Clifford Algebra and Physics

.There is a noticably higher interest in Clifford algebra among quantum physicists and pure matematicians. I am noticing more posts on Twitter, google searches and a number of new lectures on Youtube. I have to refresh this subject, making a few important reminders about the optimal choice of a Clifford algebra (there are many!) for people who are going to study the subject in physics.  

- It is important to pick an algebra that can be generated out of basic operators of uniform signature than mixed one in order to minimize the number of initial assumptions. For example Cl3,0(R) is more naturally symmetrical than Cl3,1(R) .  

- It is important that the selected Clifford algebra is “self-complexifying”, that is generates Complex number field naturally as it’s own subalgebra. Such algebras allow associating a physical interpretation to imaginary numbers in natural fashion, otherwise  Complex number field has to be introduced as a basic assumption rather than follow naturally from the algebra. There are two self-complexifying Clifford algebras of interest: Cl3,0(R) and Cl7,0(R). Their complexified isomorphs are Cl(2)xC and Cl(6)xC, respectively. The second one is more difficult to work with because of its non-associativity.

- I need to stress that the particular Clifford algebra most often employed in quantum physics lectures Cl3,1(R) to express Dirac equation and spinors is not the best choice because it is the 4-rth ‘grade’ algebra. Clifford algebras beyond grade 3 are non-associative ( a product abc evaluates differently depending on whether you evaluate ab first and right multiply by c versus first muliply bc and then left multiply by a). It is very difficult to use non-associative algebras, in particular to reduce formulae or evaluate solutions.

- I strongly recommend to use Cl3,0(R) with natural bilinear form(metrics) of Q(p,q) = pq’ + qp’ [where ()’ is quaternion conjugate]  rather than the most common Cl3,1 in physics. In Cl3,0(R) the three basic generator operators e,f,g represent 3 physical directions (for example x,y,z) while Time direction is represented by identity operator I. It is customary in algebra to write I as number 1, interchangeably. In algebra Cl3,0 all bases square to +I  that is ee=ff=gg=I=1 ( in algebra Cl3,1 e,f,g square to +I and operator h representing time axis squares to -I).  The full Cl3,0(R) algebra is 8-dimensional with the full base being{1,e,f,g,ie,if,ig,i} where i=efg (note: ef=-fe, fg=-gf, ge=-eg).

- It is important to chose the mathematical modelling tool that has the sufficient complexity such as the number of degree of freedom (dimensions) that can reproduce physics but no more than that. For example Dirac matrics spinor calculus uses 4x4 complex  matrices (gamma matrices) which have 4x4x2=32 degrees of freedom, as opposed to Cl3,0(R) spinor model which has 8 degrees of freedom. A math model which has too many degrees of freedom than needed has to have additional constrains imposed to limit the domain. Cl3,0(R) doesn’t need any additional constrains, anything that it generates has a physical counterpart and vice versa: any physical concept can be modelled using it (perhaps with the exception of gluons and quarks, which may require Cl7,0(R) but I am not 100% convinced about it).