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Omega Point (Tipler)

The Omega Point is a term used by Tulane University professor of mathematics and physics Frank J. Tipler to describe what he maintains is a necessary cosmological state in the far future of the universe. According to his Omega Point Theory, as the universe comes to an end at a singularity in a particular form of the Big Crunch, the computational capacity of the universe is capable of increasing at a sufficient rate that this computation rate is accelerating exponentially faster than the time running out. In principle, a simulation run on this universal computer can thus continue forever in its own terms, even though the universe lasts only a finite amount of proper time. Prof. Tipler states this theory requires that the current known laws of physics are true descriptions of reality, and it requires there be intelligent civilizations in existence at the appropriate time to exploit the computational capacity of such an environment.

Tipler identifies this final singularity and its state of infinite information capacity with God. The implication of this theory for present-day humans is that this ultimate cosmic computer will be able to run computer simulations of all intelligent life that has ever lived, by recreating simulations of all possible quantum brain states within the master simulation. This would manifest as a simulated reality. From the perspective of the recreated inhabitant, the states near the Omega Point would represent their resurrection in an infinite-duration afterlife, which could take any imaginable form due to its virtual nature.

Assuming that achieving the Omega Point is physically possible, Tipler says this would be accomplished by "downloaded" human consciousness in tiny quantum computers that could exponentially explore space, many times faster than biological human beings. Tipler argues that the incredible expense of keeping humans alive in space implies that flesh-and-blood humans will never personally travel to other stars. Instead, highly efficient uploads of human minds ("mind children" as Tipler calls them, they being the mental uploads of our descendants; the term is perhaps from a 1988 book of the same name by Hans Moravec) and artificial intelligences will spread civilization throughout space. This should start as early as 2100. Small spaceships under constant heavy acceleration could reach nearby stars in less than a decade. In one million years, these intelligent von Neumann probes would have completely consumed the Milky Way galaxy. In 100 million years, the Virgo Supercluster would be colonized. From that point on, the entire visible universe would be engulfed by these "mind children" as it approaches the point of maximum expansion.

History of the Omega Point Theory

Prof. Tipler has published his Omega Point Theory in a number of peer-reviewed scientific journals since 1986.[1][2][3][4][5][6][7] The first book wherein the Omega Point Theory was described was 1986's The Anthropic Cosmological Principle, written by astrophysicist John D. Barrow and Tipler.[8] The first book solely concentrating on the Omega Point Theory was Tipler's The Physics of Immortality in 1994.[9]

Physicist David Deutsch (who in 1985 pioneered the field of quantum computation by being the first person to formulate a specifically quantum computational algorithm[10]) in his 1997 book The Fabric of Reality defends the physics of Tipler's Omega Point Theory in Chapter 14: "The Ends of the Universe" (of which chapter concentrates mainly on the Omega Point Theory):[11]

I believe that the omega-point theory deserves to become the prevailing theory of the future of spacetime until and unless it is experimentally (or otherwise) refuted. (Experimental refutation is possible because the existence of an omega point in our future places certain constraints on the condition of the universe today.)

Deutsch later comments within a concluding paragraph of the same chapter regarding the synthesis of his "four strands" of fundamental reality, which includes the strengthened version of mathematician Alan Turing's theory of universal computation in the form of the Omega Point Theory:

It seems to me that at the current state of our scientific knowledge, this is the 'natural' view to hold. It is the conservative view, the one that does not propose any startling change in our best fundamental explanations. Therefore it ought to be the prevailing view, the one against which proposed innovations are judged. That is the role I am advocating for it. I am not hoping to create a new orthodoxy; far from it. As I have said, I think it is time to move on. But we can move to better theories only if we take our best existing theories seriously, as explanations of the world.

In 2007 Prof. Tipler's book The Physics of Christianity was published, which analyzes the Omega Point Theory's pertinence to Christian theology.[12] In the book Tipler identifies the Omega Point as being the Judeo-Christian God, particularly as described by Christian theological tradition, e.g., that the Omega Point cosmology when formulated in multiversal terms (of which multiverse conception isn't necessary for the physics upon which the Omega Point itself is based) is fundamentally triune in its structure: the Final Singularity (i.e., the Omega Point), the All-Presents Singularity (which Tipler states exists at all times at the edge of the multiverse), and the Initial Singularity (i.e., the beginning of the Big Bang), which Tipler identifies with the Father, the Son and the Holy Spirit, respectively (successively, the First, Second and Third Persons of the Trinity). In this book Tipler also analyzes how Jesus Christ could have performed the miracles attributed to him in the New Testament without violating any known laws of physics, even if one were to assume that we currently don't exist on a level of implementation in a computer simulation (in the case that we did, then according to Tipler such miracles would be trivially easy to perform for the society which was running the simulation, even though it would seem amazing from our perspective).

The Physics of Christianity shows a change from Prof. Tipler's earlier position within The Physics of Immortality regarding theism and Christianity. In the opening paragraph of Chapter XII: "The Omega Point and Christianity" of The Physics of Immortality, Tipler wrote the following:

To emphasize the scientific nature of the Omega Point Theory, let me state here that I am at present forced to consider myself an atheist, in the literal sense that I am not a theist. (A-theist means "not theist.") I do not yet even believe in the Omega Point. The Omega Point Theory is a viable scientific theory of the future of the physical universe, but the only evidence in its favor at the moment is theoretical beauty, for there is as yet no confirming experimental evidence for it. Thus scientifically one is not compelled to accept it at the time of my writing these words. So I do not. [Antony] Flew, among others, has in my opinion made a convincing case for the presumption of atheism. If the Omega Point Theory and all possible variations of it are disconfirmed, then I think atheism in the sense of Flew, Hume, Russell, and the other self-described atheists is the only rational alternative. But of course I also think the Omega Point Theory has a very good chance of being right, otherwise I would never have troubled to write this book. If the Omega Point Theory is confirmed, I shall then consider myself a theist.

Tipler now regards himself as a theist due to what he states have been advancements in his Omega Point Theory which occurred after the publication of The Physics of Immortality.[13][14] Namely, Tipler now says the known laws of physics—specifically, quantum mechanics, general relativity, the second law of thermodynamics, and the Standard Model of particle physics—require the existence of the Omega Point singularity in order to avoid their violation;[6][5][4] whereas in The Physics of Immortality Tipler investigated what would be necessary from the postulate that life continues forever while still keeping the analysis confined to the known laws of physics. Tipler states that these physical laws have been repeatedly confirmed by every experiment to date. According to Tipler, this constitutes a massive body of empirical evidence for the Omega Point Theory's correctness. And as indicated above, Tipler also now considers himself a Christian due to his identification of the Omega Point with the God of Christian theological tradition.

A diagram of the multiverse formulation of the Omega Point.

Outline of the physics of the Omega Point Theory

According to Prof. Tipler from a 2005 paper[6] in the journal Reports on Progress in Physics, he outlines the following reasons why he maintains the universe must end in the Omega Point in order for the known laws of physics (i.e., unitarity, the second law of thermodynamics, the Bekenstein bound, and general relativity) to be mutually consistent at all times:

Astrophysical black holes almost certainly exist, but Hawking[15] and Wald[16] have shown that if black holes are allowed to exist for unlimited proper time, then they will completely evaporate, and unitarity will be violated. Thus, unitarity requires that the universe must cease to exist after finite proper time, which implies that the universe has spatial topology S3.[17] The Second Law of Thermodynamics says the amount of entropy in the universe cannot decrease, but Ellis and Coule[18] and I[19] have shown that the amount of entropy already in the CMBR will eventually contradict the Bekenstein Bound near the final singularity unless there are no event horizons, since in the presence of horizons the Bekenstein Bound implies the universal entropy S ≤ constant × R2, where R is the radius of the universe, and general relativity requires R → 0 at the final singularity. If there are no horizons then the (shear) energy density can grow as R−6 which means that the total available energy grows as (R−6 ) R3 ~ R−3, and so the Bekenstein Bound yields E R ~ (R−3)R ~ R−2 which diverges as R−2 as R → 0 at the final singularity.[19][5] The absence of event horizons by definition means that the universe's future c-boundary is a single point,[20] call it the Omega Point. MacCallum[21] has shown that an S3 closed universe with a single point future c-boundary is of measure zero in initial data space. Barrow,[22][23] Cornish and Levin[24] and Motter[25] have shown that the evolution of an S3 closed universe into its final singularity is chaotic. Yorke et al[26][27] have shown that a chaotic physical system is likely to evolve into a measure zero state if and only if its control parameters are intelligently manipulated. Thus life (≡intelligent computers) almost certainly must be present arbitrarily close to the final singularity in order for the known laws of physics to be mutually consistent at all times. Misner[28][29][30] has shown in effect that event horizon elimination requires an infinite number of distinct manipulations, so an infinite amount of information must be processed between now and the final singularity. The amount of information stored at any time diverges to infinity as the Omega Point is approached, since S → +∞ there, implying divergence of the complexity of the system that must be understood to be controlled.

Some have pointed out that the current acceleration of the universe's expansion due to the positive cosmological constant would appear to obviate the Omega Point.[31] On this matter Prof. Tipler states[32] (see also references[6] and[5]) that baryon annihilation—which he says would be the ideal form of energy resource and rocket propulsion during the colonization of the universe—will force the Higgs field to its absolute vacuum state, resulting in the universe's collapse:

The SM provides such a mechanism, which I actually discussed in the last section of the Appendix for Scientists in (,[9] p. 515). This mechanism is the creation/destruction of baryon number by electroweak quantum tunneling. (Baryons are the heavy particles made up of quarks. Examples are neutrons and protons.) In my book, I pointed out that this mechanism would be ideal for propelling interstellar spacecraft, but I did not discuss its implications for the Higgs vacuum, a serious oversight on my part. (An oversight which invalidates the second part of my Fifth Prediction on page 149 of.[9]) If the SM is true—ALL experiments conducted to date indicate that it is (e.g.[33] and,[34] last full paragraph on p. 35)—then the net baryon number observed in the universe must have been created in the early universe by this mechanism of electroweak quantum tunneling. If the baryons were so created, then this process necessarily forces the Higgs field to be in a vacuum state that is not its absolute vacuum. But if the baryons in the universe were to be annihilated by this process, say by the action of intelligent life, then this would force the Higgs field toward its absolute vacuum, canceling the positive cosmological constant, stopping the acceleration, and allowing the universe to collapse into the Omega Point. Conversely, if enough baryons are not annihilated by this process, the positive cosmological constant will never be canceled, the universe will expand forever, unitarity will be violated, and the Omega Point will never come into existence. Only if life makes use of this process to annihilate baryons will the Omega Point come into existence.

The Omega Point and the quantum gravity Theory of Everything

In his 2005 paper[6] in the journal Reports on Progress in Physics, Prof. Tipler maintains that the correct quantum gravity theory has existed since 1962, first discovered by Richard Feynman in that year,[35] and independently discovered by Steven Weinberg and Bryce DeWitt, among others. But, according to Tipler, because these physicists were looking for equations with a finite number of terms (i.e., derivatives no higher than second order), they abandoned this qualitatively unique quantum gravity theory since in order for it to be consistent it requires an arbitrarily higher number of terms.[36] "They also did not realize that the correct quantum gravity theory is consistent only if a certain set of boundary conditions are imposed ...," writes Tipler (which includes the initial Big Bang, and the final Omega Point, cosmological singularities).[6] Tipler says that the equations for this theory of quantum gravity are term-by-term finite, but the same mechanism that forces each term in the series to be finite also forces the entire series to be infinite (i.e., infinities that would otherwise occur in spacetime, consequently destabilizing it, are transferred to the cosmological singularities, thereby preventing the universe from immediately collapsing into nonexistence). Tipler writes that "It is a fundamental mathematical fact that this [infinite series] is the best that we can do. ... This is somewhat analogous to Liouville's theorem in complex analysis, which says that all analytic functions other than constants have singularities either a finite distance from the origin of coordinates or at infinity."[37]

From the aforesaid Reports on Progress in Physics paper,[6] Prof. Tipler elaborates on the mathematics and physics of this issue, in part explained below:

So basic quantum field theory quickly forces upon us the general invariant action

This is the qualitatively unique gravitational Lagrangian picked out by quantum mechanics. Physicists do not like it because (1) it has an infinite number of (renormalizable) constants c^i_j, all of which must be determined by experiment and (2) it will not yield second order differential equations which all physicists know and love. But the countable number of constants are in effect axioms of the theory, and I pointed out in an earlier section that the Löwenheim–Skolem theorem suggests there is no real difference between a theory with a countable number of axioms and a theory with a finite number of axioms. The finite case is just easier for humans to deal with, provided the 'finite' number is a small number. Further, as Weinberg[38] has emphasized, this Lagrangian generates a quantum theory of gravity that is just as renormalizable as QED and the SM.

Since quantum field theory itself is forcing the Lagrangian (3) on us, I propose that we accept the judgement of quantum mechanics and accept (3) (and the countable number of additional terms involving the non-gravitational fields interacting with the hμν) as the actual Lagrangian of reality.

Donoghue[39] and Donoghue and Torma[40] have shown that Lagrangian (3) will not contradict experiment provided the (renormalized) values of the infinite number of new coupling constants are sufficiently small. ...

One consequence of the above Lagrangian being the true description of quantum gravity, explains Prof. Tipler, would be that so long as one is within spacetime, then one can never obtain a complete description of quantum gravity and hence of physics: there will always be infinitely more to learn and discover in the field of physics, including by requiring the use of experiment.[36] He says that physics will be able to become ever-more refined, knowledgeable and precise, but never complete (i.e., within spacetime). Only at the final singularity of the Omega Point (which is not in spacetime[20]) will the full description of physics be obtained, states Tipler.

In the same aforestated journal article, Prof. Tipler combines the above theory of quantum gravity with an extended Standard Model in order to form what he maintains is the correct Theory of Everything (TOE) describing and unifying all the forces in physics.[6][12]

Implications from string theory

If string theory is valid, it would seem to contradict the Omega Point Theory, since the Omega Point Theory requires the existence of a cosmological singularity at the end of time. Whereas, according to string theory, singularities do not actually exist because no material object can be compressed below the Planck length.[41]

See also

* John D. Barrow

* David Deutsch

o The Fabric of Reality, a book by David Deutsch.

* digital physics

* eschatology

* grand unification theory (GUT)

* quantum gravity

* resurrection of the dead

* simulated reality

* supertask

* teleology

* theory of everything (TOE)

* unified field theory

Physics books dealing with the Omega Point Theory

* Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247. Chapter I and excerpt from Chapter II. Chapter I also available here.

* David Deutsch, The Fabric of Reality: The Science of Parallel Universes—and Its Implications (London: Allen Lane The Penguin Press, 1997), ISBN 0713990619. Extracts from Chapter 14: "The Ends of the Universe," with additional comments by Frank J. Tipler; also available here and here.

* Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0198519494. 56-page excerpt available here.

* John D. Barrow and Frank J. Tipler, Foreword by John A. Wheeler, The Anthropic Cosmological Principle (Oxford: Oxford University Press, 1986), ISBN 0198519494. Excerpt from Chapter 1.


  1. ^ Frank J. Tipler, "Cosmological Limits on Computation," International Journal of Theoretical Physics, Vol. 25, No. 6 (June 1986), pp. 617-661. (First paper on the Omega Point Theory.)
  2. ^ Frank J. Tipler, "Achieved spacetime infinity," Nature, Vol. 325, No. 6101 (January 15, 1987), pp. 201-202.
  3. ^ Frank J. Tipler, "The ultimate fate of life in universes which undergo inflation," Physics Letters B, Vol. 286, Issues 1-2 (July 23, 1992), pp. 36-43.
  4. ^ a b Frank J. Tipler, "The Ultimate Future of the Universe, Black Hole Event Horizon Topologies, Holography, and the Value of the Cosmological Constant," arXiv:astro-ph/0104011, April 1, 2001. Published in Relativistic Astrophysics: 20th Texas Symposium, Austin, TX, 10-15 December 2000, edited by J. Craig Wheeler and Hugo Martel (Melville, N.Y.: American Institute of Physics, 2001), ISBN 0735400261; and in AIP Conference Proceedings, Vol. 586 (October 15, 2001), pp. 769-772.
  5. ^ a b c d Frank J. Tipler, "Intelligent life in cosmology," International Journal of Astrobiology, Vol. 2, Issue 2 (April 2003), pp. 141-148; also available here. See also here and here. Also at arXiv:0704.0058, March 31, 2007.
  6. ^ a b c d e f g h F. J. Tipler, "The structure of the world from pure numbers," Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964. (Note: citation formatting in the above-quoted passages have been modified for clarity. Typographical errors in the third quoted passage have been corrected, again for clarity.) See also here. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything," arXiv:0704.3276, April 24, 2007.
  7. ^ Frank J. Tipler, Jessica Graber, Matthew McGinley, Joshua Nichols-Barrer and Christopher Staecker, "Closed Universes With Black Holes But No Event Horizons As a Solution to the Black Hole Information Problem," arXiv:gr-qc/0003082, March 20, 2000. Published in Monthly Notices of the Royal Astronomical Society, Vol. 379, Issue 2 (August 2007), pp. 629-640.
  8. ^ John D. Barrow and Frank J. Tipler, Foreword by John A. Wheeler, The Anthropic Cosmological Principle (Oxford: Oxford University Press, 1986), ISBN 0198519494. Excerpt from Chapter 1.
  9. ^ a b c Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0198519494.
  10. ^ D. Deutsch, "Quantum theory, the Church-Turing principle and the universal quantum computer," Proceedings of the Royal Society of London; Series A, Mathematical and Physical Sciences, Vol. 400, No. 1818 (July 1985), pp. 97-117. Also available here. See also here.
  11. ^ David Deutsch, The Fabric of Reality: The Science of Parallel Universes—and Its Implications (London: Allen Lane The Penguin Press, 1997), ISBN 0713990619. Extracts from Chapter 14: "The Ends of the Universe," with additional comments by Frank J. Tipler; also available here and here.
  12. ^ a b Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247. Chapter I and excerpt from Chapter II. Chapter I also available here.
  13. ^ Sam Vincent Meddis, "Computers of the distant future," USA Today, four parts, August 3-31, 1998. Parts 1, 2, 3 and 4. See Part 1 concerning Tipler no longer being an atheist.
  14. ^ Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247; Chapter III: "Life and the Ultimate Future of the Universe," pg. 62.
  15. ^ S. W. Hawking, "Breakdown of predictability in gravitational collapse," Physical Review D, Vol. 14, Issue 10 (November 1976), pp. 2460-2473.
  16. ^ Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago: University of Chicago Press, 1994), ISBN 0226870251, Section 7.3, pp. 182-185.
  17. ^ John D. Barrow, Gregory J. Galloway and Frank J. Tipler, "The closed-universe recollapse conjecture," Monthly Notices of the Royal Astronomical Society, Vol. 223 (December 1986), pp. 835-844.
  18. ^ G. F. R. Ellis and D. H. Coule, "Life at the end of the universe?," General Relativity and Gravitation, Vol. 26, No. 7 (July 1994), pp. 731-739.
  19. ^ a b Frank J. Tipler, The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead (New York: Doubleday, 1994), ISBN 0198519494, Appendix C. "The Bekenstein Bound," pg. 410. Said Appendix is reproduced in Frank J. Tipler, "Genesis: How the Universe Began According to Standard Model Particle Physics," arXiv:astro-ph/0111520, November 28, 2001, Section 2. "Apparent Inconsistences in the Physical Laws in the Early Universe," Subsection a. "Bekenstein Bound Inconsistent with Second Law of Thermodynamics."
  20. ^ a b S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (London: Cambridge University Press, 1973), ISBN 0521200164, pp. 217-221. See also here, here, here and here.
  21. ^ Malcolm A. H. MacCallum, "On the mixmaster universe problem," Nature—Physical Science, Vol. 230 (March 1971), pp. 112-3. See also here, here and here.
  22. ^ John D. Barrow, "Chaotic behaviour in general relativity," Physics Reports, Vol. 85, Issue 1 (May 1982), pp. 1-49.
  23. ^ John D. Barrow and Janna Levin, "Chaos in the Einstein-Yang-Mills Equations," Physical Review Letters, Vol. 80, Issue 4 (January 1998), pp. 656-659. Also at arXiv:gr-qc/9706065, June 20, 1997.
  24. ^ Neil J. Cornish and Janna J. Levin, "Mixmaster universe: A chaotic Farey tale," Physical Review D, Vol. 55, Issue 12 (June 1997), pp. 7489-7510. Also at arXiv:gr-qc/9612066, December 30, 1996.
  25. ^ Adilson E. Motter, "Relativistic Chaos is Coordinate Invariant," Physical Review Letters, Vol. 91, Issue 23, Art. No. 231101 (December 2003), four pages. Also at arXiv:gr-qc/0305020, December 7, 2003.
  26. ^ Troy Shinbrot, Edward Ott, Celso Grebogi and James A. Yorke, "Using chaos to direct trajectories to targets," Physical Review Letters, Vol. 65, Issue 26 (December 1990), pp. 3215-3218.
  27. ^ Troy Shinbrot, William Ditto, Celso Grebogi, Edward Ott, Mark Spano and James A. Yorke, "Using the sensitive dependence of chaos (the 'butterfly effect') to direct trajectories in an experimental chaotic system," Physical Review Letters, Vol. 68, Issue 19 (May 1992), pp. 2863-2866.
  28. ^ Charles W. Misner, "The Isotropy of the Universe," Astrophysical Journal, Vol. 151 (February 1968), pp. 431-457.
  29. ^ Charles W. Misner, "Quantum Cosmology. I," Physical Review, Vol. 186, Issue 5 (October 1969), pp. 1319-1327.
  30. ^ Charles W. Misner, "Mixmaster Universe," Physical Review Letters, Vol. 22, Issue 20 (May 1969), pp. 1071-1074.
  31. ^ Although see Lawrence M. Krauss and Michael S. Turner, "Geometry and Destiny," General Relativity and Gravitation, Vol. 31, No. 10 (October 1999), pp. 1453-1459. Also at arXiv:astro-ph/9904020, April 1, 1999.
  32. ^ Frank Tipler, "The Omega Point and Christianity," Gamma, Vol. 10, No. 2 (April 2003), pp. 14-23; note that the foregoing version corrects character formatting errors of the versions available here, here and here. (Note: citation formatting in the above-quoted passage has been modified for clarity.) For the version in Dutch, see "Het Punt Omega en het christendom," Gamma, Jrg. 10, Nr. 2 (April 2003), pp. 14-23; also available here.
  33. ^ Frank Wilczek, "Scaling Mount Planck III: Is That All There Is?," Physics Today, Vol. 55, Issue 8 (August 2002), pp. 10-11. See also here and here.
  34. ^ Helen R. Quinn, "The Asymmetry Between Matter and Antimatter," Physics Today, Vol. 56, Issue 2 (February 2003), pp. 30-35. See also here and here.
  35. ^ Richard P. Feynman, edited by Brian Hatfield, notes taken by Fernando B. Morinigo and William G. Wagner, The Feynman Lectures on Gravitation (Reading, Mass.: Addison-Wesley, 1995), ISBN 0201627345.
  36. ^ a b Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, pp. 34-35. Chapter I and excerpt from Chapter II. Chapter I also available here.
  37. ^ Frank J. Tipler, The Physics of Christianity (New York: Doubleday, 2007), ISBN 0385514247, pp. 49 and 279. Chapter I and excerpt from Chapter II. Chapter I also available here.
  38. ^ Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations (Cambridge: Cambridge University Press, 1995), ISBN 0521550017, pp. 499 and 518-519.
  39. ^ John F. Donoghue, "General relativity as an effective field theory: The leading quantum corrections," Physical Review D, Vol. 50, Issue 6 (September 1994), pp. 3874-3888. Also at arXiv:gr-qc/9405057, May 25, 1994.
  40. ^ John F. Donoghue and Tibor Torma, "Power counting of loop diagrams in general relativity," Physical Review D, Vol. 54, Issue 8 (October 1996), pp. 4963-4972. Also released as "On the power counting of loop diagrams in general relativity," arXiv:hep-th/9602121, February 22, 1996.
  41. ^ Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (New York, N.Y.: W. W. Norton, 1999), ISBN 0393046885, pp. 252-253.


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