Algebra - Clifford, Dirac, Lorentz

[spaghetti3451]

In the representation theory of Lorentz transformations, the words Clifford algebra and Dirac algebra are used interchangeably. However, there is a distinction between the two. Indeed, the Dirac algebra is the particular Clifford algebra $Cl_{4}({\bf{C}})\equiv Cl_{1,3}({\bf{C}})$ with a basis generated by the matrices $\gamma^{\mu}$ called Dirac matrices which have the property that $\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$.

On the other hand, the spinor representation of the Lorentz algebra can be obtained using the Dirac matrices which obey the Dirac algebra. The above is a qualitative picture of the key difference between the Clifford algebra, Dirac algebra and the Lorentz algebra.

Can someone please flesh out the mathematical details?

 

[Jonathan Scott]

I think that's enough for a whole book. You can certainly find out more by Googling.

Lorentz transformations in spinor space and in four-vector space only require the Pauli Algebra or "Algebra of Physical Space", not the Dirac Algebra. David Hestenes wrote a book "Space-Time Algebra" in which he used the Dirac matrices as the basis for spacetime, but it seems that the only case in which this was actually useful was in a different way of writing the Dirac equation itself. Special Relativity can be described perfectly well using a vector algebra that is equivalent to the Pauli Algebra, which I usually call informally "Complex Four-vector Algebra" to make it easy to understand, and which William Baylis calls the "Algebra of Physical Space", for example in his book "Electrodynamics: A Modern Geometric Approach". It is also possible to write the Dirac equation in the Algebra of Physical Space (Pauli Algebra) rather than the full Dirac algebra.