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Strings and Fundamental Interactions

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Gravity, Strings and Particles

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Abstract

Why do we find in Nature only four different types of fundamental interactions (corresponding to gravitational, electromagnetic, strong, and weak nuclear forces)? And why do these forces behave just in the way we know, namely why does the electric field obey Coulomb’s law, the gravitational field Newton’s law, and so on for the other interactions?

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Notes

  1. 1.

    We will try to be as accurate and as detailed as possible, but we will limit here to a very qualitative introduction to the existing models of strings and superstrings. The readers interested in a deeper study of the physical and mathematical aspects of such models are referred to the textbooks [39–42] listed in the references.

  2. 2.

    It is the set whose elements are the possible quantum states of the considered physical system.

  3. 3.

    The rules telling us how the action of a product of operators changes if we reverse the order of the factors.

  4. 4.

    Using the invariance under general coordinate transformations we can further eliminate the “longitudinal” oscillations, and we are eventually left only with “transverse” oscillations, i.e., oscillations along the spatial directions orthogonal to the string itself.

  5. 5.

    It is the same energy already mentioned at the beginning of Sect. 3.3.

  6. 6.

    The reader interested in the exact expression of the mass spectrum for a bosonic string, and on its explicit dependence on D and N, is referred to the books [39–42] listed in the final bibliography.

  7. 7.

    As shown in the works of Ramond [44], and of Neveau and Schwarz [45].

  8. 8.

    Namely, considering the levels associated with states of vector or tensor type, noting that those states are “transverse,” that the number of independent vector components is D − 2, and imposing on those states to be massless.

  9. 9.

    The acronym GSO is due to the names of the authors, Gliozzi et al. [46].

  10. 10.

    Each spinor field can be always decomposed in two chiral components, called “right-handed” and “left-handed” (or “positive” and “negative”). These components represent opposite physical configurations in which the intrinsic angular momentum of the spinor is, respectively, parallel or antiparallel to the direction of motion. The description of a massive spinor field requires the inclusion of both chirality components. The spectrum of the superstring, however,contains massive fermions even after the selection produced by the GSO rule: they are obtained by combining together different spinors of opposite chirality, which are always present inside the massive energy levels of the spectrum.

  11. 11.

    An antisymmetric tensor field of rank n is a bosonic object represented by a variable with n indices which behaves as a tensor under the coordinate transformations of the ten-dimensional space–time, and which is characterized by the following property: it changes sign whenever any two of its indices are exchanged between themselves. In the case of an antisymmetric tensor of rank 3, for instance, we have: \(A_{\alpha \beta \gamma } = -A_{\beta \alpha \gamma } = -A_{\gamma \beta \alpha } = -A_{\alpha \gamma \beta }\).

  12. 12.

    We say that a model is affected by a “quantum anomaly” when a symmetry, which is present in the classical version of the model, is violated by the quantization procedure and thus disappears in the quantized version of the model.

  13. 13.

    It is the group representing all possible rotations that can be performed within a Euclidean space with 32 dimensions.

  14. 14.

    The name, derived from the ancient Greek language, means that we are combining things which are different and seemingly incompatible.

  15. 15.

    Fields of this type are also called “Weyl–Majorana spinors.”

  16. 16.

    The E n group is the group representing all possible translations and rotations that can take place in an n-dimensional Euclidean space.

  17. 17.

    As we shall see in Sect. 5.4.1, such a coupling constant is proportional to the exponential of the dilaton field, and is thus inverted if the sign of the dilaton is changed.

  18. 18.

    The two types of strings are actually related by a so-called duality transformation (see Sect. 5.5).

  19. 19.

    It is a compact space which has vanishing Ricci curvature, and which can be described by a complex geometric structure parametrized by three real and three imaginary coordinates.

  20. 20.

    The product of these three symmetry groups is at the ground of the so-called standard model of all fundamental interactions. The corresponding gauge fields are the photon (associated to the U 1 group), the carrier of electromagnetic interactions; the three vector bosons \({W}^{+},{W}^{-},{Z}^{0}\) (associated to the SU 2 group), the carriers of weak interactions; and the eight gluons (associated to the SU 3 group), the carriers of strong interactions.

  21. 21.

    The sector of string states with spin higher than 2 is still a largely unexplored field of research (with the exception of the precious work of a few pioneers). The importance of such higher-spin configurations and the need for their further and deeper study have been repeatedly stressed in particular by Sagnotti (see for instance his recent review paper [47]).

  22. 22.

    The Yang–Mills equations are the analogous of the Maxwell equations, written however for a non-Abelian gauge field. Unlike Maxwell’s equations they are equations of non-linear type.

  23. 23.

    This type of symmetry, which is at the core of the theory of special relativity, imposes on the electromagnetic equations to keep the same form in all inertial frames (whose coordinates are connected, indeed, by a Lorentz transformation).

  24. 24.

    They are also called “Weyl transformations.”

  25. 25.

    See for instance the textbooks [39–42] for an explicit and detailed introduction to such model.

  26. 26.

    It is the usual perturbative expansion into a series of quantum loop corrections, referred, however, to the sigma model, i.e., to a two-dimensional field theory defined on the string world sheet.

  27. 27.

    Concerning the correct physical interpretation of L S see, in particular, Sect. 5.5.

  28. 28.

    The operator representing partial derivative has dimensions of the inverse of a length. Given that the quantum corrections of order \(L_{S}^{2}\) produce field equations containing the square of the differential operator, the quantum corrections of order \(L_{S}^{4}\) will produce equations containing the fourth power of the differential operator, and so on.

  29. 29.

    Topology is the science providing a quantitative description of the global geometric property of a space, such as compactness, connectedness, continuity, and boundary.

  30. 30.

    It is defined by the integral over the whole world-sheet of a term proportional to the curvature of the world-sheet surface. The result is a constant number χ which is independent on the particular choice of coordinates, and is determined by the topological genus only.

  31. 31.

    We are considering, in particular, the contribution of a given topological configuration to the so-called total “partition function” which controls the string propagation from an initial to a final state. Such a contribution is inversely proportional to the exponential function of the Euclidean action which includes all possible string interactions.

  32. 32.

    If the dilaton is not a constant e ϕ still plays the role of a local effective coupling.

  33. 33.

    The letter T refers to the name “Target space,” which is the name used in mathematics to denote the external space in which the string is embedded.

  34. 34.

    As shown for the first time in the papers by Kikkawa and Yamasaki [48] and by Sakai and Senda [49].

  35. 35.

    As shown by the works of Tseytlin [50] and Veneziano [51].

  36. 36.

    This result characterizes what is known as “the second superstring revolution,” which took place in the mid 1990. There is also what is called “the first superstring revolution,” which dates back to the 1980, and which corresponds to the transition from string theory intended as a theory of strong interactions (with a string length of the order of the nuclear radius, L S  ∼ 10−13 cm) to a supersymmetric string theory, intended as a unified theory of all fundamental interactions (including gravity, and with a string length of the order of the Planck radius, L S  ∼ 10 L P  ∼ 10−32 cm).

  37. 37.

    The M of the name has a number of possible interpretations: it may refer to Monster (“monster theory”), or to Mother (“mother of all theories”), or to Membrane (“membrane theory”). This last interpretation is due to the fact that, switching from 10 to 11 dimensions, and adding a new spatial dimension to a one-dimensional object like a string, one obtains a two-dimensional extended object: a membrane.

  38. 38.

    As shown in a work by Witten [52]. It was just that work that, in practice, gave the start to the second string revolution.

  39. 39.

    See, e.g., the paper by Bousso and Polchinski [53].

  40. 40.

    This possible initial configuration has been originally suggested and studied by Brandenberger and Vafa [54].

  41. 41.

    For a direct check of the fact that the winding energy induces the contraction of the spatial geometry we should explicitly solve the cosmological equations describing the gravitational field produced by the considered gas of strings (see e.g., the paper by Tseytlin and Vafa [55]). Without resorting to such technical procedures, however, we can intuitively imagine that the strings wrapped around the compact dimensions behave as a “noose” which tends to tighten and to squeeze the space, counteracting its natural tendency to expansion.

  42. 42.

    It is difficult, however, to get a graphic visualization of such a situation because, to do that, we should imagine the three-dimensional compact hypersurface embedded in an external four-dimensional space (which is impossible for the ability of our mind).

  43. 43.

    If this is the case, the primordial Universe is characterized by the presence of a gas of “branes.” Such a brane-gas scenario is discussed, for instance, in a paper by Alexander et al. [56].

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  1. Dipartimento di Fisica, Università di Bari, Bari, Italy

    Maurizio Gasperini

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Gasperini, M. (2014). Strings and Fundamental Interactions. In: Gravity, Strings and Particles. Springer, Cham. https://doi.org/10.1007/978-3-319-00599-7_5

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