9.2 Composite quantum systems and tensor products

9.2 Composite quantum systems and tensor products

Concept links · terms present in this machine draft; source roles are unverified: irreducible representation · symmetry group

Consider two quantum systems, one defined by a state space ${\mathcal{H}}_1$ and a set of operators ${\mathcal{O}}_1$ on it, the second given by a state space ${\mathcal{H}}_2$ and set of operators ${\mathcal{O}}_2$ 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

$${\mathcal{H}}_T={\mathcal{H}}_1\otimes {\mathcal{H}}_2$$

with operators of the form

$$A\otimes \mathbf{Id}+\mathbf{Id}\otimes B$$

with $A\in {\mathcal{O}}_1,B\in {\mathcal{O}}_2$ . The state space ${\mathcal{H}}_T$ can be used to describe an interacting quantum system, but with a more general class of operators.

If $H$ is the state space of a quantum system, this can be thought of as describing a single particle. Then a system of N such particles is described by the multiple tensor product

$${\mathcal{H}}^{\otimes n}=\underset{\text{n times}}{\underbrace{\mathcal{H}\otimes \mathcal{H}\otimes \cdots \otimes \mathcal{H}\otimes \mathcal{H}}}$$

The symmetric group $S_n$ acts on this state space, and one has a representation $(\pi ,{\mathcal{H}}^{\otimes n})$ of $S_n$ as follows. For $\sigma \in S_n$ a permutation of the set $\{1,2,\dots ,n\}$ of $n$ elements, on a tensor product of vectors one has

$$\pi (\sigma )(v_1\otimes v_2\otimes \cdots \otimes v_n)=v_{\sigma (1)}\otimes v_{\sigma (2)}\otimes \cdots \otimes v_{\sigma (n)}$$

The representation of $S_n$ that this gives is in general reducible, containing various components with diferent irreducible representations of the group $S_n$

A fundamental axiom of quantum mechanics is that if ${\mathcal{H}}^{\otimes n}$ describes n identical particles, then all physical states occur as one dimensional representations of $S_n$ . These are either symmetric $\left({}^{64}\mathrm{boson}{\mathrm{s}}^9\right)$ or antisymmetric $\left({}^{66}\mathrm{fermion}{\mathrm{s}}^{33}\right)$ where:

Definition

A state $v\in {\mathcal{H}}^{\otimes n}$ is called

• symmetric, or bosonic $if\forall \sigma \in S_n$

$$\pi (\sigma )v=v$$

The space of such states is denoted $S^n(\mathcal{H})$

• antisymmetric, or fermionic $if\forall \sigma \in S_n$

$$\pi (\sigma )v=(-1{)}^{|\sigma |}v$$

The space of such states is denoted ${\Lambda }^n(\mathcal{H})$ ). Here |$\sigma$| is the minimal number of transpositions that by composition give $\sigma$.

Note that in the fermionic case, for $\sigma$ a transposition interchanging two particles, $\pi$ acts on the factor $\mathcal{H}\otimes \mathcal{H}$ by interchanging vectors, taking

$$w\otimes w\in \mathcal{H}\otimes \mathcal{H}$$

to itself for any vector $w\in \mathcal{H}$ . Antisymmetry requires that $\pi$ take this state to its negative, so the state cannot be non-zero. As a result, one cannot have nonzero states in ${\mathcal{H}}^{\otimes n}$ describing two identical particles in the same state $w\in \mathcal{H}$ a fact that is known as the “Pauli principle”.

While the symmetry or antisymmetry of states of multiple identical particles is a separate axiom when such particles are described in this way as tensor products, we will see later on (chapter 36) that this phenomenon instead finds a natural explanation when particles are described in terms of quantum fields.

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来源版本：2025-10-20

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