Quantum Mechanics for Mathematicians

• As a vector space overC, a basis of Cliff(n,C) is the set of elements

1 , γj, γjγk, γjγkγl, ..., γ 1 γ 2 γ 3 ···γn− 1 γn

for indicesj,k,l,··· ∈ 1 , 2 ,...,n, withj < k < l <···. To show this,

consider all products of the generators, and use the commutation relations

for theγjto identify any such product with an element of this basis. The

relationγ^2 j= 1 shows that repeated occurrences of aγjcan be removed.

The relationγjγk=−γkγjcan then be used to put elements of the product

in the order of a basis element as above.

• As a vector space overC, Cliff(n,C) has dimension 2n. One way to see
  this is to consider the product

(1 +γ 1 )(1 +γ 2 )···(1 +γn)

which will have 2nterms that are exactly those of the basis listed above.

28.2 Real Clifford algebras

We can define real Clifford algebras Cliff(n,R) just as for the complex case, by
taking only real linear combinations:

Definition(Real Clifford algebras).The real Clifford algebra innvariables is
the algebraCliff(n,R)generated over the real numbers by 1 ,γjforj= 1, 2 ,...,n
satisfying the relations
[γj,γk]+= 2δjk

For reasons that will be explained in the next chapter, it turns out that a
more general definition is useful. We write the number of variables asn=r+s,
forr,snon-negative integers, and now vary not justr+s, but alsor−s, the
so-called “signature”:

Definition(Real Clifford algebras, arbitrary signature).The real Clifford al-
gebra inn=r+svariables is the algebraCliff(r,s,R)over the real numbers
generated by 1 ,γjforj= 1, 2 ,...,nsatisfying the relations

[γj,γk]+=± 2 δjk 1

where we choose the+sign whenj=k= 1,...,rand the−sign whenj=k=
r+ 1,...,n.

In other words, as in the complex case differentγjanticommute, but only
the firstrof them satisfyγ^2 j= 1, with the othersof them satisfyingγj^2 =−1.
Working out some of the low dimensional examples, one finds:

• Cliff(0, 1 ,R). This has generators 1 andγ 1 , satisfying

γ 12 =− 1