Abstract
In Sect. 4.5, we discussed the representation of an algebra and its significance in physical applications. This significance is doubled in the case of the Clifford algebras because of their relation with the Dirac equation, which describes a relativistic spin-\({{\frac{1}{2}}}\) fundamental particle such as a lepton or a quark. With the representation of Lie groups and Lie algebras behind us, we can now tackle the important topic of the representation of Clifford algebras.
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Notes
- 1.
We avoid using the normal notation for complex numbers to be more in tune with the notation of the Clifford algebras. So, instead of z, we use
and instead of i we use e. - 2.
In physics literature, the matrices are actually labeled as σ 3, σ 1, σ 2, respectively.
- 3.
It is not exactly an isomorphism, but a homomorphism which is locally an isomorphism, but not globally. Because of the double-valuedness of the kernel of the homomorphism of Theorem 31.2.6, one can think of Spin(3,0) as two identical copies of SO(3).
- 4.
Here we are ignoring the fact that the entries of the 2-column are functions rather than numbers. The full treatment of spinors whose entries are functions requires the tensor analysis of Clifford algebras, the so-called spin bundles, a topic which is beyond the scope of this book.
- 5.
Here, we are again using the physicists’ convention of setting e 4=e 0, with
.
and instead of i we use e.
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Hassani, S. (2013). Representation of Clifford Algebras. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_31
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DOI: https://doi.org/10.1007/978-3-319-01195-0_31
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