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- 1. Ioan Muntean Indiana University-Purdue University, Fort Wayne & University of Notre Dame http://imuntean.net 1
- 2. Motivations for this talk I. Assert the role and importance of principles in quantum gravity (esp. in string theory) II. Endorse the “model-based” interpretation of string theory. III. Discuss the relation between dualities, unification and ontological fundamentalism 2
- 3. Some proposals III. Dualities as relations among models A duality principle is a general conjecture about relations among string models 3
- 4. Prospective results Duality and unification: Dualities are weaker than unification. Albeit not unificatory or explanatory in nature, dualities can: Accommodate Integrate Predict theoretical facts about the dual model. Duality is not a consequence or a condition of unification Deflationary result: Perhaps dualities are consequences of other assumptions, such as supersymmetry Dualities and structure: Dualities may unveil a deeper structure, with or without a deeper theory. (Rickles) Duality and (reductive) fundamentalism in string theory are inconsistent 4
- 5. What is a principle in physics? Several possible answers. Philosophers envisaged principles as: a) meta-laws (Lange) b) constraints on physical models of a theory c) “methodological maxims” d) as axioms (of an axiomatized system, e.g. classical mechanics. Carnap) e) as “correspondences” between the abstract mathematical formalism and the concrete empirical phenomena (Reichenbach) f) Bold conjectures, but which are not proven to be false Although all a)-f ) are germane to physics, I take here f ) as the most representative for quantum gravity. All in all, principles play a foundational role in the development and advancement of a theoretical discipline Q: In what does lie the foundation of a theory T? A: The principles that ground it. 5
- 6. Three philosophical stances How do principles ground a theory? A Principle monism: a theory T should be grounded in one principle P B Principle pluralism: a theory T is grounded in a number of independent principles {P1….Pn} C Principle deflationism: grounding a theory one principle, in certain set of principles, in a completely different set, or in no principle at all is a matter of choice. I will focus on B here. 6
- 7. A. From the ideal to reality B. Greene (1999): “Is string theory itself an inevitable consequence of some broader principle—possibly but not necessarily a symmetry principle—in much the same way that the equivalence principle inexorably leads to general relativity or that gauge symmetries lead to the nongravitational forces?” “String theory is missing a core principle” and string theorists are all “in a position analogous to an Einstein bereft of the equivalence principle” (Greene, 1999, p. 171). “there is […] no guarantee that such a fundamental principle exists, but the evolution of physics during the last hundred years encourages string theorists to have high hopes that it does.” L. Smolin (2002, 149) “string theory in its present form most likely has the same relationship to its ultimate form as Kepler's astronomy had to Newton's physics". The ideal: one symmetry principle may become The principle of quantum gravity Some candidates: “the holographic principle” `t Hooft (1993); Susskind (1995); Susskind & Lindesay (2004). The duality principle 7
- 8. Principle monism and unification Greene (2011, p. 82): “a theory based on vibrating filaments might not seem to have much in common with general relativity’s curved spacetime picture of gravity. Nevertheless, apply string theory’s mathematics to a situation where gravity matters but quantum mechanics doesn’t […] and out pop Einstein’s equations. Vibrating filaments and point particles are also quite different. But apply string theory’s mathematics to a situation where quantum mechanics matters but gravity doesn’t […] and the math of string theory morphs into the math of quantum field theory”. 8
- 9. C. Conventionalism and principles Any theory T can be grounded in a set of principles, but one can freely choose what set of conjectures constitutes the principles and what are mere consequences Example: the history of General Relativity. You can reformulate GR with no principle at all. The independent principles that ground a theory can change dramatically in time. Compare and contrast: a system in logic where we can interchange axioms with theorems and still get the same expressiveness of the system 9
- 10. B. Principle pluralism and problems The most obvious problem for the pluralist is the consistency of principles. Other less dramatic is completeness and independence Typical clash of principles in quantum gravity: principles of relativity (e.g. equivalence principle) and principles of quantum theory (superposition) Enticing results: Quantum backreaction as a logical result (inconsistency) The Weinberg-Witten (1980) theorem Semi-classical models of system are not coherent (Peres&Terno 2001) Typical solution: create a new theory with new principle(s) that explain(s) gravitation as a low energy limit of the new theory. Non-typical solution: postulate dualities that would relate classical and quantum regimes of different models. 10
- 11. First claim of this presentation Dualities do alleviate such a clash of principles between theories Dualities are conceptually weaker than unification, but can replace the unificatory power when unification is simply a “bridge too far” 11
- 12. What is a duality? Dualities can be both underestimated and overestimated. Trivial dualities are simply notational variants “Germane” dualities are inter-theoretical relations Claim: they are very rich philosophically but understudied in philosophy. Notable exceptions: E. Castellani (2009), J. Cushing (1990), Muntean (2013), D. Rickles (2009, 2011, 2013) There are several dualities in classical EM and in QFT. A duality relates two theories (or models) such that: i. One theory is “more classical” than the other (viz. “more quantum”) ii. One theory is better known than the other iii. One theory has a better explanation / prediction than the other iv. One theory is weakly coupled, the other is strongly coupled 12
- 13. Triviality and non-triviality Dualities can relate two completely different theories of the same physical system Or two completely different systems described by two different theories The classical and the quantum description of the same system A classical system to an another, quantum system A gauge theory to a gravitational theory A string theory to another string theory The weak coupling regime of a theory to its strong regime Note: Mathematical dualities or logical dualities are only remotely related to ours Equivalence between the category of sets and the category of complete atomic Boolean algebras. Conjunction and disjunction are dual 13
- 14. A classical duality A dual theory is obtained by a “duality transformation”: ur ur ur ur E B;BE It is a rotation duality in the complex vector field E+iB In QED, this symmetry signals the existence of magnetic monopoles (g). They attract each other with a force of greater than the force between two electrons Duality explanation: If there are magnetic poles, the electric charge is quantized, because: eg = 2n; 14 2 2 137
- 15. Lessons for Maxwell The “dual invariance” of the classical EM theory. ur ur ur If we define E= then Maxwell equations are: E And they are invariant to these transformation Here the conserved charge is: ur ur E = E= 15 E iB 0 i c t ur ur ur E E ia e ( ) ia q ig e q ig
- 16. Duality and explanation From the dual invariance of EM, Dirac (1931) inferred: 1. The quantization of electrical charge from 2. The existence of magnetic monopoles 1 is a strong result. It could not be explained by other means (except by a 5D compactification mechanism, Klein 1926) The other option is to take 1 as a “brute fact” of the universe 16 qg 2 mh
- 17. Bosonization of fermions in 1D For some range of the coupling constant, bosons are more useful as fundamental particle than fermions For some range of constants, fermions are fundamental. By turning the couple constants, one becomes more fundamental than the other. Does this show a common structure? 17
- 18. E-M duality and unification The dual invariance is strongly related to unification The 4-vector unifies the E and B fields. There is a deeper structure than what ordinary Maxwell equations unveil Maxwell equations are very rich in symmetries and consequently in dualities (compare to general relativity) Bianchi identity Gauge transformation Conformal symmetries (Baterman Cunningham 1909) Dual invariance Other types of symmetries NOT present in EM: Diffeomorphism Supersymmetry (SUSY) 18 μν F
- 19. The E/M duality in particle parlance EM duality relates weak and strong coupling of the same theory. In one regime, α<<1, the electron charge is “weak” compared to its dynamics, i.e. it does not interact with its own field. Electrons “barely” radiate photons. The electric field is weak. They are hard to excite (can we excite an electron?) There are magnetic poles, but they have large fields around them, they are heavy. The poles are hard to separate, spread out, composite, solitonic excitations. Structurally, they are very rich. At α>>1, the poles are fundamental and charges are heavy and rich. Charges are heavier Monopoles are more elementary Either the charge is elementary and the poles are composite, OR the poles are elementary and charges are composite. At α=1 there is no fundamentality strictly speaking. All depends on the coupling constant α. See a philosophical discussion in Castellani 2009, Rickles 2011. 19
- 20. E-M duality in QFT The EM duality as explained before does not survive quantization, but new dualities arise Montonen&Olive (1977): At different coupling constants, electric and magnetic charges exchange roles. 20 Magnetic charge = topological charge Noetherian charge The magnetic monopoles = solitons Magnetic monopoles = elementary gauge fields = elementary particles Gauge fields=solitons Weak Strong Strong Weak
- 21. Perturbative string theory The perturbative formulation of string “theory” contains the highest number of idealizations: strings are weakly coupled, the number of strings is relatively small and strings vibrate against a flat, fixed background spacetime (background dependence). The interaction term is a perturbation of the non-interactive dynamics All are problematic idealizations, but the third is the most outrageous and infamous. 21
- 22. Background dependence as the major drawback of string theory “in general, string theory, and other background-dependent approaches, are […] examples of how not to go about constructing a theory of quantum gravity” (Rickles French 2006) background independence and structuralism are “well-matched bedfellows” Can we interpret string theory as a structural metaphysics even if it is not (yet) background independent? Alternatives: talk about the promises of a background independent string theory The concept of background independence needs more philosophical work. Dualities may play the central role of smooting out some of these consequences of the idealizations 22
- 23. Whacked by the GR community “What is very frustrating is that […] string theory does not seem to fully incorporate the basic lesson of GR, which is that space and time are dynamical rather than fixed, and relational rather than absolute […] all that happens is that some strings move against this fixed background and interact with one another.” (Smolin 2001, 159) Penrose, Stachel, Woit and others would agree. But QFT is doing the same (basically) 23
- 24. Non-perturbative string theory A more realistic model of strings would assume: strings interact string that can split and join (create/annihilate strings) They have enough energy to interact with spacetime itself. In the strong coupling regime, the backreaction with spacetime is assumed and the background is not anymore fixed. 24
- 25. A naïve solution to background independence Curved spacetime is reducible to a collection of gravitons Given the Witten-Weinberg (no-go result), gravitons cannot be reduced to any known bosons, gravitation is simply not “yet another quantum field theory” Therefore you need something like strings. As gravitons are states of strings, the covariant part of the M space in string theory is in itself dynamical. 25
- 26. Four interpretations [1] A collection of theories, most likely all being aspects of a more fundamental theory (“M-theory"), which is ultimately the theory of everything (TOE) of our reality. Other theories in physics can be ultimately reduced to the TOE. [2] A collection of mathematical models of strings and branes vibrating in various types of spaces, having different symmetries and properties. These string models represent aspects of known interactions in physics: gravitation, gauge theories, black holes, etc. [3] A collection of conjectures about the relations among the string models. Some of these string models may (or may not) represent real interactions in the world; [4] A collection of conjectures about the relations between string models (as in [2]) and other theories in physics: gauge theories, gravitation, black hole thermodynamics, information theory, etc. I have some reasons to adopt [3] here and to keep an eye on [4] 26
- 27. String theories Type Spacetime dimensio ns SUSY generators chiral open strings heterotic compactific ation gauge group tachyon Bosonic (closed) 26 N = 0 no no no none yes Bosonic (open) 26 N = 0 no yes no U(1) yes I 10 N = (1,0) yes yes no SO(32) no IIA 10 N = (1,1) no no no U(1) no IIB 10 N = (2,0) yes no no none no HO 10 N = (1,0) yes no yes SO(32) no HE 10 N = (1,0) yes no yes E8 × E8 no M-theory ? 11 N = 1 no no no none no 27
- 28. Five string models There are some relevant string models in D=10 1. I: SUSY, open and closed strings, group symmetry SO(32) 2. IIA: SUSY, open and closed strings, non-chiral fermions. D-branes are the boundaries of open strings 3. IIB: SUSY, open and closed strings, chiral fermions. D-branes are the boundaries of open strings 4. Type II: 5. HO: Heterotic model: SUSY, closed strings only, right moving strings and left moving strings differ, symmetry group SO(32) 6. HE: Heterotic model: SUSY, as above, but symmetry group is E8xE8 Dualities: 1/4: I and HO 2/5: IIA and HE Witten’s conjecture (1995): all string models are related to each other by dualities. 28
- 29. Interpretation of string models 1 Empiricist: If this string model were true, what physics would look like? 2. What are the realist commitments of these models? 29
- 30. Dualities Dualities are not: Approximations of a theory by another theory Correspondences (as in “theory correspondence”) Notation variants of the same theory It is not (always) a symptom of a gauge freedom But There is representational ambiguity in dualities “Dualities can highlight which features of our ontological picture are not fundamental”. (Rickles 2011) and They may or may not point towards a deeper structure Can be related to the symmetries of a theory Hence their importance to underdetermination and scientific realism 30
- 31. A definition of dualities (Vafa) A theory is characterized by a moduli space M of the coupling constants In this space we have several regions conventionally designated as weak coupling and strong coupling. A physical system can be represented in various places of the moduli space with various observables: Q[M,O ] Two physical systems Q and Q’ are dual if: M M ';O O ' i.e. there is an isomorphism between their moduli spaces and another between their observables 31
- 32. Duality symmetries In string theory, the moduli space is very rich. Each model is characterized by a “moduli space” formed by the constants of the theory. The string coupling constant gs The topology of the manifold Other fields in the background Everything (?) The flow of theories is important but more complicated than in QFT In this moduli space a duality symmetry can relate: The weak coupling region of T1 with the strong coupling region of T2 The weak coupling region of T to the weak coupling region of the same T Interchange elementary quanta with solitons (collective excitations) Exchange what is fundamental with what is composite 32
- 33. Some known dualities 33
- 34. S-duality, informally gs is the coupling constant in string theory. But in string theory gs is a field, has its equation, changes from place to place A string can split into two strings. The probability if this process is ≈ gs. The D0-brane and “strings” can interchange the role of fundamental entity. When gs is small f-strings are fundamental, light, hard to split. D0-branes or D-1 are complicated and heavy. Their excitations are the particles (photons, gravitons) When D0 branes are light, f-string are heavier When gs=1 , f-strings and D-branes look the same. And topology is part of the moduli space. We can change dimensionality or topology as we walk in the moduli space. See (Sen 2002) for details 34
- 35. T-dualities T1 and T2 can be different (different symmetries, gauge invariants, topologies), or can be the same theory (self-duality)! Weak Weak T1 T2 35
- 36. A standard S-duality map This is not a “self-duality”. T1 and T2 are structurally different (different symmetries, gauge invariants, topologies) They can be string models or other models in QCD etc. Weak Weak T1 T2 Strong Strong 36
- 37. Rickles on dualities and unification Rickles 2011: “the dualities point to the fact that the five consistent superstring theories, that were believed to be distinct entities, are better understood as different perturbative expansions of some single, deeper theory.” (aka M-theory) For Rickles and some string enthusiasts, the five points in moduli space are representations of a single M-theory. This is the received view in the community. But…. 37
- 38. Dualities without unification He admits that “the duality relationships between the points in moduli space will hold independently of the existence of an M-theory” Rickles 2012 “in order to achieve a computable scheme for the whole of the moduli space (including regions away from the distinguished 'perturbation-friendly' points) such an underlying theory is required” Rickles 2011, my emphasis My argument is that we do not need M-theory to see the deep structure of the S-dualities One on my assumptions is to analyze these as models, not as theories Conceptually I separate the discussion on dualities from the discussion on unification 38
- 39. Non-perturbative string theory and S-duality If we knew how to relate the weak coupling to the strong coupling we would relate the non-perturbative physics to the perturbative physics. In the ADS/CFT duality this is the path to a background independent theory. 39
- 40. Dual nature of duality and conclusions We use the weak coupling sector as a calculation device, but we trust the strong coupling sector in respect of its ontology and its structure If duality is isomorphic there is no remainder in the dual strong sector that cannot be explained away from the weak coupling sector If we take dualities seriously we may need to investigate a new type of explanation based on dualities. We may want to take a throughout look at string models related to dualities (and not to “string theories”) 40
- 41. AdS/CFT duality We start from a IIB theory (D=10, SUSY with N=4, open and closed strings) (Maldacena 1998) but esp (Klebanov 2002) This theory has a 16 supersymmetric group, the smallest algebra in D=10. We take Nc parallel D3-branes close one to the other. If Ncgs ≪ 1, the low coupling regime, closed strings live in empty space and the open strings end on the D-branes and describe excitations of the D-branes. Open and closed strings are decoupled from each other. 41
- 42. Strong coupling and the YM When Ncgs ≫ 1 the gravitational effect of the D-branes on the spacetime metric is important, leading to a curved geometry and to a “black brane”. But near the horizon the strings are redshifted and have low energy. Gauge theory exists and the physics on the D-branes is nothing else than a gauge theory The physics is now described by a Yang-Mills CFT with gYM=4πgs 42
- 43. Maldacena’s duality conjecture The 4D, N=4 SUSY SU(Nc) gauge theory (Yang Mills) is dual to a IIB string theory with the AdS/CFT boundary. 43
- 44. Polchinski on branes “We start with strings in a flat background and discover that a massless closed string state corresponds to fluctuations of the geometry. Here we found first a flat hyperplane, and then discovered that a certain open string state corresponds to fluctuations of its shape. We should not be surprised that the hyperplane has become dynamical.” “Thus the hyperplane is indeed a dynamical object, a Dirichlet membrane, or D-brane for short. The p-dimensional D-brane, from dualizing 25-p dimensions, is a Dp-brane. In this terminology, the original U(n) open string theory contains n D25-branes.” A D25-brane fills space, so the string endpoint can be anywhere: it just corresponds to an ordinary Chan-Paton factor. 44
- 45. Current research (post 2003) People try to falsify the AdS/CFT. No success yet More opportunistically, others try to understand black holes through the duality as hot gases of fermions Others try to look for more realistic dualities: dS/CFT Dualities with no SUSY, but with holographic principle (T. Banks). Understand hadrons with the IIB theory (“the boomerang kid”) 45
- 46. Explanation/unification/prediction in AdS/CFT? Explanation? I do not see it Unification? Not directly, without the M-theory. Prediction? Perhaps, if we relate it to the Higgs mechanism Computational advantages? Yes, definitely! Is the holographic principle doing any explanatory/unification/predictive work? 46
- 47. Emergence in string theory Duality, when correctly interpreted, may illuminate the classical-quantum relation People believe spacetime emerges in AdS/CFT (Seiberg, Koch Murugan) In S-duality the non-perturbative aspects emerge from the perturbative aspects but we compute in the perturbative sector. Some believe the curved spacetime of general relativity is a holographic emergent construct from a quantum gauge field without gravity rather than a fundamental feature of reality. Strong emergence Any spacetime emerges from any gauge theory Weak emergence: A special type of spacetime (the IIB gravitation) emerges from a special type of gauge theory (SUSY YM in D=4) Witness we assume here that: “If a theory does not have general covariance, then the theory lacks an underlying spacetime” 47
- 48. Deflationism: ex SUSY quodlibet sequitur SUSY can be a great candidate for the deeper structure underlying S-dualities and the AdS/CFT duality. SUGR is also present in the string sector This is a an argument from symmetry There are dS/CFT dualities but no non-SUSY dualities. SUSY is part of the CFT Yang-Mills story because it is the limit of a IIB theory. I am not pluralist about SUSY, but I am pluralist/functionalist about spacetime (and background independence) 48
- 49. But symmetry is not enough in AdS/CFT In AdS/CFT we do not have a symmetry in the moduli space. It is not like a gauge orbifold in the moduli space There is something else than symmetry here. 49
- 50. References Bousso, R. (2002). The holographic principle. Reviews of Modern Physics, 74(3), 825–874. doi:10.1103/RevModPhys.74.825 Bueno, O., French, S., & Ladyman, J. (2002). On Representing the Relationship between the Mathematical and the Empirical. Philosophy of Science, 69(3), 497–518. Callender, C., & Huggett, N. (2001). Why quantize gravity (or any other field for that matter)? Philosophy Of Science, 68(3), S382–S394. Cappelli, A., Colomo, F., Di Vecchia, P., & Castellani, E. (Eds.). (2012). The Birth of String Theory. Cambridge University Press. Cartwright, Nancy, & Frigg, R. (2006). String Theory under Scrutiny. Physics World, 20(September), 14–15. Castellani, E. (2009). Dualities and intertheoretic relations. In M. Suarez, M. Dorato, & M. Redei (Eds.), Launch of the European Philosophy of Science Association. Springer. Retrieved from http://philsci-archive.pitt.edu/4679/ Dawid, R. (2007). Scientific Realism in the Age of String Theory. Physics & Philosophy, (11). Dawid, Richard. (2006). Underdetermination and Theory Succession from the Perspective of String Theory. Philosophy of Science, 73(3), 298. doi:dx.doi.org/10.1086/515415 Dawid, Richard. (2009). On the Conflicting Assessments of the Current Status of String Theory. Philosophy of Science, 76(5), 984–996. doi:10.1086/605794 Dirac, P. a. M. (1931). Quantised Singularities in the Electromagnetic Field. Proceedings of the Royal Society of London. Series A, 133(821), 60–72. doi:10.1098/rspa.1931.0130 50
- 51. References 2 Frisch, M. (2011). Principle or Constructive Relativity. Studies in History and Philosophy of Modern Physics, 42(3), 176– 183. Greene, B. (1999). The elegant universe : superstrings, hidden dimensions, and the quest for the ultimate theory. New York: W. W. Norton. Greene, B. (2011). The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos. Knopf. Gross, D., Henneaux, M., & Sevrin, A. (Eds.). (2007). The Quantum Structure of Space and Time. World Scientific Pub Co Inc. Hedrich, R. (2011, January 4). String Theory – Nomological Unification and the Epicycles of the Quantum Field Theory Paradigm. Preprint. Retrieved February 21, 2011, from http://philsci-archive.pitt.edu/8443/ Hooft, G. ’. (1974). Magnetic monopoles in unified gauge theories. Nuclear Physics B, 79(2), 276–284. doi:10.1016/0550- 3213(74)90486-6 Klein, O. (1926/1981). Quantum theory and five dimensional theory of relativity. In Modern Kaluza-Klein Theories. Menlo Park: Addison Wesley. Ladyman, J. (2007). Does Physics Answer Metaphysical Questions? Royal Institute of Philosophy Supplements, 61, 179– 201. doi:10.1017/S1358246107000197 Lange, M. (2007). Laws and Meta-Laws of Nature: Conservation Laws and Symmetries. Studies in History and Philosophy of Modern Physics, 38(3), 457–481. Morrison, M. (2007). Where have all the theories gone? Philosophy of Science, 74(2), 195–228. Norton, J. D. (1993). General Covariance and the Foundations of General Relativity: Eight Decades of Dispute. Reports of Progress in Physics, 56, 791–861. Penrose, R. (2005). The Road to Reality: a Complete Guide to the Laws of the Universe. New York: A.A. Knopf. Polchinski, J. (1996). String duality. Reviews of Modern Physics, 68(4), 1245–1258. doi:10.1103/RevModPhys.68.1245 51
- 52. References 3 Rickles, D. (2008). Quantum Gravity: A Primer for Philosophers. In D. P. Rickles (Ed.), The Ashgate Companion to Contemporary Philosophy of Physics (Ashgate Pub Co., pp. 262–365). Aldershot, Hants, England ; Brookfield, Vt., USA. Retrieved from http://philsci-archive. pitt.edu/5387/ Rickles, D. (2010). Mirror Symmetry and Other Miracles in Superstring Theory. Foundations of Physics, Online First, 1–27. doi:10.1007/s10701- 010-9504-5 Rickles, D. (2011). A philosopher looks at string dualities. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 42(1), 54–67. doi:10.1016/j.shpsb.2010.12.005 Rickles, D. (2013-forthcoming). AdS/CFT duality and the emergence of spacetime. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, online first. Retrieved from http://www.sciencedirect.com/science/article/pii/S1355219812000433 Rickles, D. P., French, S., & Saatsi, J. T. (Eds.). (2006). The Structural Foundations of Quantum Gravity. Oxford: Clarendon Press. Seiberg, N. (2006). Emergent Spacetime. hep-th/0601234. Retrieved from http://arxiv.org/abs/hep-th/0601234 Sen, A. (1998). An Introduction to Non-perturbative String Theory. arXiv:hep-th/9802051. Retrieved from http://arxiv.org/abs/hep-th/9802051 Smolin, L. (2002). Three roads to quantum gravity. New York, N.Y.: Basic Books. Smolin, L. (2006). The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. Boston, New York: Houghton Mifflin. Teh, N. (2013-forthcoming). Holography and Emergence. Thiemann, T. (2009). Loop Quantum Gravity. In D. Oriti (Ed.), Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter (pp. 169–186). Cambridge University Press. Retrieved from http://arxiv.org/abs/gr-qc/0602037 Veneziano, G. (1968). Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A, 57(1), 190–197. Weinberg, S., & Witten, E. (1980). Limits on massless particles. Physics Letters B, 96(1-2), 59–62. doi:16/0370-2693(80)90212-9 Weingard, R. (1988). A Philosopher Looks at String Theory. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 2, 95–106. Weisberg, M. (2007). Three Kinds of Idealization. Journal of Philosophy, 104(12), 639–659. Witten, E. (2005). Emergent Phenomena in Condensed Matter and Particle Physics. In Conference in honor of Sidney Coleman. Harvard University. Woit, P. (2006). Not even wrong: the failure of string theory and the search for unity in physical law. New York: Basic Books. Wüthrich, C. (2005). To quantize or not to quantize: fact and folklore in quantum gravity. Philosophy of Science, 72, 777–788. 52

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