Duality of Spacetime and Matter


In the search for a theory of quantum gravity, to combine the theories of gravity with those of quantum mechanics, an interesting theory, known as Loop Quantum Gravity (LQG) has evolved. It involves only 4 dimensions for spacetime.


To paraphrase Matthew Chalmers [PW-1]:

"There are two routes to quantum gravity. The first is to formulate general realtivity into a quantum field theory in which the gravitational force is carried by the exchange of gravitons. The problem is that gravtons carry ass and energy, which are the source of the gravitationsl field in the first place. this leads to infinities that render calculations meaningless.


"One way round this problem is to replace the idea of point-like particles with infinitesimal 1D strings (eg see Susskind [PW-3]). However, there are too many string theories, and too many dimensions (see Quevedo [PW-2]) . Since 1995, M-theory is helping theorists understand what it all means. (M-theory encompasses all the string theories, adds extended objects called  p-branes and D-branes, and adds an 11th dimension [PW-6][PW-7][Berg-1][Berg-2][DAMTP-2]).


"String theory is based on conventional quantum mechanics and assumes that spacetime is a fixed background on which particles move and interact. The second route to quantum gravity, however, starts with general relativity and involves completely rewriting quantum theory. In loop quantum gravity, there is no such thing as space, only fields. Carlo Rovelli ([PW-4] and elsewhere) arrived at a background-independent quantum field theory by treating the gravitational field in terms of closed lines or loops."


In his 1997 paper on Loop Quantum Gravity [Rov-17], Carlo Rovelli concluded:

"The most remarkable physical result obtained from loop quantum gravity is, in my opinion, evidence for a physical (quantum) discreteness of space at the Planck scale. This is manifested in the fact that certain operators corresponding to the measurement of geometrical quantities, in particular area and volume, have discrete spectrum. According to the standard interpretation of quantum mechanics (which we adopt), this means that the theory predicts that a physical measurement of an area or a volume will necessarily yield quantized results. Since the smallest eigenvalues are of Planck scale, this implies that there is no way of observing areas or volumes smaller than Planck scale. Space comes in ``quanta'' in the same manner as the energy of an oscillator. The spectra of the area and volume operators have been computed with much detail in loop quantum gravity. These spectra have a complicated structure, and they constitute detailed quantitative physical predictions of loop quantum gravity on Planck scale physics. If we had experimental access to Planck scale physics, they would allow the theory to be empirically tested in great detail.


"A few comments are in order.

The result of the discreteness of area and volume is due to Rovelli and Smolin, and appeared first in reference [186 ]. Later, the result has been recovered by alternative techniques and extended by a number of authors. In particular, Ashtekar and Lewandowski [17 ] have repeated the derivation, using the connection representation, and have completed the computation of the spectrum (adding the sector which was not computed in [186 ] The Ashtekar-Lewandowski component of the spectrum has then been rederived in the loop representation by Frittelli Lehner and Rovelli in [84 ] Loll has employed lattice techniques to point out a numerical error in [186 ](corrected in the Erratum) in the eigenvalues of the volume. The analysis of the volume eigenvalues has been performed in [77 ] where general techniques for performing these calculations are described in detail. The spectrum of the volume has then been analyzed also in [203 ] There are also a few papers that have anticipated the main result presented in [186 ]>]. In particular, Ashtekar Rovelli and Smolin have argued for a physical discreteness of space emerging from the loop representation in [23]where some of the eigenvalues of the area already appear, although in implicit form. The first explicit claim that area eigenvalues might in principle be observable (in the presence of matter) is by Rovelli in [172

The reader will wonder why area and volume seem to play here a role more central than length, when classical geometry is usually described in terms of lengths. The reason is that the length operator is difficult to define and of more difficult physical interpretations. For attempts in this direction, see [204 ]Whether this is simply a technical difficulty, or it reflects some deep fact, is not clear to me.

Area and volume are not gauge invariant operators. Therefore, we cannot directly interpret them as representing physical measurements. Realistic physical measurements of areas and volumes always refer to physical surfaces and spatial regions, namely surfaces and spatial regions determined by some physical object. For instance, I can measure the area of the surface of a certain table. In the dynamical theory that describes the gravitational field as well as the table, the area of the surface of the table is a diffeomorphism invariant quantity A, which depends on gravitational as well as matter variables. In the quantum theory, A will be represented by a diffeomorphism invariant operator. Now, as first realized in [172] it is plausible that the operator A is, mathematically, the same operator as the pure gravity area operator. This is because we can gauge fix the matter variables, and use matter location as coordinates, so that non-diff-invariant observables in the pure gravity theory correspond precisely to diff-invariant observables in the matter+gravity theory. Thus, discreteness of the spectrum of the area operator is likely to imply discreteness of physically measurable areas, but it is important to emphasize that this implication is based on some additional hypothesis on the relation between the pure gravity and the gravity+matter theories.



I have long enjoyed the beauty of the Principle of Duality. Where two systems or models are duals of each other, a more detailed knowledge of one allows you to predict features in the other, and to understand the other better. Wikipedia has an excellent article on Mathematical Duality, see http://en.wikipedia.org/wiki/Duality_%28mathematics%29, which says: “Generally speaking, dualities translate concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion.”  In Functions of a Complex Variable, Transform Methods allow the transformation of a complex topology to a simple one that is easy to analyze – an example of duals. “In analysis, frequently problems are solved by passing to the dual description of functions and operators.



Duals that I am familiar with, include:

a) the duality of acoustics with electronics,

b) the duality between electric phenomena and magnetic phenomena, see Maxwell's Equations.

The former allows you to design acoustic structures using filter theory from the field of electronics. The latter allows you to design magnetic circuits using the principles of electric (electrostatic) circuits, and vice-versa. In this latter field, electric fields become magnetic fields (and vice-versa), depending on the velocity of the observer with respect to the source of said fields.


 In quantum theory, distance is inverse to energy. Perhaps

c) distance is the dual of energy, at least at very small distances.

d) area is the dual of spin.

Certainly mass can be converted into energy, according to E = Mc2, so that

e) mass is another dual of energy.


Another type of duality is where two different models represent the same thing, as with the wave and particle natures of light. A more advanced example is the "anti-de Sitter space/Conformal Field Theory" (adS/CFT) correspondance, which is a type of duality that states that two apparently distinct physical theories are actually equivalent [Smo-12], . Perhaps Electric and Magnetic fields are another example of this type of duality, and do in fact represent the same thing.


Smolin et all postulate that the smallest possible unit of volume of spacetime is analogous to the mass of the smallest possible unit of matter, an atom; and that its surfaces, those that interface other "atoms" of space, are analogous to the valency bonds of an atom of matter.


Why volume and not length as the basis of quantum spacetime? Well firstly, the mathematics works with volume and area, and they could not get it to work with length. Secondly, volume is the "amount" of space, in the same way that mass is the "amount" of matter. Perhaps, using duality, they will discover a property of matter analogous to length in spacetime.


Spin networks are neat concepts, which follows naturally from the concept of the use of vectors to represent areas and their orientation (the vector is perpendicular to the surface it represents). The relationship between lines representing magnetic or electric fields, and the areas through which they pass, also fit nicely into spin network graph representation of space.


Spin networks, in which nodes (representing quantums of volume) interface with other nodes via lines (representing the spin or area common to the two quantums of volume), represent elements of space. Adding another dimension, time, so that the nodes become lines, and the lines become surfaces, produce a "spin foam" representation of quantum spacetime.


Roger Penrose of University of Oxford first proposed in the early 1970's that spin networks might play a role in theories of quantum gravity [Smo-1]. Penrose worked with Stephen Hawking on the early theory of Black Holes. I met him briefly when he gave a lecture on Tiling at Ottawa's National Gallery a few years ago. Smolin et al confirmed in 1994 that Penrose's intuition was correct.


Quanta of spacetime volume are of the order of one Plank length cubed. Quanta of spacetime area are of the order of one Plank length squared. A Plank length is some twenty orders of magnitude (powers of ten) smaller than an atomic nucleus. In quantum theory, distance is inverse to energy. The energy required to probe effects at the Plank scale is some fifteen orders of magnitude higher than the energy of the largest accelerators now under construction [Smo-6].


Everyone assumed that it would be impossible to probe scales that small. Everyone except one person - Giovanni Amelino-Carmelia. He (and others) realized that the Universe can be used as a laboratory to test quantum gravity effects. First, there are accelerators in distant galaxies that produced gamma burst, 10 billion years ago, particles with energies much higher than we can produce in even the largest man-made accelerators, 10 million times higher. Secondly, due to the long times and vast distances involved, effects such as the difference in speeds of these particles (speed of light is energy dependent according to LQG) can be detected over these long distances and times [Smo-1][Smo-6].


João Magueijo and Andreas Albrecht discovered that the speed of light was higher in the early universe (presumably when the energy of photons were higher) when the universe was very hot and dense, resulting in an alternative to inflationary cosmology that explains everything that inflation does [Smo-6].


One of the most popular competing theories for modeling quantum gravity is Super-string theory and its evolution - M-theory. Smolin, in his March 2003 paper [Smo-5],, compares String theory with Looped Quantum Gravity (LQG) and the results from each. In his 2001 book,  "Three Roads to Quantum Gravity" [Smo-12], , he presumably compares three theories.


There is another recent series of papers on Quantum Gravity in the Nov 2003 issue of Physics World [5].


Cambridge University have a series of non-technical papers on Quantum Gravity [DAMTP-1], [DAMTP-2], [DAMTP-3], but these are somewhat dated (1996).



You thought that space existed - not so, only fields.

You thought that light travels in straight lines, in any direction, at constant speed in any frame of reference (Special Relativity). Well, LQG shows that these assumptions are only approximations to reality.

Light travels in straight lines, except in curved spacetime (due to gravitational fields) - no, not at Planck length scales.

Light travels in any direction - no, possible directions are quantized, see Penrose etc.

Light travels at constand speed - no, not if the photon energy is extremely large. Blazar gamma-ray pulses get dispersed after billions of years, traveling billions of light-years, due to the different energies of the photons in the pulse, and thus very slightly different speeds.

Big Bounce, not Big Bang.

Recent LQG calculations by Martin Bojowald indicate that the Big Bang was actually a Big Bounce (contraction followed by an expansion) [Smo-1]. Ashtekar[Ash-10] described, in 2001, a natural resolution of the Big Bang singularity.

Speed of light dependent on energy of photon.

Quantum nature of volume, area, time, direction, ...

See Roger Penrose, "Theory of quantized directions", 1971

duality between strings and loops, Smolin June 2000


What is the spacetime dual of the E = Mc2 equation?


If such a relationship between spacetime and energy exists, then perhaps the power stations of the future will convert spacetime to energy.




Loop Quantum Gravity

Some 91 papers on Loop Quantum Gravity are available from the ArXiv archives at:



Lee Smolin,  Perimeter Institute for Theoretical Physics, Waterloo, ON; and Dept of Physics, Univ Waterloo.


The Perimeter Institute for Theoretical Physics, Waterloo, ON, where Smolin was a founding member, was founded in the fall of 1999 through the vision and leadership of Mike Lazaridis, President and Co-CEO of Research In Motion (RIM).  With his personal donationof $100 million, together with additional donations of $10 million each by fellow RIM executives Doug Fregin and Jim Balsillie. The Scientific Advisory Committee includes Sir Roger Penrose. See   http://perimeterinstitute.ca/about/fundingdonations.cfmhttp://www.perimeterinstitute.com/  and http://www.perimeterinstitute.ca


Some 63 of his most recent (2003 - 1992) papers are available from:



[Smo-1] Lee Smolin, "Atoms of Space and Time", Scientific American, January 2004, pp66-75.

[Smo-2] Fotini Markopoulou and Lee Smolin, "Quantum Theory from Quantum Gravity", 2003-11-17, 17pp,


[Smo-3], Stephon Alexander, Justin Malecki and Lee Smollin, "Quantum Gravity and Inflation", 2003-09-03, 18pp,


[Smo-4] Giovanni Amelino-Carmelia, Lee Smolin and Artem Starodubtsev, "Quantum symmetry, the cosmological constant and Plank scale phenomenology", ver 2, 2003-06-16, 19pp, http://arxiv.org/PS_cache/hep-th/pdf/0306/0306134.pdf

[Smo-5] Lee Smolin, "How Far Are We from the Quantum Theory of Gravity?", (v1: 2003-03-19), v2: 2003-04-11, 84pp,  http://arxiv.org/PS_cache/hep-th/pdf/0303/0303185.pdf

[Smo-6] Lee Smolin, "Loop Quantum Gravity", 2003-02-24, http://www.edge.org/3rd_culture/smolin03/smolin03_index.html

[Smo-7] Lee Smolin, "Quantum Gravity with a positive cosmological constant", Sept 2002, 59pp,


[Smo-8] Yi Ling and Lee Smolin, "A holographic formulation of quantum supergravity", Phys.Rev. D63 (2001) 064010, v1 2000-09-03, updated 2001-12-09, http://arxiv.org/PS_cache/hep-th/pdf/0009/0009018.pdf

[Smo-9] Lee Smolin, "A holographic formulation of quantum general relativity", Phys.Rev. D61 (2000) 084007, v2 1999-10-31, http://arxiv.org/PS_cache/hep-th/pdf/9808/9808191.pdf

[Smo-10] Lee Smolin, "The present moment in quantum cosmology: challenges to the arguments for the elimination of time", April 2001, http://arxiv.org/PS_cache/gr-qc/pdf/0104/0104097.pdf

[Smo-11] Lee Smolin, "Technical Summary of Loop Quantum Gravity", 2001-02-26, http://www.qgravity.org/loop/

[Smo-12] Lee Smolin, "Three Roads to Quantum Gravity", New York: Basic Books, 2001, [Ottawa Public Library, Nepean Centrepointe branch, call no. 530.1 Smo], originally published: London: Weidenfeld and Nicolson, 2000.

[Smo-13] Lee Smolin, "The cubic matrix model and a duality between strings and loops", June 2000, 32pp,


[Smo-14] Fotini Markopoulou and Lee Smolin, "Holography in quantum spacetime", Oct 1999, 16pp,


[Smo-15] Carlo Rovelli, Lee Smolin, "Spin Networks and Quantum Gravity", May 1995, 42pp, Phys.Rev. D52 (1995) 5743-5759, http://arxiv.org/PS_cache/gr-qc/pdf/9505/9505006.pdf , same as [Rov-21]

[Smo-16] Carlo Rovelli, Lee Smolin, "Discreteness of area and volume in quantum gravity", Nov 1994, 36pp, http://arxiv.org/PS_cache/gr-qc/pdf/9411/9411005.pdf , same as [Rov-23]

[Smo-17] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Weaving a classical geometry with quantum threads, Mar 1992, 9pp, Phys.Rev.Lett. 69 (1992) 237-240, http://arxiv.org/PS_cache/hep-th/pdf/9203/9203079.pdf , same as [Rov-24]

[Smo-18] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Gravitons and Loops", Feb 1992, 40pp, Phys.Rev D44 (1991) 1740-1755, http://arxiv.org/PS_cache/hep-th/pdf/9202/9202054.pdf , same as [Rov-26], [Ash-14] 



Amelino Camelia-Giovanni

52 of his most recent papers (2003-1996) are available from:



[ACG-1] Amelino Camelia-Giovanni, Jerzy Kowalski-Glikman, Gikanluca Mandanici, Andrea Procaccini, "Phenomenology of doubly special relativity",  http://arxiv.org/PS_cache/gr-qc/pdf/0312/0312124.pdf

[ACG-2] Amelino Camelia-Giovanni,"Planck-scale structure of spacetime and some implications for astrophysics and cosmology",  http://arxiv.org/PS_cache/astro-ph/pdf/0312/0312014.pdf

[ACG-3] Amelino Camelia-Giovanni, "Quantum-gravity phenomenology", Physics World, v16, #11, Nov 2003


[ACG-4] Amelino Camelia-Giovanni, "Quantum-gravity phenomenology", Nov 2003, 9pp, notes prepared while working on [ACG-3], http://arxiv.org/PS_cache/physics/pdf/0311/0311037.pdf

[ACG-5] Amelino Camelia-Giovanni, "Fundamental physics in space: a Quantum-Gravity perspective", Sept 2003, 20pp,


[ACG-6] Giovanni Amelino-Carmelia, "The three perspectives on the quantum-gravity problem and their implications for the fate of Lorentz symmetry", Sept 2003, 48pp,


[ACG-7] Giovanni Amelino-Carmelia, Lee Smolin and Artem Starodubtsev, "Quantum symmetry, the cosmological constant and Plank scale phenomenology", ver 2, 2003-06-16, 19pp, http://arxiv.org/PS_cache/hep-th/pdf/0306/0306134.pdf , same as [Smo-4]

[ACG-8] Giovanni Amelino-Carmelia, Claus Lammerzahl, "Quantum-gravity-motivated Lorentz-symmetry test with laser interferometers", June 2003, 17pp, http://arxiv.org/PS_cache/gr-qc/pdf/0306/0306019.pdf

[ACG-9] Giovanni Amelino-Carmelia, "Proposal of a second generation of quantum-gravity-motivated Lorentz-symmetry tests: sensitive to effects suppressed quadratically by the Planck scale", Int.J.Mod.Phys. D12 (May 2003), 1633-1640; http://arxiv.org/PS_cache/gr-qc/pdf/0305/0305057.pdf

[ACG-10] Giovanni Amelino-Carmelia, "Improved limit on quantum-spacetime modifications of Lorentz symmetry from observations of gamma-ray blazars", Dec 2002, 10pp,  http://arxiv.org/PS_cache/gr-qc/pdf/0212/0212002.pdf

[ACG-11] Giovanni Amelino-Carmelia, Francesco D'Andrea, Gianluca Mandanici, "Group velocity in noncommutative spacetime", JCAP 0309 (2003) 006, http://arxiv.org/PS_cache/hep-th/pdf/0211/0211022.pdf

[ACG-12] Giovanni Amelino-Carmelia, "Doubly-Special Relativity: first results and key open  problems", Int.J.Mod.Phys. D11 (2002) 1643, (29pp), http://arxiv.org/PS_cache/gr-qc/pdf/0210/0210063.pdf


Abhay Ashtekar, ashtekar@gravity.phys.psu.edu

59 of his most recent papers (2003-1992) are available from:



[Ash-1] Abhay Ashtekar, Badri Krishnan, "Dynamical horizons and their properties", Phys. Rev. D 68, 104030 (2003), May 2003, 44pp, http://arxiv.org/PS_cache/gr-qc/pdf/0308/0308033.pdf

[Ash-2] Abhay Ashtekar, "How black holes grow", June 2003, http://arxiv.org/PS_cache/gr-qc/pdf/0306/0306115.pdf

[Ash-3] Abhay Ashtekar, "Non-minimal couplings, quantum geometry and black hole entropy", Class.Quant.Grav. 20 (2003) 4473-4484, May 2003, 14pp, http://arxiv.org/PS_cache/gr-qc/pdf/0305/0305082.pdf

[Ash-4] Abhay Ashtekar, Alejandro Corichi, Daniel Sudarsky, "Non-minimally coupled scalar fields and isolated horizons", Class.Quant.Grav. 20 (2003) 3413-3426, May 2003, 14pp, http://arxiv.org/PS_cache/gr-qc/pdf/0305/0305044.pdf

[Ash-5] Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski, "Mathematical structure of loop quantum cosm/ology", Adv.Theor.Math.Phys. 7 (2003) 233-268, April 2003, 29pp, http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304074.pdf 

[Ash-6] Abhay Ashtekar, Stephen Fairhurst, Joshua L. Willis, "Quantum gravity, shadow states and quantum mechanics", Class.Quant.Grav. 20 (2003) 1031-1062, July 2002, 35pp, http://arxiv.org/PS_cache/gr-qc/pdf/0207/0207106.pdf

[Ash-7] Abhay Ashtekar, Badri Krishnan, "Dynamic horizons: energy, angular momentum, fluxes and balance laws", Phys.Rev.Lett. 89 (2002) 261101, July 2002, 4pp, http://arxiv.org/PS_cache/gr-qc/pdf/0207/0207080.pdf

[Ash-8] Abhay Ashtekar, Olaf Dreyer,  Jacek Wisniewski, "Isolated horizons in 2 + 1 gravity", Adv.Theor.Math.Phys. 6 (2002) 507-555, June 2002, 27pp, http://arxiv.org/PS_cache/gr-qc/pdf/0206/0206024.pdf 

[Ash-9] Abhay Ashtekar, "Quantum geometry in action: Big Bang and Black Holes", Feb 2002, 18pp,


[Ash-10] Abhay Ashtekar, "Quantum geometry and gravity: recent advances", Dec 2001, 24pp,


[Ash-11] Abhay Ashtekar, Alejandro Corichi, Daniel Sudarsky, "Hairy Black Holes, Horizon Mass and Solitons", Class.Quant.Grav. 18 (2001) 919-940, Nov 2000, 25pp, http://arxiv.org/PS_cache/gr-qc/pdf/0011/0011081.pdf 

[Ash-12] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Weaving a classical geometry with quantum threads, Mar 1992, 9pp, Phys.Rev.Lett. 69 (1992) 237-240, http://arxiv.org/PS_cache/hep-th/pdf/9203/9203079.pdf , same as [Rov-24], [Smo-17] 

[Ash-13] Abhay Ashtekar, Carlo Rovelli,  "A loop representation for the Quantum Maxwell Field", Feb 1992, 40pp, Class.Quantum.Grav. 9 (1992) 1121-1150, http://arxiv.org/PS_cache/hep-th/pdf/9202/9202063.pdf , same as [Rov-25]

[Ash-14] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Gravitons and Loops", Feb 1992, 40pp, Phys.Rev D44 (1991) 1740-1755, http://arxiv.org/PS_cache/hep-th/pdf/9202/9202054.pdf , same as [Rov-26], [Smo-18] 



Martin Bojowald,

Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany;

formerly at the Center for Gravitational Physics and Geometry, Pennsylvania State University, 104 Davey Lab, University Park, PA 16802, USA;


29 of his most recent papers (2003-1999) are available from:



[Boj-1] Martin Bojowald, Kevin Vandersloot, "Loop Quantum Cosmology and boundary proposals", Dec 2003, 18pp, http://arxiv.org/PS_cache/gr-qc/pdf/0312/0312103.pdf 

[Boj-2] Martin Bojowald, Ghanashyam Date, Kevin Vandersloot, "Homogeneous Loop Quantum Cosmology: the role of spin connection", Nov 2003, 36pp, http://arxiv.org/PS_cache/gr-qc/pdf/0311/0311004.pdf 

[Boj-3] Martin Bojowald, Ghanashyam Date, "A non-chaotic quantum Bianchi IX universe and the quantum structure of classical singularities", Nov 2003, 4pp, http://arxiv.org/PS_cache/gr-qc/pdf/0311/0311003.pdf

[Boj-4] Martin Bojowald, "Quantum Gravity and the Big Bang", CGPG-03/9-2, AEI-2003-077, Sept 2003, 6pp, http://arxiv.org/PS_cache/astro-ph/pdf/0309/0309478.pdf 

[Boj-5] Martin Bojowald, Ghanashyam Date, "Consistency conditions for fundamentally discrete theories", Class.Quantum.Grav. 21 (2004) 121-143, July 2003, 33pp, http://arxiv.org/PS_cache/gr-qc/pdf/0307/0307083.pdf

[Boj-6] Martin Bojowald, Hugo A Morales-Tecoti, "Cosmological applications of loop quantum gravity", June 2003, 42pp, http://arxiv.org/PS_cache/gr-qc/pdf/0306/0306008.pdf

[Boj-7] Martin Bojowald, "Initial conditions for a universe", Gen.Rel.Grav. 35 (2003) 1877-1883, May 2003, 7pp, http://arxiv.org/PS_cache/gr-qc/pdf/0305/0305069.pdf

[Boj-8] Martin Bojowald, Thomas Strobl, "Classical solutions for Poisson Sigma models on a Riemann surface", JHEP 0307 (2003) 002, Apr 2003, 28pp, http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304252.pdf

[Boj-9] Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski, "Mathematical structure of loop quantum cosm/ology", Adv.Theor.Math.Phys. 7 (2003) 233-268, April 2003, 29pp, http://arxiv.org/PS_cache/gr-qc/pdf/0304/0304074.pdf , same as [Ash-5] 

[Boj-10] Martin Bojowald, "Homogeneous loop quantum cosmology", Class.Quant.Grav. 20 (2003) 2595-2615, Mar 2003, 25pp, http://arxiv.org/PS_cache/gr-qc/pdf/0303/0303073.pdf

[Boj-11] Martin Bojowald, Kevin Vandersloot, "Loop Quantum Cosmology, boundary proposals, and inflation", Phys.Rev. D67 (2003) 124023,  Mar 2003, 20pp, http://arxiv.org/PS_cache/gr-qc/pdf/0303/0303072.pdf 

[Boj-12] Martin Bojowald, Alejandro Perez, "Spin foam quantization and anomalies", Mar 2003, 32pp, http://arxiv.org/PS_cache/gr-qc/pdf/0303/0303026.pdf 

[Boj-13] Martin Bojowald, Franz Hinterleitner, "Isotropic Loop Quantum Cosmology with matter", July 2002, 18pp http://arxiv.org/PS_cache/gr-qc/pdf/0207/0207038.pdf

[Boj-14] Martin Bojowald, "Inflation from quantum cosmology", Phys.Rev.Lett. 89 (2002 261301, June 2002, 4pp, http://arxiv.org/PS_cache/gr-qc/pdf/0206/0206054.pdf

[Boj-15] Martin Bojowald, "Quantization ambiguities in isotropic quantum cosmology", Class.Quant.Grav. 19 (2002) 5113-5230, June 2002, 20pp, http://arxiv.org/PS_cache/gr-qc/pdf/0206/0206053.pdf

[Boj-16] Martin Bojowald, "Isotropic loop quantum cosmology", Class.Quant.Grav. 19 (2002) 2717-2742, Feb 2002, 30pp, http://arxiv.org/PS_cache/gr-qc/pdf/0202/0202077.pdf

[Boj-17] Martin Bojowald, "The inverse scale factor in isotropic quantum geometry", Phys.Rev. D64 (2001) 084018, May 2001, 17pp, http://arxiv.org/PS_cache/gr-qc/pdf/0105/0105067.pdf

[Boj-18] Martin Bojowald, "Absence of Singularity in Loop Quantum Cosmology", Phys.Rev.Lett 86 (2001) 5227-5230, Feb 2001, 4pp, http://arxiv.org/PS_cache/gr-qc/pdf/0102/0102069.pdf

[Boj-19] Martin Bojowald, "Angular momentum in Loop Quantum Gravity", Aug 2000, 9pp, http://arxiv.org/PS_cache/gr-qc/pdf/0008/0008054.pdf

[Boj-20] Martin Bojowald, "Loop Quantum Cosmology IV: Discrete Time Evolution", Class.Quant.Grav. 18 (2001) 1071-1088, Aug 2000, 21pp, http://arxiv.org/PS_cache/gr-qc/pdf/0008/0008053.pdf

[Boj-20] Martin Bojowald, "Loop Quantum Cosmology III: Wheeler-DeWitt Operators", Class.Quant.Grav. 18 (2001) 1055-1070, Aug 2000, 19pp, http://arxiv.org/PS_cache/gr-qc/pdf/0008/0008052.pdf

[Boj-21] Martin Bojowald, "Loop Quantum Cosmology II: Volume Operators", Class.Quant.Grav. 17 (2000) 1509-1526, Oct 1999, 21pp, http://arxiv.org/PS_cache/gr-qc/pdf/9910/9910104.pdf

[Boj-22] Martin Bojowald, "Loop Quantum Cosmology I: Kinematics", Class.Quant.Grav. 17 (2000) 1489-1508, Oct 1999, 24pp, http://arxiv.org/PS_cache/gr-qc/pdf/9910/9910103.pdf



Physics World


[PW-1] Matthew Chalmers (Features ed, Phys World), "Welcome to quantum gravity",  Physics World, v16, #11, Nov 2003, http://physicsweb.org/article/world/16/11/7

[PW-2] Fernando Quevedo (DAMTP), "The string-theory landscape", Physics World, v16, #11, Nov 2003,

http://physicsweb.org/article/world/16/11/3 , same as [DAMTP-4].

[PW-3] Leonard Susskind, "Superstrings",   Physics World, v16, #11, Nov 2003, http://physicsweb.org/article/world/16/11/8

[PW-4] Carlo Rovelli, Loop Quantum Gravity", Physics World, v16, #11, Nov 2003, http://physicsweb.org/article/world/16/11/9

[PW-5] Amelino Camelia-Giovanni, "Quantum-gravity phenomenology", Physics World, v16, #11, Nov 2003, http://physicsweb.org/article/world/16/11/10 , same as [ACG-3]

[PW-6] Steven Abel and John March-Russell, "the search for extra dimensionis", Physics World, Nov 2000, http://physicsweb.org/article/world/13/11/9/2

[PW-7] Brian Green, "The Elegant Universe: superstrings, hidden dimensions and the quest for the Ultimate theory", 1999, Jonathan Cape/W W Norton, 448pp

[PW-8] John Charap (Univ London), "String Theory: simple yet elegant", a review of [PW-6], Physics World, July 1999, http://physicsweb.org/article/review/12/7/3/1


[6] John Baez, "The Quantum of Area?", Nature, v421, pp702-702, Feb 2003.


E Berghoeff

[Berg-1] E Berghoeff, "p-Branes, D-Branes and M-Branes", Nov 1996, http://arxiv.org/PS_cache/hep-th/pdf/9611/9611099.pdf 

[Berg-2] E Berghoeff, "p-Branes and D-Branes actions", July 1996, http://arxiv.org/PS_cache/hep-th/pdf/9607/9607238.pdf 





Cambridge University, Department of Applied Mathematics and Theoretical Physics (DAMTP)

Founded in 1959, the current department head since 2000 is Timothy J. Pedleyhttp://www.damtp.cam.ac.uk


[DAMTP-1] Thomas Hertog and Harvey Reall, "Quantum Cosmology", ?1996, http://www.damtp.cam.ac.uk/user/gr/public/qg_qc.html

[DAMTP-2] Carlos Herdeiro, "M-Theory, the theory formerly known as Strings", ?1996, http://www.damtp.cam.ac.uk/user/gr/public/qg_ss.html

[DAMTP-3] H Andrew Chamblin and J Ashbourn, "The Holographic Principle and M-theory", ?1996 http://www.damtp.cam.ac.uk/user/gr/public/holo/

[DAMTP-4] Fernando Quevedo (DAMTP), "The string-theory landscape",  Physics World, Nov 2003,



Scientific American


[SciAm-1] Lee Smolin, "Atoms of Space and Time", Scientific American, January 2004, pp66-75, same as [Smo-1]

[SciAm-2] formerly [9] Michael J Duff, "The Theory formerly Known as Strings", Scientific American, Feb 1998.

[SciAm-3] formerly [11] Jacob D. Bekenstein, "Information in a Holographic Universe", Scientific American, Aug 2003.

[SciAm-4] formerly [12] Neil Gehrels, Luigi Piro and Peter J T Leonard, "The Brightest Explosions in the Universe", Scientific American, Dec 2002.

[SciAm-5] formerly [13] W. Wayt Gibbs, "Ripples in Space and Time", Scientific American, April 2002.

[SciAm-6] formerly [14] Jeremiah P Ostriker and Paul J Steinhardt, "The Quintessential Universe", Scientific American, Jan 2001.


Carlo Rovelli, Luminy, Case 907, F 13288 Marseille, France. http://www.cpt.univ-mrs.fr/~rovelli/rovelli.htmlrovelli@cpt.univ-mrs.fr.

Professeur (Ire classe), Université de la Méditerranée et Centre de Physique Théorique, CNRS-Luminy, Marseille, France http://www.cpt.univ-mrs.fr ,  and with University of Rome.

Professor of History and Philosophy of Science, Departemnt of History and Philosophy of Science, Univ Pittsburg, Pittsburg, USA.


64 of his most recent papers (2003-1992) are available from:


An older lists, on his web site, and in his résumé, at

http://www.cpt.univ-mrs.fr/~rovelli/publicationlist.html, and http://www.phys.uniroma1.it/DOCS/DIR/rovelli.html

includes 2 books (1978 and 1995), 93 scientific papers (up to Sept 2000), 40 invited talks (1987-2001), 13 "Other scientific production" (1981-1989), 17 "recent" seminars (1992-1995), and 7 theses (by others) that he supervised. However, the web site is old, and there is nothing newer than Feb 2000.


[Rov-1] Carlo Rovelli, "Quantum Gravity", to be published by Cambridge University Press, 347pp, draft dated 2003-12-30 available from http://www.cpt.univ-mrs.fr/~rovelli/book.pdf, home page: http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

[Rov-2] Carlo Rovelli, "Loop quantum gravity", Physics World, Nov 2003, http://physicsweb.org/articles/world/16/11/9

[Rov-3] Florian Conrady and Carlo Rovelli, "Generalized Schroedinger equation in Euclidean field theory", Oct 2003, 25pp, http://arxiv.org/PS_cache/hep-th/pdf/0310/0310246.pdf

[Rov-4] Carlo Rovelli, "A dialog on quantum gravity", Oct 2003, 20pp, http://arxiv.org/PS_cache/hep-th/pdf/0310/0310077.pdf

[Rov-5] Florian Conrady, Luisa doplicher, Robert Oecki, Carlo Rovelli, Massimo Testa,  "The Minkowski vacuum in background independent quantum gravity", July 2003, 8pp, http://arxiv.org/PS_cache/gr-gc/pdf/0307/0307118.pdf

[Rov-6] Daniele Colosi, Carlo Rovelli, "A simple background-independent hamiltonian quantum model", June 2003, Phys.Rev D68 (2003) 104008, http://arxiv.org/PS_cache/gr-qc/pdf/0306/0306059.pdf

[Rov-7] Laurent Freidel, Etera R Livine, Carlo Rovelli, "Spectra of Length and Area in 2 +1 Lorentzian Loop Quantum Gravity", Dec 2002, 13pp, Class.Quant.Grav. 20 (2003) 1463-1478, , http://arxiv.org/PS_cache/gr-qc/pdf/0212/0212007.pdf

[Rov-8] Carlo Rovelli, SimoneSpeziale, "Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction", May 2002, 12pp, Phys.Rev. D67 (2003) 064019, , http://arxiv.org/PS_cache/gr-qc/pdf/0205/0205108.pdf

[Rov-9] Donald Marolf, Carlo Rovelli, "Relativistic quantum measurement", Mar 2002, 9pp, Phys.Rev. D67 (2002) 023510, , http://arxiv.org/PS_cache/gr-qc/pdf/0203/0203056.pdf

[Rov-10] Carlo Rovelli, "Notes for a brief history of quantum gravity", June 2000,


[Rov-11] Carlo Rovelli, "The century of the incomplete revolution: searching for general relativistic quantum field theory", Oct 1999, J.Math.Phys. 41 (2000) 3776-3800, , http://arxiv.org/PS_cache/gr-qc/pdf/9910/9910131.pdf

[Rov-12] Marcus Gaul, Carlo Rovelli, "Loop Quantum Gravity and the meaning of Diffeomorphism Invariance", Oct 1999, 52pp,  Lect.Notes Phys. 541 (2000) 277-324, , http://arxiv.org/PS_cache/gr-qc/pdf/9910/9910079.pdf

[Rov-13] Carlo Rovelli, "Quantum spacetime: what do we know?", Mar 1999, to appear in:

C Callendar, N Hugget eds, "Physics Meets Philosophy at the Planck scale", Cambridge University Press,


[Rov-14] Carlo Rovelli, "Strings, loops and others: a critical survey of the present approaches to quantum gravity", Mar 1998,


[Rov-15] Carlo Rovelli, "Incerto tempore, incertisque loci: can we compute the exact time at which a quantum measurement happens?", Feb 1998, http://arxiv.org/PS_cache/quant-ph/pdf/9802/9802020.pdf

[Rov-16] Carlo Rovelli, "A note on the formal structure of quantum constrainedsystems", Nov 1997, 4pp,


[Rov-17] Carlo Rovelli,Loop Quantum Gravity”,  Oct 1997, Living Rev.Rel. 1 (1998) 1,

http://arxiv.org/PS_cache/gr-qc/pdf/9710/9710008.pdf , http://relativity.livingreviews.org/Articles/lrr-1998-1/ ,

http://www.livingreviews.org/lrr-1998-1 , and


[Rov-18] Carlo Rovelli, "Loop Quantum Gravity and Black Hole Physics", Aug 1996, Helv.Phys.Acta 69 (1996) 582-611,


[Rov-19] Carlo Rovelli, "Black Hole Entropy from Loop Quantum Gravity", Mar 1996, 5pp, Phys.Rev.Lett. 77 (1996) 3288-3291, http://arxiv.org/PS_cache/gr-qc/pdf/9603/9603063.pdf

[Rov-20] Norbert Grot, Carlo Rovelli, Ranjeet S Tate, "Time-of-arrival in quantum mechanics", Mar 1996, 20pp, Phys.Rev. A54 (1996) 4679, http://arxiv.org/PS_cache/quant-ph/pdf/9603/9603021.pdf

[Rov-21] Carlo Rovelli, "On the Loop Representation of Quantum Gravity; a collection of papers", Pittsburg Univ, Pittsburg, 1995

[Rov-22] Carlo Rovelli, Lee Smolin, "Spin Networks and Quantum Gravity", May 1995, 42pp, Phys.Rev. D52 (1995) 5743-5759, http://arxiv.org/PS_cache/gr-qc/pdf/9505/9505006.pdf

[Rov-23] Carlo Rovelli: "Analysis of the different meaning of the concept of time in different physical theories", Il Nuovo Cimento 11013, 81 (1995)

[Rov-24] Carlo Rovelli, "Outline of a generally covariant quantum field theory and a quantum theory of gravity", Mar 1995, 33pp, J.Math.Phys. 36 (1995) 6529-6547, http://arxiv.org/PS_cache/gr-qc/pdf/9503/9503067.pdf

[Rov-25] S Frittelli, S Koshti, T Newman, C Rovelli: "Classical and quantum Faraday Lines" Physical Review D49 (1994) 6883

[Rov-26] Carlo Rovelli, Lee Smolin, "Discreteness of area and volume in quantum gravity", Nov 1994, 36pp, http://arxiv.org/PS_cache/gr-qc/pdf/9411/9411005.pdf

[Rov-27] H Morales-Tecotl, Carlo Rovelli: "Fermions in Quantum Gravity" Physical Review Letters 72, 3642 (1994)

[Rov-28] Carlo  Rovelli: "Area is the length of Ashtekar's triad field" Physical Review D 47, 1703 (1993)

[Rov-29] Carlo  Rovelli, "Statistical mechanics of gravity and thermodynamical origin of time", Classical and Quantum Gravity, 10, 1549 (1993)

[Rov-30] Carlo  Rovelli: "The statistical state of the universe" Classical and Quantum Gravity, 10, pg 1567 (1993)

[Rov-31] Carlo  Rovelli: "A generally covariant quantum field theory and a prediction on quantum measurements of geometry" Nuclear Physics B405, 797 (1993) [Rov-24] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Weaving a classical geometry with quantum threads", Mar 1992, 9pp, Phys.Rev.Lett. 69 (1992) 237-240, http://arxiv.org/PS_cache/hep-th/pdf/9203/9203079.pdf

[Rov-32] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Self duality and quantization", Journal of Geometry and Physics 8, 7 (1992)

[Rov-33] Abhay Ashtekar, Carlo Rovelli,  "A loop representation for the Quantum Maxwell Field", Feb 1992, 40pp, Class.Quantum.Grav. 9 (1992) 1121-1150, http://arxiv.org/PS_cache/hep-th/pdf/9202/9202063.pdf , same as [Ash-13]

[Rov-34] Abhay Ashtekar, Carlo Rovelli, Lee Smolin, "Gravitons and Loops", Feb 1992, 40pp, Phys.Rev D44 (1991) 1740-1755, http://arxiv.org/PS_cache/hep-th/pdf/9202/9202054.pdf , same as [Smo-18], [Ash-14]

[Rov-35] Abhay Ashtekar, Carlo Rovelli, "Connections, loops and quantum general relativity", Classical and Quantum Gravity 9, 3 (1992)

[Rov-36] Carlo  Rovelli: "What is observable in classical and quantum gravity?'' Classical and Quantum Gravity 8, 297 (1991)

[Rov-37] Carlo  Rovelli: "Quantum mechanics without time: a model" Physical Review D42, 2638 (1991)

[Rov-38] Carlo Rovelli: "Time in quantum gravity: an hypothesis" Physical Review D43, 442 (1991)

[Rov-39] Carlo  Rovelli: "Ashtekar formulation of general relativity and loop space non-perturbative Quantum Gravity: a report" Invited review paper. Classical and Quantum Gravity, 8, 1613-1675 (1991)

[Rov-40] Carlo  Rovelli, L Smolin: "Loop space representation for quantum general relativity" Nuclear Physics B331, 80 (1990)

[Rov-41] A Ashtekar, V Husain, J Samuel, C Rovelli, L Smolin: "2+1 quantum gravity as a toy model for the 3+1 theory'' Classical and Quantum Gravity 6, L185 (1989)

[Rov-42] Carlo Rovelli, "A new approach to quantum gravity", International conference on General Relativity and Cosmology, Goa, India 1988

[Rov-43] Carlo  Rovelli: "Loop space representation" In: "New perspectives in canonical gravity", A Ashtekar Bibliopolis, Naples 1988

[Rov-44] Carlo  Rovelli: "Anomalies in quantum gravity", Physics Letters B187, 88 (1987)

[Rov-45] Carlo Rovelli: "Constraint algebra in General Relativity", Il Nuovo Cimento 92B, 49 (1986)

[Rov-46] Carlo Rovelli: "Ghosts in Gravity theories with a scalar field", Il Nuovo Cimento 78B, 167 (1983)


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