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(피타고라스님이 이 페이지의 이름을 Clifford algebras and spinors로 바꾸었습니다.) |
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13번째 줄: | 13번째 줄: | ||
− | <h5> | + | <h5>spinor</h5> |
− | * | + | * consider a representation of [[Clifford algebras and spinors|Clifford algebras]] |
− | * | + | * the elements in this space are called spinors |
+ | * Spinors are classified according to Dirac, Weyl, Majorana and Weyl-Majorana spinors. | ||
+ | * applications<br> | ||
+ | ** spinor bundles | ||
+ | ** spin connections | ||
+ | ** the role of spinors in the description of the fundamental interactions between elementary particles | ||
22번째 줄: | 27번째 줄: | ||
− | <h5> | + | <h5>Pauli spinor</h5> |
+ | |||
+ | * 8-dimensional real algebra | ||
+ | * isomorphic to C(E_{3}) Clifford algebra of the Euclidean space E_{3} | ||
− | + | * http://en.wikipedia.org/wiki/Spinors_in_three_dimensions | |
+ | * spinor = a vector in two-dimensional space over complex number field | ||
+ | * Hermitian dot product is given on the vector space | ||
+ | * the space of spinors is a projective representation of the orthogonal group. | ||
30번째 줄: | 41번째 줄: | ||
− | + | <h5>Dirac matrices</h5> | |
− | + | * 16 dimensional real algebra | |
+ | * isomorphic to C(E_{3,1}) Clifford algebra of the Minkowski space E_{3,1} | ||
− | + | * <math>\gamma_{\mu}^2=\epsilon_{\mu}</math>, <math>\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=0</math><math>\epsilon_{0}=1, \epsilon_{i}=-1</math> | |
+ | * there exists unique four dimensional representation of a Clifford algebra | ||
+ | * projective representation of the Lorentz group | ||
+ | * universal covering of the Lorentz group H=SL(2,\mathbb{C}) also acts on it | ||
51번째 줄: | 66번째 줄: | ||
* [[Weyl algebra]] | * [[Weyl algebra]] | ||
+ | |||
+ | * [[Schrodinger equation]] | ||
+ | * [[Pauli equation]] | ||
+ | * [[Dirac equation]] | ||
+ | * [[spin system and Pauli exclusion principle]] | ||
+ | * [[spin structures and spinor fields]] | ||
+ | |||
+ | |||
81번째 줄: | 104번째 줄: | ||
* Lachièze-Rey, Marc. 2009. “Spin and Clifford Algebras, an Introduction”. <em>Advances in Applied Clifford Algebras</em> 19 (3-4): 687-720. doi:10.1007/s00006-009-0187-y. | * Lachièze-Rey, Marc. 2009. “Spin and Clifford Algebras, an Introduction”. <em>Advances in Applied Clifford Algebras</em> 19 (3-4): 687-720. doi:10.1007/s00006-009-0187-y. | ||
− | * [http://www.math.ucla.edu/%7Evsv/papers/ch5.pdf http://www.math.ucla.edu/~vsv/papers/ch5.pdf] | + | * [http://www.math.ucla.edu/%7Evsv/papers/ch5.pdf ][http://www.math.ucla.edu/%7Evsv/papers/ch5.pdf http://www.math.ucla.edu/~vsv/papers/ch5.pdf] |
+ | |||
+ | * Frescura, F. A. M. 1981. “Geometric interpretation of the Pauli spinor”. <em>American Journal of Physics</em> 49: 152. doi:[http://dx.doi.org/10.1119/1.12548 10.1119/1.12548.] | ||
+ | * Vivarelli, Maria Dina. 1984. “Development of spinor descriptions of rotational mechanics from Euler’s rigid body displacement theorem”. <em>Celestial Mechanics</em> 32 (3월): 193-207. doi:[http://dx.doi.org/10.1007/BF01236599 10.1007/BF01236599]. | ||
+ | * Coquereaux, Robert. 2005. “Clifford algebras, spinors and fundamental interactions : Twenty Years After”. <em>arXiv:math-ph/0509040</em> (9월 19). http://arxiv.org/abs/math-ph/0509040. | ||
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2011년 11월 19일 (토) 15:29 판
introduction
- #
- quadratic space \((V,Q)\)
- Q : non-degenerate quadratic form, defines a symmetric bilinear form \(<x,y>\)
- Clifford algebra : associative algebra generated by vectors in V with relations
- \(v^2=Q(v)\)
- \(vw+wv=2<w,v>\)
- 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
- consider a representation of Clifford algebras
- the elements in this space are called spinors
- 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
Pauli spinor
- 8-dimensional real algebra
- isomorphic to C(E_{3}) Clifford algebra of the Euclidean space E_{3}
- http://en.wikipedia.org/wiki/Spinors_in_three_dimensions
- spinor = a vector in two-dimensional space over complex number field
- Hermitian dot product is given on the vector space
- the space of spinors is a projective representation of the orthogonal group.
Dirac matrices
- 16 dimensional real algebra
- isomorphic to C(E_{3,1}) Clifford algebra of the Minkowski space E_{3,1}
- \(\gamma_{\mu}^2=\epsilon_{\mu}\), \(\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=0\)\(\epsilon_{0}=1, \epsilon_{i}=-1\)
- there exists unique four dimensional representation of a Clifford algebra
- projective representation of the Lorentz group
- universal covering of the Lorentz group H=SL(2,\mathbb{C}) also acts on it
history
- Schrodinger equation
- Pauli equation
- Dirac equation
- spin system and Pauli exclusion principle
- spin structures and spinor fields
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
- Lachièze-Rey, Marc. 2009. “Spin and Clifford Algebras, an Introduction”. Advances in Applied Clifford Algebras 19 (3-4): 687-720. doi:10.1007/s00006-009-0187-y.
- [1]http://www.math.ucla.edu/~vsv/papers/ch5.pdf
- Frescura, F. A. M. 1981. “Geometric interpretation of the Pauli spinor”. American Journal of Physics 49: 152. doi:10.1119/1.12548.
- Vivarelli, Maria Dina. 1984. “Development of spinor descriptions of rotational mechanics from Euler’s rigid body displacement theorem”. Celestial Mechanics 32 (3월): 193-207. doi:10.1007/BF01236599.
- Coquereaux, Robert. 2005. “Clifford algebras, spinors and fundamental interactions : Twenty Years After”. arXiv:math-ph/0509040 (9월 19). http://arxiv.org/abs/math-ph/0509040.
articles
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
- http://mathoverflow.net/search?q=
- http://mathoverflow.net/search?q=
- http://math.stackexchange.com/search?q=
- http://math.stackexchange.com/search?q=
- http://physics.stackexchange.com/search?q=
- http://physics.stackexchange.com/search?q=
blogs
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experts on the field