The Elegant Universe - Brian Greene [211]
7. The size of the Planck length can be understood based upon simple reasoning rooted in what physicists call dimensional analysis. The idea is this. When a theory is formulated as a collection of equations, the abstract symbols must be tied to physical features of the world if the theory is to make contact with reality. In particular, we must introduce a system of units so that if a symbol, say, is meant to refer to a length, we have a scale by which its value can be interpreted. After all, if equations show that the length in question is 5, we need to know if that means 5 centimeters, 5 kilometers, or 5 light years, etc. In a theory that involves general relativity and quantum mechanics, a choice of units emerges naturally, in the following way. There are two constants of nature upon which general relativity depends: the speed of light, c, and Newton's gravitation constant, G. Quantum mechanics depends on one constant of nature . By examining the units of these constants (e.g., c is a velocity, so is expressed as distance divided by time, etc.), one can see that the combination G/c3 has the units of a length; in fact, it is 1.616 × 10-33 centimeters. This is the Planck length. Since it involves gravitational and spacetime inputs (G and c) and has a quantum mechanical dependence () as well, it sets the scale for measurements—the natural unit of length—in any theory that attempts to merge general relativity and quantum mechanics. When we use the term "Planck length" in the text, it is often meant in an approximate sense, indicating a length that is within a few orders of magnitude of 10–33 centimeters.
8. Currently, in addition to string theory, two other approaches for merging general relativity and quantum mechanics are being pursued vigorously. One approach is led by Roger Penrose of Oxford University and is known as twistor theory. The other approach—inspired in part by Penrose's work—is led by Abhay Ashtekar of Pennsylvania State University and is known as the new variables method. Although these other approaches will not be discussed further in this book, there is growing speculation that they may have a deep connection to string theory and that possibly, together with string theory, all three approaches are honing in on the same solution for merging general relativity and quantum mechanics.
Chapter 6
1. The expert reader will recognize that this chapter focuses solely on perturbative string theory; nonperturbative aspects are discussed in Chapters 12 and 13.
2. Interview with John Schwarz, December 23, 1997.
3. Similar suggestions were made independently by Tamiaki Yoneya and by Korkut Bardakci and Martin Halpern. The Swedish physicist Lars Brink also contributed significantly to the early development of string theory.
4. Interview with John Schwarz, December 23, 1997.
5. Interview with Michael Green, December 20, 1997.
6. The standard model does suggest a mechanism by which particles acquire mass—the Higgs mechanism, named after the Scottish physicist Peter Higgs. But from the point of view of explaining the particle masses, this merely shifts the burden to explaining properties of a hypothetical "mass-giving particle"—the so-called Higgs boson. Experimental searches for this particle are underway, but once again, if it is found and its properties measured, these will be input data for the standard model, for which the theory offers no explanation.
7. For the mathematically inclined reader, we note that the association between string vibrational patterns and force charges can be described more precisely as follows. When the motion of a string is quantized, its possible vibrational states are