# From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether

and superstrings to Cantorian space–time L. Marek-Crnjac

## Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, P.O. Box 2964, SI-1001 Ljubljana, Slovenia

# Abstract

In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals.

### © 2008 Published by Elsevier Ltd.

# 1. Introduction

Many of the profound mathematical discovery and dare I say also inventions which were made by the mathematicians Arthur Cayley, Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan and Emmy Noether [1] are extremely important for high energy particles in general [2] as well as in the development of E-infinity, Cantorian space–time theory [3,4]. The present paper is dedicated to the historical background of this subject.

# 2. Arthur Cayley – beginner of the group theory in the modern way

Arthur Cayley was a great British mathematician. He helped found the modern British school of pure mathematics. Cayley (1821–1895) was born in Richmond, London. As a child he enjoyed solving complex math problems for amusement. At eighteen, he entered Trinity College, Cambridge, where he excelled in Greek, French, German and Italian, as well as mathematics. Cayley finished his undergraduate course by winning the place of Senior Wrangler and the first Smith’s prize.

His next step was to take the M.A. degree and win the Fellowship by competitive examination. His tutor was George Peacock and his private coach was William Hopkins. He continued to reside at Cambridge for four years. Because of the limited tenure of his fellowship it was necessary to choose a profession and Cayley chose law. He worked as a lawyer for 14 years but in this period of his life, Cayley produced between two and three hundred papers.

When he was 42 years old, he gave up a lucrative practice for a modest salary. He never regretted the exchange for the chair at Cambridge. He was consequently able to prove the Cayley–Hamiltonian theorem, which says that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way leading to Lie symmetry groups, as a set with a binary operation satisfying certain laws. In 1876 he published a Treatise on Elliptic Functions, which was his only book.

In 1889 the Cambridge University Press requested him to prepare his mathematical papers for publication in a collected form. They are printed in magnificent quarto volumes of which seven appeared under his own editorship. The remainder of his papers was edited by Prof. Forsyth. The Collected Mathematical Papers of thirteen quarto volumes contains 967 papers. His writings are his best monument and certainly no mathematician has ever had his monument in greater style. To the third edition of Tait’s Elementary Treatise on Quaternions, Cayley contributed a chapter entitled ‘‘Sketch of the analytical theory of quaternions.” After him are named: Cayley’s theorem, Cayley–Hamiltonian theorem, Cayley–Dickson construction in the theory of quaternions. This theory played a profound role in developing octonions and subsequently exceptional E8 Lie symmetry groups in high energy physics. [14–19] (See Fig. 1).

# 3. Felix Klein and Sophus Lie –

# Discoverer of elliptic modular groups and exceptional Lie groups which are used in super string theory

Felix Klein was a German mathematician known for his work in group theory, function theory, non-Euclidean geometry, the connection between geometry and group theory and elliptic modular groups which is now used extensively in superstrings[2,3,5].

Klein (1849–1925) was born in Du¨ sseldorf where he attended the Gymnasium and then he studied mathematics and physics at the University of Bonn. Julius Plu¨cker held Bonn’s chair of mathematics and experimental physics. Klein became his assistant and received his doctorate supervised by Plu¨cker from in 1868.

Klein’s first important mathematical discovers were made in 1870 in collaboration with Sophus Lie. They discovered the fundamental properties of the asymptotic lines on the Kummer surface. Further collaboration with Lie followed and they worked on curves invariant under group of projective transformations. This work proved to be very important for El Naschie’s theory using G. ‘t Hooft holographic principles [14–19].

Sophus Lie (1842–1899) was born in Nordfjordeid in Middle-Norway. He was taught mathematics at school by Ludwig Sylow and then attended Sylow’s lectures on group theory at the University of Christiania (Oslo) from where he graduated in 1865. In 1869 Lie went to Berlin where he met and became friends with Felix Klein. They started to work on transformation groups (See Fig. 2).

Erlangen appointed Klein professor in 1872 when he was only 23. Klein’s thesis of defining geometry as the study of the properties of a space that are invariant under a given group of transformations is known as the Erlangen Program. This program was set out in Klein’s inaugural lecture as professor at Erlangen. The Program proposed a unified approach to geometry that became the accepted view. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the Program’s definition of geometry encompassed both Euclidean and non-Euclidean geometry.

Klein devised the bottle named after him, a one-sided closed surface which cannot be constructed in Euclidean space. This is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible to construct a Klein bottle in non-Euclidean space (See Fig. 3).

Klein was pleased to be offered a chair of Munich’s Technische Hochschule in 1875. He had many excellent students as Adolf Hurwitz, Carl Runge, Max Planck, Gregorio Ricci-Curbastro and Walter von Dyck. After five years at the Technische Hohschule, Klein was appointed to a chair of geometry at Leipzig. There he had young colleagues included Walter von Dyck, Edward Study and Friedrich Engle.

Klein worked on number theory and abstract algebra, group theory, multidimensional geometry and the theory of differential equations, elliptic modular functions and automorphic functions. Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate that plane. In 1879 he looked at the action of PSL(2, 7) thought of as an image of the modular group and obtained an explicit representation of a Riemann surface. He showed that the surface was a curve in projective space, that its equation was x³y + y³ z + z³x = 0 and that its group of automorphism was PSL(2, 7) of order 168 while having (2)(168) = 336 degrees of freedom. Many years later El Naschie proved that the compactified version has 336 + 16 k 339 degrees of freedom [13]. Klein summarized his work on automorphism and elliptic modular functions in a four volume treatise written with Robert Fricke over a period of about 20 years.

Klein accepted a chair at the University of Go¨ttingen in 1886. From then until his 1913 retirement he sought to reestablish Go¨ttingen as the world’s leading mathematics research centre. Among his students was one of the founding fathers of quantum mechanics Max Born. However under Klein, Born worked on elastic buckling as mentioned in some autobiographical notes by El Naschie which also started his career working in elastic stability and buckling. His friend and colleague Sophus Lie worked as an assistant at Christiania and obtained his doctorate there. He discovered contact transformations which he applied them to extend a method for solving partial different equations. This led Lie to define what he called continuous transformation groups (now are called after him Lie groups). Lie worked at the Christiania University, but he did not have any students who where interested in his work. Klein sent his student Friedrich Engel on a mission to study under Lie and to assist him in preparation of this large unpublished theory. In 1886 Lie succeeded Klein in the chair of mathematics at Leipzig with Engel as his assistant. Unfortunately his work was overshadowed by his illness, Lie returned to Christiania in 1898 and died 1899. El Naschie considers the exceptional E8 E8 Lie symmetry group to be the blue print for the entire theory of high energy particle physics.[14â€“19].

### A three dimensional analogue of Möbius strip which is a one-sided surface. A Klein bottle does not have an inside & an outside.

# 4. Wilhelm Killing –

# Discoverer of the exceptional Lie group G2 and the Killing spinor which is used in super gravity and super string theory

Wilhelm Killing (1847–1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups and non-Euclidean geometry. Killing studied at the University of Mu¨nster and later wrote his dissertation under Karl Weierstrass and Ernst Kummer at Berlin in 1872. He became a professor at the seminary college in Braumsberg. Then he became rector of the college and chair of the town council. As a professor and administrator Killing was widely liked and respected. Finally, in 1892 he became a professor at the University of Münster. Killing invented Lie algebras independently of Sophus Lie around 1880. In fact Killing’s work was less rigorous logically than Lie’s, but Killing had much greater goals in terms of classification of groups and made a number of unproven conjectures that turned out to be true. Killing essentially classified the complex simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix. Killing also introduced the notion of a root system. He is the discoverer of the exceptional Lie algebra G2 in 1887. His root system classification showed up all the exceptional cases, but concrete constructions came later. Killing also introduced the term characteristic equation of a matrix. After Wilhelm Killing is named the Killing form, this is a symmetric bilinear form which plays a basic role in the theories of Lie groups and Lie algebras. Killing spinor indicates those twistor spinors which are also eigen spinors of Dirac’s operator. In physics, Killing spinors are used in super gravity, super string theory and the E-infinity Cantorian space–time theory. El Naschie has recently shown that the standard model is given by two sub-algebras of the E8 Lie algebra namely F4 and G2. In fact the total surface of their dimensions gives us the number of elementary particles in the standard model |F4| + |G2| = 52 + 14 = 66. On compactification El Naschie’s prediction 66 + 3 = 69 is found. According to Ji-Huan He, this means we still have 9 elementary particles missing from the standard model.[14â€“19].

# 5. Elie Cartan – inventor of spinors and spaces with torsian, which are important for relativity and quantum mechanics

Elie Cartan (1869–1951) was born in Dolomieu in Savoie and became a student at the Ecole Normale Superieure in Paris. After his doctorate, he took lecturing position in Montpellier and Lyon and became a professor in Nancy in 1903. Then he took a lecturing position in Paris in 1909 and stay there till retiring in 1942. Cartan did fundamental work in the theory of Lie groups and their geometric applications. He began by working over the foundational material on the complex simple Lie algebras tidying up the previous work by Friedrich Engeland Wilhelm Killing. He also made significant contributions to mathematical physics, differential geometry and group theory. Cartan was the inventor of spinors, spaces with torsian, a champion of projective geometries and the developer of a system of calculus called exterior differential forms. All subjects are extremely important for relativity, quantum mechanics and quantum gravity. (See Fig. 4).

# 6. Exceptional Lie groups E8, E7, E6, F4 and G2

The designation E8 comes from Wilhelm Killing and Elie Cartan’s classification of the complex simple Lie algebras which fall into four infinite families labelled An, Bn, Cn, Dn and five exceptional cases labelled E6, E7, E8, F4 and G2. The E8 algebra is the largest and most complicated of these exceptional cases. The smaller exceptional groups E7 and E6 sit inside E8. It is the basis of heterotic super string theory as well as the Cantorian E-infinity theory [3,4,6–8].

E8 has rank 8 and dimension 248. The vectors of the root system are in eight dimensions. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600. Very recently El Naschie showed that the sum of all dimensions of the Ei exceptional family plus SO(10) and SU(5) amounts to [9]:

### 284 + 133 + 78 + 45 + 24 = 528

which are the number of stats of Witten’s p = 5 Brane theory. Furthermore, when the standard model |SM| = 12 and 8 degrees of freedom of the Higgs field are added one finds 548 which is 4 times of the inverse fine structure constant of electromagnetism = 137. A similar result was recently proven by El Naschie for one and two-stein spaces [15] (See Fig. 5).

There is a Lie algebra En for every integer n 3 which is infinite dimensional if n is greater that 8. The complex Lie group E8 of dimension 248 can be considered as a simple real Lie group of dimension 496. The group E8 E8 serves as the gauge group with the dimension 496 in [9–13].

A representation of the exceptional Lie group E8 was found in the 248-dimensional math puzzle (Fig. 6) which was solved in March 2007. An international team of mathematicians has detailed a vast complex numerical ‘‘structure” which was described more than a century ago. Mapping the 248-dimensional structure, called E8, took four years of work and proceeds more data than the human genome sequence. Familiar structures such as balls and cones have symmetry in three dimensions; Lie groups are much bigger and 248-dimensional. Conceptualising, designing and running the calculations took a team of 18 mathematicians four years. The final computation took more than three day’s solid processing time on a Sage supercomputer. What came out was a matrix of linked numbers which together describe the structure of E8.

E7 has rank 7 and dimension 133, its subalgebra is SU(8) with dimension 63. The compact real form of E7 is the isometry group of a 64-dimensional Riemann manifold (Fig. 7). In string theory E7 appears as a part of the gauge group of one the versions of the heterotic string in [1–13].

E6 has rank 6 and dimension 78. The compact real form of E6 is the isometry group of a 32-dimensional Riemann manifold (Fig. 8).

The Lie algebra E6 has a F4 subalgebra which is the fixed subalgebra of an outer automorphism and a SU(3) SU(3) SU(3) subalgebra.

In grand unification theories E6 appears as a possible gauge group which after its breaking gives rise to the SU(3) SU(2) U(1) gauge group of the standard model. A way of achieving this is through breaking to SO(10) U(1) in [9].

F4 has rank 4 and dimension 52. The compact real form of F4 is the isometry group of a 16-dimensional Riemann manifold. The Weyl–Coxeter group is the symmetry group of the 24-cell (Fig. 9).

G2 has rank 2 and dimension 14. The compact form of G2 can be described as the automorphism group of the octonion algebra or as the subgroup of SO(7). The underlying real Lie algebra of complex Lie algebra G2 has dimension 28.

# 7. Emmy Noether–Noether’s theorem, the central result in theoretical physics

Emmy Noether, one of the most outstanding German–Jewish mathematicians in the field of abstract algebra, was born in Erlangen, Germany. Her father, Max Noether was a distinguished mathematician at the University of Erlangen. He was an algebraist, as was Paul Gordan who also was associated with the university and was a close friend of the Noether family. Although Erlangen did not allow woman to enrol, Emmy was able to sit in various classes. When Erlangen permitted woman to enrol in 1904, Emmy immediately enrolled as a mathematics student. She received her doctorate in 1907 under Paul Gordan. Under the influence of Ernst Fischer she passed from algorithmic aspects of Gordan’s work to the abstract axiomatic approach of Hilbert.

She moved to Goöttingen, but the University of Goöttingen refused to let her teach. Her colleague David Hilbert helped her and finally she was admitted to the faculty in the year 1919.

Emmy fled Germany in 1933, she had been forbidden from teaching undergraduate classes by the Nazi racial laws. She joined the faculty at Bryn Mawr College in the United States of America. There was her happiest and most productive period. She died in1935, at the height of her creative powers.

After her are named the Noetherian rings which have the property that every ideal is finitely generated. Her studies on abstract rings and ideals have been particularly important in the development of modern algebra. The Lasker–Noether theorem in commutative algebra is a fundamental result that describes the decomposition of ideals into primary ideals. The central result in theoretical physics is the Noether’s theorem, which expresses the one-to-one correspondence between symmetries and conservation laws. El Naschie has stated that without Emmy Noether there would be no mathematical high energy physics and no E-infinity theory (See Fig. 10).

# 8. E-infinity Cantorian space–time

The idea of hierarchy and self-similarity in science first started in cosmology. Right was the first to entertain such ideas in the 18th century. Later on the idea was repeated in the work of the Swedish scientist Swedenborg (1688–1772) and then much later and in a more mathematical way in the work of another Swedish astrophysics Charlier (1862–1934) in [6].

The work of El Naschie was done independently without any knowledge of the above starting from non-linear dynamics. The main idea of the work of El Naschie is in fact a generalisation of Einstein’s general theory of relativity.

Einstein introduced a new geometry for space–time which differs from the space–time of our sensual experience. This space–time is taken for granted to be Euclidean.

General relativity persuaded us that the Euclidean 4-dimensional space–time is an approximation, but the true geometry of the universe is a four dimensional curved manifold. In E-infinity we take a similar step and allege that space–time is far from being the smooth, flat and passive space which we use in classical physics. The mathematical tools which are needed are non-linear dynamics, the geometry of chaotic dynamics. This is fractal geometry reduced to the Cantor sets.

In E-infinity theory we admit formally infinite dimensional ‘‘real” space–time. This infinity is hierarchical and it was shown that E-infinity has formally infinitely many dimensions in [6–13]. The E-infinity space–time is constructed as an infinite number of elementary Cantor sets with all conceivable Hausdorff dimensions. All possible union and intersections form the E-infinity Cantorian space–time. This space has three dimensions, the formal dimension nf = , the topological dimension nT = 4 and the Hausdorff dimension <dc> = 4 + ³ = 4.236067977, where is the golden mean with the remarkable continued fraction representation [12]

The Hausdorff dimension is given as the expectation of a discrete gamma distribution of nf using ‘‘golden” weights [12]:

where = 0:618033989 is the golden mean [12].

The bijection formula is given by El Naschie and is an important method of lifting the Hausdorff dimension of a certain Menger–Uhryson zero dimensional set to a higher dimension. Thus we have

where = =

We obtain from the above formula the Hausdorff dimension of E-infinity space–time [6]

In other words the exact topological embedding dimension corresponding to the Hausdorff expectation dimension 4 + ³ = 4.236067977 is exact nT = 4 and that is why we perceive space–time as being exactly four dimensional at our low energy.

The exceptional Lie group E8 E8 describes the volume of E-infinity space–time while PGL(2, 7) describes the (7) holographic boundary given by Klein’s modular curve. The Klein modular curve (7) has 336 symmetries, which are the numbers of the triangles in (7). The dimension of Klein’s modular space is equal to the dimension of the simple linear group PGL(2, 7). The Lie group PGL(2, n) is defined as [13]

and we obtain for n = 7 the following result [13]

Recently El Naschie gave two remarkable derivations for the electromagnetic fine structure constant = 137 and Newton’s dimensionless gravitational constant using the holographic principle [6] and the heterotic super string theory [11]. In fact in the last three years, El Naschie generalized the transfinite version E8 E8 of exceptional Lie groups to a quantum golden field theory and used it to prove quarks confinement [14].

# 9. Conclusion

Many of the mathematical discoveries are of a great importance for high energy physics in general as well as in the development of E-infinity Cantorian space–time. The exceptional Lie group E8 E8 describes the volume of E-infinity space–time while the Lie group PGL(2, 7) describes the (7) holographic boundary by Klein’s modular curve.

We may conclude by mentioning a well known fact which nevertheless is extremely fascinating namely that the Egyptians were well acquainted with the golden mean and used it in design of their great pyramids. For instance the papyrus of Ahmes written hundred of years before the ancient Greek civilisation contain detailed account about the ‘‘Sacred” ratio used in the design of the great Giza pyramid.

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