Lecture 2

Setting - Tensors
Vν := R n , H d = H :=
d�
Vν d-fold tensor product Hilbert-s.,
ν=1
H � {(x1, . . . , x d) ↦→ U(x1, . . . , x d) ∈ R : x i = 1, . . . , n i} .
The function U ∈ H will be called an order d-tensor.
For notational simplicity, we often consider n i = n. Here
x1, . . . , x d ∈ {1, . . . , n} will be called variables or indices.
k ↦→ U(k1, . . . , k d) = (U k1,...,k d ) , k i = 1 . . . , n i .
Or in index (vectorial) notation
U = � U k1,...,k d
� ni
k i =0,1≤i≤d
dim H = n d curse of dimensions!!!
E.g. wave function Ψ(r1, s1, . . . , r N, s N)