The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [197]
18. String theory’s claim to have successfully melded quantum mechanics and general relativity rests on a wealth of supporting calculations, made yet more convincing by results we will cover in Chapter 9.
Chapter 5: Hovering Universes in Nearby Dimensions
1. Classical Mechanics: . Electromagnetism: d*F = *J;dF = 0. Quantum mechanics: . General relativity: .
2. I am referring here to the fine structure constant, e2/hc, whose numerical value (at typical energies for electromagnetic processes) is about 1/137, which is roughly .0073.
3. Witten argued that when the Type I string coupling is dialed large, the theory morphs into the Heterotic-O theory with a coupling that’s dialed small, and vice versa; the Type IIB at large coupling morphs into itself, the Type IIB theory but with small coupling. The cases of the Heterotic-E and Type IIA theories are a little more subtle (see The Elegant Universe, Chapter 12, for details), but the overall picture is that all five theories participate in a web of interrelations.
4. For the mathematically inclined reader, the special thing about strings, one-dimensional ingredients, is that the physics describing their motion respects an infinite dimensional symmetry group. That is, as a string moves, it sweeps out a two-dimensional surface, and so the action functional from which its equations of motion are derived is a two-dimensional quantum field theory. Classically, such two-dimensional actions are conformally invariant (invariant under angle-preserving rescalings of the two-dimensional surface), and such symmetry can be preserved quantum mechanically by imposing various restrictions (such as on the number of spacetime dimensions through which the string moves—the dimension, that is, of spacetime). The conformal group of symmetry transformations is infinite-dimensional, and this proves essential to ensuring that the perturbative quantum analysis of a moving string is mathematically consistent. For example, the infinite number of excitations of a moving string that would otherwise have negative norm (arising from the negative signature of the time component of the spacetime metric) can be systematically “rotated” away using the infinite-dimensional symmetry group. For details, the reader can consult M. Green, J. Schwarz, and E. Witten, Superstring Theory, vol. 1 (Cambridge: Cambridge University Press, 1988).
5. As with many major discoveries, credit deserves to be given to those whose insights laid its groundwork as well as to those whose work established its importance. Among those who played such a role for the discovery of branes in string theory are: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Hořava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard DeWitt.
6. The diligent reader might argue that the Inflationary Multiverse also entwines time in a fundamental way, since, after all, our bubble’s boundary marks the beginning of time in our universe; beyond our bubble is thus beyond our time. While true, my point here is meant more generally—the multiverses discussed so far all emerge from analyses that focus fundamentally on processes occurring throughout space. In the multiverse we will now discuss, time is central from the outset.
7. Alexander Friedmann, The World as Space and Time, 1923, published in Russian, as referenced by H. Kragh, in “Continual Fascination: The Oscillating Universe in Modem Cosmology,” Science in Context 22, no. 4 (2009): 587–612.
8. As an interesting point of detail, the authors of the braneworld cyclic model invoke an especially utilitarian application of dark energy (dark energy will be discussed fully in Chapter 6). In the last phase of each cycle, the presence of dark energy in the braneworlds ensures agreement with today’s observations