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− | + | ==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== | |
− | + | * [[Weyl algebra]] | |
− | + | * [[Schrodinger equation]] | |
+ | * [[Pauli equation]] | ||
+ | * [[Dirac equation]] | ||
+ | * [[spin system and Pauli exclusion principle]] | ||
+ | * [[spin structures and spinor fields]] | ||
− | + | ||
+ | == 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'}] | |
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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
- Schrodinger equation
- Pauli equation
- Dirac equation
- spin system and Pauli exclusion principle
- spin structures and spinor fields
노트
위키데이터
- ID : 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.[1]
- Another feature of Clifford Algebra is that it uses a coordinate-free representation.[2]
- 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]
- So if Clifford Algebra is so great, how come you have never heard of it before?[2]
- 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]
- Other products defined within Clifford algebras, such as the exterior product, are not used here.[3]
- In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.[4]
- The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations.[4]
- Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.[4]
- Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case.[4]
- Clifford algebras are used to define spinors.[5]
- By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry.[6]
- Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories.[6]
- Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone.[6]
- 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]
- 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]
- This seminar course will form an introduction to the theory of Clifford algebra, geometric algebras and their wide range of applications.[7]
- 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]
- Given a Clifford algebra, one would like to classify the modules over such an algebra, the Clifford modules.[9]
- We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself.[10]
- Clifford algebras, built up from quadratic spaces, have applications in many areas of mathematics, as natural generalizations of complex numbers and the quaternions.[11]
- 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]
- this is a very good book which provides a balanced and concise introduction to the subject of Clifford Algebras.[11]
- This construction is achieved by using the differential forms of Minkowski spacetime to realize a Clifford algebra of dimension 16.[12]
- 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]
- Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories.[13]
- 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]
- After this excursion into matrix algebras the viewer is hopefully prepared for Clifford algebras.[15]
- Consider the Clifford algebra Cl= Mat(4,) of the Minkowski space-time.[15]
- The detection of the last mistake, concerning functions in Clifford algebras, requires knowledge of idempotents, nilpotents and minimal polynomials.[15]
- There I give more counterexamples, some of them significant, and non-trivial even for experts in Clifford algebras.[15]
- 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]
- They can be realized as restrictions of representations of the even parts of Clifford algebras.[17]
- One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra.[17]
- There are infinite dimensional Clifford algebra constructions that appear in conformal field theory.[17]
- 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]
- It is of interest to anybody who wants to understand a modern description of spinors in Clifford algebra language.[18]
- Next, isomorphism-based universality of a Clifford algebra is defined.[18]
- 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]
- An example computes the reversion of a product of two (simple) mono-grade Clifford algebra elements (p-vectors).[18]
- This powerful language is based in Clifford algebra.[19]
- The relation to Clifford algebras shows up when we compute the square of for .[20]
- Clifford algebras have important applications in a variety of fields including geometry and theoretical physics.[21]
- 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]
- The universal characterization of the Clifford algebra shows that the construction of Cℓ(V,Q) is functorial in nature.[21]
- The Clifford algebra on Rp,q is denoted Cℓ p,q (R).[21]
- We begin by discussing why we should care about Clifford Algebra.[22]
- The real numbers are a subalgebra of Clifford algebra: just throw away all elements with grade > 0.[22]
- The complex numbers are another subalgebra of Clifford algebra, as discussed in reference 1.[22]
- Quaternions can be understood in terms of another subalgebra of Clifford algebra, namely the subalgebra containing just scalars and bivectors.[22]
- (one can also think of a Clifford algebra as a quantization of the exterior algebra).[23]
- This property determines the Clifford algebra uniquely up to canonical isomorphism.[23]
- ¶ Lift an invertible isometry m of the quadratic form of self to a Clifford algebra morphism.[23]
- ¶ Lift the matrix m to an algebra morphism of Clifford algebras.[23]
소스
- ↑ What's the Clifford algebra?
- ↑ 2.0 2.1 2.2 2.3 Clifford Algebra: A visual introduction
- ↑ Classification of Clifford algebras
- ↑ 4.0 4.1 4.2 4.3 Clifford algebra
- ↑ Clifford Algebra -- from Wolfram MathWorld
- ↑ 6.0 6.1 6.2 6.3 Some Applications of Clifford Algebra in Geometry
- ↑ 7.0 7.1 Clifford algebra, geometric algebra, and applications
- ↑ Clifford Algebra As a Bridge Between Discrete and Continuous Worlds
- ↑ Notes on BRST VIII: Clifford Algebras
- ↑ Clifford Algebra to Geometric Calculus - A Unified Language for Mathematics and Physics
- ↑ 11.0 11.1 11.2 Clifford algebras introduction
- ↑ Physical algebras in four dimensions. I. The Clifford algebra in Minkowski spacetime
- ↑ 13.0 13.1 Some Applications of Clifford Algebra in Geometry
- ↑ Clifford algebra as an adjunction?
- ↑ 15.0 15.1 15.2 15.3 Counterexamples in Clifford algebras
- ↑ Leopardi : A generalized FFT for Clifford algebras
- ↑ 17.0 17.1 17.2 17.3 Clifford algebra in nLab
- ↑ 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.
- ↑ Geometric algebra (Clifford algebra)
- ↑ Clifford Algebras
- ↑ 21.0 21.1 21.2 21.3 Abstract Algebra/Clifford Algebras
- ↑ 22.0 22.1 22.2 22.3 Introduction to Clifford Algebra
- ↑ 23.0 23.1 23.2 23.3 Clifford Algebras — Sage 9.2 Reference Manual: Algebras
메타데이터
위키데이터
- ID : Q674689
Spacy 패턴 목록
- [{'LOWER': 'clifford'}, {'LEMMA': 'algebra'}]
- [{'LOWER': 'clifford'}, {'LEMMA': 'algebra'}]