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3번째 줄: 3번째 줄:
 
* Clifford algebras may be thought of as quantizations (cf. quantization (physics), Quantum group) of the exterior algebra, in the same way that the [[Weyl algebra]] is a quantization of the symmetric algebra.
 
* Clifford algebras may be thought of as quantizations (cf. quantization (physics), Quantum group) of the exterior algebra, in the same way that the [[Weyl algebra]] is a quantization of the symmetric algebra.
  
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==spinor==
 
==spinor==
  
 
* Spinors are classified according to Dirac, Weyl, Majorana and Weyl-Majorana spinors.
 
* Spinors are classified according to Dirac, Weyl, Majorana and Weyl-Majorana spinors.
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*  applications<br>
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*  applications
 
** spinor bundles
 
** spinor bundles
 
** spin connections
 
** spin connections
 
** the role of spinors in the description of the fundamental interactions between elementary particles
 
** the role of spinors in the description of the fundamental interactions between elementary particles
  
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==related items==
 
==related items==
28번째 줄: 23번째 줄:
 
* [[spin system and Pauli exclusion principle]]
 
* [[spin system and Pauli exclusion principle]]
 
* [[spin structures and spinor fields]]
 
* [[spin structures and spinor fields]]
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 +
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== 노트 ==
 +
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q674689 Q674689]
 +
===말뭉치===
 +
# If you ever want to read it from an algebraic perspective, Clifford algebras appear in Jacobson's Basic Algebra volumes (one or both), and they also appear in Artin's Geometric Algebra.<ref name="ref_483011ad">[https://math.stackexchange.com/questions/261509/whats-the-clifford-algebra What's the Clifford algebra?]</ref>
 +
# Another feature of Clifford Algebra is that it uses a coordinate-free representation.<ref name="ref_1783aa37">[https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ Clifford Algebra: A visual introduction]</ref>
 +
# And the way that Clifford Algebra achieves this extraordinary Grand Unification of mathematics is by expressing algebraic concepts in the form of spatial operations on spatial structures.<ref name="ref_1783aa37" />
 +
# So if Clifford Algebra is so great, how come you have never heard of it before?<ref name="ref_1783aa37" />
 +
# Clifford Algebra was briefly rediscovered by Dirac as a superior expression of his theory of the electron, although the wider significance was not recognized.<ref name="ref_1783aa37" />
 +
# Other products defined within Clifford algebras, such as the exterior product, are not used here.<ref name="ref_0f52e470">[https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras Classification of Clifford algebras]</ref>
 +
# In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.<ref name="ref_62ded232">[https://en.wikipedia.org/wiki/Clifford_algebra Clifford algebra]</ref>
 +
# The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations.<ref name="ref_62ded232" />
 +
# Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.<ref name="ref_62ded232" />
 +
# Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case.<ref name="ref_62ded232" />
 +
# Clifford algebras are used to define spinors.<ref name="ref_bb10a191">[https://mathworld.wolfram.com/CliffordAlgebra.html Clifford Algebra -- from Wolfram MathWorld]</ref>
 +
# By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry.<ref name="ref_d06d988f">[https://www.intechopen.com/online-first/some-applications-of-clifford-algebra-in-geometry Some Applications of Clifford Algebra in Geometry]</ref>
 +
# Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories.<ref name="ref_d06d988f" />
 +
# Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone.<ref name="ref_d06d988f" />
 +
# Clifford algebra combines all the advantages of quaternion with the advantages of vector algebra and uniformly and succinctly describes the contents of geometry and physics.<ref name="ref_d06d988f" />
 +
# The best-known application of Clifford algebras is probably the "classical" theory of orthogonal maps and spinors which is used intensively in modern theoretical physics and differential geometry.<ref name="ref_77074881">[http://www.mathematik.uni-muenchen.de/~lundholm/clifford.php Clifford algebra, geometric algebra, and applications]</ref>
 +
# This seminar course will form an introduction to the theory of Clifford algebra, geometric algebras and their wide range of applications.<ref name="ref_77074881" />
 +
# We will focus on some cases where Clifford algebra manifestly shows discrete features with the hope that shedding light on some details may help a more complete scenario to come out of darkness.<ref name="ref_46584d78">[https://link.springer.com/article/10.1007/s00006-018-0884-5 Clifford Algebra As a Bridge Between Discrete and Continuous Worlds]</ref>
 +
# Given a Clifford algebra, one would like to classify the modules over such an algebra, the Clifford modules.<ref name="ref_96e2a6b1">[https://www.math.columbia.edu/~woit/wordpress/?p=1387 Notes on BRST VIII: Clifford Algebras]</ref>
 +
# We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself.<ref name="ref_260f844b">[https://www.springer.com/gp/book/9789027716736 Clifford Algebra to Geometric Calculus - A Unified Language for Mathematics and Physics]</ref>
 +
# Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions.<ref name="ref_70f8a9c0">[http://www.cambridge.org/9781107096387 Clifford algebras introduction]</ref>
 +
# This is a notable book that constitutes a valuable addition to the library of anyone interested in the study of Clifford algebras and their applications.<ref name="ref_70f8a9c0" />
 +
# this is a very good book which provides a balanced and concise introduction to the subject of Clifford Algebras.<ref name="ref_70f8a9c0" />
 +
# This construction is achieved by using the differential forms of Minkowski spacetime to realize a Clifford algebra of dimension 16.<ref name="ref_f7c913ec">[https://www.sciencedirect.com/science/article/pii/0196885883900027 Physical algebras in four dimensions. I. The Clifford algebra in Minkowski spacetime]</ref>
 +
# In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra.<ref name="ref_e0b2ca80">[https://www.preprints.org/manuscript/202002.0140/v1 Some Applications of Clifford Algebra in Geometry]</ref>
 +
# Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories.<ref name="ref_e0b2ca80" />
 +
# I would like to view the construction of the Clifford algebra as a functor from the category of quadratic vector spaces to the category of associative algebras.<ref name="ref_1bfeff52">[https://mathoverflow.net/questions/7687/clifford-algebra-as-an-adjunction Clifford algebra as an adjunction?]</ref>
 +
# After this excursion into matrix algebras the viewer is hopefully prepared for Clifford algebras.<ref name="ref_545d2c9f">[https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexamples.htm Counterexamples in Clifford algebras]</ref>
 +
# Consider the Clifford algebra Cl= Mat(4,) of the Minkowski space-time.<ref name="ref_545d2c9f" />
 +
# The detection of the last mistake, concerning functions in Clifford algebras, requires knowledge of idempotents, nilpotents and minimal polynomials.<ref name="ref_545d2c9f" />
 +
# There I give more counterexamples, some of them significant, and non-trivial even for experts in Clifford algebras.<ref name="ref_545d2c9f" />
 +
# This paper describes a relationship between fast real matrix representations of real universal Clifford algebras and the generalized Fast Fourier Transform for supersolvable groups.<ref name="ref_cf6f1f38">[https://projecteuclid.org/euclid.bbms/1110205626 Leopardi : A generalized FFT for Clifford algebras]</ref>
 +
# They can be realized as restrictions of representations of the even parts of Clifford algebras.<ref name="ref_588746b0">[https://ncatlab.org/nlab/show/Clifford+algebra Clifford algebra in nLab]</ref>
 +
# One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra.<ref name="ref_588746b0" />
 +
# There are infinite dimensional Clifford algebra constructions that appear in conformal field theory.<ref name="ref_588746b0" />
 +
# Clifford algebras are often defined using bilinear forms instead of quadratic forms (and one often sees incorrect definitions of quadratic forms in terms of bilinear forms).<ref name="ref_588746b0" />
 +
# It is of interest to anybody who wants to understand a modern description of spinors in Clifford algebra language.<ref name="ref_908f42a4">[https://journals.iucr.org/a/issues/2020/02/00/xo0142/ (IUCr) An Introduction to Clifford Algebras and Spinors. By Jayme Vaz Jr and Roldão da Rocha Jr. Oxford University Press, 2019. Paperback, pp. 256. Price GBP 34.99. ISBN 9780198836285.]</ref>
 +
# Next, isomorphism-based universality of a Clifford algebra is defined.<ref name="ref_908f42a4" />
 +
# The center of a Clifford algebra and isomorphisms of grade involution, reversion and (Clifford) conjugation are introduced, with obvious relations to corresponding notions in tensor algebra.<ref name="ref_908f42a4" />
 +
# An example computes the reversion of a product of two (simple) mono-grade Clifford algebra elements (p-vectors).<ref name="ref_908f42a4" />
 +
# This powerful language is based in Clifford algebra.<ref name="ref_1f8f1a87">[https://staff.science.uva.nl/l.dorst/clifford/index.html Geometric algebra (Clifford algebra)]</ref>
 +
# The relation to Clifford algebras shows up when we compute the square of for .<ref name="ref_b56976d7">[https://math.ucr.edu/home/baez/octonions/node6.html Clifford Algebras]</ref>
 +
# Clifford algebras have important applications in a variety of fields including geometry and theoretical physics.<ref name="ref_96fbe11e">[https://en.wikibooks.org/wiki/Abstract_Algebra/Clifford_Algebras Abstract Algebra/Clifford Algebras]</ref>
 +
# It is then straightforward to show that Cℓ(V,Q) contains V and satisfies the above universal property, so that Cℓ is unique up to isomorphism; thus one speaks of "the" Clifford algebra Cℓ(V, Q).<ref name="ref_96fbe11e" />
 +
# The universal characterization of the Clifford algebra shows that the construction of Cℓ(V,Q) is functorial in nature.<ref name="ref_96fbe11e" />
 +
# The Clifford algebra on Rp,q is denoted Cℓ p,q (R).<ref name="ref_96fbe11e" />
 +
# We begin by discussing why we should care about Clifford Algebra.<ref name="ref_2ce1ad47">[https://www.av8n.com/physics/clifford-intro.htm Introduction to Clifford Algebra]</ref>
 +
# The real numbers are a subalgebra of Clifford algebra: just throw away all elements with grade > 0.<ref name="ref_2ce1ad47" />
 +
# The complex numbers are another subalgebra of Clifford algebra, as discussed in reference 1.<ref name="ref_2ce1ad47" />
 +
# Quaternions can be understood in terms of another subalgebra of Clifford algebra, namely the subalgebra containing just scalars and bivectors.<ref name="ref_2ce1ad47" />
 +
# (one can also think of a Clifford algebra as a quantization of the exterior algebra).<ref name="ref_3beb9dbf">[https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/clifford_algebra.html Clifford Algebras — Sage 9.2 Reference Manual: Algebras]</ref>
 +
# This property determines the Clifford algebra uniquely up to canonical isomorphism.<ref name="ref_3beb9dbf" />
 +
# ¶ Lift an invertible isometry m of the quadratic form of self to a Clifford algebra morphism.<ref name="ref_3beb9dbf" />
 +
# ¶ Lift the matrix m to an algebra morphism of Clifford algebras.<ref name="ref_3beb9dbf" />
 +
===소스===
 +
<references />
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q674689 Q674689]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'clifford'}, {'LEMMA': 'algebra'}]
 +
* [{'LOWER': 'clifford'}, {'LEMMA': 'algebra'}]

2021년 2월 17일 (수) 02:16 기준 최신판

introduction

  • Clifford algebras may be thought of as quantizations (cf. quantization (physics), Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.


spinor

  • Spinors are classified according to Dirac, Weyl, Majorana and Weyl-Majorana spinors.
  • applications
    • spinor bundles
    • spin connections
    • the role of spinors in the description of the fundamental interactions between elementary particles


related items


노트

위키데이터

말뭉치

  1. If you ever want to read it from an algebraic perspective, Clifford algebras appear in Jacobson's Basic Algebra volumes (one or both), and they also appear in Artin's Geometric Algebra.[1]
  2. Another feature of Clifford Algebra is that it uses a coordinate-free representation.[2]
  3. And the way that Clifford Algebra achieves this extraordinary Grand Unification of mathematics is by expressing algebraic concepts in the form of spatial operations on spatial structures.[2]
  4. So if Clifford Algebra is so great, how come you have never heard of it before?[2]
  5. Clifford Algebra was briefly rediscovered by Dirac as a superior expression of his theory of the electron, although the wider significance was not recognized.[2]
  6. Other products defined within Clifford algebras, such as the exterior product, are not used here.[3]
  7. In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.[4]
  8. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations.[4]
  9. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.[4]
  10. Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case.[4]
  11. Clifford algebras are used to define spinors.[5]
  12. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry.[6]
  13. Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories.[6]
  14. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone.[6]
  15. Clifford algebra combines all the advantages of quaternion with the advantages of vector algebra and uniformly and succinctly describes the contents of geometry and physics.[6]
  16. The best-known application of Clifford algebras is probably the "classical" theory of orthogonal maps and spinors which is used intensively in modern theoretical physics and differential geometry.[7]
  17. This seminar course will form an introduction to the theory of Clifford algebra, geometric algebras and their wide range of applications.[7]
  18. We will focus on some cases where Clifford algebra manifestly shows discrete features with the hope that shedding light on some details may help a more complete scenario to come out of darkness.[8]
  19. Given a Clifford algebra, one would like to classify the modules over such an algebra, the Clifford modules.[9]
  20. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself.[10]
  21. Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions.[11]
  22. This is a notable book that constitutes a valuable addition to the library of anyone interested in the study of Clifford algebras and their applications.[11]
  23. this is a very good book which provides a balanced and concise introduction to the subject of Clifford Algebras.[11]
  24. This construction is achieved by using the differential forms of Minkowski spacetime to realize a Clifford algebra of dimension 16.[12]
  25. In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra.[13]
  26. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories.[13]
  27. I would like to view the construction of the Clifford algebra as a functor from the category of quadratic vector spaces to the category of associative algebras.[14]
  28. After this excursion into matrix algebras the viewer is hopefully prepared for Clifford algebras.[15]
  29. Consider the Clifford algebra Cl= Mat(4,) of the Minkowski space-time.[15]
  30. The detection of the last mistake, concerning functions in Clifford algebras, requires knowledge of idempotents, nilpotents and minimal polynomials.[15]
  31. There I give more counterexamples, some of them significant, and non-trivial even for experts in Clifford algebras.[15]
  32. This paper describes a relationship between fast real matrix representations of real universal Clifford algebras and the generalized Fast Fourier Transform for supersolvable groups.[16]
  33. They can be realized as restrictions of representations of the even parts of Clifford algebras.[17]
  34. One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra.[17]
  35. There are infinite dimensional Clifford algebra constructions that appear in conformal field theory.[17]
  36. Clifford algebras are often defined using bilinear forms instead of quadratic forms (and one often sees incorrect definitions of quadratic forms in terms of bilinear forms).[17]
  37. It is of interest to anybody who wants to understand a modern description of spinors in Clifford algebra language.[18]
  38. Next, isomorphism-based universality of a Clifford algebra is defined.[18]
  39. The center of a Clifford algebra and isomorphisms of grade involution, reversion and (Clifford) conjugation are introduced, with obvious relations to corresponding notions in tensor algebra.[18]
  40. An example computes the reversion of a product of two (simple) mono-grade Clifford algebra elements (p-vectors).[18]
  41. This powerful language is based in Clifford algebra.[19]
  42. The relation to Clifford algebras shows up when we compute the square of for .[20]
  43. Clifford algebras have important applications in a variety of fields including geometry and theoretical physics.[21]
  44. It is then straightforward to show that Cℓ(V,Q) contains V and satisfies the above universal property, so that Cℓ is unique up to isomorphism; thus one speaks of "the" Clifford algebra Cℓ(V, Q).[21]
  45. The universal characterization of the Clifford algebra shows that the construction of Cℓ(V,Q) is functorial in nature.[21]
  46. The Clifford algebra on Rp,q is denoted Cℓ p,q (R).[21]
  47. We begin by discussing why we should care about Clifford Algebra.[22]
  48. The real numbers are a subalgebra of Clifford algebra: just throw away all elements with grade > 0.[22]
  49. The complex numbers are another subalgebra of Clifford algebra, as discussed in reference 1.[22]
  50. Quaternions can be understood in terms of another subalgebra of Clifford algebra, namely the subalgebra containing just scalars and bivectors.[22]
  51. (one can also think of a Clifford algebra as a quantization of the exterior algebra).[23]
  52. This property determines the Clifford algebra uniquely up to canonical isomorphism.[23]
  53. ¶ Lift an invertible isometry m of the quadratic form of self to a Clifford algebra morphism.[23]
  54. ¶ Lift the matrix m to an algebra morphism of Clifford algebras.[23]

소스

  1. ↑ What's the Clifford algebra?
  2. ↑ 2.0 2.1 2.2 2.3 Clifford Algebra: A visual introduction
  3. ↑ Classification of Clifford algebras
  4. ↑ 4.0 4.1 4.2 4.3 Clifford algebra
  5. ↑ Clifford Algebra -- from Wolfram MathWorld
  6. ↑ 6.0 6.1 6.2 6.3 Some Applications of Clifford Algebra in Geometry
  7. ↑ 7.0 7.1 Clifford algebra, geometric algebra, and applications
  8. ↑ Clifford Algebra As a Bridge Between Discrete and Continuous Worlds
  9. ↑ Notes on BRST VIII: Clifford Algebras
  10. ↑ Clifford Algebra to Geometric Calculus - A Unified Language for Mathematics and Physics
  11. ↑ 11.0 11.1 11.2 Clifford algebras introduction
  12. ↑ Physical algebras in four dimensions. I. The Clifford algebra in Minkowski spacetime
  13. ↑ 13.0 13.1 Some Applications of Clifford Algebra in Geometry
  14. ↑ Clifford algebra as an adjunction?
  15. ↑ 15.0 15.1 15.2 15.3 Counterexamples in Clifford algebras
  16. ↑ Leopardi : A generalized FFT for Clifford algebras
  17. ↑ 17.0 17.1 17.2 17.3 Clifford algebra in nLab
  18. ↑ 18.0 18.1 18.2 18.3 (IUCr) An Introduction to Clifford Algebras and Spinors. By Jayme Vaz Jr and Roldão da Rocha Jr. Oxford University Press, 2019. Paperback, pp. 256. Price GBP 34.99. ISBN 9780198836285.
  19. ↑ Geometric algebra (Clifford algebra)
  20. ↑ Clifford Algebras
  21. ↑ 21.0 21.1 21.2 21.3 Abstract Algebra/Clifford Algebras
  22. ↑ 22.0 22.1 22.2 22.3 Introduction to Clifford Algebra
  23. ↑ 23.0 23.1 23.2 23.3 Clifford Algebras — Sage 9.2 Reference Manual: Algebras

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'clifford'}, {'LEMMA': 'algebra'}]
  • [{'LOWER': 'clifford'}, {'LEMMA': 'algebra'}]
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