Advanced prerequisite for E-infinity theory

Abstract

This is the third of a series of papers written with the primary aim of communicating necessary theoretical background knowledge required for an in-depth study of E-infinity theory. Compared to the previous two papers [El Naschie MS. Elementary prerequisites for E-infinity (Recommended background readings in nonlinear dynamics, geometry and topology). Chaos, Solitons & Fractals 2006;30(3):579–605; El Naschie MS. Intermediate prerequisites for E-infinity theory. Chaos, Solitons & Fractals 2006;30(3):622–8], the present one may be described as advanced.

Introduction

In two previous papers we introduced the absolute essential mathematical knowledge and the intermediate level background needed for a serious study of E-infinity theory [1], [2]. The present paper is concerned with relatively rather advanced subjects that are closely linked to E-infinity theory and consequently will increase the prospect of an in-depth understanding of the theory.

In what follows, six mathematical subjects will be addressed and possible connections to E-infinity theory will be touched upon.

It is a curious situation that while manifolds in smaller as well as higher dimensions than four are reasonably well understood, one had to wait for some time to resolve the situation in four dimensions. We now know that in this dimension, i.e., the dimension of Einstein’s theory as well as E-infinity expectation dimension, qualitatively different and quite subtle new phenomena occurs [3].

An obvious connection between the theory of topological four manifolds and E-infinity is that of capped gropes, the models of which are obtained by iterating a capped surface construction. The heights of these gropes are found to be the recurrence relation [4]:

$$a_n=a_{n-1}+a_{n-2}-2\text{.}$$

With initial grope height 3 we have the initial conditions

$$a_{-1}=3\text{and}{a}_0=3\text{.}$$

This is essentially the Fibonacci numbers. That means:

$$a_n=2+\frac{\sqrt{5}+3}{2\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^n+\frac{\sqrt{5}-3}{2\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^n\text{,}$$

where $(1+\sqrt{5})/2$ is the inverse of E-infinity golden mean ϕ.

We may also mention that the complexity of an immersed grope is measured in terms of words and that the maximum length is 5 while the estimate is that each step in the height raising argument may increase the diameters by a factor of 7.

Finally we refer the reader to the original work of M. Freedman and F. Quinn for the fundamental role played in 4 topological manifolds by triadic Cantor sets, the Schottky groups and compactification points at infinity, all concepts used extensively directly and indirectly in E-infinity theory [5].

Under foliation we understand a partitioning of a topological space into a disjoint sum of ‘leaves’ with a regular microscopic behaviour. From this loose definition it is intuitively obvious that foliation is potentially very relevant to E-infinity theory.

Just as we have difficulty in writing a Lagrangian for ε^(∞) in a straightforward manner, we also have obstacles in turning a foliation into a flow. This is mainly due to non-orientable singularities. Examples of such singularities are known as thorns, tripods and apples to mention only a few, all being local. The global obstacle is the labyrinth which refers to a collection of dense leaves [6].

It is also extremely important for those interested in E-infinity theory where irrational numbers and continued fractions play an import role, to realize that approximating irrational numbers using rationals is linked to the geometry of foliations. This was recognised by H. Poincaré who used irrational foliation on the torus in problems of celicital mechanics involving the three body problem. In other words E-infinity is, right from the start, involved in the theory of foliation by virtue of the expectation value of its Hausdorff dimension [5]:

$$$$

In addition foliation is linked to a paradigm of nonlinear dynamics and chaos, namely the Plykin attractor. This attractor might happen in the Diffing and van der Pol equations of mechanical, electrical and other physical systems and is the simplest example for a planar hyperbolic attractor (see Fig. 1).

Kähler manifolds are complex manifolds which are immensely useful for understanding E-infinity Cantorian spacetime from a relatively conventional view point. A simple physicist definition is given by Polchinsky in [7]:

Definition

A Kähler manifold is a complex manifold of U(n) holonomy in n complex dimensions. The most important Kähler manifold for E-infinity theory is the K3. Again we follow Polchinski by describing K3 as the unique nontrivial Calabi–Yan manifold of four real dimensions. In fact K3 is the only Kähler manifold with four real dimensions known to us and hence its importance in superstrings and E-infinity. Being topologically unique K3 still possesses complex structures and Kähler moduli. Its holonomy is that of SU(2). In K3 the moduli space parameterized by the charged hyper multiplets is the space of gauge fields with instanton number n = χ = 24.

Theorem

Let (x, J) be a K3 surface then x is simply connected with Betti numbers

$$b_2=22\text{,}{b}_2^{+}=3\text{,}{b}_2^{-}=19\text{,}$$

while the Hodge numbers are given by

$$h_2^0=h_2^0=1\text{<mi mathvariant="italic" is="true">and</mi>}{h}_1^1=20\text{.}$$

We note the similarity to E-infinity as represented by a fuzzy Kähler which is not simply connected and the corresponding Betti numbers [8]:

$$b_2=24+k\text{,}{b}_2^{+}=5+{\varphi }^3\text{<mi mathvariant="italic" is="true">and</mi>}{b}_2^{-}=19-{\varphi }^6$$

which gives $\chi =b_2^{+}+b_2^{-}+2=26+k$ where $\varphi =(\sqrt{5}-1)/2$ and k = ϕ³(1 − ϕ³).

Theorem

Let x be a K3 surface and m ⩾ 2, then the Hilbert scheme x^[m] is an irreducible complex Symplictic 2m-manifold with b₂ (x^[m]) = 23. There exists a 61-dimensional family of matrices g on x^[m] with holonomy sp^(m) and a 64-dimensional family of hyper Kähler structures (J₁, J₂, J₃, g).

The above theorem was discussed in connection with the standard model of elementary particles in earlier publications [8]. It remains only to mention that the Riemann curvature tensor g on K3 is determined by a single component R^α.

For a more detailed discussion of K3 and its application to E-infinity the reader may consult [8], [9].

It is commonly accepted, at least among theoretical physicists, that the Planck length [5]

$${\ell }_p=\sqrt{\mathit{hG}/c^3}$$

is the smallest length which can be measured in principle and that the notion of length totally looses its meaning for anything smaller. Here h is the Planck constant, G the Newton gravitational constant and c is the velocity of light. This fact seems to contradict a fundamental axiom. Accepting this limitation on measurement inevitably leads us to non-Euclidean geometry at extremely small distances similar to what we do in E-infinity theory. The subtle point is that there is a correspondence between geometry and numbers. Euclidean geometry is described by real numbers. Thus moving to non-Euclidean geometry forces us to abandon the confinement to real numbers in favour of a larger class of numbers. This is exactly the subject of P-Adic quantum mechanics [10] and is also what we have done in E-infinity theory.

It then turns out that the field of P-Adic numbers has a hierarchal structure, again exactly as in E-infinity. It is easily shown that every disc consists of a finite number of smaller discs and so on ad infinitum. This means that the field of P-Adic numbers is homomorphic to a Cantor set and thus to E-infinity Cantorian spacetime. Thus there is a natural hierarchal fractal-like structure of the field of P-Adic numbers. Even before developing the theory of P-Adic analysis, this subject was well known in mathematics.

Definition

Let Q be the field of rational numbers and the value ∣x∣ of x ∈ Q satisfy the following:

• ∣x∣ ⩾ 0, ∣x∣ = 0 ↔ x = 0,

• ∣xy∣ = ∣x∣ ∣y∣,

• ∣x + y∣ ⩽ ∣x∣ + ∣y∣.

Any function on Q with the above properties is called norm. Let p be a prime number and introduce a norm:

$$|x{|}_p$$

on the filed Q using

$$|0|=0\text{,}|x{|}_p=p^{-\gamma }\text{,}$$

where γ = γ^(x) is defined by

$$x=p^{\gamma }\frac{m}{n}$$

and the integers m and n are not divisible by p. In such a case

$$|x{|}_p$$

is called the P-Adic norm. An extremely important example of the P-Adic norm is that of $|\overline{\alpha }{|}_2$. In this case one finds [5]:

$$\Vert \overline{\alpha }=137{\Vert }_2={\Vert }_2^7+2^3+2^0{\Vert }_2=\Vert 128+8+1{\Vert }_2=1\text{.}$$

We have conjectured some time ago that the above relation is an extension of Witten’s T duality outside string theory. This is so because ${\overline{\alpha }}_{\mathrm{QG}}=1$ is the coupling of the Planck masses particle at an extremely high energy 10¹⁹ Gev. At the same time, this norm is equal to the norm of $\overline{\alpha }\cong 137$. Consequently extremely high energy is linked with the low energy of the standard model [1], [5].

In 1956 J. Nash introduced two remarkable results regarding the classical problem of Euclidean imbedding [11]. Both results are of intrinsic relevance to superstrings and E-infinity. The first result of Nash concerns the imbedding of compact manifolds [11]:

Theorem

A compact n-manifold with a c^k positive metric has a c^k isometric imbedding in any small volume of Euclidean

$$(n/2)(3n+11)$$

space provided 3 ⩽ k ⩽ ∞.

It is very interesting to note first that for a one dimensional ‘stringy’ object the imbedding dimension is

$$D(\text{Nash})=(1/2)(3+11)=7$$

which is the dimension of S⁽⁷⁾. We note that S⁽⁷⁾ has the largest possible surface area and is related to Milnor’s exotic spheres as well as the dimensionality needed for the symmetry group of the standard model SU(3) ⊗ SU(2) ⊗ U(1).

Second extending the formula beyond non-fractal dimensions we see that for the Menger sponge d_M = ℓn20/ℓn3 = 2.726833028 one finds

$$D(\text{Nash})=\frac{1}{2}\frac{\ell n20}{\ell n3}\left(\frac{3\ell n20}{\ell n3}+11\right)=26.1510092\text{.}$$

This is almost identical to the dimension of the spacetime of the transfinite string theory, namely 26.18033989. The second formula of Nash is concerned with the imbedding of non-compact manifolds [11]:

Theorem

Any Riemannian n-manifold with c^k positive metric, where 3 ⩽ k ⩽ ∞ has a c^k isometric imbedding in $\left(\frac{n^3}{2}+7n^2+\frac{5}{2}n\right)$ space, in fact in any small portion of this space. Let us consider again the case of a one dimensional string. In this case one finds

$$D(\text{Nash})=1/2+7+2.5=10$$

which is identical to the spacetime dimension of superstrings [5].

There are various connections between the mathematics of E-infinity theory and four dimensional algebra [12], [5].

Definition

A fusion algebra is a finite-dimensional associative, Unitarian, involution algebra A over $\mathbb{C}$ equipped with a distinguished basis

$$B=B_A\text{,}$$

such that the structure constants $\{N_{\mathit{xy}}^{\stackrel{̶}{Z}}:x\text{,}Y\text{,}\stackrel{̶}{Z}\in B\}$ defined by

$$\mathit{xY}=\sum _{\stackrel{̶}{Z}}{N}_{\mathit{xy}}^{\stackrel{̶}{Z}}\stackrel{̶}{Z}$$

satisfy the following conditions:

• $N_{\mathit{xy}}^{\stackrel{̶}{Z}}\in {\mathbb{Z}}_{+}(=\{0\text{,}1\text{,}2\text{,}\dots \})\text{,}\forall x\text{,}Y\text{,}\stackrel{̶}{Z}$,

• the identity denoted simply by 1 of the algebra A, belongs to B, then clearly

  $$N_{x1}^Y=N_{1x}^Y={\delta }_x^Y\text{,}\forall x\text{,}Y\in B\text{,}$$

• B_A is closed under the involution of A, hence there exists an involution $B\ni x_1\to \overline{x}\in B$ such that $\overline{x}=X^{\ast }\forall x\text{,}\in B$,

• $N_{\mathit{xy}}^{\stackrel{̶}{Z}}=N_{\overline{x}Z}^Y\forall x\text{,}Y\text{,}\stackrel{̶}{Z}\in B$.

In the case of a 4D fusion algebra with B_A = {1, α, β, ε} and the matrices corresponding to left-multiplication by the basis vectors are given by

$$\begin{array}{cc}\hfill & L_1=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\text{,}{L}_{\alpha }=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\hfill \\ \hfill & L_{\beta }=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\text{,}{L}_{\epsilon }=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\hfill \end{array}$$

The symmetry of the matrices shows that the involution is trivial when restricted to the basis B. The dimension function is given by

$$\begin{array}{cc}\hfill & d(1)=d(\epsilon )=1\text{,}\hfill \\ \hfill & d(\alpha )=d(\beta )=\varphi \text{,}\hfill \end{array}$$

where $\varphi =\frac{1+\sqrt{5}}{2}$ as in E-infinity theory [5] and the dimensional function of Connes’ non-commutative geometry [2]. We note that there is a fusion algebra corresponding to E₈ and E₆ exceptional Lie groups, but not for E₇. We also note that fusion algebra was used in connection with E-infinity analysis of the quarks mass spectrum [5].