Three Roads to Quantum Gravity

by Lee Smolin

hidden variables

  Conjectured degrees of freedom which underlie the statistical uncertainties in quantum theory. If there are hidden variables, then it is possible that the uncertainties in quantum theory are just the result of our ignorance about the values of the hidden variables and are not fundamental.

  horizon

  For each observer in a spacetime, the surface beyond which they cannot see, or receive any signals from. Examples are black hole horizons.

  information

  A measure of the organization of a signal. It is equal to the number of yes/no questions whose answers could be coded in the signal.

  knot theory

  A branch of mathematics concerned with classifying the different ways of tying a knot.

  lattice

  A space consisting of a finite number of points, with nearby points connected by lines called edges. A lattice is often, but not always, distinguished from a graph in that a lattice is a graph with a regular structure. An example of a lattice is shown in Figure 22.

  lattice theory

  A theory in which space or spacetime is considered to be a lattice.

  light cone

  All the events that can be reached by light signals travelling to the future, or coming from the past, from a single event. We may therefore distinguish between the future light cone, which contains events that can be reached by light travelling into the future, and the past light cone, which contains events that can be reached by light travelling from the past.

  link

  Two curves link in three-dimensional space if they cannot be pulled apart without passing one through the other.

  loop

  A circle drawn in space.

  loop quantum gravity

  An approach to quantum gravity in which space is constructed from the relationships between loops, originally derived by applying quantum theory to the formulation of general relativity discovered by Sen and Ashtekar.

  many-worlds interpretation

  An interpretation of quantum theory according to which the different possible outcomes of an observation of a quantum system reside in different universes, all of which somehow coexist.

  M theory

  The conjectured theory which would unify the different string theories.

  Newton’s gravitational constant

  The fundamental constant that measures the strength of the gravitational force.

  Newtonian physics

  All physical theories formulated on the pattern of Newton’s laws of motion. See classical physics, which is a synonymous term.

  non-commutative geometry

  A description of a space in which it is impossible to determine enough information to locate a point, but which can have many other properties of space including the fact that it can support a description of particles and fields evolving in time.

  past or causal past

  For a particular event, all other events that could have influenced it by sending energy or information to it.

  past light cone

  The past light cone of an event consists of all those events that could have sent a light signal to it.

  Pauli exclusion principle

  The principle that no two fermions can be put into exactly the same quantum state; named after Wolfgang Pauli.

  perturbation theory

  An approach to making calculations in physics in which some phenomena are represented in terms of small deviations from or oscillations of some stable state, or the interactions among such oscillations.

  Planck scale

  The scale of distance, time and energy on which quantum gravity effects are important. It is defined roughly by the Planck units - processes on the Planck scale take around a Planck time, which is 10-43 of a second. To observe on the Planck scale, distances of around the Planck length must be probed. This is about 10-33 of a centimetre.

  Planck’s constant

  A fundamental constant that sets the scale of quantum effects; normally denoted by h.

  Planck units

  The basic units of measure in a quantum theory of gravity. Each is given by a unique combination of three basic constants: Planck’s constant, Newton’s gravitational constant and the speed of light. Planck units include the Planck length, Planck energy, Planck mass, Planck time and Planck temperature.

  quantum chromodynamics (QCD)

  The theory of the forces between quarks.

  quantum electrodynamics (QED)

  The marriage of quantum theory with electrodynamics. It describes light and the electric and magnetic forces in quantum terms.

  quantum cosmology

  The theory that attempts to describe the whole universe in the language of quantum theory.

  quantum gravity

  The theory that unifies quantum theory with Einstein’s general theory of relativity.

  quantum theory or quantum mechanics

  The theory of physics that attempts to explain the observed behaviour of matter and radiation. It is based on the uncertainty principle and wave-particle duality.

  quantum state

  The complete description of a system at one moment of time, according to the quantum theory.

  quark

  An elementary particle which is a constituent of a proton or neutron.

  real number

  A point on the continuous number line.

  relational

  Describing a property that describes a relationship between two objects.

  relational quantum theory

  An interpretation of quantum theory according to which the quantum state of a particle, or of any subsystem of the universe, is defined, not absolutely, but only in a context created by the presence of an observer, and a division of the universe into a part containing the observer and a part containing that part of the universe from which the observer can receive information. Relational quantum cosmology is an approach to quantum cosmology which asserts that there is not one quantum state of the universe, but as many states as there are such contexts.

  relativity theory

  Einstein’s theory of space and time, comprising the special theory of relativity, which describes the causal structure of spacetime without gravity, and the general theory of relativity, in which the causal structure becomes a dynamical entity that is partly determined by the distribution of matter and energy.

  second law of thermodynamics

  The law stating that the entropy of an isolated system can only increase in time.

  spacetime

  The history of a universe, comprising all its events and their relationships.

  speed of light

  The speed at which light travels, which is known to be the maximum speed for the transmission of energy and of information.

  spin

  The angular momentum of an elementary particle which is an intrinsic property of it, independent of its motion.

  spin network

  A graph whose edges are labelled by numbers representing spins. In loop quantum gravity each quantum state of the geometry of space is represented by a spin network.

  spontaneous symmetry breaking

  The phenomena by which a stable state of a system can have less symmetry than the laws that govern the system.

  state

  In any physical theory, the configuration of a system at a specified moment of time.

  string

  In string theory, the basic physical entity, the different states of which represent the different possible elementary particles. A string can be visualized as a path or a loop that propagates through a background space.

  string theory

  A theory of the propagation and interactions of strings, in background spacetimes.

  supersymmetry

  A conjectured symmetry of elementary particle physics and string theories which asserts that bosons and fermions exist in pairs, each member of which has the same mass and interactions.

  supergravity

  An extension of Einstein’s general t
heory of relativity in which the different kinds of elementary particle are related to one another by one or more supersymmetries.

  symmetry

  An operation by which a physical system may be transformed without affecting the fact that it is a possible state or history of the system. Two states connected by a symmetry have the same energy.

  temperature

  The average kinetic energy of a particle or mode of vibration in a large system.

  thermal or thermodynamic equilibrium

  See equilibrium.

  topos theory

  A mathematical language which is appropriate for describing theories in which properties are context dependent, as in relational quantum theory.

  twistor theory

  An approach to quantum gravity invented by Roger Penrose in which the primary elements are causal processes and the events of spacetime are constructed in terms of the relationships between the causal processes.

  uncertainty principle

  A principle in quantum theory according to which it is impossible to measure both the position and momentum (or velocity) of a particle or, more generally, the state and rate of change of any system.

  wave-particle duality

  A principle of quantum theory according to which one can describe elementary particles as both particles and waves, depending on the context.

  SUGGESTIONS FOR FURTHER READING

  Here I give a brief list of sources where the interested reader can find more information about the topics discussed. More information will be available on a Website, http://www.qgravity.org.

  INTRODUCTION AND POPULAR TEXTS

  Many books aim to introduce the reader to the basic ideas of quantum theory and general relativity. They cater to all different levels, from comic books and children’s books to philosophical treatises. There are so many that the reader is advised to go to the science section of a good bookshop, look at the various books on quantum theory and relativity, read the first few pages of each and take the one you like best. The reader may also find it interesting to look at the popularizations by the inventors of these theories: Bohr, Einstein, Heisenberg and Schrödinger have all written introductions to their work for the layperson.

  My own Life of the Cosmos (Oxford University Press, New York and Weidenfeld & Nicolson, London, 1996) introduces the basic ideas of quantum theory and general relativity in Parts 4 and 5.

  Brian Greene’s The Elegant Universe (Norton, 1999) gives a very good introduction to the basic ideas of string theory and the problems it currently faces. Roger Penrose’s books, especially the Emperor’s New Mind (Oxford University Press, 1989), are a good introduction to the problem of quantum gravity and quantum black holes, emphasizing of course his own point of view.

  REFERENCE TO THE SCIENTIFIC LITERATURE

  Virtually the whole of the scientific literature on topics relevant to theoretical physics since 1991 is available in an electronic archive, which can be found at http://xxx.lanl.gov/. Note that while you generally have to have a professional affiliation to publish at this site, anyone can download and read the articles archived there. The papers of relevance to this book are mostly found in the archives hep-th and gr-qc. A search for the people mentioned below will return a list of the papers which underlie the developments described.

  Another very good source for the ideas and mathematical developments used in quantum gravity is John Baez’s Website, This Week’s Finds in Mathematical Physics, at http://math.ucr.edu/home/baez/TWF.html. He also has a nice online tutorial introduction to general relativity at http://math.ucr.edu/home/baez/gr/gr.html. The reader wanting a general introduction to the history of quantum gravity and its basic issues may find the following articles interesting: Carlo Rovelli, ‘Notes for a brief history of quantum gravity’, gr-qc/0006061; Carlo Rovelli, ‘Quantum spacetime - what do we know?’, gr-qc/ 9903045, and Lee Smolin, ‘The new universe around the next corner’, in Physics World, December 1999.

  Most of the following key references are in the xxx.lanl.gov archive. A more complete list of references is available at the Website mentioned above.

  CHAPTER 2

  The discussion of the logic of observers inside the universe is based on F. Markopoulou, ‘The internal description of a causal set: What the universe looks like from the inside’, gr-qc/9811053, Commun. Math. Phys. 211 (2000) 559-583.

  CHAPTER 3

  The consistent histories interpretation is described in R.B. Griffiths, Journal of Statistical Physics 36 (1984) 219; R. Omnes, Journal of Statistical Physics 53 (1988) 893; and M. Gell-Mann and J.B. Hartle in Complexity, Entropy, and the Physics of Information, SFI Studies in the Sciences of Complexity, Vol. VIII, edited by W. Zurek (Addison Wesley, Reading, MA, 1990). The criticisms of Kent and Dowker are found in Fay Dowker and Adrian Kent, ‘On the consistent histories approach to quantum mechanics’, Journal of Statistical Physics. 82 (1996) 1575. Gell-Mann and Hartle comment in ‘Equivalent sets of histories and multiple quasiclassical realms’, gr-qc/9404013; J. B. Hartle, gr-qc/ 9808070. The reformulation of the consistent histories formulation in terms of topos theory, which emphasizes its relational aspects, is found in C.J. Isham and J. Butterfield, ‘Some possible roles for topos theory in quantum theory and quantum gravity’, gr-qc/9910005. Other relational approaches to quantum cosmology are found in L. Crane, Journal of Mathematical Physics 36 (1995) 6180; L. Crane, in Knots and Quantum Gravity, edited by J. Baez (Oxford University Press, New York, 1994); L. Crane, ‘Categorical physics’, hep-th/9301061; F. Markopoulou, ‘Quantum causal histories’, hep-th/9904009, Class. Quan. Grav. 17 (2000) 2059-2072; F. Markopoulou, ‘An insider’s guide to quantum causal histories’, hep-th/9912137, Nucl. Phys. Proc. Suppl. 88 (2000) 308-313; C. Rovelli, ‘Relational quantum mechanics’, quant-ph/9609002, International Journal of Theoretical Physics 35 (1996) 1637; L. Smolin, ‘The Bekenstein bound, topological field theory and pluralistic quantum cosmology’, gr-qc/950806.

  CHAPTER 4

  The process formulation of quantum theory was developed first by David Finkelstein, whose work is the main inspiration for this chapter. It is described in David Ritz Finkelstein, Quantum Relativity: A Synthesis of the ideas of Einstein and Heisenberg (Springer-Verlag, 1996). Rafael Sorkin has also pioneered the exploration of the role of causality in quantum gravity.

  CHAPTERS 5-8

  This is all standard material in classical general relativity and quantum field theory. Good introductions are N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Spacetime (Cambridge University Press, 1982); and Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994).

  CHAPTERS 9 AND 10

  There are several expositions of loop quantum gravity at a semi-popular or semi-technical level. They include Carlo Rovelli, ‘Loop quantum gravity’, gr-qc/9710008, Carlo Rovelli, ‘Quantum spacetime: what do we know?’, gr-qc/9903045; L. Smolin in Quantum Gravity and Cosmology, edited by Juan Perez-Mercader et al. (World Scientific, 1992); L. Smolin, ‘The future of spin networks’, in The Geometric Universe (1997), edited by S.A. Huggett et al. (Oxford University Press, 1998), gr-qc/9702030. The book by Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, 1996) describes their approach to the subject.

  The mathematically rigorous approach to loop quantum gravity is presented in Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao and Thomas Thiemann, ‘Quantization of diffeomorphism invariant theories of connections with local degrees of freedom’, Journal of Mathematical Physics 36 (1995) 6456, gr-qc/9504018; Abhay Ashtekar, Jerzy Lewandowski, ‘Quantum field theory of geometry’, hep-th/ 9603083; and T. Thiemann, ‘Quantum spin dynamics I and II’, gr-qc/ 9606089, gr-qc/9606090, Classical and Quantum Gravity 15 (1998) 839, 875.

  The original references for the Ashtekar-Sen formalism are in A. Sen, Physics. Letters B119 (1982) 89; International Journal; of Theoretical Physics 21 (1982) 1; A. Ashtekar, Physical Rev
iew Letters 57 (1986) 2244; A. Ashtekar, Physical. Review D36 (1987) 1587.

  CHAPTER 11

  This is all standard material in string theory, to which Brian Greene’s The Elegant Universe (Norton, 1999) is an excellent introduction. The best textbook is J. Polchinksi, String Theory (Cambridge University Press, 1998).

  CHAPTER 12

  The original references for the holographic principle are Gerard ’t Hooft, ‘Dimensional reduction in quantum gravity’, gr-qc/9310006, in Salanfestschrift, edited by A. Alo, J. Ellis, S. Randjbar-Daemi (World Scientific, 1993); and Leonard Susskind, ‘The world as a hologram’, hep-th/9409089, Journal of Mathematical Physics 36 (1995) 6377. Ideas closely related to the holographic principle were presented earlier by L. Crane in ‘Categorical physics’, hep-th/9301061 and hep-th/9308126 in Knots and Quantum Gravity, edited by J. Baez (Oxford University Press, 1994); L. Crane, ‘Clocks and categories: is quantum gravity algebraic?’ Journal of Mathematical Physics 36 (1995) 6180, gr-qc/ 9504038.

  The Bekenstein bound was proposed in J.D. Bekenstein, Lettere Nuovo Cimento 4 (1972) 737, Physical Review D7 (1973), 2333; Physical Review D9 (1974) 3292. Ted Jacobson’s paper deriving general relativity from the Bekenstein bound and the laws of thermodynamics is ‘Thermodynamics of spacetime: the Einstein equation of state’, gr-qc/9504004, Physical Review Letters 75 (1995) 1260. The derivation of the Bekenstein bound in loop quantum gravity is in L. Smolin, ‘Linking topological quantum field theory and nonperturbative quantum gravity’, gr-qc/9505028, Journal of Mathematical Physics 36 (1995) 6417. Another very promising version of the holographic principle was proposed by Rafael Bousso in ‘A covariant entropy conjecture’, hep-th/9905177, Journal of High-Energy Physics, 9907 (1999) 0004; R. Bousso, ‘Holography in general space-times’, hep-th /9906022, Journal of High-Energy Physics 9906 (1999) 028. A related theorem was proved in E. Flanagan, D. Marolf and R. Wald, hep-th/ 9908070. F. Markopoulou and I proposed a background independent version in ‘Holography in a quantum spacetime’, hep-th/9910146. In ‘The strong and weak holographic principles’, hep-th/0003056 I review the arguments for and against the different versions of the principle.