## Mathematics (11)

### Towards a Universal Probability Theory, Part I

Written by Mlakár Katalin*Introduction Andrei Khrennikov's Contextual Probability and the Two Slit Experiment*

Quoted **Khrennikov**:

“One of problems was of purely mathematical character. The standard probabilistic formalism based on Kolmogorov’s axiomatics, 1933, was a fixed context formalism. This conventional probabilistic formalism does not provide rules of operating with probabilities calculated for different contexts. However, in quantum theory we have to operate with statistical data obtained for different complexes of physical conditions, contexts. In fact, this context dependence of probabilities as the origin of the superposition principle was already discussed by W. Heisenberg; unfortunately, only in quite general and rather philosophic framework.”^{1}

**Contents**

- Khrennikov's contextualism
- What's next?
- Feynman and the two slit experiment
- Appendix

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^{1}**Andrei Khrennikov**, *„**Contextual viewpoint to quantum stochastics”; **https://arxiv.org/pdf/hep-th/0112076.pdf*

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**The Ambiguity of Geometric Algebra in the Models of Spacetime**

**Contents**

1. Memo

1.1. Multivectors in GA_{3}

1.2. Conjugations in GA_{3}

1.2.1. Clifford conjugation

1.2.2. Pauli conjugation

1.2.3. The role of complex imaginary unit in the Clifford and Pauli conjugations

2. Minkowski spacetime generated in GA_{1,3}

2.1. Bivektor algebra

2.2. Lorentz transformation

3. Spacetime model in classic GA_{3}: the scalar element as a time representation

4. Summary

1. Appendix

Pauli algebra and matrices

2. Appendix

Gamma or Dirac matrices, Dirac algebra

3. Appendix

Lorentz transformation

a. Einstein description with coordinates

b. Applying hyperbolic functions

c. Usage of hyperbolic numbers to describe space time

References

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**Conjugation concepts in GA _{3}**

**Contents**

- Memo
- Involutions: reversion and conjugation in GA
_{3}- Reversion
- Magnitude or modulus
- Conjugation

- Comments

Appendix

Pauli algebra

**References**

„**David Hestenes**,*Space–Time Algebra”***David Hestenes**, „*New Foundations for Classical Mechanics”***Chris Doran**&**Anthony Lasenby,**„*Geometric Algebra for Physicist”***Stephen Gull, Anthony Lasenb****y, Chris Doran**,*„Imaginary Numbers are not Real — the Geometric Algebra of Spacetime”***http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf**

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*Subjective and unfinished thoughts on GA dimensions*

In everyday life we ordinarily use the term dimension as spatial size. In sciences, than the mathematics and physics, the term of the dimension is used in a diverse, different sense.

**Contents**

- Dimension and grade in GA
- Unit of measurement as dimension
- Qualitative infinite as dimension

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### CA, GA and the Two-element Numbers – Secondly

Written by Mlakár Katalin**With Exciting Dénouement**

**Contents**

- Introduction
- Geometric Algebra of the Plane
- The Algebra of 3-Space
- Conclusions

**Abstract by a quotation**

*„These considerations all indicate that our present thinking about quantum mechanics is infested with the deepest misconceptions. We believe, with David Hestenes, that geometric algebra is an essential ingredient in unravelling these misconceptions.”*

^{1}

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^{1} See **Stephen Gull, Anthony Lasenby, Chris Doran**, „Imaginary Numbers are not Real — the Geometric Algebra of Spacetime”;

**http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf**

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### The Clifford Algebra, the Geometric Algebra and the Two-element Numbers

Written by Mlakár Katalin**Abstract**

The geometric algebra (GA) is a Clifford algebra (CA) of a vector space over the field of real numbers. A similar concept of complex numbers can be derived from the GA. It is very instructive to compare, as the various authors deduce one of the two-element numbers from CA, or from GA.

**Contents**

- David Hestenes
- Garret Sobczyk
- Doran-Lasenby
- Stefan Ulrich
- Summary

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*Desiderata*

**Content**

- The notion of number
- Desiderata to the extension of the numbers
- Summary

**Abstract**

Although the numbers are yet considered as the simplest abstractions of mathematics, it is increasingly taking shape is a picture in which the numbers will the most complex and exciting concept in mathematics. The geometric algebra is a probable candidate for the new number system, but desiderata of the author is unlike the criteria of the geometric algebra in three points:

- The number line of real numbers is orthogonal to the space-dimensions according to the definition of scalar product.
- As the fundamental operation of arithmetic the product is commutative.
- The space-dimensions evolve from time-dimension which is represented by number line of real numbers.

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### Not space and time, but space-time has to use in the probability theory

Written by Mlakár Katalin**Two-element numbers as space-time elements of probability theory**

**Abstract**

- Introductory
- The probability theory and the space-time
- Different interpretations and problems of the concept of probability
- The
**Bertrand**paradox and his explanation - Relative frequency concept and the paradoxes
**Kolmogorov**’s probability theory

- The
- Epilog

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### The way from the natural numbers through the real numbers to the two-element numbers

Written by Mlakár Katalin**The role of the concept of computability in physics**

*“It would require infinite precision for the coordinates of a phase-space point- i.e. all the decimal places!- in order for it to make sense to say that the point is non-computable. (A number described by a finite decimal is always computable.) A finite portion of .a decimal expansion of a number tells us nothing about the computability of the entire expansion of that number. But all physical measurements have a definite limitation on how accurately they can be performed, and can only give information about a finite number of decimal places. Does this nullify the whole concept of 'computable number' as applied to physical measurements?” (Roger Penrose, The Emperor’s New Mind – Concerning Computers, Minds, and The Laws of Physics, Chapter 5; The classical World, Phase space)*

**Contents**

1. The natural numbers and the qualitative characteristic of infinite

2. The positional numeral systems and the two-element numbers

3. “Actualization” of the potentially infinite – examples in physics

4. Epilogue

*Appendix*

* *

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### The Infinity which is hiding in the Geometric Algebra

Written by Mlakár Katalin**Content**

The infinity and the two-element numbers

Infinity differently – a heuristic approach

Infinity described with homogeneous coordinates

The infinity and the geometric algebra

The complex and hyperbolic numbers as two-dimensional Clifford algebras - examples from the literature

The hiding infinity

**Abstract**

Types of geometric algebra generated by vectors of 1-dimensional vector space are not at all exhibit poor structure. These geometric algebras have very exciting basic structure as they are represented by two-element-numbers which are the complex, the parabolic (dual) and the hyperbolic numbers. On the other hand, special types of infinities – related to the continuum hypothesis – are modeled by the two-element numbers. So it can be said that vectors of 1-dimensional vector space, which generates the above mentioned geometric algebras are nothing more than the infinite extensions of scalars of the geometric algebras.

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#### More...

In the geometric algebra^{1} there are some problems, which not found in the literature, and I would like to share my thoughts about these now. I restrict my examination to those geometric algebras, where the starting point is a finite dimensional Euclidean vector space and the real numbers. In the Euclidean vector space an internal product is defined with an outer product generalized by Grassmann. After that, a so-called geometric product, which will be the basic operation in the geometric algebra, is defined.

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^{1} See for example the book of **Chris Doran** - **Anthony Lasenby** titled *Geometric Algebra for Physicists*;**http://www.cambridge.org/nl/academic/subjects/physics/theoretical-physics-and-mathematical-physics/geometric-algebra-physicists?format=PB**

or an article titled *„Imaginary Numbers are not Real — the Geometric Algebra of Spacetime**” by Gull-*

**Lasenby-Doran**

**http://www.researchgate.net/publication/226188504_Imaginary_numbers_are_not_realThe_geometric_algebra_of_spacetime**

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