Editing Spinor (section)

=== Component spinors ===

Given a vector space ''V'' and a quadratic form ''g'' an explicit matrix representation of the Clifford algebra {{math|Cℓ(''V'', ''g'')}} can be defined as follows. Choose an orthonormal basis {{math|''e''<sup>1</sup> ... ''e''<sup>''n''</sup>}} for ''V'' i.e. {{math|1=''g''(''e''<sup>''μ''</sup>''e''<sup>''ν''</sup>) = ''η''<sup>''μν''</sup>}} where {{math|1=''η''<sup>''μμ''</sup> = ±1}} and {{math|1=''η''<sup>''μν''</sup> = 0}} for {{math|''μ'' ≠ ''ν''}}. Let {{math|1=''k'' = ⌊''n''/2⌋}}. Fix a set of {{math|2<sup>''k''</sup> × 2<sup>''k''</sup>}} matrices {{math|''γ''<sup>1</sup> ... ''γ''<sup>''n''</sup>}} such that {{math|1=''γ''<sup>''μ''</sup>''γ''<sup>''ν''</sup> + ''γ''<sup>''ν''</sup>''γ''<sup>''μ''</sup> = 2''η''<sup>''μν''</sup>1}} (i.e. fix a convention for the [[gamma matrices]]). Then the assignment {{math|''e''<sup>''μ''</sup> → ''γ''<sup>''μ''</sup>}} extends uniquely to an algebra homomorphism {{math|Cℓ(''V'', ''g'') → Mat(2<sup>''k''</sup>, <math>\Complex</math>)}} by sending the monomial {{math|''e''<sup>''μ''<sub>1</sub></sup> ⋅⋅⋅ ''e''<sup>''μ''<sub>''k''</sub></sup>}} in the Clifford algebra to the product {{math|''γ''<sup>''μ''<sub>1</sub></sup> ⋅⋅⋅ ''γ''<sup>''μ''<sub>''k''</sub></sup>}} of matrices and extending linearly. The space <math>\Delta = \Complex^{2^k}</math> on which the gamma matrices act is now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the [[Pauli matrices|Pauli sigma matrices]] gives rise to the familiar two component spinors used in non relativistic [[quantum mechanics]]. Likewise using the {{math|4 × 4}} Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic [[quantum field theory]]. In general, in order to define gamma matrices of the required kind, one can use the [[Weyl–Brauer matrices]].

In this construction the representation of the Clifford algebra {{math|Cℓ(''V'', ''g'')}}, the Lie algebra {{math|'''so'''(''V'', ''g'')}}, and the Spin group {{math|Spin(''V'', ''g'')}}, all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2<sup>''k''</sup> complex numbers and is denoted with spinor indices (usually ''α'', ''β'', ''γ''). In the physics literature, such [[abstract indices|indices]] are often used to denote spinors even when an abstract spinor construction is used.