The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory

by Brian Greene

  Perturbative approach, Perturbative method. See perturbation theory.

  Phase. When used in reference to matter, describes its possible states: solid phase, liquid phase, gas phase. More generally, refers to the possible descriptions of a physical system as features on which it depends (temperature, string coupling constant values, form of spacetime, etc.) are varied.

  Phase transition. Evolution of a physical system from one phase to another.

  Photoelectric effect. Phenomenon in which electrons are ejected from a metallic surface when light is shone upon it.

  Photon. Smallest packet of the electromagnetic force field; messenger particle of the electromagnetic force; smallest bundle of light.

  Planck energy. About 1,000 kilowatt hours. The energy necessary to probe to distances as small as the Planck length. The typical energy of a vibrating string in string theory.

  Planck length. About 10-33 centimeters. The scale below which quantum fluctuations in the fabric of spacetime would become enormous. The size of a typical string in string theory.

  Planck mass. About ten billion billion times the mass of a proton; about one-hundredth of a thousandth of a gram; about the mass of a small grain of dust. The typical mass equivalent of a vibrating string in string theory.

  Planck's constant. Denoted by the symbol h, Planck's constant is a fundamental parameter in quantum mechanics. It determines the size of the discrete units of energy, mass, spin, etc. into which the microscopic world is partitioned. Its value is 1.05 × 10-17 grams-cm/sec.

  Planck tension. About 1039 tons. The tension on a typical string in string theory.

  Planck time. About 10-43 seconds. Time at which the size of the universe was roughly the Planck length; more precisely, time it takes light to travel the Planck length.

  Primordial nucleosynthesis. Production of atomic nuclei occurring during the first three minutes after the big bang.

  Principle of equivalence. Core principle of general relativity declaring the indistinguishability of accelerated motion and immersion in a gravitational field (over small enough regions of observation). Generalizes the principle of relativity by showing that all observers, regardless of their state of motion, can claim to be at rest, so long as they acknowledge the presence of a suitable gravitational field.

  Principle of relativity. Core principle of special relativity declaring that all constant-velocity observers are subject to an identical set of physical laws and that, therefore, every constant-velocity observer is justified in claiming that he or she is at rest. This principle is generalized by the principle of equivalence.

  Product. The result of multiplying two numbers.

  Proton. Positively charged particle, typically found in the nucleus of an atom, consisting of three quarks (two up-quarks and one down-quark).

  Quanta. The smallest physical units into which something can be partitioned, according to the laws of quantum mechanics. For instance, photons are the quanta of the electromagnetic field.

  Quantum chromodynamics (QCD). Relativistic quantum field theory of the strong force and quarks, incorporating special relativity.

  Quantum claustrophobia. See quantum fluctuations.

  Quantum determinism. Property of quantum mechanics that knowledge of the quantum state of a system at one moment completely determines its quantum state at future and past moments. Knowledge of the quantum state, however, determines only the probability that one or another future will actually ensue.

  Quantum electrodynamics (QED). Relativistic quantum field theory of the electromagnetic force and electrons, incorporating special relativity.

  Quantum electroweak theory. See electroweak theory.

  Quantum field theory. See relativistic quantum field theory.

  Quantum fluctuation. Turbulent behavior of a system on microscopic scales due to the uncertainty principle.

  Quantum foam. See spacetime foam.

  Quantum geometry. Modification of Riemannian geometry required to describe accurately the physics of space on ultramicroscopic scales, where quantum effects become important.

  Quantum gravity. A theory that successfully mergers quantum mechanics and general relativity, possibly involving modifications of one or both. String theory is an example of a theory of quantum gravity.

  Quantum mechanics. Framework of laws governing the universe whose unfamiliar features such as uncertainty, quantum fluctuations, and wave-particle duality become most apparent on the microscopic scales of atoms and subnuclear particles.

  Quantum tunneling. Feature of quantum mechanics showing that objects can pass through barriers that should be impenetrable according to Newton's classical laws of physics.

  Quark. A particle that is acted upon by the strong force. Quarks exist in six varieties (up, down, charm, strange, top, bottom) and three "colors" (red, green, blue).

  Radiation. The energy carried by waves or particles.

  Reciprocal. The inverse of a number; for example, the reciprocal of 3 is 1/3, the reciprocal of ½ is 2.

  Relativistic quantum field theory. Quantum-mechanical theory of fields, such as the electromagnetic field, that incorporates special relativity.

  Resonance. One of the natural states of oscillation of a physical system.

  Riemannian geometry. Mathematical framework for describing curved shapes of any dimension. Plays a central role in Einstein's description of spacetime in general relativity.

  Schrödinger equation. Equation governing the evolution of probability waves in quantum mechanics.

  Schwarzschild solution. Solution to the equations of general relativity for a spherical distribution of matter; one implication of this solution is the possible existence of black holes.

  Second law of thermodynamics. Law stating that total entropy always increases.

  Second superstring revolution. Period in the development of string theory beginning around 1995 in which some nonperturbative aspects of the theory began to be understood.

  Singularity. Location where the fabric of space or spacetime suffers a devastating rupture.

  Smooth, Smooth space. A spatial region in which the fabric of space is flat or gently curved, with no pinches, ruptures, or creases of any kind.

  Space-tearing flop transition. See flop transition.

  Spacetime. A union of space and time originally emerging from special relativity. Can be viewed as the "fabric" out of which the universe is fashioned; it constitutes the dynamical arena within which the events of the universe take place.

  Spacetime foam. Frothy, writhing, tumultuous character of the spacetime fabric on ultramicroscopic scales, according to a conventional point-particle perspective. An essential reason for the incompatibility of quantum mechanics and general relativity prior to string theory.

  Special relativity. Einstein's laws of space and time in the absence of gravity (see also general relativity).

  Sphere. The outer surface of a ball. The surface of a familiar three-dimensional ball has two dimensions (which can be labeled by two numbers such as "latitude" and "longitude," as on the surface of the earth). The concept of a sphere, though, applies more generally to balls and hence their surfaces, in any number of dimensions. A one-dimensional sphere is a fancy name for a circle; a zero-dimensional sphere is two points (as explained in the text). A three-dimensional sphere is harder to picture; it is the surface of a four-dimensional ball.

  Spin. A quantum-mechanical version of the familiar notion of the same name; particles have an intrinsic amount of spin that is either a whole number or half a whole number (in multiples of Planck's constant), and which never changes.

  Standard model of cosmology. Big bang theory together with an understanding of the three nongravitational forces as summarized by the standard model of particle physics.

  Standard model of particle physics, Standard model, Standard theory. An enormously successful theory of the three nongravitational forces and their action on matter. Effectively the union of quantum chromodynamics and the electroweak theory.r />
  String. Fundamental one-dimensional object that is the essential ingredient in string theory

  String coupling constant. A (positive) number that governs how likely it is for a given string to split apart into two strings or for two strings to join together into one—the basic processes in string theory. Each string theory has its own string coupling constant, the value of which should be determined by an equation; currently such equations are not understood well enough to yield any useful information. Coupling constants less than 1 imply that perturbative methods are valid.

  String mode. A possible configuration (vibrational pattern, winding configuration) that a string can assume.

  String theory. Unified theory of the universe postulating that fundamental ingredients of nature are not zero-dimensional point particles but tiny one-dimensional filaments called strings. String theory harmoniously unites quantum mechanics and general relativity, the previously known laws of the small and the large, that are otherwise incompatible. Often short for superstring theory.

  Strong force, Strong nuclear force. Strongest of the four fundamental forces, responsible for keeping quarks locked inside protons and neutrons and for keeping protons and neutrons crammed inside of atomic nuclei.

  Strong force symmetry. Gauge symmetry underlying the strong force, associated with invariance of a physical system under shifts in the color charges of quarks.

  Strongly coupled. Theory whose string coupling constant is larger than 1.

  Strong-weak duality. Situation in which a strongly coupled theory is dual—physically identical—to a different, weakly coupled theory.

  Sum-over-paths. Formulation of quantum mechanics in which particles are envisioned to travel from one point to another along all possible paths between them.

  Supergravity. Class of point-particle theories combining general relativity and supersymmetry.

  Superpartners. Particles whose spins differ by ½ unit and that are paired by supersymmetry.

  Superstring theory. String theory that incorporates supersymmetry.

  Supersymmetric quantum field theory. Quantum field theory incorporating supersymmetry.

  Supersymmetric standard model. Generalization of the standard model of particle physics to incorporate supersymmetry. Entails a doubling of the known elementary particle species.

  Supersymmetry. A symmetry principle that relates the properties of particles with a whole number amount of spin (bosons) to those with half a whole (odd) number amount of spin (fermions).

  Symmetry. A property of a physical system that does not change when the system is transformed in some manner. For instance, a sphere is rotationally symmetrical since its appearance does not change if it is rotated.

  Symmetry breaking. A reduction in the amount of symmetry a system appears to have, usually associated with a phase transition.

  Tachyon. Particle whose mass (squared) is negative; its presence in a theory generally yields inconsistencies.

  Thermodynamics. Laws developed in the nineteenth century to describe aspects of heat, work, energy, entropy, and their mutual evolution in a physical system.

  Three-brane. See brane.

  Three-dimensional sphere. See sphere.

  Time dilation. Feature emerging from special relativity, in which the flow of time slows down for an observer in motion.

  T.O.E. (Theory of Everything). A quantum-mechanical theory that encompasses all forces and all matter.

  Topologically distinct. Two shapes that cannot be deformed into one another without tearing their structure in some manner.

  Topology. Classification of shapes into groups that can be deformed into one another without ripping or tearing their structure in any way

  Topology-changing transition. Evolution of spatial fabric that involves rips or tears, thereby changing the topology of space.

  Torus. The two-dimensional surface of a doughnut.

  Two-brane. See brane.

  Two-dimensional sphere. See sphere.

  Type I string theory. One of the five superstring theories; involves both open and closed strings.

  Type IIA string theory. One of the five superstring theories; involves closed strings with left-right symmetric vibrational patterns.

  Type IIB string theory. One of the five superstring theories; involves closed strings with left-right asymmetric vibrational patterns.

  Ultramicroscopic. Length scales shorter than the Planck length (and also time scales shorter than the Planck time).

  Uncertainty principle. Principle of quantum mechanics, discovered by Heisenberg, that there are features of the universe, like the position and velocity of a particle, that cannot be known with complete precision. Such uncertain aspects of the microscopic world become ever more severe as the distance and time scales on which they are considered become ever smaller. Particles and fields undulate and jump between all possible values consistent with the quantum uncertainty. This implies that the microscopic realm is a roiling frenzy, awash in a violent sea of quantum fluctuations.

  Unified theory, Unified field theory. Any theory that describes all four forces and all of matter within a single, all-encompassing framework.

  Uniform vibration. The overall motion of a string in which it moves without changes in shape.

  Velocity. The speed and the direction of an object's motion.

  Vibrational mode. See vibrational pattern.

  Vibrational pattern. The precise number of peaks and troughs as well as their amplitude as a string oscillates.

  Vibration number. Whole number describing the energy in the uniform vibrational motion of a string; the energy in its overall motion as opposed to that associated with changes in its shape.

  Virtual particles. Particles that erupt from the vacuum momentarily; they exist on borrowed energy, consistent with the uncertainty principle, and rapidly annihilate, thereby repaying the energy loan.

  Wave function. Probability waves upon which quantum mechanics is founded.

  Wavelength. The distance between successive peaks or troughs of a wave.

  Wave-particle duality. Basic feature of quantum mechanics that objects manifest both wavelike and particle-like properties.

  W bosons. See weak gauge boson.

  Weak force, Weak nuclear force. One of the four fundamental forces, best known for mediating radioactive decay.

  Weak gauge boson. Smallest bundle of the weak force field; messenger particle of the weak force; called W or Z boson.

  Weak gauge symmetry. Gauge symmetry underlying the weak force.

  Weakly coupled. Theory whose string coupling constant is less than 1.

  Winding energy. The energy embodied by a string wound around a circular dimension of space.

  Winding mode. A string configuration that wraps around a circular spatial dimension.

  Winding number. The number of times a string is wound around a circular spatial dimension.

  World-sheet. Two-dimensional surface swept out by a string as it moves.

  Wormhole. A tube-like region of space connecting one region of the universe to another.

  Z boson. See weak gauge boson.

  Zero-dimensional sphere. See sphere.

  * * *

  References and Suggestions for Further Reading

  Abbot, Edwin A. Flatland: A Romance of Many Dimensions. Princeton: Princeton University Press, 1991.

  Barrow, John D. Theories of Everything. New York: Fawcett-Columbine, 1992.

  Bronowski, Jacob. The Ascent of Man. Boston: Little, Brown, 1973.

  Clark, Ronald W. Einstein, The Life and Times. New York: Avon, 1984.

  Crease, Robert P., and Charles C. Mann. The Second Creation. New Brunswick, N.J.: Rutgers University Press, 1996.

  Davies, P. C. W. Superforce. New York: Simon & Schuster, 1984.

  Davies, P. C. W., and J. Brown, eds. Superstrings: A Theory of Everything? Cambridge, Eng.: Cambridge University Press, 1988.

  Deutsch, David. The Fabric of Reality. New York: Allen Lane, 199 7.


  Einstein, Albert. The Meaning of Relativity. Princeton: Princeton University Press, 1988.

  ——. Relativity. New York: Crown, 1961.

  Ferris, Timothy Coming of Age in the Milky Way New York: Anchor, 1989.

  ——. The Whole Shebang. New York: Simon & Schuster, 1997.

  Fölsing, Albrecht. Albert Einstein. New York: Viking, 1997.

  Feynman, Richard, The Character of Physical Law. Cambridge, Mass.: MIT Press, 1995.

  Gamow, George. Mr. Tompkins in Paperback. Cambridge, Eng.: Cambridge University Press, 1993.

  Gell-Mann, Murray. The Quark and the Jaguar. New York: Freeman, 1994.

  Glashow, Sheldon. Interactions. New York: Time-Warner Books, 1988.

  Guth, Alan H. The Inflationary Universe. Reading, Mass.: Addison-Wesley, 1997.

  Hawking, Stephen. A Brief History of Time. New York: Bantam Books, 1988.

  Hawking, Stephen, and Roger Penrose. The Nature of Space and Time. Princeton: Princeton University Press, 1996.

  Hey, Tony, and Patrick Walters. Einstein's Mirror. Cambridge, Eng.: Cambridge University Press, 1997.

  Kaku, Michio. Beyond Einstein. New York: Anchor, 1987.

  ——. Hyperspace. New York: Oxford University Press, 1994.

  Lederman, Leon, with Dick Teresi. The God Particle. Boston: Houghton Mifflin, 1993.

  Lindley, David. The End of Physics. New York: Basic Books, 1993.

  ——. Where Does the Weirdness Go? New York: Basic Books, 1996.

  Overbye, Dennis, Lonely Hearts of the Cosmos. New York: HarperCollins, 199 1.

  Pais, Abraham. Subtle Is the Lord: The Science and the Life of Albert Einstein. New York: Oxford University Press, 1982.

  Penrose, Roger. The Emperor's New Mind. Oxford, Eng.: Oxford University Press, 1989.

  Rees, Martin J. Before the Beginning. Reading, Mass.: Addison-Wesley, 1997.

  Smolin, Lee. The Life of the Cosmos. New York: Oxford University Press, 1997.

  Thorne, Kip. Black Holes and Time Warps. New York: Norton, 1994.

  Weinberg, Steven. The First Three Minutes. New York: Basic Books, 1993.