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 theory 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
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
.
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.
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 Review 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.
CHAPTER 13
The view of the relationship between loop quantum gravity and string theory is based on L. Smolin, ‘Strings as perturbations of evolving spin networks’, hep-th/9801022; L. Smolin, ‘A candidate for a background independent formulation of M theory’, hep-th/9903166; L. Smolin, ‘The cubic matrix model and a duality between strings and loops’, hep-th/ 006137.
There is an extensive literature on black holes in both string theory and loop quantum gravity. A sample of string theory papers is: A. Strominger and C. Vafa, Physics Letters B379 (1996) 99, hep-th/9601029; C.V. Johnson, R.R. Khuri and R.C. Myers, Physics Letters B378 (1996) 78, hep-th/9603061; J.M. Maldacena and A. Strominger, Physical Review Letters 77 (1996) 428, hep-th/9603060; C.G. Callan and J.M. Maldacena, Nuclear Physics B472 (1996) 591, hep-th/9602043; G.T. Horowitz and A. Strominger, Physical Review Letters 77 (1996) 2368, hep-th/9602051.