Quantum Gravity Convergence

Figure 15-05b

"The history of physics is closely linked to the unification of seemingly disparate phenomena. (See Figure 15-05a.) Each stage of unification in turn advanced a new theoretical framework, which provides a deeper understanding of nature. (see Figure 15-05b.) The followings provides a history of the development of physics and the many unifications as depicted in Figure 15-05a. It also contains a brief description of the theoretical concept at each stage.



Figure 15-06a Global and Local Transformations



Figure 15-06b Local Guage Field Invariance

The modern theory of elementary particles depends heavily on the concept of gauge invariance, which was used to describe some changes that do not have any effect on observation. For example, if the electric voltage throughout the circuit is raised uniformly by the same amount, there would not be any observable effect. Such instantaneous global change seems to be somewhat un-natural as it takes a definite time to signal the change. Theory with local gauge invariance is more realistic, but much more stringent. The difference between a global and local transformation is illustrated pictorially in Figure 15-06a (just for visualization purpose - it's not the gauge variety). Since local gauge symmetry can change in a different way at every point, the only way in which theory can be kept invariant under such general changes is for certain forces to constrain the allowed motions. That something turns out to be the gauge bosons mentioned earlier in Figure 15-03. For example, in the electromagnetic interaction a local disturbance can be considered as a two dimensional rotation of the quantum field (which is usually a complex function2) in an "internal space". The photon (the gauge boson) is the response to restore the "appearance" (see Figure 15-06b), which signifies mathematically the invariance of the "Action" under this internal rotation.

Theoretical physicists are fond of putting similar objects together called a group. For the case of electromagnetic interaction, there is only one kind of objects -- the two dimensional internal rotation. The different rotational displacements form a group, this particular group is called U(1). The symbol U indicates that the transformation (the internal rotation) is unitary, which preserves the normalization (probability). This U(1) group has the property that the internal rotation operations are commutative -- mathematicians call such kind of group an Abelian group. Similar gauge invariance exists for the strong and weak interactions, the "internal rotation" depends on more than one parameter in these cases. Group of objects can be formed from these generalized "rotational displacements". However, these elements are no longer commutative. Such groups are called non-Abelian. The gauge theory for the U(1) is called Quantum Electrodynamics (QED).



Figure 15-05p

According to Lee Smolin, there are three roads leading to the domain of qunatum gravity:

Hawking Radiation - By combining the uncertainty principle in quantum theroy and the general relativity concept of black holes, Stephen Hawking showed that black holes are not completely black, but would actually radiate. As a bonus, he also found new connections between gravity and thermodynamics, i.e., entropy (and hence information) is equal to 1/4 of the area (in Planck units) of the black hole's event horizon. Thus, the amount of information that can be contained in that region is finite. This implies that the world must be discrete on the Planck scale, for were it continuous any region could contain an infinite amount of information. See Hawking Radiation, Relativity in the appendix for more details.

Superstring theory - It started out as a generalization of quantum field theory where instead of point particles, string-like objects (with size ~ the Planck length = 10-33 cm) propagate in a fixed spacetime background. Although string theory had its origins in the study of quark confinement and not of quantum gravity, it was soon discovered that the string spectrum contains the graviton, and that "condensation" of certain vibration modes of strings is equivalent to a modification of the original background. It is suggested that in the "yet to be formulated" fundamental superstring theory such as the M theory, the background dependent apporach should be replaced by a scheme involving dynamic space time. See section on "Superstrings" for more details.

Loop Quantum Gravity - One reason why it has taken so long to construct a quantum theory of gravity is that all previous quantum theories were background dependent. The problem of how to build a quantum theoretic description of a world in which space and time are nothing but networks of relationships was solved only 20 years ago in the 1980s. The relational theory that resulted is called loop quantum gravity. According to this theory, space is made of discrete units in size of the Planck length. In contrast to ordinary geometry, a given region cannot have a volume which is arbitrarily big or small - instead, the volume must be one of a finite set of numbers (similar to some of the quantized quantities in quantum theory). Similarly, time is also made of discrete units called the Planck time (10-43 sec). The theory offers more than a picture: it makes precise predictions about what would be observed were it possible to probe the geometry of space at distances as short as the Planck scale. See section on "Quantum Foam and Loop Quantum Gravity" for more details.

A few of the other approaches to quantum gravity will turn out to play significant roles in the final synthesis. Among them will be the twistor theory and the non-commutative geometry. They will provide essential insights into the nature of the quantum geometry of space-time. Quantum gravity will emerge as a more fundamental theory since it will possess more explanatory and predictive powers. Figure 15-05p shows the relationship between quantum gravity and the other branches of physics at the limit of the various universal constants, where the gravitational constant G is associated with gravity, the Planck constant is for quantum, and the velocity of light c comes with special relativity. In quantum gravity, all the fundamental units are expressed in terms of G, , and c: Planck length = (G/c3)1/2 = 1.62x10-33 cm, Planck time = (G/c5)1/2 = 5.39x10-44 sec, Planck mass = (c/G)1/2 = 2.17x10-5 gm, Planck energy = (c5/G)1/2 = 1.22x1019 Gev, and Planck temperature = (c5/GkB2)1/2 = 1.42x1032 oK, where kB is the Boltzmann's constant, which relates energy to absolute temperature on the Kelvin scale.



Figure 15-21 Manifold

Manifold, Vacuum Energy, Multiverse, and Eternal Inflation

"The possibility of decay from one stable vacuum to another suggests a radical new picture of the universe. Figure 15-23a depicts the large dimension space in color. The blue region represents an universe originally sitting in the minimum vacuum energy A (as shown in Figure 15-22). Decay of a flux line in A creates a different manifold, which tunnels to a new minimum in B (see Figure 15-22 and the red bubble in Figure 15-23a). Then decay of another flux line in B creates another manifold, which tunnels to another new minimum in C (see Figure 15-22 and the green bubble in Figure 15-23a), and so on ad infinitum. The whole universe is therefore a foam of expanding bubbles within bubbles, each with its own laws of physics.



Figure 15-23a Eternal Inflation

Such scenario is referred to as "eternal inflation". Extremely few of the bubbles are suitable for the formation of complex structures such as galaxies and life. The observable universe is a small region within one of these bubbles as shown in Figure 15-23a. The Big Bang was just the beginning of a new manifold within an older universe.

A superstring theory research in 2007 indicates that there may be a way to arrive at an unique universe without invoking the anthropic principle. It turns out that in spite of previous assumption, the myriad manifolds (ways to compactify the extra dimensions) can transform into each other. As shown in Figure 15-23b there are only a few universes with simple manifold and low Euler number, which is related to the dimensionality and the numbers of holes or handles for the manifold. It is also found that the several manifolds in the sparsely populated tip of the diagram (in Figure 15-23b) seem to correspond to universes like our own. This means that the universe might have started out completely differently and been transformed, through a series of transitions, from one manifold to another, ending up at the tip.




Figure 15-23b Universes

Perhaps the universe is minimizing something through an unknown mechanism. Further research is required to identify how the more complex universes trickle down to become the one we live in today.

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 mechanics.



Figure 15-28 Quantum Time

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 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 15-29a Test

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 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 network.
...
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



Figure 15-31 Quantum Gravity Convergence

The diagram in Figure 15-31 shows the convergence of quantum effect and gravity toward a point - the apex. The straight line on the left is a plot of mass against Compton wavelength =h/mc, which is related to Compton scattering. It appears also in quantum field equations to define the length scale of the quantum process. In the present context, it can be interpreted as the appearance of pair-creation with large quantum fluctuation in momentum (~ mc) resulting in position uncertainty (xh/p) about the order of a Compton wavelength. Individual object does not exist in region below that line as the single particle description is no longer applicable. The straight line on the right is a plot of mass against the Schwarzschild radius rs=2GM/c2. Objects cannot be accessed in region below that line as it would be wrapped inside the event horizon (since r < rs). All objects exist only within the region bound by these two lines. Table 15-03 lists some of the objects within or at the border. The two lines converge at the apex where both quantum effect and gravity become important. The object at the apex seems to be related to the Planck scale at the very beginning of the universe as shown in Table 15-04.
http://universe-review.ca/F15-particle.htm#manifold