Representations

In abstract algebra, we may find another way of describing the uncertainty principle. We first define the algebra generated by the operators appeared in the preceding subsection.

$$A_n(k)=k\langle x_1,x_2,\dots,x_n,\partial_1,\partial_2,\dots,\partial_n \rangle$$

We call it the Weyl algebra over $k$ . Here, $k$ is the field $\mathbb{C}$ of complex numbers in the physics context, but may well be a domain of characteristic 0 .

In general, including the case where the characteristic of the ground field $k$ is non zero (or even the case where $k$ is an arbitrary ring), we define as follows.

DEFINITION 6.1   Let $n$ be a positive integer. A Weyl algebra $A_n(k)$ over a commutative ring $k$ is an algebra over $k$ generated by $2 n$ elements $\{\gamma_1,\gamma_2,\dots,\gamma_{2 n}\}$ with the ``canonical commutation relations'' (CCR)

$$[\gamma_i \gamma_j]=h_{i j} \qquad (1\leq i,j \leq 2 n).$$

Where $h$ is a non-degenerate anti-Hermitian $2 n \times 2 n$ matrix of the following form.

$$(h)= \begin{pmatrix} 0 & -1_n \\ 1_n & 0 \end{pmatrix}.$$

In what follows, $h$ will always mean the matrix above. we denote by $\bar{h}$ the inverse matrix of $h$ .

LEMMA 6.2   Any element $a$ of $A_n(k)$ is written uniquely as

$$\sum_{i_1,i_2,i_3,\dots,i_{2n}} a_{i_1,i_2,i_3,\dots, i_{2n}} \gamma_1^{i_1} \gamma_2^{i_2} \gamma_3^{i_3} \dots \gamma_{2 n}^{i_n}$$

Then the fact is:

LEMMA 6.3   Assume $k$ is a field of characteristic zero. Then the Weyl algebra $A_n(k)$ is simple (that means, has no proper two-sided ideals). There exists no finite dimensional representation of $A_n(k)$

PROOF.. Let $\mathfrak{a}$ be a non trivial two-sided ideal of $A_n(k)$ . We take a non zero element $x \in \mathfrak{a}$ with the lowest degree when expressed as a polynomial of $\gamma$ .

$$x=\sum_{I} x_I\gamma^I$$

Then it is easy to see that the commutator $[\gamma_i,x]$ has the degree lower than $x$ , and that one of the commutators is non zero unless $x$ is a constant.

From the manner we choose the element $x$ , we deduce that $x$ should be a non zero constant in $\mathfrak{a}$ . That means,

$$\mathfrak{a}=A_n(k).$$

This is contrary to the assumption that $\mathfrak{a}$ is non trivial.

$\qedsymbol$

When the characteristic of the base field $k$ is not zero, things are different. We shall see this in the next section.

Before that, we make an easy explanation for the latter part of the Lemma above. Let

$$\alpha: A_n(k)\to M_d(k)$$

be a finite dimensional representation. Then taking a trace of the CCR relations we obtain

$$0= \operatorname{tr}{[\alpha(\gamma_{n+i}) \alpha( \gamma_i)]} = ... ...ratorname{tr}{\alpha([\gamma_{n+i} \gamma_i])} =\operatorname{tr}(\alpha(1))=d$$

which is absurd.