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 is extremely short at 4x10-33 cm.

(a) Big Bang - The universe began from a very tiny space and high energy. (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.

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

(e) Superstring Theory - A theory involves Planck size strings and spin-2 graviton. (f) Loop Quantum Gravity - A theory incorporates quantum uncertainty and granular nature of space-time at Planck Scale.


The diagram in Figure 15-33 shows the convergence of quantum effect and gravity toward a point. 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 (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 Individual object does not exist in region below that line as the single particle description is no longer applicable.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 a point where both quantum effect and gravity become important.

Figure 15-33 Quantum Gravity Convergence

Figure 15-33 also shows some characteristic scales of the universe.

Object	Mass (gm)	Compton Wavelength (cm)	Size (cm)
Electron	9.1x10-28	3.8x10-11	~ 10-16, Pair Creation at High Energy
Proton	1.67x10-24	2x10-14	~ 10-13, Composite Particle
Higgs Boson	1.25x10-22	10-16	Excitation from Higgs Field, Unstable
Buckyball (C60)	1.2x10-21	2.8x10-17	~ 10-7
Protein	~ 10-19	3.2x10-19	~ 10-6
Virus	~ 10-8	3.5x10-30	~ 10-5
Object	Mass (gm)	Schwarzschild Radius (cm)	Size (cm)
Earth	6x1027	0.9	6x108
Sun	2x1033	4x104	2x1011
Neutron Star	1034	2x105	106
Cygnus X-1	2x1034	4x105	Stellar Binary + Black Hole
Sgr A*	8x1040	1.2x1013	~ 6.4x1012 Milky Way 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

Giving the original definition of Planck length L_PL, all the other Planck units can be derived by invoking two fundamental principles :


• Planck time - it is related to quickest crossing of 1 unit Planck length L_PL according to Special Relativity, i.e., t_PL = L_PL/c.
• Planck energy - the fluctuation of energy is dictated by the Uncertainty Principle t E giving E_PL = /t_PL.
• Planck Mass - M_PL = E_PL/c².
• Planck Temperature - E_PL/k_B, where k_B is Boltzmann's constant

		Planck Length - LPL = (G/c3)1/2 = 1.62x10-33 cm. Planck Time - tPL = (G/c5)1/2 = 5.39x10-44 sec. Planck Energy - EPL = (c5/G)1/2 = 1.22x1019 Gev. Planck Mass - MPL = (c/G)1/2 = 2.17x10-5 gm. Planck Temperature - TPL = (c5/GkB2)1/2 = 1.42x1032 oK.
Figure 15-34 White Hole	Figure 15-35 Evolution of Cosmic Entropy	Since the Schwarzschild Radius rs = 2GMPL/c2 = 2LPL for a Planck scale entity, it is a black hole, while the distance to horizon DH = c tPL = LPL, and the energy density ~ 10114 erg/cm3.



It turns out that there are some interesting characteristics for Planck scale black hole. For example, as the mass of the black hole is dissipated by Hawking radiation, the black hole evaporation time is given by :

t_ev = 5120G²m³/(c⁴).

Thus, t_ev = 8.6x10^-40 sec for a black hole with m = M_PL. 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".



Another example : 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 (see "Origin of Time"). 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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