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Elementary Particles and the World of Planck Scale

Realm of Planck Scale

Soon after Max Planck introduced the Planck constant h in 1899 to account for the spectrum of blackbody radiation, he realized that the only way to construct a unit of length out of h = 6.625x10-27 erg-sec, the velocity of light c = 3x1010 cm/sec, and the gravitational constant G = 6.67x10-8 cm3/sec2-gm, is LPL = (Gh/c3)1/2. This is the now famous Planck length. Since h is related to Quantum Theory, c and G emerge from the Special and General Relativity respectively, it implies that any length scale in a Quantum Gravity Theory would involve LPL, which fittingly is extremely short at 4x10-33 cm.

Subsequently, it transpires that the realm of Planck scale would involve entities with size of the order 10-33 cm and associated with high energy phenomena at 1019 Gev. Such environment is considered to occur at the moment of Big Bang - a most favorable theory for the creation of this universe. Theoretically, both quantum theory and general relativity are required to formulate a correct description - hence the many theories of "Quantum Gravity". All of them have evaded observational confirmation so far and current technology does not generate such high energy to probe at such tiny region. Some examples are shown in Figure 15-32, they are more or less conjectures as explained briefly below.
Realm of Planck Scale
  • (a) Big Bang - The universe began from a very tiny space and high energy if the theory is correct.
  • (b) Black Hole - Big Bang may initiated from a black hole of Planck mass and radius at density of ~ 1093 gm/cm3. It is about 1080 times the inner core density of the neutron star, and is sometimes identified to be the dark energy.
  • (c) Creation of Matter - Matter and anit-matter were created from quantum fields following the Big Bang. They are super-heavy and do not exist in the current epoch anymore.
  • (d) Quantum Foam - Space at Big Bang is granular obeying the uncertainty principle.
  • (e) Superstring Theory - A theory involves Planck size strings and spin-2 graviton.

Figure 15-32 Realm of Planck Scale [view large image]

The diagram in Figure 15-33 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
Convergence 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 (x/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.

Figure 15-33 Quantum Gravity Convergence [view large image]

Object Mass (gm) Compton Wavelength (cm) Size (cm)
Photon (Red Light) 3.4x10-33=E/c2 6.65x10-5 Elementary Particle, Boson
Electron 9.1x10-28 2.4x10-10 Elementary Particle, Fermion
Proton 1.67x10-24 1.3x10-13 Composite Particle
Buckyball (C60) 1.2x10-21 1.8x10-16 ~ 10-7
Protein ~ 10-19 2x10-18 ~ 10-6
Virus ~ 10-8 2.2x10-29 ~ 10-5
Object Mass (gm) Schwarzschild Radius (cm) Size (cm)
Earth 6x1027 0.9 6x108
Neutron Star 1034 2x105 106
Cygnus X-1 2x1034 4x105 Stellar Black Hole
SgrA* 8x1039 1.2x1012 Galactic Black Hole
3C273 4x1042 6x1014 Quasar Black Hole
Observable Universe 4x1055 6x1027 1028 (very close to form a Black Hole)

Table 15-03 Convergence of Quantum Effect and Gravity

By equating the size and mass of the two lines at the intersection, i.e., L = h/mc = 2/mc = 2Gm/c2, from which a formula can be derived for the Planck mass m = MPL = (c/G)1/2 (by neglecting a factor of ) giving the modern version of the Planck scale units :
White Hole Planck Entropy
  • Planck Mass - MPL = (c/G)1/2 = 2.17x10-5 gm.
  • Planck Energy - EPL = MPLc2 = (c5/G)1/2 = 1.22x1019 Gev.
  • Planck Temperature - TPL = EPL/kB = (c5/GkB2)1/2 = 1.42x1032 oK, where kB is the Boltzmann's constant.
  • Planck Length - LPL = GMPL/c2 = (G/c3)1/2 = 1.62x10-33 cm.
  • Planck Time - tPL = LPL/c = (G/c5)1/2 = 5.39x10-44 sec.

Figure 15-34 White Hole

Figure 15-35 Evolution of Cosmic Entropy

    Since the Schwarzschild Radius rs for a Planck size entity having the Planck mass is rs = 2LPL, it is a black hole unless its mass is bumped up to greater than 2MPL.

As the mass of the black hole is dissipated by Hawking radiation, the black hole evaporation time is given by :

tev = 5120G2m3/(c4).

Thus, tev = 8.6x10-40 sec for a black hole with m = MPL. That is, it vanishes instantly in a puff of radiation having the appearance of a white hole spitting out matter-energy as shown in Figure 15-34. See an example in "Quantum Inflation".

The direction of cosmic evolution of entropy always tends to increase or at least to remain constant as shown in Figure 15-35. According to the scenario of backward tracing, the universe would be a dense pack of particles with maximum entropy at the beginning. It is not clear then what makes this trend of increasing entropy if it was at its maximum already. It turns out that there would not be any problem to explain the origin of low entropy at Big Bang if it starts from a black hole.
Quantum Gravity Derivation of black hole entropy shows that the absolute value of information | I | = 1.2x1066 R2 bits, where R is the Schwarzschild radius (note that the entropy S = (kBln2) | I |, with the Boltzmann constant kB = 1.38x10-16 erg/Ko). If the radius of the black hole R = LPL, then | I | = 3, which agrees with the no hair theorem - only three number of parameters or three degrees of freedom (mass, angular momentum, and electric charge) is required to describe a black hole completely. So the entropy was at its minimum with S = 3x(kBln2) in Big Bang. It increases upward forever since then as expected by the 2nd law of thermodynamics.

Figure 15-36 Quantum Gravity

Figure 15-36 shows the relationship between quantum gravity and the other branches of physics at the limit of the various universal constants. In other word, the various theories at the perimeter are just the special cases of the more general theory of quantum gravity - hence it is often referred to as the Theory of Everything.

This view of considering quantum gravity as the all-encompassing theory is markedly different from the convergence of two effective theories expressed in Figure 15-33. According to the latter view point, quantum gravity is necessary and feasible only at the Planck scale. An example is the "Quantization of the Friedmann Equation", which concerns with only the event at the Big Bang. The problems with the "Theory of Every thing" perhaps stems from the incompatible nature of quantum theory and general relativity beyond the Planck scale.

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