planck Scale

Quantum Foam and Loop Quantum Gravity

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 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 15-24a) 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. In other word, it is impossible to determine the coordinates precisely at any one time. This is essentially the extension of the uncertainty principle in quantum
Quantum Foam VLTI mechanics. Thus, at the small scale, the usual notion of space has broken down. However, it is found that large pieces of relativity theory, quantum theory and particle physics can be carried over into such a world. In the last few years theoretical physicists have been surprised to discover that both loop quantum gravity and superstring theory describe worlds in which the geometry is noncommutative, which can be used now as a new language to compare the two theories. Observationally, it is suggested in 2006 that the VLTI (Vey Large Telescope Interferometer to be completed in a few

Figure 15-24a Quantum Foam
[view large image]

Figure 15-24b VLTI [view large image]

years on Mount Paranal, Chile) can be used to detect the slightly different paths caused by the quantum foam (Figure 15-24b).

A theory to describe space with the size of the Planck length has been developed sicne the 1950s. It is called theory of Loop Quantum Gravity (LQG), which postulates that the minimum linear size of space is of the order of the Planck length. Space with larger expanse are built upon this
Quantum Space Spin Network lowest size such that the area and volume are quantized as shown in Figure 15-25. In the LQG formulation, the length is not the fundamental attribute. The theory is based on quantized angular momentum, which corresponds to an oriented area element (see an example in "Curvilinear Motion"). Thus the area is more fundamental than the length. There is a nonzero absolution minimum volume about 10-99 cm3, 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.

Figure 15-25 Quantum Space [view large image]

Figure 15-26 Spin Network

Description of a quantum of space can be simplified by representing the volume with a dot or node, and the area (enclosing the volume) with a line perpendicular to the face (see Figure 15-26 a and b). The numbers for the node or line (in Figure 15-26 b) indicate the size of the volume or area. In this case, the quantum of the volume has eight units of the cubic Planck length. Figure 15-26 c and d show the connection of two volumes and its representation in nodes and lines. The network in Figure 15-27 shows the connection of many discrete volumes; it is called the "spin network". Particles (real and virtual), such as electrons, correspond to certain types of nodes, which are represented by adding more labels on the nodes. Field, such as the electromagnetic field, are represented by additional labels on the lines of the graph. For example, the eigenvalue of the quantized area, i.e., the link between two nodes and essentially to the total angular momentum (see Figure 15-26c,d), is approximately given by the formula at the bottom of Figure 15-25, where j is an integer or 1/2 integer spin quantum number associated with the link.

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
Computer Model Quantum Spacetime shown in Figure 15-27. 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 15-28, 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 15-28 shows a connected group of three volume quanta merge to become a single one. Figure 15-27 is a computer model of a quantum spacetime, showing the evolution of the spin network. It portrays the strong fluctuation caused by the uncertainty principle.

Figure 15-27 Spin Network Evolution [view large image]

Figure 15-28 Quantum Time

    Predictions and Tests:

  1. 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.
  2. 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
  3. Test quantum gravity theories. 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 through the spin

    Figure 15-29a Test [view large image]

    network. The discrete nature of space causes higher-energy gamma rays to travel slightly faster than lower-energy ones. The difference is tiny, but its effect steadily
    accumulating during the rays' billion-year voyage. If a burst's gamma rays arrive at Earth in slightly different times according to their energy, that would be evidence for loop quantum gravity (see Figure 15-29a). The GLAST satellite, which is scheduled to be launched in 2005, will have the required sensitivity for this experiment.
    Gamma-ray Observatories In the summer of 2007, it is reported that the MAGIC Gamma-ray Telescope has measured a 4-minute time difference between the arrival of high and low-energy gamma rays released at the same time in a flare from the Markarian 501 galaxy, some half a billion light years away. However, the order seems to be the reverse, it is the lower-energy photons that arrive earlier. Currently in 2009, there are 5 gamma-ray telescopes (Figure 15-29b) scanning the sky across a wide range of energy. In addition, there will be a neutrino observatory at the South Pole to check if distant neutrinos arrive here in equal number of all flavours as predicted by the interaction with quantum foam. Also see "remarkable GRB130427A".

    Figure 15-29b Gamma-ray Observatories

    For those quantum-gravity theories which predict the variation of photon speed v with photon energy E, there is an approximate formula to express the relation at E << EPlank = 1.22x1019Gev:
    v c [1 + (E/EQG,n)n]
    where n is a model-dependent integer, and EQG,n is the quantum-gravity energy scale (for a particular model n) at which the quantum-gravity effect "turns on". For n = 1, this formula can be reduced further to:
    t = (D/c) (E/EQG,1)
    where D is the distance to the source, t is the difference in arrival time corresponding to the difference in photon energy E. Observations of the GRB090510 gamma-ray bursts by the Fermi Gamma-ray Space Telescope measured no discernible speed variation in a range of energy from 35 Mev to 31 Gev within the interval 0.50 - 1.45 sec. Since GRB090510 exhibits a red shift of z = 0.9, the corresponding comoving distance is 1.3x1028cm, thus a lower limit of EQG,1 = 1.2 EPlank is obtained from the formula. It means that there is no variation of photon speed with quantum-gravity model involving such energy scale (or inversely involving the smallest unit of length ~ LPlanck/1.2) .
  4. Another possible effect of discrete spacetime involves very high energy cosmic rays. It was predicted that cosmic-ray protons with an energy greater than 3x1019 ev would scatter off the cosmic microwave background that fills space and should therefore 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 AGASA observations hold up, and if no other explanation is found, then it may turn out that the discreteness of space has already been detected.
  5. When the spin network is tied in a braid, it forms something like a particle. This entity is stable, and it can have electric charge and handedness. Some of the different braids match known particles as shown in Figure 15-30, where a complete twist corresponds to +1/3 or
  6. Braided Space-time -1/3 unit of electric charge (denoted by a red dot) depending on the direction of the twist. It is found that the configuration can be stabilized from space-time quantum fluctuations by considering each quantum of space as a bit of quantum information. In short, the universe is

    Figure 15-30 Braided Space-time [view large image]

    treated as a giant quantum computer which shows that a collections of qubits are far more robust than an individual one. Thus after 20 years, loop quantum gravity finally makes some connection to particle physics.
  7. 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.
  8. 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). It has been shown already that Newton's law can emerge from quantized space.
  9. Unlike the superstring theory, 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).
  10. There is no link between the loop quantum theory and the superstring theory. While the supporters of the former stress the shortcoming of relying on a pre-defined space-time frame (in the superstring theory), and thus will not provide an adequate description of gravitation at small scale; the supporters of the superstring theory point out that the interaction between gravitons and other particles is inconsistent in loop quantum theory. It is suggested that both camps perceive only a small aspect of the whole thing - like the blind men and the elephant. It is suggested that string theory and loop quantum gravity are each part of a single theory. Each is correct, in the sense that it describes to a good approximation in a certain limited domain. The string theory is able to provide a framework for the graviton as mode of excitation of a string, while the loop quantum theory prescribes a way to make a background independent theory. Thus, each solves part of the problem. But each also has limits, which prevent it from forming the basis for a complete theory of nature. It is regrettable that physicists work exclusively only in one group or the other, criticizing each other instead of learning from each other.
  11. Application of LQG to cosmology predicts that the universe did not arise from nothing in a Big Bang. Instead it grew from the collapse of a pre-existing universe that bounced back from a very high density state. This model would suppress the production of gravitational waves at cosmological scales, and that there would be no such imprint in the CMBR. See more in Loop Quantum Cosmology.
By 2008, loop quantum gravity is not the only alternative in the quest for an ultimate theory combining general relativity and quantum theory (so-called "theory of everything"). There are at least 4 other scenarios as shown in Table 15-02 below.


Table 15-02 Theories of Everything [view large image]

    More Theories of Everything emerge as reported in the 29 August 2013 issue of Nature and shown in Figure 15-31 below :

  1. Gravity as Thermodynamics - Some research works are purported to link gravity with thermodynamics including Newton's law of gravitation and Einstein's equation. The idea was motivated by the Hawking radiation and the relationship between entropy and area of the event horizon (of a blackhole).

  2. As discussed previously in "Loop Quantum Gravity".

  3. Causal Sets - This theory postulates that the building blocks of space-time are simple mathematical points connected by links pointing from past to future. It predicts an accelerating universe before the discovery of dark energy.

  4. See Table 15-02 above for "Causal Dynamical Triangulations".

  5. Holograpy - The current theory is a variation of the original "Holographic Space-time".
Theories, More

Figure 15-31 More Theories of Everything [view large image]

See "Un-relativistic Theory" for a novel formulation of quantum gravity, and the LQG by Lee Smolin.

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