Quantum Mechanics for Mathematicians

In terms of the matrixB, the bilinear form is computed as

B(v,v′) =

(

v 1 ... vd

)







B 11 ... B 1 d

Bd 1 ... Bdd













v 1 ′

vd′





=v·Bv′

9.6 Symmetric and antisymmetric multilinear forms

The symmetric bilinear forms lie inS^2 (V∗) ⊂V∗⊗V∗and correspond to
symmetric matrices. Elements ofV∗give linear functions onV, and one can
get quadratic functions onVfrom elementsB∈S^2 (V∗) by taking

v∈V→B(v,v) =v·Bv

Equivalently, in terms of tensor products, one gets quadratic functions as the
product of linear functions by taking

(α 1 ,α 2 )∈V∗×V∗→

1

2

(α 1 ⊗α 2 +α 2 ⊗α 1 )∈S^2 (V∗)

and then evaluating atv∈V to get the number

1

2

(α 1 (v)α 2 (v) +α 2 (v)α 1 (v)) =α 1 (v)α 2 (v)

This multiplication can be extended to a product on the space

S∗(V∗) =⊕nSn(V∗)

(called the space of symmetric multilinear forms) by defining

(α 1 ⊗···⊗αj)(αj+1⊗···⊗αn) =P+(α 1 ⊗···⊗αn)

≡

1

n!

∑

σ∈Sn

ασ(1)⊗···⊗ασ(n) (9.3)

One can show thatS∗(V∗) with this product is isomorphic to the algebra
of polynomials onV. For a simple example of how this works, takevj∈V∗to
be thejth coordinate function. Then the correspondence between monomials
invjand elements ofS∗(V∗) is given by

vnj↔(vj⊗vj⊗···⊗vj)

︸ ︷︷ ︸

n-times

(9.4)

Both sides can be thought of as the same function onV, given by evaluating
thejth coordinate ofv∈Vand multiplying it by itselfn-times.
We will later find useful the fact thatS∗(V∗) and (S∗(V))∗are isomorphic,
with the tensor product

(α 1 ⊗···⊗αj)∈S∗(V∗)