scholarly journals Real Clifford Algebras and Their Spinors for Relativistic Fermions

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 168
Author(s):  
Stefan Floerchinger
Keyword(s):  
Clifford Algebra ◽  
Clifford Algebras ◽  
Spin Groups ◽  
Mathematical Structures ◽  
Matrix Algebras ◽  
Space And Time ◽  
The Relationship ◽  
Real Complex

Real Clifford algebras for arbitrary numbers of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relationship between mathematical structures and applications to describe relativistic fermions is emphasized throughout.

scholarly journals Explicit isomorphisms of real Clifford algebras

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
N. Değırmencı ◽  
Ş. Karapazar
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Matrix Algebra ◽  
Clifford Algebras ◽  
Explicit Construction ◽  
The Other ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Other Hand ◽  
Nondegenerate Quadratic Form

It is well known that the Clifford algebraClp,qassociated to a nondegenerate quadratic form onℝn (n=p+q)is isomorphic to a matrix algebraK(m)or direct sumK(m)⊕K(m)of matrix algebras, whereK=ℝ,ℂ,ℍ. On the other hand, there are no explicit expressions for these isomorphisms in literature. In this work, we give a method for the explicit construction of these isomorphisms.

Hume’s System of Space and Time

2010 ◽  
Vol 13 (1) ◽  
pp. 76-89 ◽  
Author(s):  
Angela M. Coventry
Keyword(s):  
Fundamental Principle ◽  
Space And Time ◽  
The Relationship

Hume’s account of the origin and nature of our ideas of space and time is generally thought to be the least satisfactory part of his empiricist system of philosophy. The main reason is internal in that the account is judged to be inconsistent with Hume’s fundamental principle for the relationship between senses and cognition, the copy principle. This paper defends Hume against the inconsistency objection by offering a new systematic interpretation of Hume on space and time and illuminating more generally the role of the copy principle in his philosophy. Humes Theorie des Wesens und des Ursprungs unserer Vorstellungen von Raum und Zeit wird generell zu den am wenigsten befriedigenden Teilen seiner empiristischen Philosophie gezählt. Der Hauptgrund dafür ist werkimmanent: Die Raum- Zeit-Theorie einerseits und Humes „copy principle“ andererseits – d.h. dasjenige Fundamental-Prinzip, das die Relation zwischen unseren Sinnen und unserem Denken regelt – werden als miteinander inkonsistent erachtet. Dieser Beitrag bietet eine neue, systematische Interpretation der Raum-Zeit-Lehre Humes und eine umfassendere Darstellung der Rolle des „copy principles“ in seiner Philosophie an. Auf diese Weise wird Hume gegen den Vorwurf der Inkonsistenz verteidigt.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

scholarly journals Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1373
Author(s):  
Louis H. Kauffman
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Clifford Algebras ◽  
Discrete Systems ◽  
Complex Numbers ◽  
Majorana Fermions ◽  
Matrix Algebras ◽  
Dirac Equations ◽  
Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

scholarly journals OBSERVAÇÕES CRÍTICAS SOBRE UMA RESENHA DE GUIDO IMAGUIRE DA OBRA: ESTRUTURA E SER. UM QUADRO REFERENCIAL TEÓRICO PARA UMA FILOSOFIA SISTEMÁTICA

2012 ◽  
Vol 40 (126) ◽  
pp. 43
Author(s):  
Lorenz B. Puntel
Keyword(s):  
Mathematical Structures ◽  
Justified Belief ◽  
Philosophical Language ◽  
Systematic Philosophy ◽  
The Status ◽  
Important Argument ◽  
The Aesthetic ◽  
Definition Of ◽  
The Relationship ◽  
Ontological Theory

Este artigo responde pormenorizadamente às críticas feitas por G. Imaguire em sua resenha do livro indicado no título (= ES). Trata-se principalmente de nove temas respectivamente teses de caráter central para a concepção exposta no livro. O presente artigo analisa cada um destes temas, em parte corrigindo erros de apresentação e de interpretação e em todos os casos respondendo às objeções de Imaguire. Trata-se dos seguintes temas/teses: (1) Para esclarecer o estatuto das sentenças filosóficas, ES propõe uma teoria dos três operadores que explicitam o caráter de sentenças: são estes o operador teórico, o operador prático e o operador estético. O artigo esclarece o sentido exato desta teoria. (2) ES apresenta uma nova definição de saber/conhecimento em oposição direta à já famosa definição “knowledge is true justified belief” articulada por E. Gettier. (3) ES defende uma concepção de orientação ontológica das estruturas formais fundamentais (lógicas e matemáticas); estas são esclarecidas. (4) Em ES é exposta e defendida uma nova concepção de ontologia em perfeita conformidade com a semântica de uma linguagem filosófica transparente; esta ontologia exclui o conceito de “substância” e critica o uso do conceito de “objeto”. (5) ES expõe uma nova teoria semântico-ontológica da verdade que tem como consequência um relativismo moderado da verdade. (6) ES formula um argumento muito especial contra o fisicalismo; o artigo explica pormenorizadamente este argumento. (7) A concepção exposta em ES afirma que o cristianismo, em virtude do caráter racional e teórico da teologia que o explicita, constitui, em oposição a outras religiões, uma temática com prioridade de importância e atenção para o filósofo sistemático. Neste artigo esta tese é explicada e defendida contra interpretações erradas. (8) O oitavo tema é a grande questão posta pelo conceito de mundo no contexto das relações entre teorias filosóficas e teorias científicas. O artigo esclarece uma série de mal-entendidos a respeito deste grande tema. (9) Finalmente, com relação a um argumento-chave que ES apresenta para fundamentar a tese que, por razões sistemáticas, se deve admitir uma dimensão absolutamente necessária do Ser, o artigo demonstra que a resenha comete um muito grave erro de interpretação, baseando neste erro uma crítica infundada ao argumento. O artigo esclarece extensamente o argumento, suas pressuposições e suas consequências.Abstract: This article is a detailed answer to G. Imaguire’s criticisms of the book Structure and Being: A Theoretical Framework for a Systematic Philosophy (hence referred as ES). Imaguire focuses on nine topics that are central to the book. The present article analyses each one of these theses, sometimes correcting errors made and misrepresentations introduced by Imaguire, and in all cases, responding to Imaguire’s objections. The theses are the following: (1) In order to clarify the status of theoretical sentences occurring in philosophical works, ES presents a theory about the three operators that make explicit the statuses of three mutually irreducible kinds of sentence: the theoretical operator, the practical operator, and the aesthetic operator. (2) ES offers a new definition of knowledge in significant opposition to the now-famous definition formulated by E. Gettier, “knowledge is true justified belief.” (3) ES defends an ontologically oriented conception of the fundamental formal (logical and mathematical) structures. (4) In ES, a new ontology is propounded in strong conformity with the semantics of a transparent philosophical language. This ontology rejects the category of substance and criticizes the widely used concept of object. (5) ES presents a completely new semantico-ontological theory of truth. One of its consequences is a moderate relativism with respect to truth. (6) ES presents a unique argument against physicalism; this article elaborates on it. (7) ES considers the phenomenon of religion and states that, due to its rational and theoretical theology, Christian religion, in opposition to other religions, provides a uniquely promising resource for philosophical considerations. (8) ES extensively thematizes the concept of world in connection with the problem of the relationship between philosophy and science. (9) Finally, ES develops the main features of a theory of Being as such and as a whole. ES offers especially an important argument on behalf of the thesis that the universal dimension of Being must be conceived of as two-dimensional: as the dimension of absolutely necessary Being and the dimension of contingent beings. This article reconstructs the exact meaning of the argument and explains its presuppositions and consequences.

scholarly journals Conclusion

2021 ◽  
pp. 93-98
Author(s):  
Rosita Fibbi ◽  
Arnfinn H. Midtbøen ◽  
Patrick Simon
Keyword(s):  
Racial Discrimination ◽  
Future Research ◽  
Space And Time ◽  
The Relationship

AbstractThis chapter briefly summarizes the content of the book, emphasizing how the impressive breadth of research reveal a worrying picture of enduring discrimination in immigrant-receiving societies across space and time, suggesting the contour of troubling “three P’s” in contemporary European societies: that ethno-racial discrimination appears to be pervasive, perpetuating, and persistent. The chapter also revisits the relationship between theories of integration and discrimination, and it concludes by pointing out promising for future research on discrimination.

The Moral World

2020 ◽  
pp. 228-240
Author(s):  
Christopher J. Insole
Keyword(s):  
Highest Good ◽  
Concept Of God ◽  
The Third ◽  
Space And Time ◽  
Third Phase ◽  
Moral World ◽  
The Relationship ◽  
The Way

This chapter shows how central it is, for Kant, that the concept of God only comes downstream from, and after, the possibility of belief in the ‘moral world’. This moral world is the realm of freedom, wherein autonomy is possible. Only if (deterministic) space and time do not go ‘all the way down’, are freedom, and autonomy, possible. If space and time are ‘things-in-themselves’, Kant asserts, ‘then freedom cannot be saved’ (A536/B564). Only if there is a dimension of reality beyond mechanism, is end-setting, and so autonomy, and the highest good possible. Not even God could achieve the highest good in a universe without end-setting, and without freedom, because this universe would be a sort of ‘desert’ with no ‘inner value’. The sequence of thought we find, both in the second Critique, and in other texts is this: first of all, Kant identifies a need for happiness in proportion to virtue; then Kant identifies the obstacle to the realization of such happiness, which is the mechanistic and deterministic structure of nature; and then Kant moves to the solution, which involves leaning into the realm of freedom, which realm includes God. The significance of the third phase in the progression of thought (the realm of freedom) has not been sufficiently considered, it is argued, when considering the Kant’s ‘moral proof’, and the relationship, for Kant, between morality, the highest good, and God.

Infinity

2018 ◽  
Author(s):  
A.W. Moore
Keyword(s):  
Nineteenth Century ◽  
Conscious Awareness ◽  
Mathematical Study ◽  
Space And Time ◽  
Divine Perfection ◽  
To Come ◽  
The Relationship

The infinite is standardly conceived as that which is endless, unlimited, immeasurable. It also has theological connotations of absoluteness and perfection. From the dawn of civilization, it has held a special fascination: people have been captivated by the boundlessness of space and time, by the mystery of numbers going on forever, by the paradoxes of endless divisibility and by the riddles of divine perfection. The infinite is of profound importance to mathematics. Nevertheless, the relationship between the two has been a curiously ambivalent one. It is clear that mathematics in some sense presupposes the infinite, for instance in the fact that there is no largest integer. But the idea that the infinite should itself be an object of mathematical study has time and again been subjected to ridicule. In the nineteenth century this orthodoxy was challenged, with the advent of ‘transfinite arithmetic’. Many, however, have remained sceptical, believing that the infinite is inherently beyond our grasp. Perhaps their scepticism should be trained on the infinite itself: perhaps the concept is ultimately incoherent. It is certainly riddled with paradoxes. Yet we cannot simply jettison it. This is why the paradoxes are so acute. The roots of these paradoxes lie in our own finitude: it is self-conscious awareness of that finitude which gives us our initial sense of a contrasting infinite, and, at the same time, makes us despair of knowing anything about it, or having any kind of grasp of it. This creates a tension. We feel pressure to acknowledge the infinite, and we feel pressure not to. In trying to come to terms with the infinite, we are trying to come to terms with a basic conflict in ourselves.

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