13.1 The Heisenberg Lie algebra

13.1 The Heisenberg Lie algebra

Concept links · terms present in this machine draft; source roles are unverified: vector space · Lie algebra · Lie bracket

In either the position or momentum space representation the operators $P_j$ and $Q_j$ satisfy the relation

$$[Q_j,P_k]=i{\delta }_{jk}1$$

Soon after this commutation relation appeared in early work on quantum mechanics, Weyl realized that it can be interpreted as the relation between operators one would get from a representation of a $2d+1$ dimensional Lie algebra, now called the Heisenberg Lie algebra. Treating first the $d=1$ case, we define:

Definition (Heisenberg Lie algebra, $d = 1 )

$.TheHeisenbergLiealgebra${ \mathfrak { h } } _ { 3 }$isthevectorspace$\mathbf { R } ^ { 3 }$withtheLiebracketdefinedbyitsvaluesonabasis$( X , Y , Z )b y$

$$[X,Y]=Z,[X,Z]=[Y,Z]=0$$

Writing a general element of ${\mathfrak{h}}_3$ in terms of this basis as $xX+yY+zZ.$ and grouping the $x,y$ coordinates together (we will see that it is useful to think of the vector space ${\mathfrak{h}}_3$ as ${\mathbf{R}}^2\oplus \mathbf{R})$ , the Lie bracket is given in terms of the coordinates by

$$\left[\left(\left(\genfrac{}{}{0ex}{}{x}{y}\right),z\right),\left(\left(\genfrac{}{}{0ex}{}{x^{\prime }}{y^{\prime }}\right),z^{\prime }\right)\right]=\left(\left(\genfrac{}{}{0ex}{}{0}{0}\right),x{y}^{\prime }-y{x}^{\prime }\right)$$

Note that this is a non-trivial Lie algebra, but only minimally so. All Lie brackets of $Z$ with anything else are zero. All Lie brackets of Lie brackets are also zero (as a result, this is an example of what is known as a “nilpotent” Lie algebra).

The Heisenberg Lie algebra is isomorphic to the Lie algebra of 3 by 3 strictly upper triangular real matrices, with Lie bracket the matrix commutator, by the following isomorphism:

$$X\leftrightarrow \left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),Y\leftrightarrow \left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),Z\leftrightarrow \left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$ $$\left(\left(\genfrac{}{}{0ex}{}{x}{y}\right),z\right)\leftrightarrow \left(\begin{array}{ccc}0& x& z\\ 0& 0& y\\ 0& 0& 0\end{array}\right)$$

and one has

$$\left[\left(\begin{array}{ccc}0& x& z\\ 0& 0& y\\ 0& 0& 0\end{array}\right),\left(\begin{array}{ccc}0& x^{\prime }& z^{\prime }\\ 0& 0& y^{\prime }\\ 0& 0& 0\end{array}\right)\right]=\left(\begin{array}{ccc}0& 0& x{y}^{\prime }-x^{\prime }y\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$

The generalization of this to higher dimensions is:

Definition (Heisenberg Lie algebra)

The Heisenberg Lie algebra ${\mathfrak{h}}_{2d+1}$ is the vector space ${\tilde{\mathbf{R}}}^{2d+1}={\tilde{\mathbf{R}}}^{2d}\oplus$ R with the Lie bracket defined by its values on a basis $X_j,Y_j,Z(j=1,\dots d)$ by

$$[X_j,Y_k]={\delta }_{jk}Z,[X_j,Z]=[Y_j,Z]=0$$

Writing a general element as $\begin{array}{r}{\sum }_{j=1}^d{x}_j{X}_j+{\sum }_{k=1}^d{y}_k{Y}_k+zZ\end{array}$ , in terms of coordinates the Lie bracket is

$$\left[\left(\left(\genfrac{}{}{0ex}{}{\mathbf{x}}{\mathbf{y}}\right),z\right),\left(\left(\genfrac{}{}{0ex}{}{{\mathbf{x}}^{\prime }}{{\mathbf{y}}^{\prime }}\right),z\right)\right]=\left(\left(\genfrac{}{}{0ex}{}{0}{0}\right),\mathbf{x}\cdot {\mathbf{y}}^{\prime }-\mathbf{y}\cdot {\mathbf{x}}^{\prime }\right)\text{\hspace{1em}(13.1)}$$

This Lie algebra can be written as a Lie algebra of matrices for any $d$. For instance, in the physical case of $d=3$ , elements of the Heisenberg Lie algebra can be written

$$\left(\begin{array}{ccccc}0& x_1& x_2& x_3& z\\ 0& 0& 0& 0& y_1\\ 0& 0& 0& 0& y_2\\ 0& 0& 0& 0& y_3\\ 0& 0& 0& 0& 0\end{array}\right)$$

Chapter contents · Next section: 13.2 The Heisenberg group · 中文版本

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