Mathematical indices are mapped to physical indices which are themselves mapped to nets, such that the nets can be used for describing the four forces. The consideration of quaternions and octonions leads to forces, which can describe the effect of dark matter and dark energy. Singularities are described by disconnected nets of infinite cardinality.

Leseprobe

Für Aaron und alle unsere Lieben.

Table of Contents

PART A: Mathematics ... 4

I. Basic Structure ... 4

II. Number Systems ... 5

III. Functions ... 20

IV. Functionals ... 21

V. Cardinal Arithmetic ... 23

VI. Measure Theory ... 25

VII. Functional Analysis ... 28

VIII. The Structure of Functions and Functionals ... 32

IX. Stochastic Analysis ... 35

X. Atiyah-Singer Index Theorem ... 40

XI. Net Theory ... 45

PART B: Physics ... 56

XII. Definition of Elementary Physical Structures ... 56

XIII. Derivation of Composed Physical Structures ... 60

XIV. Physical Measurements and Conservation Laws ... 71

XV. Special Relativity and Quantum Mechanics ... 77

XVI. Complementary Coordinates ... 82

XVII. The Structure of Information and Causality Spaces ... 86

XVIII. Inner and Outer Structures of Vector Spaces ... 91

XIX. Standard Model of Particle Physics and Cosmology ... 105

XX. Standard Model of Physics ... 128

XXI. Observations and Experiments... 168

Bibliography ... 169

PART A: Mathematics

I. Basic Structure

0 Sets

"Under a set we shall understand an aggregation of distinguishable elements

which are merged to a whole. (Georg Cantor, 1895)."

Hence, we have:

= {

}

The elements of are distinguishable but not ordered.

Two sets are equal if they have the same elements [1].

The building of a set proceeds in two steps. First, we build a multeity

, an

accumulation, a range. Then, this multeity is merged to a unity, to a set [1].

Following definitions are useful.

= {}

The empty set does not comprise any element.

: ( ) = { }

The power set is the set of all subsets of . If the set has elements, then the

power set has

2 elements.

The set shall be called infinite, if a true subset

exists which can be

mapped in a bijective manner to ; hence, if there is a

with | | = | | [1].

A set shall be called finite, if it is not infinite [1].

Two sets

are of equal cardinality | | = | | if they can be mapped in a

bijective manner to each other. The power set of a set is always of bigger cardinality

that the set itself [1].

II. Number Systems

1 The Natural Numbers: The Set

The natural numbers are formed from the set . The idea is that all elements of the

set shall have an ordering, giving each element of a "name"/a "number". The

natural numbers define a set

in which one element 0 is distinguished (the first

element); and a self-mapping

: (

) exists, such that the

following axioms apply [2]:

( 1)

( 2)0 ()

( 3)

, =

The set

starts with the point 0 and proceeds with the successor, with its successor,

and so forth. We can therefore characterize the natural numbers with an index:

= ( + 1) - ( )

This index exists for each natural number, since the difference between a successor

and its predecessor is always

1. But this index does not exist for the elements of the

set , since said elements are not ordered, such that no subtraction can be carried

out. Hence, this index is a characteristic of the natural numbers.

The successor function can be applied in an infinite manner to the natural numbers.

The limit number of

which cannot be reached by a finite number of steps shall be

called . The cardinality of

shall be called

[1].

The successor function defines one direction, namely from an element towards its

successor. Therefore, the natural numbers define a structure with a single direction,

hence an open structure.

2 The Integers: The Set

The natural numbers allow an addition in every case, but do not allow the inverse

operation, the subtraction, in any case. In order to remove this disadvantage, the

natural numbers are extended to the integers. Every integer can be defined as the

difference

( - ) of two natural numbers , [2]. The integer ( - ) shall be written

as a pair (

, ) [2]. We consider the following relation on × : ( , )~( , ) +

= + ( ( - = - ) [2].

The integers are defined as equivalence classes of the relation

~ [2]. The integers

set up a commutative group with respect to the addition [2]. The integers set up an

integrity ring (commutative, zero division free ring with one-element) [2]. Every

integer

( , ) possesses an inverse integer ( , ) [2]. The integers define a set [2].

We can therefore characterize the integers with an index:

= ( - ) - ( - )

This index exists for each integer

( - ), since we can always build an inverse

integer

( - ). But this index does not exist for the elements of the set , since said

elements do not allow negative numbers, such that the index cannot be always

computed. Hence, this index is a characteristic of the integers.

The cardinality of

is the same like the cardinality of [1]. Hence, has also the

cardinality

. The integers possess, like the natural numbers, one direction given by

the addition operation of the successor function. The integers possess, contrary to

the natural numbers, also the opposite direction given by the subtraction operation

which can be always computed in

but not in . Hence, the integers define a

structure with two opposite directions, hence a closed structure.

The integers (and also the natural numbers) possess the following ordering type.

, <

( , )

, <.

( , )

, < :

= sup( )

= inf( )

< .

is then the successor of in

, <, written = + 1. is then the predecessor of

, <, written = - 1. Figure 1 shows the structure of a step/jump.

Figure 1: Step/Jump of a Linear Ordering

3 The Rational Numbers: The Set

The integers allow an addition, a subtraction, and a multiplication in every case, but

do not allow a division in any case. In order to remove this disadvantage, the

integers are extended to the rational numbers. Every rational number can be

described as a quotient of two integers

, [2]. The rational number shall be

written as a pair (

, ) [2]. We consider the following relation on

× \{0}: ( , )~( , ) = ( = ) [2].

The rational numbers are defined as equivalence classes of the relation

~ [2]. By

using the relation

: , () the set is mapped in an isomorph

manner to the subring

() [2]. The field is the smallest field containing as a

subring [2]. Every rational number

( , ) possesses an inverse rational number ( , )

[2]. We can therefore characterize the rational numbers with an index:

= -

This index exists for each rational number , since we can always build an inverse

rational number . But this index does not exist for the elements of the set

, since

said elements do not allow floating point numbers, such that the index cannot be

always computed. Hence, this index is a characteristic of the rational numbers.

The cardinality of

is the same like the cardinality of [1]. Hence, has also the

cardinality

The rational numbers possess the following ordering type [1].

, <

( ) , <

Gaps of a linear ordering: Shall

, <define a linear ordering, and shall ( , ) define

a cut in

, <. ( , ) is named a gap in , < if sup( ) does not exist. Figure 2

shows the structure of a gap.

Figure 2: Gap of a Linear Ordering

4 The Real Numbers: The Set

The rational numbers allow an addition, a subtraction, a multiplication, a division, and

an exponentiation in every case, but do not allow the root extraction and the

extraction of logarithms in any case. Also transcendental numbers like or cannot

be defined by rational numbers. In order to remove these disadvantages, the rational

numbers are extended to the real numbers. This extension is performed by Dedekind

cuts [2].

A Dedekind cut is [1]: Shall

, < define a linear ordering. A Dedekind cut in , <

is a pair

, with the following properties:

( ) , , = , =

( )

( ) Sup( )

The supremum

( ) exists for each rational number. The supremum

( ) does

not exist for irrational numbers. Here, we have a gap which is closed by the

Dedekind cut. Otherwise stated, the Dedekind cuts close all gaps of

and lead to

the set

which is complete [2]. We can define the set in the following manner [1]:

Every non-empty upper bound subset of

possesses a supremum (smallest

upper bound); there is hence a

(1)

(2)

is called the supremum of . We have therefore

( ). We can therefore

characterize the real numbers with an index:

( )

(

)

This index exists for each real number, since we can always define a supremum with

a Dedekind cut. But this index does not exist for the elements of the set

, since said

elements do not always possess a supremum, such that the index cannot be always

computed. Hence, this index is a characteristic of the real numbers.

The cardinality of

is greater than the cardinality of [1]. As proved in [1], the

cardinality of

is the power set of the cardinality of ; hence the cardinality of is

= 2

. Hence,

has the cardinality .

The real numbers possess the following ordering type [1].

, <

( ) , <

Separable means: Between two real numbers there is always a rational number.

Hence, the real numbers possess a dense subset. For a dense subset we have:

Shall

. is named dense in , if: for all , with < there is a

with

< < . Figure 3 shows the structure of a continuity.

Figure 3: Continuity of a Linear Ordering

5 The Complex Numbers: The Set

The real numbers allow an addition, a subtraction, a multiplication, a division, an

exponentiation, a root extraction and an extraction of logarithms in any case, but do

not always lead to a solution of algebraic equations. Indeed, when said solution

demands the extraction of a square root from a negative real number, then the real

numbers do not allow a solution. In order to remove this disadvantage, the real

numbers are extended to the complex numbers. This extension is performed in the

following manner [2].

The set

× of all ordered real number pairs ( , ) defines by the natural

addition (

, ) + ( , ) (

) an Abel group. We define a

multiplication in

× in the following manner [1]:

( , ) ( , ) (

)

(1,0) is the one-element. (0,1) is the imaginary unit with = -1 [2]. We

have then:

(

) ( + ) ((

) + (

))

is called the essential real unit.

is called the essential complex unit.

Hence, we can write any complex number as

The set

is not ordered. We can define [2] a conjugation × ,

with

= +

and with

= - as an automorphism which maps in itself, and which

maps into the second, in principal equal, zero point

- of

+ 1 = 0. We can

therefore characterize the complex numbers with an index:

= ( ) = -

This index exists for each complex number including the essential complex unit,

since we can always build a complex conjugate for any such complex number. But

this index does not exist for the elements of the set

, since said elements do not

possess an imaginary unit, such that the index cannot be computed. Hence, this

index is a characteristic of the complex numbers.

The cardinality of

is the same like the cardinality of [1]. Hence, has also the

cardinality

6 The Quaternions: The Set

The complex numbers allow the solution of any algebraic equation by introducing

one imaginary unit. The question arises, if one can generalise the complex numbers

by introducing further imaginary units. The first generalisation leads to the

quaternions [2]. We shall define the algebra

of quaternions as: In the four

dimensional

vector space the standard basis is defined by the ordered real

quadruples:

= (1,0,0,0);

= (0,1,0,0);

= (0,0,1,0);

= (0,0,0,1). is the one-

element. The nine products

, = 1,2,3 are defined by the following

multiplication table of Figure 4.

Figure 4: Multiplication Table of Quaternions

We have:

= -1. We also have

is called the essential quaternion unit. Hence, we can write any quaternion as

, , ,

The Hamiltonian algebra

is an associative division algebra [2]. We have

- [2]. We can therefore characterize the quaternions with an index:

= ( , ) = -

This index exists and is not zero for each quaternion including at least two complex

units, since said quaternions do not commute. But this index is always zero for the

elements of the set

, since said elements always commute. Hence, this index is a

characteristic of the quaternions.

The cardinality of

is the same like the cardinality of [1]. Hence, has also the

cardinality

7 The Octonions: The Set

The quaternions are the first generalization of the complex numbers. The second

generalization leads to the octonions [2]. For elements of

× we define a product

[2]:

, ( , ) ( , )

The set defines an alternative division algebra [2]. The 49 products

, = 1 7 are defined by the following multiplication table of Figure 5.

is called the essential octonion unit. Hence, we can write any octonion as

, , , , , , ,

We have 7 quaternionic groups in the octonions:

( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ).

Figure 5: Multiplication Table of Octonions

We have:

( )

= -1. The alternative algebra

implies [2]:

: ( ) = ( )

We have

( ) = -( ) [2]. We can therefore characterize the

octonions with an index:

: :

= ( , , ) = ( ) - ( )

, ,

This index exists and is not zero for each octonion including units from different

quaternionic groups, since said octonions do not associate. But this index is always

zero for the elements of the set

, since said elements always associate. Hence, this

index is a characteristic of the octonions.

The four algebras

, , ,

are the only division algebras; hence algebras

without zero divisors [2].

The cardinality of is the same like the cardinality of

[1]. Hence, has also the

cardinality

8 The Sedenions: The Set

The generalization of the octonions leads to the sedenions. The set of the

sedenions is built by the product of

× by [3]:

, ( , ) ( , )

The 225 products

, = 1 15 are defined by the following multiplication

table depicted in Figure 6 [3].

is called the essential sedenion unit. Hence, we can write any sedenion as

+ +

, , , , , , , , , , , , , , ,

We have octonionic groups in the sedenions.

Figure 6: Multiplication Table of Sedenions

The multiplication of sedenions is not commutative, not associative, and not

alternative [2]. Said multiplication is only power associative,

= ( ) ( ) , and

flexible, (

, , ) = ( ) - ( ) = 0 [2]. The multiplication of sedenions

possesses zero divisors depicted in Figure 7 [3].

Figure 7: The 84 Zero Divisors of the Sedenions

Sedenions set up a non-commutative Jordan algebra [2]. It is proven that the space

of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact

form of the exceptional Lie group [3]. We can therefore characterize the

sedenions with an index:

: :

: = ( , , ) = ( ) - ( )

This index exists and is not zero for each sedenion including units from different

octonionic groups, since said sedenions are not alternative. But this index is always

zero for the elements of the set , since said elements are always alternative.

Hence, this index is a characteristic of the sedenions.

Remark 1: If

= 0 such that ( ) = 0, for , , hence if is a zero

divisor, we have

0, since is never a zero divisor according to Figure 7. Then,

we also have ( )

0, since ( ) is also never a zero divisor according to

Figure 7. Hence, the

: is always existing and not zero for every

sedenion including units from different octonionic groups, even if

is a zero

divisor.

The cardinality of is the same like the cardinality of

[1]. Hence, has also the

cardinality

If we further generalize the sedenions to higher dimensional complex numbers, no

new algebraic structures occur [2]. We can finally define an index

which is zero for all numbers:

= ( , , ) = ( ) - ( ) = 0

, , , , , , , ,

9 Summary of the Structures

, , , , , ,

The structures of the sets

, , , , , ,

are summarized in Figure 8 below.

Figure 8: The structures of the sets

, , , , , ,

The form, ordering type, and operator define three independent features, in the

sense that they can be combined independently from one another. Hence, we can

use sedenions with rational coefficients which are always positive.

The elements of a single feature cannot be, however, combined. Indeed, we cannot

combine an open with a closed form, since they are excluding each other. Once we

have selected the form, its index (counter

, orientator

) is decided upon and

cannot be changed.

Also, we cannot combine a discrete, a dense, and a continuous ordering type with

one another, since they are also excluding each other. Once we have selected the

ordering type, its index (orientator

, interleaver

, complementator

) is decided

upon and cannot be changed.

Finally, once we have selected how many dimensions a number has, the operators

leading to its index (complementator

, conjugator

, commutator

, associator

, alternator ) are decided upon and cannot be changed.

Hence, we can derive 30 different number types from

, , , , , ,

, see

Table 1.

Number Description

Natural Number

Integer

Positive Rational Number

Rational Number

Positive Real Number

Real Number

Complex Number with Natural

Coefficients

Complex Number with Integer

Coefficients

Complex Number with Positive Rational

Coefficients

Complex Number with Rational

Coefficients

Complex Number with Positive

Coefficients

Complex Number

Quaternion with Natural Coefficients

Quaternion with Integer Coefficients

Quaternion with Positive Rational

Coefficients

Quaternion with Rational Coefficients

Quaternion with Positive Coefficients

Quaternion

Octonion with Natural Coefficients

Octonion with Integer Coefficients

Octonion with Positive Rational

Coefficients

Octonion with Rational Coefficients

Octonion with Positive Coefficients

Octonion

Sedenion with Natural Coefficients

Sedenion with Integer Coefficients

Sedenion with Positive Rational

Coefficients

Sedenion with Rational Coefficients

Sedenion with Positive Coefficients

Sedenion

Table 1: 30 Different Number Types

III. Functions

10 The Functions and the Set

We define a function in the following manner [1]:

, ,

: ( , )

( , )

(

)

, ,

, , , , , , , ,

Functions are therefore single valued relations. A relation is hence a function

exactly when for each

( ) there is exactly one with ( , ) (

( )).

We write then: ( )

( , ) .

Let and define sets. The cardinality of is smaller or equal than the cardinality of

, | |

| |, if [1]:

( )

We further have:

| | = | |:

: : :

;

A real function is a

: . The set of all functions shall be named . We have [1]:

|| < | |

|| = | ()|

| | = | ()|

Hence, the cardinality of is greater than the cardinality of

[1]. As proved in [1],

the cardinality of is the power set of the cardinality of

; hence the cardinality of

= 2

. Hence, has the cardinality

IV. Functionals

11 The Functionals and the Set

A continuous linear mapping between normed spaces

, is named continuous

operator [4]. If the range of the mapping is the scalar field, we call the mapping a

functional instead of operator.

: :

The spaces

, are function spaces ; hence, their elements are functions. Hence,

the continuous operators/functionals are functions operating on functions. The space

( , )

= = of the continuous linear functionals of a normed space

is called the dual space of , is forming the set of all functionals, and is called .

We have [4]:

|| =

|| = | ()| = = 2

| | = | ()| = = 2

| | = | ( )| = = 2

Hence, the cardinality of is greater than the cardinality of [1]. As proved in [1],

the cardinality of is the power set of the cardinality of ; hence the cardinality of

= 2

. Hence, has the cardinality

A functional can be written as [4]:

( ) =

( ) ( )

If we further generalize the mapping process, such that we define

operators/functionals operating between functionals of the spaces , then we still

have the principle of using operators/functionals operating on functionals defined by

a combination of functions; and this does not add something significantly different to

the principle of functionals.

Otherwise stated, we begin with defining functions by operating on numbers, and

end with defining operators/functionals by operating on functions. Hence, the sets

and complement the sets of the number systems

, , , , , , , . Now we

therefore have the relevant set sequence:

, , , , , , , , , .

This set sequence has four cardinalities, see Table 2.

Set

Cardinality

Table 2: The four cardinalities of the numbers, functions, and functionals.

By the mentioned generalization of the mapping process, we can easily achieve a

higher cardinality

with each generalization step , although said generalization

does not add something significantly different to the principle of functionals. Hence,

we can define a limit cardinality

which is never reached by applying said

generalization steps in a finite manner. The set comprising all said generalization

steps including the limit cardinality

shall be called

. Hence, we generalize Table

2 including

to Table 3.

Set

Cardinality

Table 3: The five cardinalities of the numbers, functions, functionals, and the limit

cardinality

. This gives then the set sequence:

, , , , , , , , , ,

V. Cardinal Arithmetic

12 Cofinalities, Covering Numbers, and the Index

Until now, we have operated with numbers, functions, functionals/operators, and

have defined a limit cardinality

. Now, we shall define some structures of the

infinite cardinalities.

First, we define the cofinality, , of a linear ordering [1].

, <

, <,

( , <)

( , <) =

{| || ,

, <}.

The cofinality of a linear ordering is hence the minimal number of steps which we

need to reach the end of the linear ordering [1].

Next we define filters and ultrafilters [1].

( ).

, :

( ) ,

( )

, ,

( ) , , .

( ) - .

Now, we can use the cofinality principle to derive covering numbers for infinite

cardinalities [16].

( , , , )

< ,

&| | < (

)

&| | < &

We always assume

, , > 1, > 1, [

= &

< ]

so that

( , , , ) is well defined [16].

The covering number can then be used to define a pseudo power,

, of infinite

cardinalities [16]:

( ) =

( , , , 2)

( ) =

< & ( ) =

( ) =

In this way we know the structure of power operations with infinite sets.

Knowing the structure of power operations, we can now proceed to

mentioned in

IV.11 as an infinite step generalization of functionals.

, =

()

() <

()

( ) <

() | ()|

This formula is proved in [16]. This formula is further refined in [16] to:

()

() | ()|

Setting

= () + (), this formula above leads to Shelah's famous formula [16]:

( ) < (

)

Hence, even in the case where the functionals are generalized to

and power

operations on

are considered, there is a limit cardinality covering ( ), namely

. The set comprising the set

including the covering of

shall be called

. Hence, we generalize Table 3 including

to Table 4.

Set

Cardinality

Table 4: The six cardinalities of the numbers, functions, functionals, the limit

cardinality

, and its covering

. This gives then the set sequence:

, , , , , , , , , ,

The power formula can be refined in the following manner according to [16].

( ) <

Hence, if, for example,

< & ( ) =

( ) < . This special case is of

great importance, since the cardinalities

, , ,

are the cardinalities of

the natural numbers, the real numbers, the functions, and the functionals,

respectively, which play a crucial role for finite and infinite structures at the same

time. Thus we have proved the following Theorem 1.

Theorem 1: The cardinality

defines an index characterizing the finite structures

of the natural numbers, the real numbers, the functions, and the functionals,

including their infinite extensions with power operations.

VI. Measure Theory

13 Algebras and Measures

Until now, we have defined sets with different mathematical properties and with

different cardinalities. Now, we shall refine these properties by introducing measures

on the sets. First we define a algebra [17]:

(i)

( )

Next, we define a pre-measure [17].

[0, +].

() = 0

( )

(

) =

( ) is called content.

Every pre-measure defined on a algebra of a set

is named measure on

[17].

The pre-measure , defined on the ring

of -dimensional figures in is called

Lebesgue-pre-measure in

and is written [17].

The elements of the algebra generated by the system

of half-open intervals in

are called Borel-sets of the space [17].

Every countable subset of

is a Lebesgue-Borel-null-set [17].

A real function defined on

is named -elementary function (or not negative step

function) if said function is not negative, is -measureable, and takes only a finite

number of values [17].

We define an integral in the following manner [17].

( )

(

)

A numerical function on

is called -integrable if said function is -measurable

and if the integrals

and

define real numbers [17]. We have [17]:

Shall define a property, such that for every point

it is defined if possesses

said property or not. We shall say " -almost points

possess the property " or

" is valid -almost everywhere on

", if there is a -null set such that all points

N possess the property [17].

A null set is defined as [17]:

andN

and(N) = 0

Shall (

, ) define a measure space, and shall and define measures on . We

write

if is -continuous [17]. Hence, if the second measure is continuous

with respect to the first measure , we write

We call singular with respect to (or -singular) and write

if a null set

exists with ( )

= 0 and ( ) = 0 [17]. Hence, if the first measure is zero with

respect to , and the second measure is zero with respect to

, we write .

We immediately see that the relation

is symmetric in and . The sets and

interchange their roles during the transition . Therefore, the measures and

on with

are called singular with respect to each other, or mutual singular

[17].

We have in general [17]:

( ) = ( ) + ( )

: ( ) = ( )

( ) = 0

In this case, the measure is carried by a -null set.

The unit measure or Dirac-measure is defined as [17]:

( ) = 1,

The terms -continuity and -singularity are opposite to each other. For the

Lebesgue-Borel-measure and for every Dirac-measure on

we have

( ) [17].

VII. Functional Analysis

14 Spaces and Dual Spaces

Until now, we have introduced measures on sets. Now, we shall define measures on

spaces on which functions and functionals are operating.

We will consider vector spaces of the fields

= and = . We will call said

vector spaces vector spaces. We first define a semi-norm and a norm [4].

: [0, )

( ) ( , ) = | | ( ) , ,

( ) ( + ) ( ) + ( ) ,

. ( )

( ) ( ) = 0 = 0,

( )

Norms will also be depicted by

. . The pair ( , ) is called a semi-normed or

eventually a normed space. A normed space (

, . ) possesses a natural metric [4]:

( , ) = - ,

A sequence ( ) of elements of a (semi-) normed space is called a Cauchy-

sequence if [4]

> 0 ( ) ,

( )

A metric space in which every Cauchy-sequence converges is called complete. A

complete normed space is called Banach-space [4].

The following three norms are of big importance:

| |,

| | ,

= max

,...,

| |.

The complete semi-normed vector space

( ) can be defined by using the

Lebesgue integral defined in the previous chapter 13. The interval

can be

open, semi-open, or closed, limited or unlimited. is the Lebesgue measure. For

1 we define [4]:

( ) =

: :

| |

< ,

| |

( )

The space

( ) is, as already said, a complete semi-normed vector space. The

limits in this space are not uniquely determined; they are determined only modulo the

elements of the core of the semi-norm

, namely

= { : = 0

} [4]. It is therefore obvious to identify functions

which coincide almost everywhere. The appropriate mathematical procedure is

hence to replace functions with their equivalence classes [ ], in the corresponding

quotient vector space

( ) = ( )/

We have then the following Lemma [4]:

( , .

)

( ) = { :

= 0}

( )[ ] =

( )

Normed spaces have the following properties [4]:

( ) dim < ,

( )

= { : 1}

;

( )

Hence, normed spaces have a finite dimensionality!

A metric (or topological) space is called separable, if said space possesses a

countable dense subset [4]. is called dense in when

= . Hence, the set is

dense when each nonempty open set contains a point of . Every point in is limit

of a sequence of [4].

In [4] it is proved that:

[ , ]

[ , ], ,

( [ , ], . )

As already stated in IV.11 a continuous linear mapping between normed spaces

is named continuous operator [4]. If the range of the mapping is the scalar field, we

call the mapping a functional instead of operator.

: :

A continuous operator

: fulfills hence one of the equivalent conditions [4]:

( ) lim

= , lim

( )

> 0

( )

( ) = { :

}

Next we define an operator norm [4].

( ) =

( , ),

( )

( , )

( , );

= .

In the next step dual spaces are defined [4].

( , )

Hence, if the space is defined by vectors in

, then the dual space has the

cardinality

. If the space is defined by vectors containing functions, then the dual

space has the cardinality

Hilbert spaces are Banach spaces with a scalar product as an additional structure

[4].

. , . : ×

(

) :

( ) + , = , + , ,

( ) , = , , ,

( ) , = , ,

( ) , 0

( ) , = 0 = 0

In Hilbert spaces we have the Cauchy-Schwartz-inequality [4]:

. , . .

:| , | , , , .

For a Hilbert space with infinite dimensions we have [4]:

( )

A resolvent set and a spectrum are [4]:

( ) ( ) = { :( - )

( )}

( )

: ( ) ( ),

: =

( ): = ( - )

( )

( ) =

( )

The following structure of infinite dimensional spaces is useful [4]:

( ). :

( )

0 ( ).

( ) (

)

( ) {0}

( )

( ) {0}

ker( - )

= ( ) ( )

( ) ( ),

( ) ( )

( )

ker( - )

( - )

( ) ( ).

( ) ( )

VIII. The Structure of Functions and Functionals

15 Quarkonial Representation

Until now, the structure of spaces has been described. Now, we shall refine this

description by structuring the function spaces. We ask for quarkonial representations

and Taylor series of functions, distributions/functionals/operators, and function

spaces. (Distributions are a class of linear functionals that map a set of test functions

into the set of real numbers.) Then, in 16, we shall ask for spectral representations

by Fourier and Laplace transformations.

For holomorphic functions we have the Taylor series of a function

( ) [19]:

( ) =

( )

is a non-negative

function in

with compact support such that

{ ( - ):

} is a resolution of unity in . Let

( ) =

( ) where

and

. This representation can be refined to a quarkonial representation [19]:

( ) =

This resembles, at least formally, the Weierstrassian approach to holomorphic

functions (in the complex plane), combined with the wavelet philosophy: translations

- where and dyadic dilations 2 where [19].

Hence, the idea is to describe a function or distribution/functional/operator locally as

a sum of simple functions or distributions/functionals/operators which include

translations and dilations.

Let

( ) be a complex valued function in a domain of the complex plane . There

are two versions of conditions under which

( ) is called holomorphic [19]:

(i) The Riemannian (or qualitative) approach:

( ) is holomorphic in if it is complex

differentiable at any point

[19].

(ii) The Weierstrassian (or quantitative) approach:

( ) is holomorphic in if it can

be expanded locally in a Taylor series at any point

[19].

Function spaces are usually defined in the Riemannian spirit. In [19] this standard is

complemented by a Weierstrassian approach.

Let (

) be the Schwartz space of all complex valued, rapidly decreasing, infinitely

differentiable function on

. By ( ) we denote its topological dual, the space of

all tempered distributions on

[19].

In [19] it is shown that the coefficients

can be represented as:

= 2

| |

( ) ( )

, ,

Hence, the coefficients

can be represented as pairings in (

) and ( ) [19].

Otherwise stated, the space of functions (

)of cardinality , and the space of

distributions/functionals/operators

( ) of cardinality jointly define the

coefficients

of the quarkonial decomposition.

16 Spectral Representation

According to [14] we can approximate a function

( ) by:

( ) =

( )

with

( ) =

( )

The function ( ) is called the Fourier transform of the function ( ). The

coordinates and are inverse to one another, in the sense that if

= defines a

portion of space, then

= defines the spatial frequency [14]. The function ( )

defines the spectrum of the function ( ) [14].

The broader the width of ( ) is, the narrower the width of ( ) is, and vice versa

[14].

Hence, the idea is to describe a function globally as an integral of spectral functions,

considering the entire spectrum from

- to .

The Fourier transform has the peculiarity that the exponent in the integral is pure

imaginary (

). This is very useful when examining the spectrum of a function,

since it shows which frequencies are present. But a pure imaginary exponent does

not allow a damping factor which is, however, important in physical processes. In

order to consider damping, the Fourier transform can be generalized to the Laplace

transform [34].

According to [34] we can approximate a function

( ) by:

( )

< 0

> 0,

with

( ) =

( )

The function ( ) is called the Laplace transform of the function ( ). ( ) exists if

| ( )|

> 0,

> 0

> 0 and if | ( )|

> 0 [34]. Since possesses a real and an imaginary part, real damping

factors can be jointly considered with imaginary spectral factors.

IX. Stochastic Analysis

17 Stochastic and Deterministic Processes

Until now, we have focused on spaces and described their structures. Now, we shall

concentrate on changes in spaces, said changes describing deterministic and

stochastic evolutions.

Basis for our investigation will be the probability space [18]

(; ; )

a sample space which is the set of all possible outcomes; a set of events, where

each event is a set containing zero or more outcomes; and the probability measure

function. A measurable space

( ; )

is defined by the non empty set and its -algebra [18].

Stochastic processes describe the evolution (in time) of a phenomenon under an

additional probabilistic dependence [18]. They are hence functions of two different

variables which strongly differ from one another with respect to their character: a

deterministic parameter of a set ; and a probabilistic parameter of a probability

space

(; ; ) [18]. A stochastic process can be formally defined as [18]:

(; ; )

;

( ; )

(; ; )

= ( )

(

)

:(; ; ) ( ; ); .

:(; ; )

( ) ( ) =

( )

We observe that the path mapping

is itself

/ -measurable. Hence, this path

mapping defines a probabilistic variable from which one can recover the entire

process by

, [18]. All stochastic information of the process is hence

gained from the distribution

of the path mapping [18].

Ende der Leseprobe aus 170 Seiten - nach oben

Details

Titel: General Index Theory: Its Mathematical and Physical Structures
Veranstaltung: -
Autor: Dr. Alexander Mircescu (Autor:in)
Erscheinungsjahr: 2018
Seiten: 170
Katalognummer: V423592
ISBN (eBook): 9783668691780
ISBN (Buch): 9783668691797
Sprache: Englisch
Schlagworte: Quantum Field Theory General Relativity Black Holes Universe Dark Matter Dark Energy Elementary Particles

Arbeit zitieren: Dr. Alexander Mircescu (Autor:in), 2018, General Index Theory: Its Mathematical and Physical Structures, München, GRIN Verlag, https://www.grin.com/document/423592

General Index Theory: Its Mathematical and Physical Structures

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