Quantum Mechanics for Mathematicians

byξj,j= 1,...,n, satisfying the relations

ξjξk+ξkξj= 0

is called the Grassmann algebra.

Note that these relations imply that generators satisfyξ^2 j = 0. Also note
that sometimes the Grassmann algebra product ofξjandξkis denotedξj∧ξk.
We will not use a different symbol for the product in the Grassmann algebra,
relying on the notation for generators to keep straight what is a generator of
a conventional polynomial algebra (e.g.,qjorpj) and what is a generator of a
Grassmann algebra (e.g.,ξj).
The Grassmann algebra is the algebra Λ∗(V∗) of antisymmetric multilinear
forms onV discussed in section 9.6, except that we have chosen a basis ofV
and have written out the definition in terms of the dual basisξjofV∗. It is
sometimes also called the “exterior algebra”. This algebra behaves in many
ways like the polynomial algebra onRn, but it is finite dimensional as a real
vector space, with basis

1 , ξj, ξjξk, ξjξkξl, ···, ξ 1 ξ 2 ···ξn

for indicesj < k < l <··· taking values 1, 2 ,...,n. As with polynomials,
monomials are characterized by a degree (number of generators in the product),
which in this case takes values from 0 only up ton. Λk(Rn) is the subspace of
Λ∗(Rn) of linear combinations of monomials of degreek.

Digression(Differential forms). Readers may have already seen the Grass-
mann algebra in the context of differential forms onRn. These are known to
physicists as “antisymmetric tensor fields”, and given by taking elements of the
exterior algebraΛ∗(Rn)with coefficients not constants, but functions onRn.
This construction is important in the theory of manifolds, where at a pointx
in a manifoldM, one has a tangent spaceTxMand its dual space(TxM)∗. A
set of local coordinatesxjonMgives basis elements of(TxM)∗denoted bydxj
and differential forms locally can be written as sums of terms of the form

f(x 1 ,x 2 ,···,xn)dxj∧···∧dxk∧···∧dxl

where the indicesj,k,lsatisfy 1 ≤j < k < l≤n.

A fundamental principle of mathematics is that a good way to understand a
space is in terms of the functions on it. What we have done here can be thought
of as creating a new kind of space out ofRn, where the algebra of functions
on the space is Λ∗(Rn), generated by coordinate functionsξjwith respect to a
basis ofRn. The enlargement of conventional geometry to include new kinds
of spaces such that this makes sense is known as “supergeometry”, but we will
not attempt to pursue this subject here. Spaces with this new kind of geometry
have functions on them, but do not have conventional points since we have seen
that one can’t ask what the value of an anticommuting function at a point is.