Quantum Mechanics for Mathematicians

the state of the measurement apparatus, thought of as lying in a state space
Happaratus. The laws of quantum mechanics presumably apply to the total
systemHsystem⊗Happaratus, with the counter-intuitive nature of measurements
appearing due to this decomposition of the world into two entangled parts: the
one under study, and a much larger for which only an approximate description
in classical terms is possible. For much more about this, a recommended reading
is chapter 2 of [75].

9.4 Tensor products of representations

Given two representations of a group a new representation can be defined, the
tensor product representation, by:

Definition(Tensor product representation of a group).For(πV,V)and(πW,W)
representations of a groupG, there is a tensor product representation(πV⊗W,V⊗
W)defined by
(πV⊗W(g))(v⊗w) =πV(g)v⊗πW(g)w

One can easily check thatπV⊗W is a homomorphism.
To see what happens for the corresponding Lie algebra representation, com-
pute (forXin the Lie algebra)

π′V⊗W(X)(v⊗w) =

d

dt

πV⊗W(etX)(v⊗w)t=0

=

d

dt

(πV(etX)v⊗πW(etX)w)t=0

=

((

d

dt

πV(etX)v

)

⊗πW(etX)w

)

t=0

+

(

πV(etX)v⊗

(

d

dt

πW(etX)w

))

t=0

=(π′V(X)v)⊗w+v⊗(π′W(X)w)

which could also be written

πV′⊗W(X) = (π′V(X)⊗ (^1) W) + ( (^1) V⊗π′W(X))

9.4.1 Tensor products ofSU(2) representations

Given two representations (πV,V) and (πW,W) of a groupG, we can decom-
pose each into irreducibles. To do the same for the tensor product of the two
representations, we need to know how to decompose the tensor product of two
irreducibles. This is a fundamental and non-trivial question, with the answer
forG=SU(2) as follows:

Theorem 9.1(Clebsch-Gordan decomposition).
The tensor product(πVn 1 ⊗Vn 2 ,Vn^1 ⊗Vn^2 )decomposes into irreducibles as

(πn 1 +n 2 ,Vn^1 +n^2 )⊕(πn 1 +n 2 − 2 ,Vn^1 +n^2 −^2 )⊕···⊕(π|n 1 −n 2 |,V|n^1 −n^2 |)