"클리퍼드 대수"의 두 판 사이의 차이

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(피타고라스님이 이 페이지의 이름을 Clifford algebras and spinors로 바꾸었습니다.)
13번째 줄: 13번째 줄:
 
 
 
 
  
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<h5>Pauli matrices</h5>
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<h5>spinor</h5>
  
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* 8-dimensional real algebra
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* consider a representation of [[Clifford algebras and spinors|Clifford algebras]]
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* C(E_{3}) Clifford algebra of the Euclidean space E_{3}
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* the elements in this space are called spinors
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* Spinors are classified according to Dirac, Weyl, Majorana and Weyl-Majorana spinors.
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*  applications<br>
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** spinor bundles
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** spin connections
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** the role of spinors in the description of the fundamental interactions between elementary particles
  
 
 
 
 
22번째 줄: 27번째 줄:
 
 
 
 
  
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<h5>Dirac matrices</h5>
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<h5>Pauli spinor</h5>
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* 8-dimensional real algebra
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* isomorphic to C(E_{3}) Clifford algebra of the Euclidean space E_{3}
  
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* http://en.wikipedia.org/wiki/Spinors_in_three_dimensions
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* spinor = a vector in two-dimensional space over complex number field
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* Hermitian dot product is given on the vector space
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*  the space of spinors is a projective representation of the orthogonal group.
  
 
 
 
 
30번째 줄: 41번째 줄:
 
 
 
 
  
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<h5>Dirac matrices</h5>
  
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* 16 dimensional real algebra
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* isomorphic to C(E_{3,1}) Clifford algebra of the Minkowski space E_{3,1}
  
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* <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
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* universal covering of the Lorentz group H=SL(2,\mathbb{C}) also acts on it
  
 
 
 
 
51번째 줄: 66번째 줄:
  
 
* [[Weyl algebra]]
 
* [[Weyl algebra]]
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* [[Schrodinger equation]]
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* [[Pauli equation]]
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* [[Dirac equation]]
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* [[spin system and Pauli exclusion principle]]
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* [[spin structures and spinor fields]]
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 +
 
  
 
 
 
 
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.
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* [http://www.math.ucla.edu/%7Evsv/papers/ch5.pdf http://www.math.ucla.edu/~vsv/papers/ch5.pdf]
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* [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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*  
 
  
 
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>

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}

 

 

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

 

 

related items

 

 

encyclopedia

 

 

books

 

 

 

expositions
  • 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

 

 

 

question and answers(Math Overflow)

 

 

 

blogs

 

 

experts on the field

 

 

links
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