Quantum Mechanics for Mathematicians

9.2 Composite quantum systems and tensor prod-

ucts

Consider two quantum systems, one defined by a state spaceH 1 and a set of
operatorsO 1 on it, the second given by a state spaceH 2 and set of operatorsO 2.
One can describe the composite quantum system corresponding to considering
the two quantum systems as a single one, with no interaction between them, by
taking as a new state space

HT=H 1 ⊗H 2

with operators of the form

A⊗Id+Id⊗B

withA∈O 1 ,B∈O 2. The state spaceHTcan be used to describe an interacting
quantum system, but with a more general class of operators.
IfHis the state space of a quantum system, this can be thought of as
describing a single particle. Then a system ofNsuch particles is described by
the multiple tensor product

H⊗n=H⊗H⊗···⊗H⊗H︸ ︷︷ ︸
n times
The symmetric groupSnacts on this state space, and one has a representa-
tion (π,H⊗n) ofSnas follows. Forσ∈Sna permutation of the set{ 1 , 2 ,...,n}
ofnelements, on a tensor product of vectors one has

π(σ)(v 1 ⊗v 2 ⊗···⊗vn) =vσ(1)⊗vσ(2)⊗···⊗vσ(n)

The representation ofSnthat this gives is in general reducible, containing var-
ious components with different irreducible representations of the groupSn.
A fundamental axiom of quantum mechanics is that ifH⊗ndescribesniden-
tical particles, then all physical states occur as one dimensional representations
ofSn. These are either symmetric (“bosons”) or antisymmetric (“fermions”)
where:

Definition.A statev∈H⊗nis called

• symmetric, or bosonic if∀σ∈Sn

π(σ)v=v

The space of such states is denotedSn(H).

• antisymmetric, or fermionic if∀σ∈Sn

π(σ)v= (−1)|σ|v

The space of such states is denotedΛn(H). Here|σ|is the minimal number

of transpositions that by composition giveσ.