Geometry, Nonlinear Analysis and Theoretical Physics

Transcription

1 Geometry, Nonlinear Analysis and Theoretical Physics Harvard University August,

2 The purpose of this talk is to demonstrate a close connection between three great subjects mentioned in the title. The best way to do so is to explain how our concept of spacetime evolved in the history and how geometers and analysts helped the physicists to pave the way. 2

3 Einstein: Subtle is the Lord, but malicious he is not. Geometric problems that reveal true nature of space-time are always exciting and rewarding, no matter how difficult they are. 3

4 In the past three thousand years, our concept of space-time evolves according to our understanding of nature. Geometry is the basic tool for such investigations. It is often hard to distinguish geometry from physical nature. Over the years, development of geometry has shown deep influence in our understanding of space-time. On the other hand, new concept of space-time always gives breakthroughs in geometry. 4

5 In ancient days, people believed that space was static and flat. This was partially due to the limitation of our understanding of geometry. We shall describe how our concept of spacetime evolves according to our understanding of geometry. 5

6 Euclidean Geometry. Aristotle ( BC) Euclid ( BC) Aristotle thought that there are four basic elements that made up the universe while Democritus proposed the theory of atoms. In mathematics, axioms for geometry was formulated by Euclid. Such a unique development in geometry is perhaps based on the belief of the Greek philosopher that the building blocks of nature should be elegant and simple. 6

7 There are two most basic theorems: 1 Pythagoras theorem: For the right angle triangle, c 2 = a 2 + b 2 c a b 2 The sum of inner angles of triangle is π. The first theorem is the most important statement of geometry. Up to now, modern geometry demands this statement to be true infinitesimally. The second theorem is a statement that amounts to the fact that the plane is flat and has no curvature. 7 It is equivalent to the following statement which was observed by Legendre.

8 Euclid s Fifth Postulate: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines if produced indefinitely meet on that side on which the angles are less than two right angles. Gauss ( ), Bólyai ( ) and Nikolai Ivanovich Lobachevski ( ) independently discovered hyperbolic geometry: two dimensional surfaces with constant negative curvature. This discovery led to the solution that the Euclid fifth postulate is independent of the other postulates. 8

9 In addition to its contribution to the foundation of mathematics, the discovery of hyperbolic geometry shows us a new abstract geometry that is not intuitively clear in day to day life. In hyperbolic geometry, the sum of inner angles mines π is equal to the curvature of the surface times its area of the triangle. This was generalized by Gauss to an integral formula of curvature. This formula is called Gauss-Bonnet formula and plays fundamental role in modern geometry and topology as it relates local quantity (curvature) to a global topological quantity (the Euler number). 9

10 Analytic geometry An important breakthrough in geometry is due to Rene Descartes whose invention of analytic geometry allows geometric figures to be described by Cartesian coordinate. Geometry and Algebra is unified through the introduction of coordinate system. It would be difficult to solve classical problems such as trisecting an angle without using algebra (through Galois theory). Analytic geometry allows us to consider geometric figures without using the classical axioms of Euclid. Only with coordinate system, we can visualize and calculate higher dimensional geometric figures. 10

11 Calculus and nonlinear analysis There is a great deal of limitation for geometric objects built solely by planes and spheres. However, the concept is completely changed once we know how to construct curved geometric objects by infinite process of approximation. This was invented by Archimedes who, for example, computed areas cut by straight lines with parabola. This infinite process is the seed for the invention of Calculus which was achieved by Newton and Leibniz much later. 11

12 Combination of analytic geometry and calculus allow Newton and astronomers to do extensive calculation for motion of celestial bodies. Description of these bodies is based on a single coordinate system. Newton thought that space is static and time is independent of space. 12

13 Newton( ) Leibniz( ) Newton (Philosophiae Naturalis Principia Mathematica): Absolute space, in its own nature and with regard to anything external, always remains similar and unmovable. Relative space is some movable dimension or measure of absolute space, which our senses determine by its position with respect to other bodies, and is commonly taken for absolute space. Newton s idea was challenged by Leibniz ( ). 13

14 The great success of calculus and Newtonian mechanics kept physicists and mathematicians busy until the nineteenth century. Euler was the major contributor to geometry in this period. He was the founder of calculus of variation and combinatorial topology. Newtonian mechanics initiated the study of nonlinear dynamics of a system of ordinary differential equations. Many of the deep questions are still unsolved up to now. Euler derived equations of motions of fluid, equations for soap bubbles. They became the foundations of modern nonlinear analysis. 14

15 Gauss and Riemannian geometry When classical geometers (such as Euler) described a surface in three space, they found two directions at each point where curvatures should be measured. For example, on the cylinder, one is the direction along the circle and the other is along the straight line. Gauss (1827) found that the product of these two curvatures has remarkable property. It is the same even if we deform the surface as long as we do not stretch it. It is called the Gauss curvature. Gauss found a beautiful formula for the curvature in terms of the metric of the surface. The nonlinear expression is a building block for modern nonlinear analysis. 15

16 Intrinsic geometry of Gauss: Gauss published this theorem in his Disquisitiones generales circa superficies curvas where he distinguished the inner properties of a surface, that is the geometry experienced by small flat bugs living in the surface, from its outer properties, the way that it embeds in a higher dimensional space. He said that this inner property is most worthy of being diligently explored by geometers. The inner geometry is what we called intrinsic geometry. 16

17 Lobachevsky ( ) Gauss noticed that when the Gauss curvature is a positive constant the surface is a piece of the sphere. When the Gauss curvature is minus one, it is a piece of the hyperbolic surface, which was developed by Lobachevsky and himself. 17

18 Gauss: I am becoming more and more convincing that the necessity of our geometry cannot be proved, at least not by human reason nor for human reason. Perhaps in another life we will be able to obtain insight into the nature of space which is now unattainable. Gauss: Until then we must place geometry not in the same class with arithmetic which is purely a priori, but with mechanics. 18

19 Riemannian geometry: While Gauss described his geometry for two dimensional surface. it was Georg Friedrich Bernhard Riemann ( ) who presented intrinsic geometry for higher dimensional manifolds in his Goöttingen inaugural lecture: Über die Hypothesen, welche der geometrie zu Grunde liegen. Riemann formally introduced the concept of an abstract space which is defined by some infinitesimal form of a metric. The concept of Gauss curvature was then given by a proper meaning. This is an important event as we have finally formulated the concept of a space with no reference to flat Euclidean (linear) space. 19

20 In his address, Riemann mentioned the influence of Newtonian mechanics and physics on his thinking in the formulation of the abstract space. Riemann s new discovery did radically change mathematician s view of geometry. It was followed by Christoffel, Ricci, Levi-Civita, Beltrami. They developed calculus on manifolds: tensor calculus. 20

21 When Riemann created his geometry, he had some idea of its relation to Newtonian Mechanics and he knew that any meaningful geometric quantities should be independent of choice of coordinate systems. The concept of Riemann curvature tensor was introduced for that purpose. Nowadays we say that the group of diffeomorphism provides gauge symmetry for geometry. 21

22 Relativity: Einstein s view of spacetime Einstein E. Mach The first constructive criticism on Newtonian absolute space was by Austrian Philosopher Ernst Mach( ). He put forth the hypothesis that there is influence of the mass of the earth and the other celestial bodies to determine the inertial frames. This is called Mach s principle. 22

23 Poincare ( ) Lorentz ( ) Minkowski ( ) By 1905, special relativity was found, by Einstein (with contribution by Poincare, Lorentz and Minkowski). A very important fact is that space and time are unified (under Lorentz transformation). In 1908, Minkowski said that from now on, we cannot discuss space and time independently. 23

24 However, Newton s basic concept of gravity, action at a distance, is inconsistent with the basic principle of special relativity that information cannot travel faster than light. In 1907, Einstein introduced the principle of equivalence of gravitation and inertia. He realized the importance of relative motion to gravity and there is similarity of gravity with electricity and magnetism. 24

25 Gravity has the effect to make physical body accelerate. According to special relativity, when we measure length, it will stretch according to the velocity if the measurement is parallel to the velocity vector. However, the length will not change if the direction is perpendicular to the velocity vector. The changes of the metric according to direction and point gives rise to a metric tensor according to Riemannian geometry. 25

26 Grossmann ( ) Einstein tried for about ten years to combine this fundamental concept of general relativity with Newton s theory of gravity. One of the reason it took some time for him to achieve such a goal was his lack of knowledge of mathematics of abstract space. Until he was informed by his friend Grossmann, he did not realize the powerful meaning of Riemann s curvature tensor, which he later used to describe gravitational force. The equivalence principle of physics requires the laws of gravity to be independent of choice of coordinate systems. 26

27 The quantity given by Riemann has exactly this property. Trace of the curvature tensor is called Ricci tensor which was developed by Ricci. Einstein found that he could construct from the Ricci tensor the kind of tensor he needed to satisfy the classical conservation law of matter. (Bianchi found the identity that Einstein needed.) So the abstract space of Riemann is exactly suitable for description of gravity. (Hilbert found the Hilbert action, which is the integral of scalar curvature, to be the action principle behind the Einstein equation.) 27

28 The gravitational field must be identified with the 10 components of the metric tensor of Riemannian space-time geometry. He presented to the Prussian Academy of Sciences a series of papers in which he worked out the metric tensor and calculated the gravitational deflection of light and the precession of the perihelia of Mercury. This was summarized in the foundation of the general theory of relativity in Annalen der Phys

29 Hence space-time fits beautifully well in the framework of Riemannian geometry. The effect of gravity can be expressed by curvature. Geometry and gravity cannot be distinguished any more. Since space and time is now merged together, the universe is dynamically driven by gravity and is no more static. When celestial bodies move, the geometry and topology of space-time move according to the velocity of light. This solves the puzzle of contradiction arose in Newtonian mechanics and special relativity. 29

30 Symmetry dictates dynamics in physics and mathematics Maxwell Faraday Besides the inspiration of Mach principle, Einstein observed the importance of symmetry in formulating dynamical equations in physics. The fact that Maxwell equations admit Lorentz group as group of symmetry is a foundation for special relativity. The derivation of Einstein equation is based on the invariance of coordinate change. Conservation laws in physics are linked to conserved quantity of continuous30groups of symmetries.

31 S. Lie ( ) F. Klein ( ) E. Cartan ( ) On the other hand, mathematicians realized the importance of symmetries much earlier. S. Lie introduced concept of continuous group. F. Klein outlined his famous Erlargen program in They believed that geometry should be dictated by global group of symmetries. This is a beautiful principle. Except that most manifolds have no global symmetries whatsoever. It was not until E. Cartan s introduction of connection on fiber bundle that this problem was solved. Global symmetries will act on the fiber bundle that is built on the manifold, instead 31 on the manifold itself.

32 Grassmann ( ) In the process, Cartan developed exterior differential calculus based on Grassmann s work on exterior algebra and Frobenius work on integrability conditions for solving differential equations. These works of Cartan initiated modern Gauge theory and modern theory of superspace. 32

33 The revolutionary concept of symmetry had great impacts on the development of geometry. E. Cartan and H. Weyl classified Lie groups, the finite dimensional groups of symmetries. This development has given mathematician a powerful tool to understand detail structure of local geometry. It is remarkable that Cartan found the concept of spinors by analyzing the representations of spin groups, without motivation from quantum mechanics. 33

34 From local geometry to global geometry The basic laws of physics are governed by differential equations and local in nature. Based on such local laws and certain boundary conditions, we can predict global behavior of spacetime, such as the origin of the universe or black hole singularities. The development of geometry followed such a path. The most common path to go from local to global geometry is the global theory of differential equations. 34

35 Lefschetz R. Bott Many important ideas were invented. Most notable is the calculus of variation such as the theory of Morse. The idea to study global behavior of extremal points of a smooth function or a vector field is very powerful. It was used by Poincare, Morse, Hopf, Lefschetz, Bott and Smale to understand global topology of algebraic varieties, symmetric spaces, higher dimensional Poincare conjecture and the classification of smooth manifolds. 35

36 The analysis behind such theory is global theory of ordinary different equations. In the last twenty years, we see a new development on critical point theory based on differential equations of multi-variables. Witten and Floer enriched the theory by including higher dimensional analytic objects and the theory have been powerful in supersymmetric quantum field theory and geometry. 36

37 Many methods were created in constructing global geometric structures over manifolds. As long as the equation is of elliptic type, we have reasonable good hopes. However, when it consists of large nonlinear system of elliptic equations, we still have great difficulties to understand the singular behavior of the solution. 37

38 The idea to dynamically drive geometric object to a desirable one is old but powerful. For example, when we put heat at one point, heat will spread out instantaneously. This idea was used to give elegant proof of the existence of harmonic forms relating analysis to topology of space. We simply put heat along a submanifold and spread heat out to get harmonic forms. Another example is to move rubber band in a manifold. When it is driven by its curvature, it will settle down to a geodesic. 38

39 The concept of renormalization flow is also related to symmetry and it has great import in modern physics and mathematics. D. Friedan was the first one to study it in quantum gravity. In order for the spacetime to admit a conformally invariant sigma model, the Ricci tensor of the spacetime is zero. The flow to such a conformally invariant theory is the Ricci flow which drives a metric along the negative direction of the Ricci tensor. 39

40 Independently, Richard Hamilton was interested in creating Einstein metric by working on the same flow. His work is deep. He did detail analysis on the development of singularities of this flow. It has been one of the most powerful tool in understanding global geometry in the past twenty five years. Part of the theory was used by Perelman to prove the Poincare conjecture, by providing some additional key ideas. The Hamilton-Perelman proof of the Poincare conjecture can be considered as one of the crowning achievement of geometric analysis as was developed by geometers in the past thirty years. 40

41 R. Hamilton G. Perelman There are still many interesting problems in nonlinear dynamics that are far away from being understood. The work of Hamilton-Perelman gave the first nontrivial understanding of singularities naturally created without assuming symmetries of the solution. 41

42 From linear theory to nonlinear theory Joseph Fourier ( ) A very important property of linear equation is the superposition principle: A linear combination of solutions of the equation is still a solution. This principle allows powerful method of integral transform to be used to solve the equation. Fourier s assertion that any wave can be decomposed into basic standing waves is spectacular. Its generalization is influential in practically every branch of 42 mathematics.

43 Pontryagin S. Chern M. Atiyah I. Singer It is amazing that spectral theory of linear operator was developed before the creation of quantum mechanics. Singular integral operator was developed in the fifties to such an extent that linear elliptic system are well understood. The great achievement was the proof of the theorem of Atiyah-Singer where index of an elliptic operator is related to the topology of the operator through the concept of Chern class or Pontryagin classes. 43

44 The achievements in nonlinear theory is much more limited. It is always admirable to see the ability of physicists to write down exact solutions for nonlinear equations. Two notable examples are the KdV equations where detail behavior of solutions were found and the work of Kerr on explicit solution of rotating black hole. However, it is clear that we will not be able to find exact solutions for nonlinear equations in general. Hence perturbation results or numerical solutions have been proposed. 44

45 For perturbation method, a parameter is adjusted to be small in order for the method to work. A recent success in string theory was the ability to find duality between different quantum field theories based on ideas of supersymmetry. It was found that, in many interesting circumstance, theory with large parameter is exactly the same as the perturbation theory of other exactly solvable model. This powerful idea then gives solution to many important quantum field theory with large coupling constants. 45

46 A great triumph of this idea of string theory is a solution of some hundred year old problems in algebraic geometry (mentioned by Hilbert). The counting of number of rational curves in an algebraic manifold had been very difficult. The principle of duality gave a formula and this was finally proved independently by Lian-Liu-Yau and Givental. 46

47 Numerical methods are of course important. But it lacks of full generality in understanding the nonlinear problem. It is often non accurate in general. It should be guided by a priori estimates of the nonlinear equations. This is the basis for theoretical understanding of nonlinear dynamics. 47

48 The success of Hamilton on the Ricci flow depends on his ability to derive a priori estimates. Some of his ideas went back to the work of Li-Yau on heat equation. The idea is to understand solitary wave of the equation and derive inequalities based on quantities constructed from the solitary wave. The philosophy is that among all the solution of a parabolic system, many important inequalities become equalities for certain extremal solutions such as the solitary wave solution. This idea has been successful for many system of parabolic type. 48

49 However, equations in nature usually consist of mixed types. For general nonlinear system, it is a major challenge to identify regions where the type of equations is important. 49

50 Quantum Mechanics and unified theory of Physics The great discovery of quantum mechanics led to understanding of particles in high energy physics. It requires spinors and gauge theory to understand the basic building block of forces in nature. However, these concepts also appeared in geometry. In fact, E. Cartan studied the theory of fiber bundle (twisted space) before physicists. Herman Weyl introduced abelian gauge theory for electromagnetism while Yang-Mills introduced non-abelian gauge theory. Strong and weak interactions are dictated by such gauge 50 theories.

51 The success of quantum field theory also changed our understanding of geometry of space-time. However, there is still great difficulty to understand space-time when the radius is below planck scale. Quantum physics contradicts general relativity in small scale. It is clear that space should not be made up of points only. The very difficult task of quantizing gravity led physicists to build many different models. This is part of the ambitious goal of unifying all forces in nature (a dream that Einstein wanted to accomplish). 51

52 Birth of string theory: Physicist Veneziano found that Γ functions of Euler can be related to functions appeared in describing phenomena of strong force. Afterwards, Nambu, Nielsen and Susskind suggested that if fundamental particles are strings instead of points, Γ function can indeed be derived in the theory of strong interaction. However, the tremendous success of standard model show that this interpretation is not needed. 52

53 M. Green J. Schwarz Thus string theory was quiet for a long time although Scherk and Schwarz did propose string theory to include strong force and gravity in this period. The first breakthrough occured in 1984 when Green and Schwarz found that in a consistent quantized string theory (with supersymmetries), there are only two gauge groups SO(32) and E 8 E 8 and space-time is ten dimensional. With the anomaly cancellation, the field theory converge in one loop term. 53

54 T. Kaluza ( ) O. Klein ( ) Kaluza-Klein model: What we observe in the real world is four dimensional, we need a mechanism to reduce ten dimensional string theory to an effective four dimensional theory. This was provided by Kaluza-Klein who discovered such mechanism right after the discovery of general relativity. Kaluza and Klein looked at the five dimensional space by thickening standard space-time by a circle. A good example is a line thickened to be a circular cylinder. If the circle is very tiny, we may think of that cylinder as a straight line. 54

55 Kaluza-Klein consider the equation for geometry (with no matter) on this thickened space-time and conclude that when the circle is very small. The vacuum Einstein equation on five-dimensional spacetime will give rise to a coupled system of Maxwell equation with the four-dimensional gravitational equation on the four-dimensional space-time. Hence it gives a unified theory of gravity with electricity and magnetism. Einstein thought highly of this theory. Unfortunately there was an extra scalar particle that does not have a good physical meaning at the time when it was introduced. 55

56 Geometry of string theory For a consistent quantized string theory, spacetime is ten-dimensional. We would like to imitate what Kaluza-Klein did by looking at the four-dimensional spacetime thickened with a six-dimensional space. This six-dimensional space would be tiny. String theorists believe that when the energy is very high, there is a symmetry relating bosons and fermions. Such a correspondence is called supersymmetry. It provides parallel spinors on the six-dimensional internal space. 56

57 Calabi-Yau space: The requirement of space-time to be supersymmetric gives a strong constraint on this six dimensional space. It must be a vacuum space with a complex structure. This was discovered by Candelas, Horowitz, Strominger and Witten. It turns out that such space was constructed by me many years ago. This class of space is called Calabi-Yau space. The construction of the Calabi-Yau space is based on analysis of nonlinear elliptic equations where one needs to study detail estimates of the metrics. In the last twenty years, a great deal of activities have been devoted to the study of such space. It has been a major place for interactions for string theory and mathematics. 57

58 Calabi-Yau phenomenology: Calabi-Yau Space Based on the Kaluza-Klein idea, one could compute fundamental constants for particles (Yukawa Coupling) by the topology and complex structure of these manifolds. 58

59 In principle, wave functions ψ(x, y) for Dirac operators on R 3,1 M are written as ψi (x) ψ i (y), where ψ i and ψ i are eigenfunctions of Dirac operators on R 3,1 and M respectively. When M is very small, the non-zero spectrum of Dirac operator is very large. They should not be observable and we are only interested in harmonic spinors on M. For Calabi-Yau manifolds, the Dirac spinors can be identified with the Hodge groups of M. The Yukawa Couplings can be computed in terms of geometry of intersections products of elements in the Hodge groups. These are informations that are computable by topology. 59

60 The theory would be great as it can be used to compute fundamental constants in particle physics. Unfortunately there were large classes of Calabi-Yau manifolds and we need to pick the right model. A way to constrain the number of such models is the following. One knows that the number of family of Fermions is half of the Euler number of the manifold. Experiments show that this number is equal to three and so the Euler number of the manifold must be equal to ±6. There are only a couple of such manifolds. 60

61 I constructed such a manifold by taking a quotient of the space defined by the following equations: zi 3 = 0, wi 3 = 0, z i w i = 0. i i This manifold was used by B. Greene in his thesis to study string phenomiology. Greene et al demonstrated that such a model is consistent with standard model. i 61

62 Cosmic strings, black holes and quantum geometry: Based on the inputs of string theorists, we now know quite a bit about the mathematics of Calabi-Yau spaces. One hopes that eventually one can compute fundamental constants in particle physics in terms of these spaces. We may construct models of cosmology or black holes based upon continuous evolution among these spaces. Perhaps one can observe fundamental strings through cosmology. In any case, the fundamental question of building a spacetime that can incorporate quantum physics, general relativity and geometry is the most fascinating problem in the future. 62

63 Chuang Tzu ( BC) Chuang Tzu said: heaven and earth were born together with me and the myriad things are one with me. ( Translated by Victor Mair, university of Hawaii press, 1994 ) 63