Quantum Mechanics for Mathematicians

taking values in some tensor product of copies of the tangent space and its dual
space. The simplest tensor fields are vector fields, functions taking values in
the tangent space. A more non-trivial example is the metric tensor, which takes
values in the dual of the tensor product of two copies of the tangent space.

9.1 Tensor products

Given two vector spacesVandW(overRorC), the direct sum vector space
V⊕Wis constructed by taking pairs of elements (v,w) forv∈V,w∈W, and
giving them a vector space structure by the obvious addition and multiplication
by scalars. This space will have dimension

dim(V⊕W) = dimV+ dimW

If{e 1 ,e 2 ,...,edimV}is a basis ofV, and{f 1 ,f 2 ,...,fdimW}a basis ofW, the

{e 1 ,e 2 ,...,edimV,f 1 ,f 2 ,...,fdimW}

will be a basis ofV⊕W.
A less trivial construction is the tensor product of the vector spacesVand
W. This will be a new vector space calledV⊗W, of dimension

dim(V⊗W) = (dimV)(dimW)

One way to motivate the tensor product is to think of vector spaces as vector
spaces of functions. Elements

v=v 1 e 1 +v 2 e 2 +···+vdimVedimV∈V

can be thought of as functions on the dimVpointsej, taking valuesvjatej. If
one takes functions on the union of the sets{ej}and{fk}one gets elements of
V⊕W. The tensor productV⊗Wwill be what one gets by taking all functions
on not the union, but the product of the sets{ej}and{fk}. This will be the
set with (dimV)(dimW) elements, which we will writeej⊗fk, and elements
ofV⊗W will be functions on this set, or equivalently, linear combinations of
these basis vectors.
This sort of definition is less than satisfactory, since it is tied to an explicit
choice of bases forV andW. We won’t however pursue more details of this
question or a better definition here. For this, one can consult pretty much any
advanced undergraduate text in abstract algebra, but here we will take as given
the following properties of the tensor product that we will need:

• Given vectorsv∈V,w∈Wwe get an elementv⊗w∈V⊗W, satisfying
  bilinearity conditions (forc 1 ,c 2 constants)

v⊗(c 1 w 1 +c 2 w 2 ) =c 1 (v⊗w 1 ) +c 2 (v⊗w 2 )

(c 1 v 1 +c 2 v 2 )⊗w=c 1 (v 1 ⊗w) +c 2 (v 2 ⊗w)