20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy: Difference between revisions

::The Clifford algebra is constructed directly from the first vector space, by formally/symbolically multiplying vectors and simplifying these expressions according to the rule: <math> v\cdot v = -q(v)1 </math> where q is the length/quadratic form of the vector that the rotations must preserve, or utilizing the polarization identities of quadratic forms: <math> 2q(v,w)=q(v+w)-q(v)-q(w)\rightarrow v\cdot w + w\cdot v = -2q(v,w) </math>. The two-input bilinear form <math> q(\cdot ,\cdot ) </math> is (often) concretely the dot product, and the original quadratic form is recovered by plugging in the same vector twice <math> q(v,v) </math>. The intermediate mathematical structure which keeps track of the formal products of the vectors is known as the tensor algebra <math> T(V)</math> and notably does not depend on <math> q </math>. It helps to interpret the meaning of the unit <math> 1 </math> in the relation as another formally adjoined symbol. Then, the resulting Clifford algebra with the given modified multiplication rule/relation depending on <math> q </math> is denoted by <math> \mathcal{Cl}(V,q)=T(V)/(v\cdot v-q(v)1) </math> as the quotient of <math> T(V) </math> by the subspace of expressions which we want to evaluate to 0.

::The Spin group is then found within the Clifford algebra,