Chapter 9

Tensor Products,

Entanglement, and

Addition of Spin

If one has two independent quantum systems, with state spacesH 1 andH 2 ,
the combined quantum system has a description that exploits the mathematical
notion of a “tensor product”, with the combined state space the tensor product
H 1 ⊗ H 2. Because of the ability to take linear combinations of states, this
combined state space will contain much more than just products of independent
states, including states that are described as “entangled”, and responsible for
some of the most counter-intuitive behavior of quantum physical systems.
This same tensor product construction is a basic one in representation the-
ory, allowing one to construct a new representation (πW 1 ⊗W 2 ,W 1 ⊗W 2 ) out of
representations (πW 1 ,W 1 ) and (πW 2 ,W 2 ). When we take the tensor product of
states corresponding to two irreducible representations ofSU(2) of spinss 1 ,s 2 ,
we will get a new representation (πV 2 s 1 ⊗V 2 s 2 ,V^2 s^1 ⊗V^2 s^2 ). It will be reducible,
a direct sum of representations of various spins, a situation we will analyze in
detail.
Starting with a quantum system with state spaceHthat describes a single
particle, a system ofnparticles can be described by taking ann-fold tensor
productH⊗n=H⊗H⊗···⊗H. It turns out that for identical particles, we
don’t get the full tensor product space, but only the subspaces either symmetric
or antisymmetric under the action of the permutation group by permutations
of the factors, depending on whether our particles are “bosons” or “fermions”.
This is a separate postulate in quantum mechanics, but finds an explanation
when particles are treated as quanta of quantum fields.

Digression.When physicists refer to “tensors”, they generally mean the “ten-
sor fields” used in general relativity or other geometry-based parts of physics,
not tensor products of state spaces. A tensor field is a function on a manifold,