String Theory III

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time." (John D. Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"A five-dimensional space is not a strange deformation of ordinary space, one that only mathematicians can see, but a place where numbers are collected in ordered sets. When string theorists talk of the eleven dimensions required by their latest theory, they are not encouraging one another to search for eight otherwise familiar spatial dimensions that have somehow become lost. They are saying only that for their purposes, eleven numbers are needed to specify points. Where they are is no one’s business." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005) 

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"String theory was not invented to describe gravity; instead it originated in an attempt to describe the strong interactions, wherein mesons can be thought of as open strings with quarks at their ends. The fact that the theory automatically described closed strings as well, and that closed strings invariably produced gravitons and gravity, and that the resulting quantum theory of gravity was finite and consistent is one of the most appealing aspects of the theory." (David Gross, "Einstein and the Search for Unification", 2005)

"Like many a maturing beauty, string theory has gotten rich in relationships, complicated, hard to handle and widely influential. Its tentacles have reached so deeply into so many areas in theoretical physics, it’s become almost unrecognizable, even to string theorists." (K C Cole, "The Strange Second Life of String Theory", Quanta Magazine", 2016) [source] 

"String theory today looks almost fractal. The more closely people explore any one corner, the more structure they find. Some dig deep into particular crevices; others zoom out to try to make sense of grander patterns. The upshot is that string theory today includes much that no longer seems stringy. Those tiny loops of string whose harmonics were thought to breathe form into every particle and force known to nature (including elusive gravity) hardly even appear anymore on chalkboards at conferences." (K C Cole, "The Strange Second Life of String Theory", Quanta Magazine", 2016) [source]

On Manifolds V (Geometry III)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. [...] Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. [...] the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)