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The Internal and External Problems of String Theory: A Philosophical View

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Abstract

String theory is at the moment the only advanced approach to a unification of all interactions, including gravity. But, in spite of the more than 30 years of its existence, it does not make any empirically testable predictions, and it is completely unknown which physically interpretable principles could form the basis of string theory. At the moment, “string theory” is no theory at all, but rather a labyrinthic structure of mathematical procedures and intuitions. The only motivations for string theory consist in the mutual incompatibility of the standard model of quantum field theory and of general relativity as well as in the metaphysics of the unification program of physics, aimed at a final unified theory of all interactions, including gravity. The article gives a perspective on the problems leading to and resulting from this situation.

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Notes

  1. A systematical introduction to string theory can be found in Polchinski (2000), (2000a), Kaku (1999), Lüst and Theisen (1989) and Green et al. (1987). For more recent developments, see especially Lerche (2000), Schwarz (2000), Dienes (1997) and Vafa (1997). The early development of string theory is reflected in a commented collection of original articles: Schwarz (1985). Greene (1999) gives a recommendable popular introduction.

  2. See also Woit (2002) and Schroer (2006).

  3. Cf. Hedrich (2002) and (2002a).

  4. Cf. Weinberg (1992).

  5. Anyway, if the ambitions of string theory will turn out to be successful or not, the theory and its development will be of eminent interest for historians of science, and especially for sociologists of science.

  6. There are probably many more good reasons for a philosophical examination of string theory than those mentioned in the following.

  7. But a philosophical investigation of incomplete theories like string theory or even the complete field of approaches to quantum gravity has to be much more flexible and open-minded than the traditional methodologies of philosophy of science might suggest:

    ‘Quantum gravity' primarily refers to an area of research, rather than a particular theory of quantum gravity. Several approaches exist, none of them entirely successful to date. Thus the philosopher's task, if indeed she has one, is different from what it is when dealing with a more-or-less settled body of theory such as classical Newtonian mechanics, general relativity, or quantum mechanics. In such cases, one typically proceeds by assuming the validity of the theory or theoretical framework and drawing the ontological and perhaps epistemological consequences of the theory, trying to understand what it is that the theory is telling us about the nature of space, time, matter, causation, and so on. Theories of quantum gravity, on the other hand, are bedeviled by a host of technical and conceptual problems, questions, and issues which make them unsuited to this approach. However, philosophers who have a taste for a broader and more open-ended form of inquiry will find much to think about. (Weinstein (2005) 2)

  8. String theory reproduces general relativity as well as gauge invariances, possibly those of the standard model of quantum field theory, as low-energy approximations; general relativity comes with the phenomenologically correct parameter values, if the string length and tension are assumed to lie in the order of magnitude of the Planck scale: a further indication for the context of quantum gravity.

  9. At the time of their discovery it wasn't completely clear for string theorists that the dynamics of spin-2 states necessarily lead to the reproduction of general relativity and of the phenomenology of gravitation as a low-energy approximation:

    […] with appropriate caveats, general relativity is necessarily recovered as the low-energy-limit of any interacting theory of massless spin-2 particles propagating on a Minkowski background, in which the energy and momentum are conserved […]. (Butterfield and Isham (2001) 59)

    […] a general result that goes back to Feynman: any theory of an interacting spin two massless particle must describe gravity. So string theory must reproduce gravitational physics. (Giddings (2005) 6)

    Weinberg (1995), in his discussion of covariant quantum gravity, showed that, in the vacuum case, one can derive the equivalence principle and general relativity from the Lorentz-invariance of the spin-2 quantum field theory of the graviton: the spin-2 theory is equivalent to general relativity and follows from the quantum theory. The upshot of this is that any theory with gravitons is a theory that can accommodate general relativity (in some appropriate limit). This analysis forms the basis of string theory's claim that it is a candidate theory of quantum gravity: since there is a string vibration mode corresponding to a massless spin-2 particle, there is an account of general relativity […]. (Rickles (2005) 9)

  10. Additionally, also by pure chance, it promises to give a nomologically unified description of all interactions. But without the least experimental results which would make a unified theory necessary, the only concrete motivation for a unification (beside metaphysical considerations) consists in the mutual incompatibility of general relativity and quantum theory; and the desire to overcome this incompatibility is sufficient at best as a motivation for a conceptual unification: a theory of quantum gravity or quantum geometry like e.g. Loop Quantum Gravity (see below). It does not make necessary a nomological unification of all interactions in the sense of string theory.

  11. The equivalence of inertial and gravitational mass is one of the fundaments of general relativity. General relativity presupposes this empirical fact in the determination of its theory structure. It does not explain it in any way. But a fundamental theory should explain why the inertial mass (a quantity of motion) is equivalent to the gravitational mass (the "charge" of gravitation).

  12. String theory does at least give an explanation for the existence and for the number of particle generations. The latter is determined by the topology of the compactified additional spatial dimensions of string theory; their topology determines the structure of the possible oscillation spectra. The number of particle generations is identical to half the absolute value of the Euler number of the compact Calabi-Yau topology. But, because it is completely unclear which topology should be assumed for the compact space, there are no definitive results. This ambiguity is part of the vacuum selection problem; there are probably more than 10100 alternative scenarios in the so-called string landscape (see below). Moreover all concrete models, deliberately chosen and analyzed, lead to generation numbers much too big. (There are phenomenological indications that the number of particle generations can not exceed three. String theory admits generation numbers between three and 480.)

  13. Cf. Hedrich (2006) as well as (2005), (2005a) and (2007a). Cf. also Banks et al. (2003); Dine (2004); Douglas (2003), Susskind (2003), (2004) and (2005).

  14. Cf. Hedrich (2006).

  15. Cf. Smolin (2001), (2003), (2005), Rovelli (1998), (2004) and Ashtekar (2005).

  16. Cf. Hedrich (2006).

  17. As Lee Smolin tells us, this is actually no great surprise:

    But is unification enough of a criteria to pick out the right theory? By itself it cannot be, for there are an infinite number of symmetry algebras which have the observed symmetries as a subalgebra. (Smolin (2005) 26)

  18. Cf. Hedrich (1990), (1995), (1998), (1998a), (1999), (2001) and (2002b).

  19. Cf. Cartwright (1994), (1999) as well as (1983) and (1989).

  20. Considerations with regard to the Bekenstein-Hawking entropy of black holes lead to the idea that a discrete structure (and finite information densities) should be assumed for the most fundamental level of nature. Cf. Bekenstein (2000) and (2001), Jacobson (1999), Jacobson et al. (2005) and Sorkin (2005). In the context of quantum gravity, the mathematics of the continuum could well lead to theoretical artifacts without any descriptive content.

  21. See above.

  22. Cf. Bekenstein (2000), Bousso (2002), Horava (1999), Markopoulou and Smolin (1999), Susskind (1995), ‘t Hooft (1993) and (2000).

  23. Cf. Wheeler (1979), (1983) and (1989).

  24. Cf. Nielsen (1983), Frogatt and Nielsen (1991) and Nielsen et al. (1994).

  25. A survey of the spectrum of already existing alternative approaches to quantum gravity can be found in Rovelli (1998).

  26. Cf. Rovelli (2004).

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Correspondence to Reiner Hedrich.

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Research for this paper was supported by the Deutsche Forschungsgemeinschaft under the project FA 261/7-1 "Vereinheitlichung in der Physik durch den Superstring-Ansatz: Wissenschaftstheoretische und naturphilosophische Analyse". Preliminary research was carried out during my stay (January–April 2002) as a Visiting Fellow at the Center for Philosophy of Science of the University of Pittsburgh. I am grateful to both institutions as well as to Brigitte Falkenburg.

More details are to be found in Hedrich (2007).

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Hedrich, R. The Internal and External Problems of String Theory: A Philosophical View. J Gen Philos Sci 38, 261–278 (2007). https://doi.org/10.1007/s10838-007-9048-3

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